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Step‐wise pulling protocols for non-equilibrium dynamics

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Step‐wise pulling protocols for non-equilibrium dynamics

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Content STEP-WISE PULLING PROTOCOLS FOR NON-EQUILIBRIUM DYNAMICS By Van Anh Ngo A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2014 Copyright 2014 Van Anh Ngo ii Dedication To my wife Thu Hương, and my daughter Bảo Lan for their constant love and endurance during one of the most struggling time of my life. To my parents Hồng Loan and Ngọc Vượng, my sister Ngọc Linh, and my uncle Phong. iii Acknowledgements I would like to give special thanks to my advisor, Prof. Stephan Haas, for his invaluable support. Without his support, I would have likely quit PhD program in Physics. My struggling in almost three years of PhD seemed unbearable, but I have regained a tremendous amount of joy from doing research, discussing ideas and chatting with Stephan. He is simply awesome. I would like to acknowledge Prof. Aii-chiro Nakano for his absolute kindness and constant encouragements, especially during the time that I wrote the first and single- author paper. He is the one who taught and showed me how to write a scientific paper in a beautiful, simple, and clear language. For a good starting point in my scientific career, I am deeply indebted to him. I have been also lucky enough to have a chance to learn from and work with Prof. Robert Farley for a couple of years. I am so grateful to have such a wonderful mentor, who has been patiently teaching me very details of experiments in molecular biology. Thanks to him, I love biological functions, structural characteristics of proteins, and known and unknown mechanisms of protein channels, and so on. I would like to thank other scientists with whom I have chance to work such as Profs. Rosa Di Felice, Darko Stefanovski, Rajiv Kalia, and Priya Vashishta. Especially, I have learned a lot from Prof. Rosa Di Felice in writing papers and her insights. Last but not least, among my friends, Ken-ichi Nomura, Amit Choubey, Mohammad Vedadi, and many others have shaped my positive thoughts about sciences, iv about how to get the most fun out of sciences with the least stress. Laughing and encouragements have been always among us. I am so grateful to have them as friends. v Table of Contents Chapter Page I Introduction: Mechanical Work, Free Energy and Temperature ............................... 1 II Stepwise Pulling Protocols for Free Energy Evaluation ........................................... 7 II.1 Introduction .................................................................................................. 7 II.2 Theory ........................................................................................................... 12 II.3 Protocols ....................................................................................................... 17 II.4 Implementation and Testing ......................................................................... 20 II.4.1 Sequential Pulling Protocol (SPP) ............................................... 20 II.4.2 Parallel Pulling Protocol (PPP) .................................................................... 23 II.4.3 Effects of Spring Constants k ....................................................................... 27 II.4.4 Effects of Relaxation τ ................................................................................. 29 II.5 Discussions and Conclusions 30 III Demonstration of Jarzynski’s Equality in Open Quantum Systems Using a Stepwise Pulling Protocol ......................................................................................... 38 III.1 Introduction .................................................................................................. 38 III.2 Theory ........................................................................................................... 42 III.3 Testing .......................................................................................................... 51 III.3.1 Control Parameter λ as the Center of a Harmonic ....................................... potential . 51 III.3.2 Control Parameter λ as the Spring Constant of a vi Harmonic Applied Potential ......................................................................... 55 III.4 Discussions and Conclusions ........................................................................ 58 IV Non-‐equilibrium Approach to Thermodynamics Using Jarzynski’s Equality and Diagonal Entropy ................................................................................ 63 IV.1 Introduction .................................................................................................. 63 IV.2 Theory ........................................................................................................... 66 IV.3 Examples ...................................................................................................... 69 IV.3.1 Example 1: Simple Harmonic Oscillator ..................................................... 69 IV.3.2 Example 2: One-dimensional Hardcore Boson Lattice ................................ 74 IV.4 Discussions and Conclusions ........................................................................ 77 V Non-Equilibrium Dynamics Determine Ion Selectivity In The KcsA Channel ...................................................................................................................... 80 V.1 Introduction .................................................................................................. 80 V.2 Materials and Methods ................................................................................. 84 V.3 Results .......................................................................................................... 93 V.3.1 Free-energy Profiles ..................................................................................... 93 V.3.2 Location of Stable Ion Binding Sites ........................................................... 96 V.3.3 Movement of Ions between Stable Binding Sites ........................................ 98 V.3.4 Differential Dehydration of Na + And K + By The Selectivity Filter ........................................................................................... 101 V.3.5 Structural Rearrangement of The Selectivity Filter in Response to Na + And K + .............................................................................. 103 vii V.3.6 Estimation of Channel Conductance ............................................................ 107 V.4 Discussions .................................................................................................. 109 VI Is The G-‐quadruplex an Effective Nanoconductor for Ions? ...................................... 118 VI.1 Introduction .................................................................................................. 118 VI.2 Methodology ................................................................................................. 123 VI.3 Results and Discussions ................................................................................ 128 VI.3.1 Binding Sites ................................................................................ 128 VI.3.2 Water Molecules in the Channel .................................................. 131 VI.3.3 Fluctuations of Bases around Ions ............................................................... 135 VI.3.4 Free-energy Profiles ..................................................................................... 137 VI.3.5 Conductance of G-quadruplex ..................................................................... 141 VI.4 Final Remarks ............................................................................................... 142 VII Supercrystals of DNA-Functionalized Gold Nanoparticles: a Million- Atom Molecular Dynamics Simulation Study ............................................................ 145 VII.1 Introduction .................................................................................................. 145 VII.2 Methodology ................................................................................................. 148 VII.3 Results and Discussions ................................................................................ 151 VII.4 Conclusions .................................................................................................. 165 VIII Molecular Mechanism of Flip-flop in Triple-layer Oleic-Acid Membrane: Correlation Between Oleic Acid and Water ............................. 166 VIII.1 Introduction .................................................................................................. 166 VIII.2 Methodology ................................................................................................. 169 viii VIII.3 Results and Discussions ................................................................................ 171 VIII.3.1 Triple-layer Structure ................................................................................... 171 VIII.3.2 Oleic Acid Flip-flopping and Water Traversing Membrane ..................................................................................................... 179 VIII.4 Conclusions .................................................................................................. 186 Bibliography 188 Appendices Page A Single Pulling Step 208 B Series of Pulling Steps 210 C Algorithm for Work Distribution Construction 213 D The Relation between ˆ O total and ˆ W 216 E The Recursion Relation 218 F Contribution of Ground States to Free-energy Changes 220 ix List of Tables Table V.1: Z-Coordinates (Å) for Stable Positions for Na + and K + Ions After Each Pulling Step (Δλ = 1 Å) ............................... 90 Table VII.1: Quantities computed for BCCS and FCCS ...................................... 164 x List of Figures I.1 Work distribution functions of unfolding and refolding an RNA molecules at different pulling rates. The thick black line indicates the free-energy diference between the unfolding and folding states . ............................................................................... 4 II.1 Schematic illustration for (a) original parallel pulling (arrowed arcs) protocol (b) our proposed sequential (series of circles) and parallel pulling (arrowed arcs) protocols. A reaction coordinate, x o , represents any state without virtual harmonic potentials. The arcs with arrows represent parallel pulling trajectories. A and B are initial and final configurations, respectively. N p is a number of trajectories. The circles represent sets of states or distributions of the reaction coordinate. Shaded areas are mutual overlapping states. Index i denotes the ordered number of pulling and runs from 1 to s, where indices 1 and s also represent the initial and final configurations, respectively. ....................................................................................... 18 II.2 (a) Normalized distribution of x i . The numbers indicate the pulling steps i. (b) Configurations of deca-alanine at λ = 13, 25 and 33 Å. The two biggest balls are fixed and pulled ends. (c) Dimensionless quantity γ i 2 versus λ. .................................................. 21 II.3 (a) Normalized work distribution. The numbers indicate the pulling steps i. (b) Mechanical work (squares) and free energies versus λ. PMF (crosses), ΔF JE (empty circles), ΔF G (triangles) and ΔF fluct (dots) are free energies computed by the PMF method, Eq. (12), Eq. (13) and Eq. (14), respectively. ............................................................................... 22 xi II.4 (a) Free energies versus λ. ΔF fluct (dots) are in the SPP with Δλ = 0.5 Å. ΔF fluct (triangles) and ΔF JE (squares) are in the PPP with Δλ = 1 Å. (b) Accumulating forces ΔF fluct /(λ s – λ 1 ) = k (λ i − x i )/s i=1 s−1 ∑ versus λ in both protocols. The force at s = 1 is set to zero. ......................................................... 24 II.5 Free energy profiles of (a) ΔF fluct and (b) ΔF JE with different spring constants k=1.0 (dots), 7.2 (empty boxes) and 50.0 (triangles) kcal/mol/Å 2 . (c) Normalized distributions of x i at step i = 20 (gray) and 21 (black) in the three cases. The insets in (a) are zoomed at the minimum and kink positions. The inset in (c) shows γ i 2 versus λ. ........................................................................................................ 25 II.6 (a) Free energies ΔF fluct (dots), ΔF JE (full triangles) and ΔF com = (ΔF fluct +ΔF JE )/2 (dashed line) at λ = 26 Å versus relaxation time τ. (b) Normalized work distributions at λ = 26 Å in a range from 0 to 25 kcal/mol at τ = 0.4 (dots), 0.6 (empty boxes) and 4.0 (empty triangles) ns. The inset in (b) shows the overall work distributions at the three values of τ. ................................................. 28 III.1 Schematic diagram for a step-wise pulling protocol: (a) control parameter λ, and (b) expectation value of the applied potential versus time. ........................................................................................ 41 III.2 Schematic diagram for optimal transitions. The parabolic curves represent an applied harmonic potential at the (i – 1)th and ith pulling steps. The red-shaded upper and green-shaded lower areas are the probabilities ψ E i−1 (x i−1 ) 2 and ψ E i (x i ) 2 , respectively. The blue-dashed arrow indicates an optimal transition from E i–1 (green- dashed lower line) to E i (red-dashed upper line). The region denoted by Δx is the overlap between ψ E i−1 (x i−1 ) 2 and ψ E i (x i ) 2 . ............................................... 50 xii III.3 (a) Normalized work distributions. The numbers (2 and 10 for the highest and lowest curves, respectively) indicate the ith-pulling steps at s = 11, and n = 0. (b) Free energy profiles at s = 11, and n = 0. (c) Free-energy changes ΔF(λ 1 ,λ s ) versus number n of the eigenstates and eigenvalues. (d) Free-energy changes ΔF(λ 1 ,λ s ) versus increment Δλ = λ s /(s – 1) at n = 10. All data here are evaluated at a = 1.......................................................................................................... 53 III.4 (a) Free-energy profiles (solid lines) at s = 11, and for different reduced temperatures a=ω /2k B T. The numbers indicate the values of a. The dashed line is the average work 〈W〉 along the pathway. (b) Free-energy ΔF(λ 1 ,λ s=11 ), standard deviation 〈(W–〈W〉) 2 〉 1/2 of work distributions, and average work 〈W〉 ≈ 0.274 ω /2 versus log 2 (a). ........................................................................... 54 III.5 (a) Normalized work distribution at s = 2. (b) Normalized work distribution function at s = 11. (c) Free-energy profiles. The work distributions and free-energy profiles are computed at a 0 =ω 0 /k B T =0.1. ......................................................................... 56 IV.1 Protocol of multiple quenches to generate a set of diagonal density operators Ω i ≡Ω λ i (Δλ) . The parameter λ denotes an external control, which can tune λ at will. Each circle and square is associated with the Hamiltonian H(λ i ). Here, the ground states are chosen as initial states for Examples 1 and 2, but not restricted for more general cases as long as the characteristic temperature is the same for all quenches. .......................................... 66 IV.2 (a) Temperature T (in units of ω /2 ) versus y=mωΔλ 2 /8 computing via Eq. (5). (b) Diagonal entropy S (in units of k B ) versus T in a quantum harmonic oscillator. T B and S B are the temperature and entropy of a non-interacting boson system computed by ω /ln(1+ n −1 ) xiii and [(1+〈n〉)ln(1+〈n〉) – 〈n〉ln〈n〉], respectively. Here, 〈n〉 is the expectation value of the number operator a λ + a λ . The red (A → B) and black (A → C) arrows present the transitions from an out-of-equilibrium state to the equilibrium states. The transition A → B has 〈n〉 = y, and A → C has T B = T. ................................................................................. 70 IV.3 (a) Distribution of reaction coordinate at λ = 0 and (b) free-energy profile at T = 0.35, Δλ = 0.6935. (c) Distribution of reaction coordinate at λ = 0 and (d) free-energy profile at T = 3.52, Δλ = 4.0. The distributions at other values of λ are identical to (a) and (c). Free energy change ΔF, T, and λ are in units of ω /2 ,ω /k B , and  /mω , respectively. DE-JE represents a free-energy profile computed by Diagonal Ensemble and Jarzynski’s Equality. The error bars are the standard deviations of the work distribution functions. ........................................................................ 73 IV.4 (a) Temperature T (in unit of J /k B ) versus Δλ 2 . (b) Diagonal entropy S (in units of k B ) versus T. (c) Distribution of the center-of-mass x of the superfluid during the time-evolution of ψ(t,λ i ) following a quench with λ i = 14. The distributions for other values of λ are almost identical. (d) Free-energy profile in a lattice of Hard-Core bosons trapped in a harmonic potential versus λ (Example 2). ......................................................... 75 V.1 Schematic diagram for a step-wise pulling protocol. (A) Control parameter λ, and (B) expectation value of an applied potential versus time, U(z, λ) = k(z – λ) 2 /2. Coordinate z is the position of an ion along the z-direction. The pulling in the direction of increasing λ is called forward; the opposite direction of pulling is called backward. (C) Comparison between profiles for PMF (blue) and free energy changes (red) computed using our method. ....................................... 86 xiv V.2 Free energy profiles for Na + and K + ions in the KcsA channel. (A) Free energy difference (ΔF) for Na + and K + ions after each pulling step (λ) compared to λ = 0. (B) Magnitude of the difference between Na + and K + ion free energy differences (ΔΔF) at each position of λ shown in (A). (C) The convergence of ΔΔF at λ = 11 Å during relaxation time t. The uncertainty for computing free-energy profiles is about 1.0 kcal/mol. ............................................................................................. 94 V.3 Histogram showing the frequency (probability) of finding the Na + and K + ions at each position (z) in the selectivity filter during step-wise pulling. The histogram is construct from 25 pulling-step simulations for each ion. Sites S0- S4 correspond to sites of stable K + binding in the selectivity filter of KcsA. Stable Na + positions are located in the plane of the carbonyl oxygen atoms that separate stable K + binding sites. Note that Na + ions are not stable within the selectivity filter at positions beyond the S2-S3 junction until they emerge from the channel at S0. Z = 0 corresponds to the center of mass of the protein-lipid-water system, within the central water-filled cavity in the transmembrane domain of the channel. ............................................. 96 V.4 Force (kcal/mol/Å) on Na + (circles) or K + (squares) ions in the z-direction as they are pulled step-wise through the KcsA channel. Zero on the z-coordinate axis represents the center of mass of the system. Site S4 in the selectivity filter occurs approximately at z = 7.1 Å, site S3 approximately at z = 10.4 Å, site S2 approximately at z = 13.6 Å, and site S1 approximately at z = 17.0 Å. Histograms of the z positions of each of the two ions were created from all different values of λ (Figure 2). Univariate normal mixture decomposition implemented in STATA (STATA Corp., State College, TX) was used to determine the number of Gaussians that contribute to the complex distribution of each ion distribution in xv Figure 3. The mean of each Gaussian was associated with a stable position for the ions along the z-axis of the KcsA channel. The 95% confidence interval for each Gaussian was used to partition the force, f(z), to an equal number of segments of normal distributions. Means and standard deviations (SD) for the force were calculated for each segment. Mean values of f(z) are shown in the plot with SD estimates in both z-coordinate and f(z). We define the probability of transition from one stable position to another, P(z i → z i+1 ), as the probability that the force in the current position, f(z i ), is equal to or larger to the f(z i+1 ) in the subsequent position, P(f(z i ) ≥ f(z i+1 )). Thus, we assume that the transitions between to stable positions of the ions can occur by diffusion or by attraction to the subsequent position without any energy cost. For transitions between two stable points with small SD around the mean f(z), we consider that the probability of transition to the subsequent position is 1 if the mean f(z i ) ≥ f(z i+1 ). Subsequently, the probability for the ion of going back once it reaches stable position z i+1 is 0. For transitions between stable position of the ion where either the mean for the force, f(z i ), or the subsequent f(z i+1 ) has a small SD and the other one has a large SD, we calculate the probability P(f(z i ) ≥ f(z i+1 )) by focusing on the f(z) with the large SD. Within this given segment for f(z) we calculate the probability as sum of the observations that satisfy the inequality, f(z i ) ≥ f(z i+1 ), divided by the total number of observations within that segment. Finally, for transitions between stable positions of the ion with coordinate z and a f(z) with very large SD, the probability of transition between two positions, P(f(z i ) ≥ f(z i+1 )), will be represented with P(f(z i ) ≥ f(z i+1 ) ∩ f(z i+1 ) ≤ f(z i )) = P(f(z i ) ≥ f(z i+1 )) * P(f(z i+1 ) ≤ f(z i )). To calculate the two probabilities first find the observations in both segments that satisfy the inequality f(z i ) ≥ f(z i+1 ) and then we divide the respective number by the total number of xvi observations in the given segment. Furthermore, the above formula implies that we consider the two events independent. ..................................................................... 99 V.5 The number of water molecules within 3 Å of Na + and K + ions as they are pulled incrementally through the selectivity filter of KcsA from the cytoplasmic side of the membrane (left) to the extracellular surface of the membrane (right). On the x-axis, z = 0 corresponds to the center of mass of the system in the vestibule of the channel. The first data point on the left is a position in the vestibule below site S4. Site S4 is located at z ≈ 7 Å. Values shown are means ± SD. ........................................................ 103 V.6 Positions of carbonyl oxygen atoms for amino acids T75-G79 in the KcsA selectivity filter during step-wise pulling of either Na + (top row) or K + (bottom row) from the vestibule toward the extracellular surface of the membrane. The y-axis is the frequency (probability) of finding the oxygen atoms at the position shown on the x-axis. Zero in the middle of the x-axis corresponds to the position of the atoms observed in the crystal structure. The carbonyl atoms corresponding to each amino acid in the tetrameric selectivity filter are shown in a single color, and each amino acid is represented by a different color (T75, black; V76, green; G77, blue; Y78, red; G79, light blue). ............................................. 105 V.7 Calculated values of conductance (G) for a K + ion pulled through the KcsA channel using a step-wise pulling protocol. Values of λ are the center of the harmonic pulling potential, and values between λ = 7 Å and λ = 17 Å occur when the center of the pulling potential is within the selectivity filter of KcsA. A negative value of G simply implies negative work (see Materials and Method) corresponding to the attraction (not resistance) from the selectivity filter at a pulling step. But for all pulling steps, the total conductance is always positive xvii because the total work and diffusion coefficients are always positive. ...................................................... 109 VI.1 Snapshot of the G-quadruplex in the presence of K + ions, after 10-ns equilibration. The quadruplex is visualized in a ribbon mode that highlights the backbone; water molecules (O red; H white) and potassium ions (green) are visualized as ball&stick. A planar position is defined as the crossing point of a G-quartet plane and the z-axis, while a cage-like position (S1 to S8) is the center of any two successive G-quartets along the z-axis. The three- dimensional structure is rendered by VMD. 33 ................................. 123 VI.2 (a) Histograms of ion’s positions along the z- direction in all pulling steps. The histograms are the combination of all separated normalized g i (z) for s = 25 pulling steps for each ion. (b) Averaged position 〈z〉 i = ∑ z g i (z)z of ions along the z-direction. ................................................................................. 128 VI.3 Three-dimensional structures of the investigated systems at different stages of the simulations, illustrating the axial motion of Na + (orange, a-e), K + (green, f-j) and NH + 4 (blue&white, k-o) ions in the GQ channel. The snapshots are at 5 ns of each pulling step. ....................................... 130 VI.4 Positions of water molecules and ions along the axis of motion at various stages of the dynamical simulations, pruned from three- dimensional snapshots of the entire systems. (a- g) Sub-system extracted from the simulation with Na + ions (orange spheres). (h-n) Sub- system extracted from the simulation with K + ions (green spheres). (o-t) Sub-system extracted from the simulation with NH + 4 ions (blue&white spheres). The arrows indicate directions of hydrogen bonds starting from oxygen and ending at hydrogen atoms. The snapshots are at 5 ns of each pulling step. ....................................... 133 xviii VI.5 Three-dimensional structures of portions of the simulated systems at selected snapshots in a ball&stick representation, to visualize the escape of water molecules from the GQ core. Water molecules that abandon the channel are represented in black. Na + and K + ions are at λ = –6 and –4 Å, respectively. Na + and K + ions are represented as orange and green spheres, respectively. Carbon, oxygen, nitrogen and hydrogen atoms are shown as cyan, red, blue and white spheres, respectively. ....................................................... 135 VI.6 Normalized histogram of RMSD evaluated over the four nearest-neighbor G-quartets to a given ion (16 guanine bases), with respect to the equilibrated structure. The RMSD is evaluated on the heavy atoms of the bases, excluding the backbone. Since we do not allow shrinking of the channel, the fluctuations do not represent the overall stability of the GQ due to different ions. The RMSDs are collected over 5000 frames during 5 ns for each λ. ................................................ 136 VI.7 Free-energy profiles for the motion of different ionic species in the GQ channel d[G 9 ] 4 , as a function of the parameter λ that scans the axis of the quadruplex. ............................................................................ 137 VII.1 (a) Sequence and atomistic structure of a DNA connecting two neighbor AuNPs via hexanethiol molecules. Here, C is Cytosine, G is Guanine, and hydrogen atoms are not shown. Blue letters represents sticky bases. (b) Body- centered-cubic (bcc) and (c) face-centered- cubic (fcc) supercrystals consisted of AuNP, DNA, hexanethiol, water and Na + . Yellow is AuNP and hexanethiol. Red ribbons are DNA. Green is Na + . Water is not shown. ................................................... 149 VII.2 Normalized histogram of hexanethiol lengths. The histograms are averaged over the last 100 frames (2 ns). ................................................................................... 152 xix VII.3 (a) Root mean square deviations (RMSDs) of a single DNA and all DNA molecules with respect to their B-DNA structure. (b) RMSDs of 4bp and 6bp duplexes with respect to their B-DNA structure. (c) Normalized distributions of base-base stacking distance in BCCS (thick line) and FCCS (thin line) during the last 2 ns. Every pair of successive bases in the same DNA strand is collected to measure the distance between the centroids of the 6- membered rings. ............................................................................... 155 VII.4 Histograms of number of hydrogen bonds per DNA in the BCCS (a) and FCCS (b). .............................................. 156 VII.5 Number of ions within 15 Å from DNA versus time. ................................................................................................. 157 VII.6 Radial distribution functions of (a) ion-ion and (b) ion-phosphorous. ........................................................................ 158 VII.7 Stress tensor σ xx versus strain ε xx . The error bars are 0.002 GPa for BCCS and 0.0016 GPa for FCCS. Young’s modulus Y is computed by ∂σ xx /∂ε xx . ........................................................................................... 163 VIII.1 Snapshots of the oleic acid (OA) membrane during simulations. (A) Initial crystalline membrane in γ-phase. (B) OA bilayer in NVT. (C-E) OA membrane in NPAT. (F) OA membrane in isotropic NPT. Red, white and cyan colors are oxygen, hydrogen and carbon atoms, respectively. Water is not shown. (G) Four water molecules surrounded by OAs. Hydrogen atoms bonded to carbon atoms are not shown. ........................................................................................ 172 VIII.2 (A) Snapshot of the OA membrane during simulations. Red (initially at the bottom) and blue (initially at the top) spheres are COOH head-groups. Cyan lines are carbon chains. (B) Snapshot of water in the system. Red and white spheres are oxygen and hydrogen atoms, respectively. (C) Density of water and COOH xx along the z-direction with a bin width of 2 Å. The data are averaged over 500 ns. .................................................. 174 VIII.3 Mean Square Displacements (MSD) of OAs in the xy-plane and z-normal to the membrane. The positions of carbon atoms of COOH head- groups are used to compute the MSDs. ........................................... 175 VIII.4 (A) Lateral pressure (P xx + P yy )/2 and normal pressure (–P zz ). (B) Stress profile π(z) = (P xx +P yy )/2 – P zz along the z-direction normal to the membrane with a bin width of 1 Å. ........................................... 177 VIII.5 Histograms of flip-flop (A) and traversing (B) durations of OAs and water, respectively. ....................................... 179 VIII.6 Accumulated numbers (NE) of water and OA molecules completely migrating across the membrane versus time (t finish ) (see Sec. II). ..................................... 180 VIII.7 (A-F) Snapshots of OA-flip-flopping and water-traversing events. The flip-flopping molecule is highlighted in blue. Its head-group and tail are denoted by balls and sticks, respectively. The surrounding water molecules of the head-group are found within a search of 3.5 Å. The traversing water molecule is highlighted in magenta. Red, cyan and white are oxygen, carbon and hydrogen atoms, respectively. The starting time for these events is 71.4 ns. The numbers denote the durations after the starting time. ...................................................................... 181 VIII.8 Averaged number of water molecules around COOH per flip-flopping OA versus z during migration. The bin width of slaps along the z- direction is 1 Å. Water molecules are counted within 3.5 Å of COOH through out 500 ns in each slap. No OA molecule is found for |z| > 36 Å ....................................................................................................... 183 xxi Abstract The fundamental laws of thermodynamics and statistical mechanics, and the deeper understandings of quantum mechanics have been rebuilt in recent years. It is partly because of the increasing power of computing resources nowadays, that allow shedding direct insights into the connections among the thermodynamics laws, statistical nature of our world, and the concepts of quantum mechanics, which have not yet been understood. But mostly, the most important reason, also the ultimate goal, is to understand the mechanisms, statistics and dynamics of biological systems, whose prevailing non- equilibrium processes violate the fundamental laws of thermodynamics, deviate from statistical mechanics, and finally complicate quantum effects. I believe that investigations of the fundamental laws of non-equilibrium dynamics will be a frontier research for at least several more decades. One of the fundamental laws was first discovered in 1997 by Jarzynski, so-called Jarzynski’s Equality. Since then, different proofs, alternative descriptions of Jarzynski’s Equality, and its further developments and applications have been quickly accumulated. My understandings, developments and applications of an alternative theory on Jarzynski’s Equality form the bulk of this dissertation. The core of my theory is based on stepwise pulling protocols, which provide deeper insight into how fluctuations of reaction coordinates contribute to free-energy changes along a reaction pathway. We find that the most optimal pathways, having the largest contribution to free-energy changes, follow the principle of detailed balance. This is a glimpse of why the principle of detailed balance xxii appears so powerful for sampling the most probable statistics of events. In a further development on Jarzynski’s Equality, I have been trying to use it in the formalism of diagonal entropy to propose a way to extract useful thermodynamic quantities such temperature, work and free-energy profiles from far-from-equilibrium ensembles, which can be used to characterize non-equilibrium dynamics. Furthermore, we have been applied the stepwise pulling protocols and Jarzynski’s Equality to investigate the ion selectivity of potassium channels via molecular dynamics simulations. The mechanism of the potassium ion selectivity has remained poorly understood for over fifty years, although a Nobel Prize was awarded to the discovery of the molecular structure of a potassium-selective channel in 2003. In one year of performing simulations, we were able to reproduce the major results of ion selectivity accumulated in fifty years. We have been even boldly going further to propose a new model for ion selectivity based on the structural rearrangement of the selectivity filter of potassium-selective KcsA channels. This structural rearrangement has never been shown to play such a pivotal role in selecting and conducting potassium ions, but effectively rejecting sodium ions. Using the stepwise pulling protocols, we are also able to estimate conductance for ion channels, which remains elusive by using other methods. In the light of ion channels, we have also investigated how a synthetic channel of telemeric G- quadruplex conducts different types of ions. These two studies on ion selectivity not only constitute an interesting part of this dissertation, but also will enable us to further explore a new set of ion-selectivity principles. xxiii Beside the focus of my dissertation, I used million-atom molecular dynamics simulations to investigate the mechanical properties of body-centered-cubic (BCCS) and face-centered-cubic (FCCS) supercrystals of DNA-functionalized gold nanoparticles. These properties are valuable for examining whether these supercrystals can be used in gene delivery and gene therapy. The formation of such ordered supercrystals is useful to protect DNAs or RNAs from being attacked and destroyed by enzymes in cells. I also performed all-atom molecular dynamics simulations to study a pure oleic acid (OA) membrane in water that results into a triple-layer structure. The simulations show that the trans-membrane movement of water and OAs is cooperative and correlated, and agrees with experimentally measured absorption rates. The simulation results support the idea that OA flip-flop is more favorable than transport by means of functional proteins. This study might provide further insight into how primitive cell membranes work, and how the interplay and correlation between water and fatty acids may occur. xxiv Preface I have been thinking about the concepts in thermodynamics, statistical mechanics and quantum mechanics for a long time since I started solving the first problem of heat transferred between two bottles of different-temperature water in a secondary school. But, I truly entered the field of thermodynamics and statistical mechanics in an unexpected way. It started with Jarzynski’s Equality. I was not supposed to alternatively prove Jarzynski’s Equality, but to use it in pulling two gold-nanoparticles apart, which are bonding via hydrogen bonds of two single-stranded DNAs attached to the gold- nanoparticles via hexanethiol molecules. I was partly not satisfied with that pulling protocol for using Jarzynski’s Equality, but mostly I wanted to know how relaxation times have effects on the convergence of free-energy profiles computed from Jarzynski’s Equality. That idea ended up into my first single-author paper. I have continued a journey to unfold the powers of Jarzynski’s Equality and stepwise pulling protocols since then. That journey has awarded me with great pleasures. To help readers to understand my journey of discovering Jarzynski’s Equality, and its further development and applications, the contents of this dissertation is organized as follows: Chapter I serves as my reflection on the fundamental laws and concepts in thermodynamics, statistical mechanics, and quantum mechanics. I pose some challenging questions, part of which I have been trying to answer in the following chapters. Some questions, I believe, remain not yet fully answered. And the biggest questions are still xxv ahead for anyone, who dares to break down to the deepest meanings of the fundamental laws found and not yet fully established. Chapter II is my first and single-author paper, which has opened up a number of questions and led to a series of papers. It deals with an alternative proof of Jarzynski’s Equality, which allows us to compute free energy differences from distributions of work. In molecular dynamics simulations, the traditional way of constructing work distributions is to perform as many pulling simulations as possible. But reliable work distributions are not always produced in a finite number of simulations. The computational cost of using JE is not less than other commonly used methods such as Thermodynamic Integration and Umbrella Sampling methods. Here we first show a different proof of JE based on the idea of step-wise pulling procedures that is efficient in computing free energies by using JE. The key point in our proof is that the processes of turning-on/off a harmonic potential to perform work are described by double Heaviside functions of time. We then show that the distributions of work performed by the potential can be easily generated from the distributions of a reaction coordinate along a pathway. Based on the proof, we propose sequential and parallel step-wise pulling protocols for generating work distributions that require suitable relaxation time at each pulling step. The criterion for reliable work distributions is that there must be sufficient mutual overlaps between the adjacent distributions of the reaction coordinate along the pathway. We arrive at an alternative formula (besides JE) to compute free energy differences from the averaged values of the reaction coordinate. The combination of JE and the alternative formula provides a viable way to determine the accuracy of computed free energy differences. For the stretching of xxvi a deca-alanine molecule, our approach requires 21 parallel simulations and relaxation time as small as 0.4 ns for each simulation to estimate free energy differences with an uncertainty of about 13%. Based on the theory presented in the previous chapter, in Chapter III, I present a generalization of Jarzynski’s Equality, applicable to quantum systems, relating discretized mechanical work and free-energy changes. The theory is also based on a step- wise pulling protocol. We find that work distribution functions can be constructed from fluctuations of a reaction coordinate along a reaction pathway in the step-wise pulling protocol. We also propose two sets of equations to determine the two possible optimal pathways that provide the most significant contributions to free-energy changes. We find that the transitions along these most optimal pathways, satisfying both sets of equations, follow the principle of detailed balance. We then test the theory by explicitly computing the free-energy changes for a one-dimensional quantum harmonic oscillator. This approach suggests a feasible way of measuring the fluctuations to experimentally test Jarzynski’s Equality in many-body systems, such as Bose-Einstein condensates. Two unanswered questions in Chapter III are that how we can maintain temperature and whether heat baths maintaining temperature for systems are necessary. In other words, can we estimate temperature for a set of non-equilibrium processes regardless of heat baths? This is mainly addressed in Chapter IV. In this chapter, we combine the formalisms of diagonal entropy and Jarzynski’s Equality to study the thermodynamic properties of closed quantum systems. Applying this approach to a quantum harmonic oscillator, the diagonal entropy offers a notion of temperature for xxvii closed systems away from equilibrium, and allows computing free-energy profiles. We also apply this approach to a hard-core boson lattice model, and discuss measures how to estimate temperature, entropy and measure work distribution functions. This technique offers a path to investigate non-equilibrium quantum dynamics by means of performing work through a series of quenches. To show how novel the step-wise pulling protocols and Jarzynski’s Equality are in practice, I apply them to investigate ion selectivity of ion channels in Chapter V. The ability of biological ion channels to conduct selected ions across cell membranes is critical for the survival of both animal and bacterial cells. Numerous investigations of ion selectivity have been conducted over more than 50 years, yet the mechanisms whereby the channels select certain ions and reject others are not well understood. Here we report a new application of Jarzynski’s Equality to investigate the mechanism of ion selectivity using non-equilibrium molecular dynamics simulations of Na + and K + ions moving through the KcsA channel. The simulations show that the selectivity filter of KcsA dynamically adapts and responds to the presence of the ions with structural rearrangements that are different for Na + and K + . These structural rearrangements facilitate entry of K + ions into the selectivity filter and permeation through the channel, and rejection of Na + ions. A mechanistic model of ion selectivity by this channel based on the results of the simulations relates the structural rearrangement of the selectivity filter to the differential dehydration of ions and multiple-ion occupancy and describes a mechanism to efficiently select and conduct K + . Estimates of the K + /Na + selectivity ratio and steady state ion conductance for KcsA from the simulations are in good quantitative xxviii agreement with experimental measurements. This model also accurately describes experimental observations of channel block by cytoplasmic Na + ions, the “punch through” relief of channel block by cytoplasmic positive voltages, and the knock-on mechanism of ion permeation. Chapter VI shows another application of the stepwise pulling protocol in molecular dynamics simulations to identify how a G-quadruplex selects and conducts Na + , K + and NH + 4 ions. By estimating the minimum free-energy changes of the ions along the central channel via Jarzynski’s Equality, we find that the G-quadruplex selectively binds the ionic species in the following order: K + > Na + > NH + 4 . This order implies that K + optimally fits the channel. However, the features of the free-energy profiles indicate that the channel conducts Na + best. These findings are in fair agreement with experiments on G-quadruplexes and reveal a profoundly different behavior from the prototype potassium-ion channel KcsA, which selects and conducts the same ionic species. We further show that the channel can also conduct a single file of water molecules and deform to leak water molecules. We propose a range for the conductance of the G-quadruplex. In Chapter VII, I use million-atom molecular dynamics simulations to study body- centered-cubic (BCCS) and face-centered-cubic (FCCS) supercrystals of DNA- functionalized gold nanoparticles, which are solvated with water and neutralized with sodium ions. The two supercrystals contain 2.77 and 5.05 million atoms. Having large numbers of DNAs and hexanethiols attached to 3nm-diameter gold nanoparticles, we observe smooth changes of the averaged DNA structures over simulation time. We find xxix that after 10 ns the DNA structures are different from the canonical B-DNA structures in terms of root mean square deviations, base-base stacking structures, and hydrogen bonds. We also examine ion distributions around DNAs, and estimate the melting-temperature increases for the supercrystals from the ion distributions, which are ΔT BCCS = 12.9 K and ΔT FCCS = 8.0 K. The radial distribution functions for the correlation between ions and DNA show that ions bind stronger in BCCS than FCCS. This correlation explains the higher melting-temperature increase in BCCS, and supports that there are more entropic effects in FCCS than BCCS. We also report the Young’s and bulk moduli of the supercrystals, which resemble those of water. The Possion ratios for both supercrystals (~ 0.39) are close to the ideal value (= 1/3). In Chapter VIII, I perform all-atom molecular dynamics simulations to study a pure oleic acid (OA) membrane in water that results into a triple-layer structure. We compute the pressure profiles to examine the hydrophobic and hydrophilic regions, and to estimate the surface tension (≈ 34.5 mN/m), which is similar to those of lipid membranes. We observe that the membrane of OAs having a large diffusion coefficient (0.4×10 –7 cm 2 /s) along the normal to the membrane is an ideal model to study oleic acid flip-flop. In the model, the membrane contains a middle layer serving as an intermediate for water and OAs to easily migrate (flip-flop) from one to other leaflets. Water molecules surrounding OA head-groups help to reduce the barriers at the hydrophobic interface to trigger flip-flop events. Within 500 ns we observe 175 flip-flop events of OAs and 305 events of water traversing the membrane. The ratio of water passing rate (k H2O = 0.673 ns –1 ) to OA flip-flop rate (k OA = 0.446 ns –1 ) is 3/2. The ratio of the totally correlated xxx water-OA events to the totally uncorrelated water-OA events, n cor /n uncor , is also 3/2. The probability of the totally and partially correlated events is 69%. The results indicate that the trans-membrane movement of water and OAs is cooperative and correlated, and agrees with experimentally measured absorption rates. They support the idea that OA flip-flop is more favorable than transport by means of functional proteins. This study might provide further insight into how primitive cell membranes work, and how the interplay and correlation between water and fatty acids may occur. Finally, Appendices are details of some derivations in Chapters II and III. 1 CHAPTER I Introduction: Mechanical Work, Free Energy and Temperature Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensive to everyone. -Albert Einstein- A classical definition of work in thermodynamics (1) is “the external forces applied to a body can do work on it, which is determined, according to the general rules of mechanics, by the products of these forces and the displacements which they cause.” Work W on a targeted system and absorbed heat Q are thermodynamic quantities, which are used to describe a total change of the internal energy, ΔE = W + Q, (I.1) which is the first law of thermodynamics. The law holds when the targeted system, environments and any external objects are classical and in equilibrium processes. The term ‘equilibrium’ is used to demonstrate any processes with infinitesimally slow changes of measurable physical quantities in macroscopically large systems. This law has been used for several centuries because it has produced a lot of useful information and passed so many tests in macroscopic systems. So the definitions of work and heat appear 2 naturally to every student and scientist. Few were dare to try to redefine those definitions and the first law of thermodynamics, and so the other laws of thermodynamics. However, the definitions do not allow us to compute work and heat in quantum mechanics, because the applied forces are not directly used in quantum mechanics, and because temperature and heat are not easy concepts to grasp through the fundamental equations of quantum mechanics. Thus, simply based on the definition of ‘equilibrium’, the fundamental law of thermodynamics is inapplicable in quantum mechanics because of the fast time scales (~ less than picoseconds) and microscopic dimensions. Furthermore, the fundamental law [Eq. (I.1)] might be also violated in biological systems because of adaptability and the time scales (~ nanoseconds to microseconds) in biochemical processes. It appears that work performed in biological might be in non-equilibrium processes but yield high efficiency. For example, the efficiency of auto engines is about 25–63% (2), which obeys the second law of thermodynamics, but the efficiency of molecular motors converting chemical energy into mechanical work through conformation changes and displacements can be 100% as in F 1 -ATPase (3, 4). Technically, W was never considered to have a distribution in statistical mechanics because such a distribution was meaningless. Any values of work larger than the minimum value of W contain no statistical meaning related to equilibrium processes. They simply imply non-equilibrium and irreversible processes. Only the minimum value of W does matter and can produce correct changes of thermodynamic quantities, such as free-energy changes ΔF (constant volume and temperature) and thermodynamic potential ΔΦ (constant temperature and pressure). Free-energy changes are more important than 3 work because it contains entropic effects, which are essential in many biological functions. However, the determination of free-energy changes is usually a challenge. Taking into account the facts that non-equilibrium work is important, and distribution functions of work might contain useful information, if any new laws of non- equilibrium dynamics can address those facts, they will significantly advance our understandings of any biological processes, and offer deeper insights into the relation among quantum mechanics, statistical mechanics and thermodynamics. Jarzynski (5) and Crooks (6) have first discovered such the first powerful laws for classical non- equilibrium dynamics (see following Chapters). Experiments have confirmed Jarzynski’s Equality and Crook’s Fluctuation Theorem in classical systems (7). Figure I.1 shows distributions of work in forward and backward pulling trajectories of unfolding and refolding an RNA molecule. Although the experiments were performed at different pulling rates (1.5 to 20 pNs –1 ) indicating non-equilibrium processes, they produce the same free-energy difference between the folding and unfolding states. An interesting opposition to the interpretation of the experimental results by Cohen and Mauzerall (8) argued that the uncertainty in measuring averaged work in those work distribution functions is only about 5%, thus it is not convincing to state that Jarzynski’s Equality and Crook’s fluctuation theorem hold in non-equilibrium processes. The debate went viral when some showed that the derivation of Jarzynski’s Equality was problematic (9), and Jarzynski’s Equality did not offer a efficient technique to compute free-energy changes (10). 4 Despite encountering such opposition at first, the fundamental laws of non- equilibrium dynamics have been quickly developed and applied in many disciplines. Main focuses of further developing more general ideas and principle for non-equilibrium dynamics are on (i) whether such laws hold in quantum mechanics, (ii) how to encode and extract information in non-equilibrium processes, and (iii) how to efficiently applied those laws to understanding biological systems. Most parts of this dissertation deal with items (i) and (iii). Figure I.1 Work distribution functions of unfolding and refolding an RNA molecules at different pulling rates. The thick black line indicates the free- energy difference between the unfolding and folding states. 5 The statistical mechanics, which can be used to shed direct insights into the concepts of heat, temperature, free energies and other thermodynamic quantities, bare a similarity with quantum mechanics. Both the statistical mechanics and quantum mechanics have the concept of probability of measuring physical quantities or randomness. On one hand, quantum mechanics is usually considered as the first level of randomness, which emerges from the dual particle-wave property of a microscopic system. On the other hand, statistical mechanics appear at the second level to account for the macroscopic observations on the microscopic system, which are assumed to inherit randomness. When combining both the statistical mechanics and quantum mechanics a measurable physical quantity is computed as A = Tr ˆ A ˆ Ω Tr ˆ Ω , (I.2) where ˆ Ω is a density operator characterizing the probability distribution of quantum states, and ˆ A is a physical operator. If the system is said in canonical distributions, ˆ Ω is equal to exp(−H /k B T), where H is a Hamiltonian operator, T is temperature and k B is Boltzmann’s constant. The questions that have been not yet fully answered are (1) how do quantum mechanics give rise to the exponential distribution and (2) how does temperature coefficient enter the exponential distribution? Temperature contains randomness and probability, so does a wave function of quantum mechanics. So can we compute temperature simply based on a single-wave function? The answer is trivially NO because a particle can be at different temperatures while it is still trapped in a ground state. 6 In classical statistical mechanics, temperature is computed via total kinetic energies. It emerges from the chaotic property of macroscopically large systems in an equilibrium state. In a formalism developed by Srednicki (11), temperature is computed from total energy, which is averaged in a quantum eigenstate. He hypothesized that every eigeinstate gives rise to thermalization into a thermal state at a certain value of temperature. If all energies computed from all eigenstates form a narrow spectrum, one can construct canonical ensembles from quantum states in micro-canonical ensembles, i.e., based on this hypothesis, temperature naturally enters exp(−H /k B T). However, self- thermalization of closed quantum systems remains an open question, and the meanings of work, free-energy and temperature in quantum mechanics are still unclear. 7 CHAPTER II Stepwise Pulling Protocols For Free Energy Evaluation II.1 Introduction In 1997, Jarzynski (5) showed that the free energy change from an initial configuration A to a final configuration B can be evaluated from finite-time non-equilibrium measurements by ΔF A → B = –β − 1 ln〈exp(–βW), (II.1) where β = 1/k B T, T is temperature, k B is Boltzmann constant, W is applied work and 〈…〉 denotes the average over all possible trajectories in which work is performed. Those configurations are associated with external control parameters λ. For simplicity, a system can be characterized by a single control parameter λ that is used to monitor pathways of a reaction coordinate. If using a harmonic potential to perform work on the system, parameter λ can be the center of the harmonic potential. The center-of-mass position x trapped by the potential in the direction along λ is considered as a reaction coordinate. By varying λ from λ 1 (corresponding to configuration A) at time t 1 to λ s (corresponding to configuration B) at time t s , one generates a trajectory in which W is measured from a force-versus-extension curve: W exp = f λ λ 1 λ s ∫ δλ, (II.2) 8 where fλ is an applied force measured at a value of λ whose increment is δλ. The work definition in Eq. (II.2) was used to validate Jarzynski’s equality (JE) in experiments of stretching a single biological molecule (7, 12, 13). Since the control parameter λ is varied, the Hamiltonian H(z,λ(t)) of the system accordingly changes with time, where z ≡ (q, p) denotes a point in the phase space with q and p being coordinates and momenta. According to Jarzynski, work W can be evaluated from the Hamiltonian by W H = ∂H(z,λ(t))/∂t [ ] dt t 1 t s ∫ (5, 12, 14–18). If one expresses W as H(z s ,λ s )−H(z 1 ,λ 1 ) , it is straightforward to prove JE by taking the average of exp(– βW) over the canonical ensemble of H(z 1 ,λ 1 )and applying the canonical transformation for initial z 1 to final z s . For stochastic processes, H(z s ,λ s )−H(z 1 ,λ 1 ) = W exp + Q st , where the discretized form of W exp is given by H(λ i+1 ,x i )−H(λ i ,x i ) [ ] i=1 s−1 ∑ and Q st is heat. To prove JE in this case where W is replaced with the discretized W exp , one must take into account the detailed balance for transition probabilities (16, 19). To further ascertain the theoretical validity of JE, Crooks (6) provided a general approach, so-called the Crooks fluctuation theorem in which JE can be considered as a special case. The Crooks fluctuation theorem states that the distribution of work ρ F (+W) in a forward process A→B and that of work ρ R (−W) in a reverse process A←B satisfy the symmetry relation, ρ F (+W)/ρ R (−W) = exp[β(W−ΔF A → B )]. Multiplying both sides of the relation with ρ R (−W)exp(−βW) and integrating over W, we obtain Eq. (II.1). Another approach to prove JE was realized by Hummer and Szabo (20–24). Their proof is based on the observation of phase space evolution established by Feynman-Kac theorem of statistical 9 mechanics, exp[−βΔF(t s )] = 〈exp(−βW H )〉 which is Eq. (II.1). Hummer and Szabo pointed out that reconstructing free energy profiles from JE must take into account the initial positions of an applied potential. Their conclusion related to the initial potential energy, however, cannot drawn from the former approaches. We report here that there is an alternative approach to prove JE (Section II.2). We also arrive at the same conclusion as of Hummer and Szabo (Section II.5). In order to use Eq. (II.1) for evaluating the free energy change, one has to construct work distributions from all possible force-versus-extension curves. The remarkable feature of JE is that work distributions can be computed from non- equilibrium processes. Because of this feature, JE provides a powerful tool to compute free energy changes in molecular dynamics simulations in which non-equilibrium processes are often encountered. The main challenge of using JE in molecular dynamics simulations is how to generate reliable work distributions in an efficient way (10, 14, 25– 29). It is computationally expensive to generate all possible trajectories for sampling rare small values of work W that dominate in the average of exp(–βW). One scheme to overcome the computational difficulty is to implement the Potential of Mean Force (PMF) method developed by Park et al. (25, 28) in steered molecular dynamics simulations (30). In the PMF method W is defined as −vk (x−λ 1 −vt)dt 0 t s ∫ , where v is a guiding velocity of a harmonic potential with spring constant k and x is a reaction coordinate. In this definition, parameter λ is linear with time t, i.e., λ ~ vt. In experiments, this definition of work is valid to estimate the free energy 10 change by Eq. (II.1). In molecular dynamics simulations, this work is observed to bias free energy changes. The PMF method remedies the biasing problem by utilizing the second order cumulant expansion (25, 28, 31). The PMF requires that k must be sufficiently large for the expansion. Then the computed free energy differences (FEDs) are unbiased even with a finite number of trajectories. Though the PMF method has been used, the efficiency and accuracy of computing FEDs in complicated systems remain unsatisfactory. The efficiency is defined as computational cost of evaluating FEDs for given error tolerance. Rodriguez-Gomez and Darve (10, 32) showed in the simulations of transfer of fluoromethane across a water- hexane interface that the accuracy of the PMF method is poor in comparison with their adaptive biasing force method. Bastug et al. (33, 34) showed that FEDs computed from a finite number of pulling trajectories are biased. The cost of evaluating the FEDs using the PMF method is higher than that of using Umbrella Sampling with Weighted Histogram Analysis method (35–37). Oberhofer et al. (26) analyzed that the efficiency of simulations based on JE cannot compete with the ones of the Umbrella Sampling and Thermodynamic Integration (31, 38) methods. Recently, Hernandez and colleagues (39) developed an adaptive steered molecular dynamics for improving the accuracy of the PMF method. In this adaptive scheme, instead of applying JE for single pulling trajectories they applied JE in series of shorter steps of the pulling trajectories. At each c-th step, a system is allowed to relax for a certain amount of time before performing next pulling step. The configuration for next pulling step is chosen so that it minimizes the work difference, ΔF c −W α c (t) , where ΔF c 11 and W α c (t) are free energy change and work done at c-th step in α-th trajectory at time t, respectively. This scheme essentially suggests that rare small work distributions can be generated by breaking a long pathway into shorter ones and letting systems relax to minimize the biasing effects of an applied potential on free energy changes. Similar to the original PMF method, this scheme fails if work distributions are not trivial Gaussian functions; hence the second-order cumulant expansion cannot be used. In this article we propose a different approach to prove JE and to compute free energy changes more efficiently than the PMF methods. We take the definition of W H as a starting point and treat control parameter λ and time t in different manners. We introduce double Heaviside functions of time t, divide a pathway into a series of steps and take into account relaxation time at each step. The Heaviside functions are used to describe the procedures of turning-on/off a harmonic potential to perform work. A series of steps and relaxation time are crucial to generate rare small values of work. The approach results in an identity 〈exp(−βW H )〉 = 1 [Eq. (II.10)] that helps to prove JE, ΔF A → B = –β − 1 ln〈exp(– βW exp )〉 [Eq. (II.12)]. Our theory also suggests an approximation to compute FEDs (besides JE) from the averaged values of a reaction coordinate [Eq. (II.14)]. This approximation resembles the Thermodynamic Integration. Based on the theory, we propose sequential and parallel step-wise pulling protocols to reconstruct work distributions. We show that it is straightforward to produce work distributions from the distributions of a reaction coordinate without any adaptive scheme. Based on the simulation results in a test case, we provide a criterion that requires 12 sufficient overlaps between the adjacent distributions of a reaction coordinate for ensuring reliable work distributions. In comparison with the original and adaptive PMF methods, our scheme of generating work distributions and computing FEDs does not require large values of k and the assumption of Gaussian distributions of work. Given that relaxation time is long enough (several ns) and the number of discretized pulling steps is adequate, FEDs can be accurately computed. Our simulation results show that the parallel-pulling protocol speeds up free energy calculations with an acceptable accuracy. Our article is organized as follows: we present the theory in Section II.2; Section II.3 highlights two methods based on the theory to compute FEDs and describes sequential and parallel pulling protocols; the theory is tested in the simulations of unfolding a deca-alanine molecule in Section II.4; Section II.5 provides discussions on the theory and methods. II.2 Theory Let us consider a system of N particles described by a time-independent Hamiltonian H o (  p 3N ,  r 3N ), where  p 3N and  r 3N are the 3N dimensional momenta and coordinates, respectively. The system has a canonical ensemble with temperature T. An external harmonic potential U(x,λ 1 ) = k(x – λ 1 ) 2 /2 can be applied to a set of trapped particles, whose center-of-mass position along x-direction is defined as a reaction coordinate x. Here λ 1 is the center of the harmonic potential (control parameter) and k is the spring constant. We suppose that there is no coupling between the harmonic potential with the 13 particles’ momenta. Then our partition functions Z(λ i ,k) (i ≥ 1) can be simply expressed in terms of spatial coordinates (  r 3N−1 ,x) for calculating free energy differences (FEDs). From time t 0 to t 1 , we apply the harmonic potential to the system. One can in principle take the infinity limit for t 0 and t 1 . As a result, the coupling Hamiltonian has the following formula: H(λ 1 ,x)=H o (  r 3N−1 ,x)+ k 2 (x−λ 1 ) 2 θ(t−t 0 )θ(t 1 −t), (II.3) where θ(t) is a Heaviside step function. Here we have omitted momenta in H o for simplicity. Then the work applied to the system from a specific state at t = t 0 to a final state is computed as W H = ∂H(λ 1 ,x) ∂t dt t 0 t 1 ∫ = k 2 [(x 0 −λ 1 ) 2 −(x 1 −λ 1 ) 2 ], (II.4) where x 0 is an initial value at time t 0 of the reaction coordinate belonging to the ensemble withH o (  r 3N−1 ,x 0 ) and x 1 is any final value of the reaction coordinate at t 1 ≥ t 0 belonging to the ensemble with H(λ 1 ,x 1 ). The applied force exerted on the reaction coordinate along x-direction at time t 1 is f(x 1 ,λ 1 ) = –∂U(x 1 ,λ 1 )/∂x 1 . Given that the canonical ensemble of x 1 is generated after turning on the harmonic potential, one can take the average of exp(–βW H ) (β = 1/k B T) over the canonical ensemble (see Appendix A) to obtain exp[β k 2 (x 1 −λ 1 ) 2 ] (x 1 ,λ 1 ,k) =exp[βΔF(λ 1 ,k)], (II.5) 14 where (x 1 ,λ 1 ,k) represents all possible points with Hamiltonian H(λ 1 ,x 1 ) in phase space and ΔF(λ i=1 ,k) = F(λ i=1 ,k)–F 0 = –β − 1 ln[Z(λ i=1 ,k)] + β − 1 ln[Z(0)]. The partition functions are Z(λ i ,k) = ∫ d  r 3N−1 dx i exp[–βH(λ i ,x i )] and Z(0) = ∫ d  r 3N−1 dx 0 exp[–β H o (  r 3N−1 ,x 0 ) ]. ΔF(λ i=1 ,k) is the FED between the configurations with and without the harmonic potential. Alternatively, given the canonical ensemble of x 0 the average can be taken over all x 0 instead of x 1 to evaluate the same FED: ΔF(λ 1 ,k)=−β −1 ln exp[−β k 2 (x 0 −λ 1 ) 2 ] (x 0 ,k=0) , (II.6) where (x 0 ,k=0) represents all possible points with Hamiltonian H o (  r 3N−1 ,x 0 ) in phase space. From Eqs. (II.5) and (II.6) along with inequality 〈exp(–βW H )〉 ≥ exp(–β〈W H 〉) (40), we have the lower and upper bounds to ΔF(λ 1 ,k): k 2 (x 1 −λ 1 ) 2 x 1 ,λ 1 ,k ( ) ≤ΔF(λ 1 ,k)≤ k 2 (x 0 −λ 1 ) 2 (x 0 ,k=0) . (II.7) Now, we perform a series of steps with λ i (i = 1, 2…s) to pull the reaction coordinate by moving the center of the harmonic potential from λ 1 to λ s . At each step with λ i the system is relaxed from time t i–1 to t i . Then the center is instantaneously shifted from λ i to λ i+1 at time t i . As a result, the total work (see Appendix B) is W total = k 2 (x i−1 −λ i ) 2 −(x i −λ i ) 2 " # $ % i=1 s ∑ = k 2 (x o −λ 1 ) 2 −(x s −λ s ) 2 " # $ % +W mech , (II.8) where mechanical work W mech in each generated trajectory is defined by 15 W mech = k 2 (λ i+1 −λ i )( i=1 s−1 ∑ λ i+1 +λ i −2x i )≈ f λ δλ λ 1 λ s ∫ , (II.9) where fλ = ∂U(x,λ)/∂λ for a sufficiently small increment δλ. When s is equal to 1, W mech is zero and Eqs. (II.5) or (II.6) should be used to estimate FEDs. Equation (II.9) can be expressed as H(λ i+1 ,x i )−H(λ i ,x i ) [ ] i=1 s−1 ∑ that was used to prove Jarzynski’s equality (JE) in stochastic processes (16, 19). It should be noted that the canonical ensembles with H(λ i ,x i ) for different λ i are independent of one another. Hence, one can take average of exp(–βW total ) over all the canonical ensembles of x 0 , x 1 .…x s at the same time to arrive at the following identity: exp(−βW total ) (x 0 ,x 1 ,...,x s ) =1. (II.10) Alternatively we first average exp(–βW total ) over the ensemble of x 0 with H o (  r 3N−1 ,x 0 ) and the ensemble of x s with H(λ s ,x s )=H o (  r 3N−1 ,x s )+U(x s ,λ s ) to obtain exp(−βW total ) x 0 ,x s =exp β F(λ s ,k)−F(λ 1 ,k) [ ] { } ×exp(−βW mech ). (II.11) We then average the left hand side of Eq. (II.11) over all the rest of x i and make use of the equality Eq. (II.10) to arrive at exp(−βW mech ) FVE =exp[−βΔF JE (λ s ,λ 1 ,k)], (II.12) where ΔF JE (λ s ,λ 1 ,k) = F(λ s ,k) – F(λ 1 ,k) with F(λ i ,k) = –(1/β)ln[Z(λ i ,k)] and 〈…〉 FVE is the average over all values of W mech measured from force-versus-extension (FVE) curves (detailed derivation of Eq. (II.12) is given in Appendix B). Equation (II.12) is identical to JE, Eq. (II.1). 16 It is noted that the distributions of x i have the property of randomness due to thermal fluctuations. According to the probability theory of independent random numbers (41), W mech ’s distribution ρ(W mech ) is proportional to ρ(x 1 )ρ(x 2 )...ρ(x s–1 ) where ρ(x i ) are x i ’s distributions. The relation between ρ(W mech ) and ρ(x i ) means that ρ(W mech ) can be constructed if a set of ρ(x i ) is known. Since the reaction coordinate is trapped by the potential, ρ(x i ) can be approximated as exp[–βk(x i – 〈x i 〉) 2 /(2γ i 2 )], where 〈x i 〉 is the averaged position of the reaction coordinate at i th pulling step and γ i 2 is kσ i 2 /k B T with σ i equal to the standard deviation of x i ’s distribution. Thus ρ(W mech ) can be expressed in terms of x i , 〈x i 〉 and γ i 2 with i from 1 to s–1. Given the distribution ρ(W mech ) and based on Eq. (II.12) we derive a simple relationship between Gaussian-approximated FED ΔF G (λ s ,λ 1 ,k) and 〈x i 〉 (see Appendix B): ΔF G (λ s ,λ 1 ,k)= k(λ s −λ 1 ) 2 2s (1−γ i 2 ) i=1 s−1 ∑ s +k(λ s −λ 1 ) (λ i − x i ) i=1 s−1 ∑ s . (II.13) The first term in Eq. (II.13) vanishes as s goes to infinity, i.e., infinitely slow pulling limit. Then the FED ΔF G (λ s ,λ 1 ,k) can be determined from the second term, ΔF fluct (λ s ,λ 1 ,k)=k(λ s −λ 1 ) (λ i − x i ) i=1 s−1 ∑ s , (II.14) which only depends on the average of the differences between λ i and 〈x i 〉. In that limit, the right hand side of Eq. (II.14) becomes the Thermodynamic Integration 17 ∂H(λ,x)/∂λ λ dλ λ 1 λ s ∫ . This limit implies that the required relaxation time τ i = t i – t i–1 at each λ i can be arbitrarily small. Moreover, if all γ i 2 are unity, ΔF G (λ s ,λ 1 ,k) can also be determined from ΔF fluct (λ s ,λ 1 ,k) even with a finite value of s. γ i 2 equal to unity indicates that x i ’s distributions resemble the canonical distributions of the system (~ exp[–βH(λ i , x i )]). The condition for γ i 2 equal to unity is satisfied when τ i is large (the assumption for canonical distributions). These two limiting cases of τ i suggest that the first term in Eq. (II.13) is not important and Eq. (II.14) can be used to evaluate FEDs with finite s and finite τ i . II.3 Protocols Based on the theory in the previous section, we propose two methods for free energy calculations: (i) from the distributions of mechanical work W mech that is defined by Eq. (II.9) used for JE, Eq. (II.12); (ii) from averaged values of 〈x i 〉 used for Eq. (II.14). According to JE, or Eq. (II.12), it is necessary to generate as many pulling trajectories as possible to construct work distributions. As illustrated by Fig. II.1a, a conventional protocol is that N p pulling trajectories from configuration A to configuration B are performed in parallel. Unfortunately, there has been no criterion to determine a minimum of N p . Equation (II.5) supports that a free energy difference (FED) ΔF(λ i ,k) can be calculated when a complete canonical ensemble of x i is sampled regardless of pathways from x o (42). But using Eq. (II.5) requires large relaxation time τ i . 18 Alternatively, according to the expression of W mech [Eq. (II.9)] work distributions can be produced from the distributions of x i . This expression implies that rare small values of work are not available if the increments λ i+1 –λ i are larger than the magnitude of x i ’s fluctuations, which is defined as the distance from the center of x i ’s distribution to its right end. Therefore, the adjacent distributions of x i must mutually overlap to generate rare small values of work. Figure II.1 Schematic illustration for (a) original parallel pulling (arrowed arcs) protocol (b) our proposed sequential (series of circles) and parallel pulling (arrowed arcs) protocols. A reaction coordinate, x o , represents any state without virtual harmonic potentials. The arcs with arrows represent parallel pulling trajectories. A and B are initial and final configurations, respectively. N p is a number of trajectories. The circles represent sets of states or distributions of the reaction coordinate. Shaded areas are mutual overlapping states. Index i denotes the ordered number of pulling and runs from 1 to s, where indices 1 and s also represent the initial and final configurations, respectively. Guided by the argument we propose two protocols (illustrated in Fig. II.1b) to construct reliable work distributions with a small number of pulling trajectories and small values of τ i : 19 (i) Sequential pulling protocol (SPP): given a pathway that can be characterized by a single control parameter λ (the center of a harmonic potential) in a system, we assign λ 1 to the configuration A, λ s to the configuration B. We then divide the pathway into s–2 intermediate configurations (steps) that are characterized by different λ i . We let the harmonic potential pull the reaction coordinate by sequentially assigning λ to be λ i with i from 1 to s in a single simulation. At each step corresponding to λ i , the system is relaxed for time τ i to collect distributions of x i . All possible values of x i are used to generate work distributions whose algorithm is given in Appendix C. (ii) Parallel pulling protocol (PPP): given a pathway and a set of different λ i corresponding to s targeted configurations as in (i), s pulling simulations are independently carried out to pull the reaction coordinate from a state x o to the targeted configurations. In each simulation, the center of a harmonic potential λ is independently assigned to be λ i . Each simulation is run for adequate relaxation time τ i to collect distributions of x i corresponding to λ i . The same procedure to construct work distributions from x i is given in Appendix C. Once work distributions are constructed, it is straightforward to evaluate FEDs based on JE. By measuring all averaged values of x i corresponding to λ i , FEDs are also estimated by Eq. (II.14). 20 II.4 Implementation and Testing In this section we test the two protocols on an exemplary system: helix-coil transition of deca-alanine in vacuum at T = 300 K (see Ref. (25) for more details of simulation setups). NAMD2 package (43) and CHARMM22 force fields (44) are used here. We generate work distributions and compute free energy profiles by Eqs. (II.12) and (II.14) in the sequential and parallel pulling protocols. We also investigate the effects of spring constants k and relaxation time τ i = τ for all pulling steps. II.4.1 Sequential Pulling Protocol (SPP) The simulation setups are the same as in Ref. (25). One end of the molecule is kept fixed at the origin. The other end is sequentially pulled by a harmonic potential having the spring constant k = 7.2 kcal/mol/Å 2 (we use this potential through the paper otherwise mentioned). The position along the pulling direction of the pulled end is considered as a reaction coordinate. We increase λ by Δλ = 0.5 Å from λ 1 = 13 to λ s = 33 Å (s = 41). At each step with λ i (i = 1, 2…s) the system is relaxed for τ = 10 ns. We record the position of the pulled end x i every 0.1 ps at each relaxation step. Some of distributions x i are shown in Fig. II.2a. Figure II.2b shows the initial (λ = 13 Å), transition (λ = 25 Å) and final (λ = 33 Å) configurations. We use Least Square Fitting method to obtain γ i 2 = kσ i 2 /k B T as shown in Fig. II.2c, where σ i is equal to the standard deviation of x i ’s distributions and k B is Boltzmann constant. The dimensionless values of γ i 2 fall in the range of 1.1 to 1.4 and are peaked at λ = 26.5 Å. 21 We divide the ranges of x i and W mech into small bins to construct work distributions (see Appendix C). The range for x i is from 0 to 50.0 Å with a bin width δx = 0.001 Å. The range for W mech is from -150 to 150.0 kcal/mol with a bin width δW = 0.01 kcal/mol. The work distributions change as λ increases (Fig. II.3a). Plugging these distributions and measured 〈x i 〉 into Eqs. (9), (12), (13) and (14), we compute applied mechanical work W mech , ΔF JE , ΔF G and ΔF fluct as functions of λ, respectively. The referenced free energy for those free energy differences (FEDs) is F(λ 1 ,k) that is used through out the paper. λ and k are omitted in the functions for a simple notation. The Potential of Mean Force (PMF) method (25) is used to generate the exact PMF profile with the pulling speed of v = 0.1 Å/ns. Figure II.2 (a) Normalized distribution of x i . The numbers indicate the pulling steps i. (b) Configurations of deca-alanine at λ = 13, 25 and 33 Å. The two biggest balls are fixed and pulled ends. (c) Dimensionless quantity γ i 2 versus λ. Figure II.3b shows that the profiles of ΔF JE and ΔF fluct are in a good agreement with the PMF profile. At the minimum position (λ ~ 15 Å) ΔF JE is the same as that of the PMF profile (~ –2.1 kcal/mol) that is smaller than ΔF fluct by 0.5 kcal/mol. At λ ≥ 25 Å (kink position), the values of the PMF profile are larger than ΔF fluct by 1.0 kcal/mol, but 22 less than ΔF JE (≥ 16 kcal/mol) by 0.5 kcal/mol. Free energies ΔF G are smaller than the others because the effects of finite s and τ cause a significant contribution to ΔF G . The smaller values of ΔF G and the agreement between the profile of ΔF fluct and the PMF profile confirm the argument at the end of Section II.2 that the first term in Eq. (II.13) is negligible. Figure II.3 (a) Normalized work distribution. The numbers indicate the pulling steps i. (b) Mechanical work (squares) and free energies versus λ. PMF (crosses), ΔF JE (empty circles), ΔF G (triangles) and ΔF fluct (dots) are free energies computed by the PMF method, Eq. (II.12), Eq. (II.13) and Eq. (II.14), respectively. The variance of ΔF JE can be estimated by σ W 2 /Q+β 2 σ W 4 /2(Q−1) (10, 45), where σ W is the standard deviations of work distributions and Q is a number of bins that have non-zero values. Then the standard deviations of ΔF JE are about 6% of ΔF JE . But the uncertainty of ΔF fluct cannot be computed as (k B T/k) 1/2 γ i i=1 s−1 ∑ / (λ i − x i ) i=1 s−1 ∑ that are unreasonably large (> 100%). Since ΔF fluct is an approximation from ΔF JE , its uncertainty should be comparable to that of ΔF JE . The uncertainty of ΔF JE can be reduced when Q is larger. But we observe that the difference between ΔF JE and ΔF fluct is not changed at 23 larger values of Q. This suggests that the difference can be chosen as an uncertainty of the FEDs (ΔF fluct and ΔF JE ) if it is larger than that of ΔF JE . It is noted that the highest value of γ i 2 occurs at the same position where the free energy profiles are observed to be a kink (λ ~ 26.5 Å). The variation of γ i 2 indicates how the approximated widths of x i ’s distributions relatively change along a pathway. Therefore, any changes of γ i 2 can be used to examine how the overlapping portions of x i ’s distributions vary. II.4.2 Parallel Pulling Protocol (PPP) The agreement among the FEDs computed by the exact PMF method, ΔF fluct [Eq. (II.14)] and ΔF JE [Eq. (II.12)] suggests that one can compute FEDs if the values of 〈x i 〉 and their distributions are available in any pulling protocols. To verify this observation, we perform 21 10ns-simulations in parallel of stretching the molecule from the same initial state x o . In the simulations 21 values of λ i are assigned to be 13, 14…33 Å (Δλ = 1 Å and s = 21). Each simulation characterized by a single value of λ i is independent of the others. In each simulation we record values of x i in every 0.1 ps for constructing work distributions. We use τ = 10 ns. 24 Figure II.4 (a) Free energies versus λ. ΔF fluct (dots) are in the SPP with Δλ = 0.5 Å. ΔF fluct (triangles) and ΔF JE (squares) are in the PPP with Δλ = 1 Å. (b) Accumulating forces ΔF fluct /(λ s – λ 1 ) = k (λ i − x i )/s i=1 s−1 ∑ versus λ in both protocols. The force at s = 1 is set to zero. The computed profile of ΔF fluct (triangles) is plotted in Fig. II.4a together with ΔF fluct (dots) obtained in Section II.4.1. Figure II.4a shows that ΔF fluct in both SPP and PPP have the same minimum FED. For λ > 15 Å, the profile of ΔF fluct in the PPP is gradually shifted below the one in the SPP by an amount of 1.0 kcal/mol. With the data of x i we generate work distributions and evaluate ΔF JE as also plotted in Fig. II.4a. The minimum value of ΔF JE is slightly smaller than those of ΔF fluct . The values of ΔF JE in the PPP are 2.0 kcal/mol larger than those of ΔF fluct in the SPP at λ ≥ 25 Å. The uncertainty of ΔF JE is about 6%. 25 Figure II.5 Free energy profiles of (a) ΔF fluct and (b) ΔF JE with different spring constants k=1.0 (dots), 7.2 (empty boxes) and 50.0 (triangles) kcal/mol/Å 2 . (c) Normalized distributions of x i at step i = 20 (gray) and 21 (black) in the three cases. The insets in (a) are zoomed at the minimum and kink positions. The inset in (c) shows γ i 2 versus λ. We observe that the separation between the profiles of ΔF JE and ΔF fluct becomes smaller as Δλ is changed from 1.0 Å (see Fig. II.4a) to 0.5 Å (see Fig. II.3b). The more accurate profile of ΔF fluct in the SPP lies between the profiles of ΔF JE and ΔF fluct in the PPP. This suggests that the ΔF JE and ΔF fluct in the PPP can be combined to estimate an accurate free energy profile of the system. The free energies of the combined profile can be defined as (ΔF JE +ΔF fluct )/2. Subsequently, the uncertainty of these free energies would be equal to (ΔF JE − ΔF fluct )/2 that is at most 1.5 kcal/mol (~ 10%) as Δλ = 1.0 Å. In terms 26 of computational cost, the PPP is two times faster and more efficient than the SPP given large computational resources. It is worth comparing the averaged applied forces in both SPP and PPP. The factor € k (λ i − x i )/s i=1 s−1 ∑ in Eq. (II.14) can be interpreted as averaged accumulating forces. Figure II.4b shows that the accumulating forces in both the pulling protocols are almost the same. In order to examine the effect of the magnitude of increment Δλ on the free energies, we collect 11 distributions of x i with Δλ = 2 Å (s = 11, λ i = 13, 15…33 Å) out of the 21 distributions to construct corresponding work distributions. As a result, the FEDs at λ = 25 Å computed from this data set using JE and Eq. (II.14) are about 31.8 and 11.4 kcal/mol, respectively. With these data we overestimate the FED if using JE and underestimate it if using Eq. (II.14). These FEDs clearly indicate that the data set with Δλ = 2 Å cannot be used to construct correct distributions of work and measure a sufficient history of 〈x i 〉. This observation is consistent with the implication of the expression of W mech that the increments Δλ = λ i – λ i–1 should not exceed the magnitude of x i ’s fluctuations, which is defined as the distance from the center of x i ’s distribution to its right end; otherwise rare small values of W mech are not available. Figure II.2a shows that the magnitude of x i ’s fluctuations is about 1.0 Å. Consequently Δλ should not be greater than 1.0 Å. It means that s can be finite but not arbitrarily small. 27 II.4.3 Effects of Spring Constant k In order to investigate how FEDs vary in response to different spring constants k, we perform two sequential pulling simulations in which k = 1 and 50 kcal/mol/Å 2 , Δλ = 0.5 Å and τ = 10 ns (46). We observe that the sets of 〈x i 〉 in all the three cases are not the same. In the case of k = 1 kcal/mol/Å 2 〈x i 〉 are not as perfectly linear with λ as in the other cases. But the free energy profiles of ΔF fluct as plotted in Fig. II.5a are in a good agreement with one another. The minimum FEDs are –0.9, –1.6 and –1.9 kcal/mol for k = 1, 7.2 and 50 kcal/mol/Å 2 , respectively. At λ ≥ 25 Å, the three curves converge and shift by less than 1.0 kcal/mol from each other with the same order as at the minimum position (see the insets in Fig. II.5a). The smallest spring constant results in the higher values of the FEDs at the minimum and kink positions (λ ~ 15 and 26 Å), while the strongest one gives rise to the smaller values. With the uncertainty of 1.0 kcal/mol, we confirm that those important values evaluated by Eq. (II.14) are independent of spring constants in the range from 1 to 50 kcal/mol/Å 2 , even though there is noticeable lowering of the FEDs at k = 1 kcal/mol/Å 2 along the pathway from the minimum to the kink positions. 28 Figure II.6 (a) Free energies ΔF fluct (dots), ΔF JE (full triangles) and ΔF com = (ΔF fluct +ΔF JE )/2 (dashed line) at λ = 26 Å versus relaxation time τ. (b) Normalized work distributions at λ = 26 Å in a range from 0 to 25 kcal/mol at τ = 0.4 (dots), 0.6 (empty boxes) and 4.0 (empty triangles) ns. The inset in (b) shows the overall work distributions at the three values of τ. As computed by JE (47), the free energy profiles of ΔF JE with k = 1 and 7.2 kcal/mol/Å 2 agree with each other, whereas the profile with k = 50 kcal/mol/Å 2 is noticeably higher than the others (see Fig. II.5b). To explain the significant distinction among the profiles, we look at the distributions of x i at step i = 20 and 21 (λ ~ 23 Å) where the transition is about to occur. Figure II.5c shows that the two x i ’s distributions at k = 1 kcal/mol/Å 2 have the largest mutual overlapping, whereas at k = 50 kcal/mol/Å 2 they have much less mutual overlapping. The values of γ i 2 in the inset of Fig. II.5c indicate how x i ’s distributions relatively change along the pathway. The smaller values of γ i 2 /k indicate narrower widths of x i ’s distributions since the same increment Δλ = 0.5 Å is used. The properties of those distributions suggest an explanation for the distinction based on the expression of W mech in Eq. (II.9). With large spring constants, the harmonic potential confines the reaction coordinate in such narrow regions that its distributions have little mutual overlapping. Consequently, large k reduces the number of rare small 29 values of work, thus increases the magnitude of computed FEDs. We notice that the values of ΔF JE can also noticeably increase if τ is too small. Therefore it is essential to investigate the effects of τ on the FEDs. II.4.4 Effects of Relaxation Time τ We repeat the PPP in Section II.4.2 by reducing relaxation time τ from 10 ns to 0.01 ns to optimize required simulation time for the test case. Spring constant k = 7.2 kcal/mol/Å 2 and increment Δλ = 1.0 Å (s = 21) are used in the simulations. We evaluate FEDs ΔF fluct and ΔF JE at λ = 26 Å (kink position). Here we consider those FEDs as functions of τ. Figure II.6a shows how the FEDs change in response to variation of τ. The curve of ΔF JE is noticeably higher than the curve of ΔF fluct that starts converging at τ = 0.4 ns. The convergence of ΔF fluct is due to the strong applied harmonic potential that makes the corresponding sets of 〈x i 〉 not essentially change even at τ = 0.4 ns. But ΔF JE does not converge as smoothly as ΔF fluct does. The values of ΔF JE have small peaks at τ = 0.6 and 6.0 ns. These peaks indicate some important changes in the work distributions. For example, in Fig. II.6b the work distribution at τ = 0.6 ns in a range from 0 to 25 kcal/mol is smaller than those at τ = 0.4 and 4.0 ns. These values of work have major contributions to ΔF JE . The work having smaller distributions gives rise to larger ΔF JE . Nevertheless, the inset of Fig. II.6b shows no visible deviation among their overall distributions. In addition, we recall that the separation between the profiles of ΔF JE and ΔF fluct reduces as we change Δλ from 1.0 Å to 0.5 Å (see Figs. II.3b and II.4a). It means that 30 ΔF JE and ΔF fluct give us an upper bound and a lower bound to the FED, respectively. This suggests that an accurate value of the FED at λ = 26 Å would be somewhere between the ΔF JE and ΔF fluct as shown by the dashed line in Fig. II.6a. We define a combined FED as ΔF com = (ΔF JE + ΔF fluct )/2. This combined FED starts converging at τ = 0.4 ns. Therefore, the uncertainty of ΔF com is equal to (ΔF JE – ΔF fluct )/2 or ~ 2/16 ~ 13% at τ = 0.4 ns, which is larger than that of ΔF JE (~ 8%). II.5 Discussions and Conclusions The theory contains a different treatment of the correlation between control parameter λ and time t. We suggest that λ in general should rather be a time-independent parameter characterizing perturbed states. Finite relaxation time τ is required to let a system evolve into those perturbed states. To account for the overall change of parameter λ with time, one can use double Heaviside functions of time t. The introduction of the double Heaviside functions describes the physics of how gradually a system absorbs W H from an applied harmonic potential over time (see Section II.2). The use of a single Heaviside function of time does not correctly reflect the process of performing work by using harmonic potentials (9, 48, 49). It should be clearly noted here that our definition of work by Eq. (II.9) agrees with the integral expression of work, −vk (x−λ 1 −vt)dt 0 t s ∫ or in general ∂H(z,λ)/∂λ [ ] (dλ /dt)dt t 1 t s ∫ where v is a guiding velocity, k is spring constant, λ 1 is an initial fixed value of λ, x is a reaction coordinate, H(z,λ) is a total Hamiltonian with z 31 being a point in the phase space, and λ is here a function of time. The discretized expression of the work in Eq. (II.9) is the same as the one used in stochastic processes (16, 19). Using the discretized expression we proved JE without adopting the concept of detailed balance for transition probabilities. For a large number of pulling steps, λ can be expressed as λ 1 + vt in Eq. (II.9) to recover the integral expression of work used in the steered molecular dynamics simulations (25, 28, 39). The resulting relationship between Hamiltonian-related work W total = W H = ∂H(z,λ(t))/∂t [ ] dt t 1 t s ∫ and W mech is given by Eq. (II.8). The theory suggests that W total = W H ≠ ∂H(z,λ) /∂λ [ ] (dλ /dt)dt=W t 1 t s ∫ ≡ W mech . As indicated by the identity, Eq. (II.10), W total should not appear in Eq. (II.1) as W. This identity suggests that one can add any heat bath effects (8) to systems, which do not influence applied potentials, and normalize the effects to unity. Then the free energy changes can be always computed from the distributions of W mech in non-equilibrium processes regardless of heat bath effects. Jarzynski and Crooks using different approaches arrived at the same conclusion (5, 6, 14–19, 49). The identity appears to not agree with the expression derived by Hummer and Szabo (23) using Feynman-Kac theorem, exp[−βΔF(t s )]=〈exp(−β [∂H(z,t)/∂t]dt 0 t s ∫ )〉 where 〈…〉 denotes the average over all possible trajectories. The difference stems from our introduction of the double Heaviside functions to the coupling Hamiltonian H(z,t) (Section II.2). It is noted that whenever an applied harmonic potential is turned on, the free energy of a system always increases by ΔF(λ 1 ,k) as indicated by the inequalities (II.7). 32 We define G(λ s ,λ 1 ) = ΔF JE (λ s ,λ 1 ,k) – ΔF(λ 1 ,k) as a state free energy function. From Eqs. (II.5) and (II.12), G(λ s ,λ 1 ) can be written as G(λ s ,λ 1 )=−β −1 ln{〈exp[−β( f dx x 0 x 1 ∫ −U(x 0 ,λ 1 ))]〉 (x 1 ,λ 1 ,k) ×〈exp(−β f dλ λ 1 λ s ∫ )〉 FVE }, (II.15) where f is –∂U(x,λ 1 )/∂x, x 0 is an initial position of the reaction coordinate and x 1 is a position of the reaction coordinate at the first pulling step. Equation (II.15) looks very similar to the most important result derived by Hummber and Szabo, i.e., the expression of Eq. 45 th in Ref. (23). The first average is equivalent to the average over all first-time- turning-on-potential procedures. Nevertheless, there is a slight difference between the second averages of Eqs. (II.15) and 45 th . In the average 〈…〉 FVE the mechanical work is an integral taken over λ instead of x for all force-versus-extension (FVE) curves (as noted by Jarzynski (18)). One more note here is that the Eq. (II.15) is exact whereas Eq. 45 th is an approximation. To some extent, the similarity between the two equations suggests that Eq. (II.15) can be used to reconstruct free energy landscapes in the same way as proposed by Hummer and Szabo. It means that the initial positions of an applied potential must be taken into account to reconstruct free energy landscapes. We have shown that it is possible to construct work distributions from the distributions of a reaction coordinate (Sections II.3 and II.4). An acceptable agreement of ΔF JE (λ s ,λ 1 ,k) with the exact PMF profile (Figs. II.3b and II.4a) confirms the validity of the sequential and parallel pulling protocols. In the protocols, one first has to choose an ordered set of targeted configurations that can be characterized by different values of λ. 33 One then independently or sequentially turns on the harmonic potential to drive the system into those targeted configurations from any chosen state x o . The protocols do not require any additional task in choosing a typical structure for each pulling step as implemented in the adaptive steered molecular dynamics simulations (39). All distributions of a reaction coordinate are equally treated to reconstruct work distributions. A number s of pulling steps and an amount of relaxation time τ are decisive to generate rare small work distributions. In terms of computational cost, the parallel pulling simulations (Section II.4.2) are two times faster than the sequential ones (Section II.4.1) and have the computational cost equal to performing 100 trajectories with pulling speed v = 10 Å/ns using the PMF method (25). In the PMF method, a finite number (N p ) of parallel forward-pulling trajectories, as illustrated in Fig. II.1a, would produce free energy profiles that are likely dependent of pulling speeds (25, 28, 33, 34). It has been elusive to guarantee that the pulling protocol produces rare states that have a major contribution to work distributions (or what should be a minimum of N p ?). If work distributions are not Gaussian functions, the PMF method fails. Kofke et al. (50, 51) suggested that the overlap of the work distributions constructed in forward- and reverse-pulling procedures should be adequate to ensure the convergence of exponential-work averages in Eq. (II.1). In other words, one might have to perform simulations of reverse pulling (folding) for examining the accuracy of work distributions constructed in forward pulling (un-folding) simulations. It is possible to carry out a few reverse-pulling trajectories in experiments and use the Crooks fluctuation 34 theorem to examine work distributions (6, 12, 29, 52). But in simulations, such reverse- pulling trajectories can be expensive. Based on the theory and the simulation results in Section II.4, we propose a criterion that the overlaps between successive x i ’s distributions should be appropriately comparable to their standard deviations, as schematically described in Fig. II.1b and plotted in Fig. II.5c. The more the adjacent distributions overlap, the more rare small values of work are available. If the sequential and parallel step-wise pulling protocols meet the criterion, one can carry out only forward-pulling simulations to produce reliable work distributions. These work distributions do not depend on pulling speeds v. The theory also suggests that one can combine trajectories with different pulling speeds, which are generated by the PMF method. Because the data [x(t 1 ) and x(t 2 )] of any two trajectories with v 1 and v 2 belong to the same control parameter λ 1 + v 1 t 1 (or the same perturbed state), where v 1 t 1 = v 2 t 2 . These data play the same role in the proposed protocols. In addition, we have shown that averaged values of a reaction coordinate can be used for estimating free energy differences (FEDs) based on Eq. (II.14). In the limit of slow pulling, i.e., infinitely small Δλ = (λ s –λ 1 )/s, Eq. (II.14) reduces to the well-known Thermodynamic Integration (31, 38, 53, 54) and is similar to the Adaptive Biasing Force (ABF) equations (10, 32): ΔF fluct (λ s ,λ 1 ,k) Δλ→0 # → ## ∂H(λ,x) ∂λ λ 1 λ s ∫ λ dλ=− f a (x) λ dλ= λ 1 λ s ∫ ΔF ABF , (II.16) where –〈f a (x)〉λ are averaged adaptive forces exerted on a reaction coordinate and ΔF ABF is the FED computed by the ABF method. In the ABF method one has to collect all possible 35 external forces from s windows in comparison with only s – 1 ones as in Eq. (II.14). Hence, the difference between ΔF fluct (λ s ,λ 1 ,k) and ΔF ABF at the same finite increment Δλ is Δλ〈f s 〉 ≈ –Δλ f a (x) λ s where 〈f s 〉 [in Eq. (II.9)] is the final averaged force measured at λ = λ s . The error of the ABF method is proportional to Δλσ f a s(1+2κ)δt /τ , where € σ f a is the standard deviation of f a (x), κ is the correlation length (54) of the forces and δt is a time-step for collecting forces. Accordingly, with a sufficiently large number of available data, the error of ΔF fluct (λ s ,λ 1 ,k) should be proportional to Δλ | f s |+σ f λ s(1+2κ)δt /τ ( ) , where σ f λ is the standard deviation of fλ. The ABF method does not require specifying any forms of applied forces that are adaptive or constrained along pathways. In our method [Eq. (II.14)], no constraint is imposed on applied forces. Importantly, the behaviors of the averaged accumulating forces (Fig. II.4b) in the sequential and parallel pulling protocols are almost the same. Moreover, τ can be small (~ns) to collect a set of 〈x i 〉 for free energy calculations (Fig. II.6a). These two results suggest that it is possible to estimate FEDs by performing parallel-pulling protocols to measure 〈x i 〉 or averaged accumulating forces without imposing any constraints on external forces. The introduction of Eq. (II.14) for estimating FEDs turns out to be useful to examine the reliability of work distributions used in JE, Eq. (II.12). As shown in Section II.4, the accuracy of FEDs depends upon a number s of pulling steps, relaxation time τ and spring constant k. A viable method to estimate the accuracy is to compare FEDs 36 evaluated by both JE and Eq. (II.14). Given finite s and τ, JE might overestimate FEDs and Eq. (II.14) underestimates FEDs (Figs. II.4 and II.6). Equation (II.14) is used in the test case for an acceptable accuracy at many values of spring constant k (Fig. II.5a) and relaxation time τ as small as 0.4 ns. FEDs computed by JE are unbiased when k is small and τ is large (Figs. II.5b and II.6a). Using both equations, one can estimate FEDs that are equal to ΔF com (λ s ,λ 1 ,k) = [ΔF JE (λ s ,λ 1 ,k) + ΔF fluct (λ s ,λ 1 ,k)]/2 with an uncertainty equal to [ΔF JE (λ s ,λ 1 ,k) – ΔF fluct (λ s ,λ 1 ,k)]/2. By checking the convergence of ΔF com (λ s ,λ 1 ,k) with respect to τ, one can confirm the accuracy of computed FEDs (Fig. II.6a). In conclusion, we provided an alternative proof of Jarzynski’s equality (JE). The key point in our proof is that the processes of turning-on/off a harmonic potential to perform work are described by double Heaviside functions of time. The important results of the proof [Eqs. (II.9), (II.12) and (II.15)] are consistent with the established theories on JE. Our theory also suggests a formula [Eq. (II.14)] to evaluate free energy differences from averaged values of a reaction coordinate. Our contributions to computational studies of JE are: (i) Work distributions are simply constructed from the distributions of a reaction coordinate. (ii) We proposed sequential and parallel step-wise pulling protocols to generate work distributions. (iii) To use Eqs. (II.12) (JE) and (II.14) with a finite number of pulling steps and reasonably small relaxation time, the mutual overlapping range between the adjacent distributions of a reaction coordinate must be comparable to their standard deviations. 37 The more the adjacent distributions overlap, the more rare small values of work are available. (iv) The combination of Eqs. (II.12) and (II.14) can be used to estimate free energy changes with an uncertainty equal to half the difference between them. (v) We showed in a test case that our method requires 21 parallel simulations and relaxation time as small as 0.4 ns for each simulation to estimate free energy profiles with an uncertainty of about 13%. 38 CHAPTER III Demonstration of Jarzynski’s Equality in Open Quantum Systems Using a Step-Wise Pulling Protocol III.1 Introduction Jarzynski’s Equality (JE) is well-known for closed systems, both in classical and quantum mechanics (5, 18, 55–57). The JE describes a relation between applied work W and free-energy changes [see Eq. (III.1)]. For closed quantum systems it does not require any heat bath to maintain temperature. As a result, work W is evaluated as E m (t f ) – E n (0), where E m (t f ) is the m-th eigenvalue of a final Hamiltonian ˆ H(t f ) at time t f , and E n (0) is the n-th eigenvalue of an initial Hamiltonian ˆ H(t= 0) (56). In contrast, for open quantum systems W cannot be expressed in terms of simple energy differences, due to the presence of heat baths. To extend the JE for open systems, Crooks (19) developed a theory that considers the relation between discretized mechanical work W = [H(x i ,λ i+1 )−H(x i ,λ i )] i=1 s−1 ∑ and free-energy changes ΔF(λ 1 ,λ s ) for open stochastic classical systems, whose evolution follows the principle of detailed balance. Here the systems are characterized by Hamiltonian H(x,λ); x i denotes a reaction coordinate x at the i th -discretized step; λ is a control parameter; and s is the number of discretized steps. The relation is 39 exp[−ΔF(λ 1 ,λ s )/k B T] = 〈exp(−W/k B T)〉, (III.1) where k B is Boltzmann’s constant, T is temperature, and 〈…〉 indicates the average over all possible values of W. To further examine the connection between work distributions and free-energy changes for general open classical systems, Crooks formulated a fluctuation theorem, which relates the distributions of work ρ F (+W) in forward processes with ρ R (−W) in reverse processes via ρ F (+W)/ρ R (−W) = exp(β[W−ΔF(λ 1 ,λ s )]) (6). This theorem is more universal than the JE, because one can obtain the JE by multiplying both sides with ρ R (−W)exp(−βW) and integrating over W. Later, Campisi et al. (58) extended the applicability of the JE and the fluctuation theorem for open and arbitrarily strong- coupling quantum systems. However, the expression of work W = [H(x i ,λ i+1 )−H(x i ,λ i )] i=1 s−1 ∑ has not yet been shown to be essential for open quantum systems. In other words, the possibility of constructing work distribution functions by using the explicit discretized form of W in quantum mechanics has not been studied. Since the major issue of the JE is how to perform and measure work distribution functions (59), this would shed new light on how the JE works in quantum systems. Recently, one of us (60) presented a proof for the relation between the discretized mechanical work W = [U(x i ,λ i+1 )−U(x i ,λ i )] i=1 s−1 ∑ and free-energy changes in open classical systems without using the principle of detailed balance. This proof is based on a step-wise pulling protocol (see Fig. III.1), in which an applied potential U(x,λ) is used to perform work W in a step-wise manner. To implement the step-wise pulling protocol, a 40 double Heaviside functions of time t, θ(t – t i–1 )θ(t i – t), was used, where the index i denotes a pulling step. For a relaxation time t i–1 – t i the applied potential is U(x i ,λ=λ i ), in which x i varies during the relaxation time. By introducing the double Heaviside functions into a coupled Hamiltonian H(t), one can verify that 〈exp(−∫[∂H(t)/∂t]dt/k B T)〉 = 1, which helps to prove the JE, Eq. (III.1). If H(t) does not contain double Heaviside functions of time t, one can instead consider Hummer-Szabo’s proof (20, 21, 23, 24) for the JE. Although these two proofs are different, Ngo, Hummer, and Szabo arrived at similar approaches for reconstructing free-energy landscapes, which take into account the initial positions of an applied potential. The main advantage of using a step-wise pulling protocol is that it allows to obtain work distribution functions by measuring thermal-fluctuation distributions of x i . These distributions of x i can be generated with a finite number s of pulling steps, and reasonably small relaxation times (t i – t i–1 ), hence utilizing non-equilibrium pulling processes. Finite relaxation times are required to allow the system to evolve into states, which are used to generate distributions of trajectories along a reaction pathway. Here, rare trajectories corresponding to small values of work are found to yield the dominant contributions in Eq. (III.1), when computing free-energy changes. They are present in Eq. (III.1) if overlaps between successive distributions of fluctuating x i are larger than the standard deviations of their distribution functions. 41 Figure III.1. Schematic diagram for a step-wise pulling protocol: (a) control parameter λ, and (b) expectation value of the applied potential versus time. In this article, we extend Ngo’s proof (60) to quantum systems (Sec. II). Specifically, we define a reference free energy for the free-energy changes and obtain the average on the right-hand side of Eq. (III.1) in terms of operators. This quantum version of the JE can be applied to any quantum systems, as long as an external potential with a control parameter λ is applicable. As an example, using a harmonic potential to perform work, we show how discretized mechanical work enters Eq. (III.1) for quantum systems. As a result, work distribution functions can be generated from eigenstates and eigenvalues of a coupled Hamiltonian operator, or in general thermal and quantum fluctuation distributions of a reaction coordinate. From an explicit expression of the work distribution function, we obtain two sets of equations to determine which transition pathways provide the dominant contributions to free-energy changes. The pathways satisfying both sets of equations follow the principle of detailed balance. We test this 42 theory on a quantum harmonic oscillator (Sec. III.3). Finally, we discuss some important consequences of this theory (Sec. III.4). III.2 Theory Let us consider a stationary system of N particles described by a time-independent Hamiltonian ˆ H 0 ( ˆ p 3N , ˆ r 3N−1 , ˆ x) , where ˆ p 3N and ˆ r 3N−1 , ˆ x are 3N momentum and position operators, and the system is coupled to a heat bath at temperature T. Here ˆ x is a one- dimensional reaction coordinate operator, which can be coupled to an external potential operator ˆ U( ˆ x,λ), where the parameter λ controls the center of the applied potential. We define a pre-defined classical pathway in which λ is changed from λ 1 → λ 2 → λ 3 … → λ s , where s ≥ 2 is the number of pulling steps (see Fig. III.1). For each pull, parameterized by λ i , the coupled system is allowed to relax. The relaxation time τ i = t i − t i–1 is chosen sufficiently large to equilibrate the system following each instantaneous pulling step. This is called a step-wise pulling protocol. To describe the coupling between ˆ U( ˆ x,λ) and ˆ H 0 ( ˆ p 3N , ˆ r 3N−1 , ˆ x), we use double Heaviside functions in time θ(t− t i−1 )θ(t i − t). Then, at the ith-pulling step the coupled Hamiltonian can be written as ˆ H( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ i ) = ˆ H 0 ( ˆ p 3N , ˆ r 3N−1 , ˆ x)+ ˆ U( ˆ x,λ i )θ(t−t i−1 )θ(t i −t). For sufficiently large τ i , ˆ H( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ i ) acts as a quasi-time-independent operator. Therefore, we can assume 43 that a canonical ensemble with Hamiltonian ˆ H( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ i ) exists for each pulling step. The general coupled Hamiltonian can be written as ˆ H( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ 1 ,...,λ s )= ˆ H 0 ( ˆ p 3N , ˆ r 3N−1 , ˆ x)+ ˆ U( ˆ x,λ i )θ(t−t i−1 )θ(t i −t) i=1 s ∑ . (III.2) Next, we show how to extract mechanical work from the general coupled Hamiltonian. Let us formally define an operator ˆ O total and the mechanical work ˆ W operator in the step-wise protocol as ˆ O total = ∂ ˆ H( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ 1 ,λ 1 ...λ s ) ∂t dt t 0 t s ∫ , (III.3a) ˆ W = δ ˆ W i i=1 s−1 ∑ = [ ˆ U( ˆ x,λ i+1 )− ˆ U( ˆ x,λ i )] i=1 s−1 ∑ , (III.3b) where δ ˆ W i = ˆ U( ˆ x,λ i+1 )− ˆ U( ˆ x,λ i ). (See Appendix D for the relation between ˆ O total and ˆ W ). Let us now denote x i to be the eigenvalues of the operator ˆ x for the ith-pulling step from t i–1 to t i . Then, for a trajectory x 1 → x 2 → x 3 … → x s–1 we can write the expression of the expectation value W (without hat) as follows: W = [U(x i ,λ i+1 )−U(x i ,λ i )] i=1 s−1 ∑ . (III.3c) We will now show how this expression for W can be used to generate a work distribution function using s – 1 distributions of fluctuating x i to evaluate the average in the Jarzynski’s Equality Eq. (III.1) for quantum systems. 44 To derive Jarzynski’s Equality Eq. (III.1), we have to specify how to take averages for pre-defined classical pathways, and define a reference free energy. Firstly, we define C(λ 1 ,…,λ s ) as a function of only (λ 1 ,…,λ s ): C(λ 1 ,...,λ s )={Z 0 Z i i=1 s ∏ } −1 ×Tr X 0 e −β ˆ H 0 ( ˆ p 3N ,ˆ r 3N−1 , ˆ x) e −β ˆ U( ˆ x,λ 1 ) × Tr X i e −β[ ˆ H 0 ( ˆ p 3N ,ˆ r 3N−1 , ˆ x)+ ˆ U( ˆ x,λ i )] e −βδ ˆ W i { } i=1 s−1 ∏ ×Tr X s e −β ˆ H 0 ( ˆ p 3N ,ˆ r 3N−1 , ˆ x) e +β ˆ U( ˆ x,λ s ) , (III.4) where here β is 1/k B T; Tr denotes a trace over a complete set of states in the Hilbert space; X 0 represents the complete set of states for the initial Hamiltonian ˆ H 0 ( ˆ p 3N , ˆ r 3N−1 , ˆ x) ; X i represents the complete set of states at the ith-pulling step for Hamiltonian ˆ H i ( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ i )= ˆ H 0 ( ˆ p 3N , ˆ r 3N−1 , ˆ x)+ ˆ U( ˆ x,λ i ). The partition function before pulling is Z 0 = Tr X 0 exp[−β ˆ H 0 ( ˆ p 3N , ˆ r 3N−1 , ˆ x)]=exp[−βF 0 ], where F 0 is the free energy without the applied potential. The partition function Z i is Tr X i exp[−β ˆ H i ( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ i )]=exp[−βF(λ i )], where F(λ i ) is the free energy at the ith-pulling step. Henceforth, we omit ˆ p 3N , ˆ r 3N−1 in the Hamiltonian operators to simplify the notation. We then define the reference free energy F ref (λ 1 ,λ s ) as F ref (λ 1 ,λ s )=−β −1 ln Tr X 0 e −β ˆ H 0 ( ˆ x) e −β ˆ U( ˆ x,λ 1 ) C(λ 1 ,...,λ s ) " # $ $ × Tr X s e −β[ ˆ H 0 ( ˆ x)+ ˆ U( ˆ x,λ s )] e +β ˆ U( ˆ x,λ s ) Z 0 & ' ( ( . (III.5) Substituting the traces over X 0 and X s in Eq. (III.4) by using Eq. (III.5), we obtain 45 exp[−βΔF(λ 1 ,λ s )]= exp −β[F(λ s )−F ref (λ 1 ,λ s )] ( ) = Tr X i ˆ Ω i i=1 s−1 ∏ exp(−βδ ˆ W i ), (III.6) which is a quantum-mechanical analogue of the JE. Here, ˆ Ω i = exp[−β ˆ H i ( ˆ p 3N , ˆ r 3N−1 , ˆ x;λ i )]/Z i are density operators. In Eq. (III.6) F ref (λ 1 ,λ s ) depends on λ s . In other words, F ref (λ 1 ,λ s ) varies with respect to s. Note that in classical mechanics C(λ 1 ,…,λ s ) is identical to unity (60), and F ref (λ 1 ,λ s ) is identical to F(λ 1 ) since the initial Hamiltonian commutes with the applied potential. In quantum mechanics, we will explicitly examine below how ΔF(λ 1 ,λ s ) and F ref (λ 1 ,λ s ) vary in a one-dimensional harmonic oscillator as one keeps λ s − λ 1 unchanged and increases s at various temperatures (see Sec. III.3). To illustrate that Eq. (III.6) is the operator expression for the JE Eq. (III.1), we consider a harmonic potential ˆ U( ˆ x,λ)=k(x−λ) 2 / 2 to perform work. Then, the expectation value of the mechanical work W following Eq. (III.3c) is simply a linear function of all x i , which denotes the eigenvalues of ˆ x at the ith-pulling steps. Suppose that at the ith-pulling step there is a complete set of states {⏐E i 〉} in Hilbert space, which are eigenstates of Hamiltonian ˆ H i ( ˆ x;λ i ) with eigenvalues E i . In the position representation, each state ⏐E i 〉 is related to a wave function  r 3N−1 ,x i E i . Then, the probability of finding the reaction coordinate at x i in the domain of spatial fluctuations V i during the ith-pulling step is given by ψ E i (x i ) 2 = d  r 3N−1  r 3N−1 ,x i E i 2 ∫ . By using s – 46 1 sets of the states and probabilities ψ E i (x i ) 2 , and the traces in Eq. (III.6) can be written as dx i i=1 s−1 ∏ ψ E i (x i ) 2 e −β E i i=1 s−1 ∑ e −βW i=1 s−1 ∏ V 1 ,V 2 ...V s−1 ∫ E 1 ,E 2 ...E s−1 ∑ Z i i=1 s−1 ∏ ( ) = dWρ(W;E 1 ,E 2 ...E s−1 )e −βW ∫ E 1 ,E 2 ...E s−1 ∑ = e −βW , (III.7a) where W is given by Eq. (III.3c), the sums run over all possible values of E 1 , E 2 …E s−1 , and ρ(W;E 1 ,E 2 ...E s−1 )= Z i i=1 s−1 ∏ ( ) −1 × dx i i=1 s−1 ∏ ψ E i (x i ) 2 exp(−β E i ) i=1 s−1 ∑ i=1 s−1 ∏ V 1 ,V 2 ...V s−1 ∫ ×δ W− [U(x i ,λ i+1 )−U(x i ,λ i )]) i=1 s−1 ∑ ( ) , (III.7b) is a quantum work distribution function along an energy pathway characterized by (E 1 , E 2 …E s−1 ). Thus, we can re-write Eq. (III.6) as exp[−βΔF(λ 1 ,λ s )] = ∫dWρ(W)exp(−βW)= 〈exp(−βW)〉, (III.8) where ρ(W)= ρ(W;E 1 ,E 2 ...E s−1 ) E 1 ,E 2 ...E s−1 ∑ is the total work distribution function. In Eq. (III.8), ρ(W) only exists for s > 1, and ∫dWρ(W) is equal to unity (61). Equations (III.6-8) suggest that if s – 1 complete sets of states {⏐E i 〉} and eigenvalues E i of the coupled Hamiltonian are known, work distribution functions can be constructed to compute ΔF(λ 1 ,λ s ). We can also express the total work distribution function as 47 ρ(W)= dx i f (x i )δ W− [U(x i ,λ i+1 )−U(x i ,λ i )]) i=1 s−1 ∑ ( ) V i ∫ i=1 s−1 ∏ , (III.9) where f i (x i )= ψ E i (x i ) 2 exp(−βE i )/Z i E i ∑ are the distribution functions of x i . In computational studies, f i (x i ) is generated by sampling the quantum-thermal fluctuations of x i , or by direct computation of the eigenstates and eigenvalues. For long enough relaxation times, f i (x i ) can be approximated as exp[−βk(x i −〈x i 〉) 2 /2] to obtain ΔF app = F(λ s )−F ref (λ 1 ,λ s )=kΔλ (λ i − x i ) i=1 s−1 ∑ , (III.10) where 〈x i 〉 is the average value of x i at the ith pulling step (60). For a sufficiently large number of pulling steps, the right-hand side of Eq. (III.10) becomes the Thermodynamic Integral (62), ∂ ˆ H ∂λ λ dλ λ 1 λ s ∫ , where ... λ is the average at each value of λ. The expressions in Eqs. (III.7) imply that the work distribution function ρ(W) is a sum of all possible quantum work distribution functions ρ(W;E 1 ,E 2 …E s–1 ). Thus, the pre- defined classical pathway λ 1 → λ 2 → λ 3 … → λ s is a sum of all possible quantum pathways. A quantum pathway is defined as {E 1 → E 2 → E 3 … → E s–1 and x 1 → x 2 → x 3 … → x s–1 }. Based on this idea, we aim to identify those energy pathways E 1 → E 2 → E 3 … → E s–1 which have the largest contribution to the free-energy change, given a set of (x 1 , x 2 ...x s–1 ). We also wish to determine which spatial pathways x 1 → x 2 → x 3 … → x s–1 have the largest contribution to the free-energy change, given a set of (E 1 , E 2 …E s–1 ). The first answer is to propose a possible picture of phase transitions or chemical reactions in 48 terms of an energy diagram. The second one is to provide insight into how chemical reactions occur in the spatial domain. To answer these questions, we use the variational principle for functionals (63). The variational principle allows one to find optimal trajectories, which maximize or minimize an integral. It is noted that the left-hand side of Eq. (III.7a) contains the multi- dimensional integrals over s – 1 variables x i , one-dimensional integral over W and sums (~ integrals in the continuum limit) over s – 1 variables E i . Given (E 1 , E 2 …E s–1 ), the function under the integrals over variables x i can be expressed as G 1 (E i ,ψ(x i ,E i ) 2 ,W,x i ), where E i , ψ(x i ,E i ) 2 ≡ψ E i (x i ) 2 and W are considered as functionals. Using the variational principle, we obtain ∂lnψ(x i ,E i ) 2 ∂E i =β(1+ ∂W ∂E i ), (III.11a) where i runs from 1 to s – 1. Similarly, given (x 1 , x 2 ...x s–1 ) we express the function under the sums over variables E i as G 2 (ψ(x i ,E i ) 2 ,W,E i ,), where x i and W are considered as functionals. Using the variational principle, we obtain another set of equations ∂lnψ(x i ,E i ) 2 ∂x i =β ∂W ∂x i . (III.11b) The solutions to Eq. (III.11a) are the spatial trajectories of optimal transitions x 1 → x 2 → x 3 … → x s–1 given (E 1 , E 2 …E s–1 ). The solutions to Eq. (III.11b) correspond to the energy trajectories of optimal transitions E 1 → E 2 → E 3 … → E s–1 , given (x 1 , x 2 ...x s–1 ). The trajectories satisfying both sets of Eqs. (11a-11b) yield the optimal contributions to the 49 free-energy change. Integrating the above equations for a transition from the (i–1)th- to ith-pulling steps, we arrive at ψ E i (x i ) 2 ψ E i−1 (x i ) 2 =e β(E i −E i−1 +W i −W i−1 ) , (III.12a) ψ E i (x i ) 2 ψ E i (x i−1 ) 2 =e β(W i −W i−1 ) , (III.12b) where W i = [U(x j ,λ j+1 )−U(x j ,λ j )] j=1 j=i ∑ . For the optimal trajectories, we combine the two sets of equations to obtain ψ E i (x i−1 ) 2 ψ E i−1 (x i ) 2 =e β(E i −E i−1 ) , (III.13) which resembles the detailed balance equations P i−1→i /P i−1←i = exp(−βE i−1 )/exp(−βE i ). In the detailed balance equations, P i−1→i is a forward transition probability for the system at E i–1 moving into E i , and P i−1←i is a reverse transition probability for the system at E i returning to E i–1 . Given m pairs of (x i–1 , x i ) satisfying Eq. (III.13), the sums of ψ E i (x i−1 ) 2 over m values of x i–1 and of ψ E i−1 (x i ) 2 over m values of x i are proportional to P i−1→i and P i−1←i , respectively. Figure (III.2) illustrates an optimal transition (denoted by the blue arrow) based on Eq. (III.13), which is analogous to the Franck-Condon principle(64–66). An optimal transition from E i–1 to E i > E i–1 occurs, if ψ E i (x i−1 ) 2 >ψ E i−1 (x i ) 2 , which indicates that the probability 50 for the particle at E i is more preferable than at E i–1 . It happens in the region Δx, which denotes the overlap between ψ E i−1 (x i−1 ) 2 and ψ E i (x i ) 2 . For example, in Ref. (67) the overlap Δx among the electronic orbitals of two reactants is assumed in an oxidization- reaction theory, which involves electron transfers in solution. If Δx is equal to zero, transitions not satisfying Eq. (III.13) are unlikely to occur. Consequently, the trajectories not satisfying Eq. (III.13) contribute much less than the optimal ones. Therefore, the optimal trajectories have the maximum contributions to the free-energy change. Figure III.2. Schematic diagram for optimal transitions. The parabolic curves represent an applied harmonic potential at the (i – 1)th and ith pulling steps. The red-shaded upper and green-shaded lower areas are the probabilities ψ E i−1 (x i−1 ) 2 and ψ E i (x i ) 2 , respectively. The blue-dashed arrow indicates an optimal transition from E i–1 (green-dashed lower line) to E i (red-dashed upper line). The region denoted by Δx is the overlap between ψ E i−1 (x i−1 ) 2 and ψ E i (x i ) 2 . To complete this discussion, we rewrite Eq. (III.8) in terms of dominant contributions to the free energy change. Using the principle of detailed balance, 51 Jarzynski and Crooks (19, 68) derived the JE to compute a free-energy change ΔF S for stochastic processes. In our theory, ΔF S exists as a term on the right-hand side of Eq. (III.8). For the optimal pathways following either Eq. (III.12a) or (III.12b) we consider them as deterministic because we can control work to induce reactions. We define the free-energy change for these pathways as ΔF D and the free-energy change for the most optimal pathways as ΔF OP . Since the most optimal pathways have a contribution to both ΔF S and ΔF D , we write the total free-energy change as e −βΔF(λ 1 ,λ s ) =e −βΔF S +e −βΔF D −e −βΔF OP +e −βΔF B , (III.14) where ΔF B is due to biased pathways, which have a small contribution to the total free- energy change. If a sampling of reaction pathways does not capture any optimal pathways, which can be tested by Eqs. (III.12-13), then ΔF(λ 1 ,λ s ) is highly biased. III.3 Testing III.3.1 Control Parameter λ as the Center of a Harmonic Potential To test Eqs. (III.6-8), we compare ΔF(λ 1 ,λ) with the analytical free-energy changes by explicit calculation for an one-dimensional quantum harmonic oscillator. The non- perturbed Hamiltonian is given by ˆ H 0 ( ˆ p, ˆ x)= ˆ p 2 /2m+kˆ x 2 /2, and a potential operator ˆ U( ˆ x,λ)=k( ˆ x−λ) 2 / 2. is applied. The eigenvalues of the coupled Hamiltonian ˆ H( ˆ p, ˆ x;λ)= ˆ H 0 ( ˆ p, ˆ x)+ ˆ U( ˆ x,λ)are E n = ω(n+1/2)+kλ 2 /4 , where n is an integer and ω = (2k/m) 1/2 . The corresponding eigenstates are ψ n (x,λ)= (1/2 n n!) mω /π 52 ×exp[−mω(x−λ /2) 2 /2]H n ((x−λ /2) mω /) , where the Hermite polynomials are H n (y) = (−1) n exp(y 2 )∂ n exp(−y 2 )/∂y n . The analytical free-energy at each value of λ is F(λ) = (ω /2)a −1 ln(e a −e −a )+kλ 2 /4, where the reduced temperature is given by a=ω /2k B T. We consider pulling protocols in which the values of λ are changed from λ 1 = 0 to λ s = 1×(/mω) 1/2 in increments of Δλ = λ s /(s – 1), where s is the number of pulling steps. The free-energy changes ΔF(λ 1 ,λ) computed from Eq. (III.8) are tested for s ∈ {2- 11, 21}, n numbers of eigenstates, eigenvalues ∈ {0-10, 20, 50}, and a = 2 l , with l being integers, -4 ≤ l ≤ 4. The work distribution functions for each pulling step are evaluated by using the recursion relation (see Appendix E) ρ i (W)=Q i dwρ i−1 (w)f i−1 (λ i−1 + Δλ 2 − W −w kΔλ ) ∫ , (III.15) where ρ i (W) is the normalized work distribution at the ith-pulling step (i > 1), f i (x i )= ψ j (x i ,λ i ) 2 j=0 n ∑ exp[−βE j (λ i )], Q i is the normalization factor of ρ i (W), and ρ 2 (W) = Q 2 f 1 [λ 1 +Δλ/2 – W/kΔλ]. To illustrate these work distribution functions in quantum mechanics, we show them at a = 1, n = 0, and s = 11 in Fig. III.3(a). The plotted work distribution functions have the highest peak at s = 2, and shift to the right with lower peaks as s increases. From the work distribution functions, we compute the free-energy profile of ΔF(λ 1 ,λ) [shown in Fig. III.3(b)]. ΔF(λ 1 ,λ) at a = 1, n = 0, and s = 11 perfectly fits with the exact free- 53 energy profile, ΔF Target = F(λ) – F(0) = (ω /2)(λ / /mω) 2 /4 . One can verify the perfect fit by analytically carrying out the integrations in Eq. (III.7a) (see Appendix F). By increasing n, ΔF(λ 1 ,λ 11 ) converges to 0.241136 at n = 7 [see Fig. III.3(c)]. Note that F ref (λ 1 ,λ s ) in Eq. (III.5) does not have the same expression as F(λ 1 ) because of the non- zero commutation between the initial Hamiltonian and the applied potential operator. To examine the effect of the non-zero commutation, we show the dependence of ΔF(λ 1 ,λ s ) on Δλ in Fig. III.3(d). For sufficiently large Δλ (0.5λ s to 1.0λ s ), the non-zero commutation in Eq. (III.5) becomes significant. As Δλ decreases, the commutation becomes negligible; hence ΔF(λ 1 ,λ s ) approaches ΔF Target as a linear function of Δλ, – 0.0884Δλ /  /mω + 0.2501. This consistency suggests that the definition of F ref (λ 1 ,λ s ) [Eq. (III.5)] for the step-wise pulling protocol is valid at reduced temperature a = 1. Figure III.3. (Color online) (a) Normalized work distributions. The numbers (2 and 10 for the highest and lowest curves, respectively) indicate the ith-pulling steps at s = 11, and n = 0. (b) Free energy profiles at s = 11, and n = 0. (c) Free- energy changes ΔF(λ 1 ,λ s ) versus number n of the eigenstates and eigenvalues. (d) Free-energy changes ΔF(λ 1 ,λ s ) versus increment Δλ = λ s /(s – 1) at n = 10. All data here are evaluated at a = 1. 54 Figure III.4. (a) Free-energy profiles (solid lines) at s = 11, and for different reduced temperatures a=ω /2k B T. The numbers indicate the values of a. The dashed line is the average work 〈W〉 along the pathway. (b) Free-energy ΔF(λ 1 ,λ s=11 ), standard deviation 〈(W–〈W〉) 2 〉 1/2 of work distributions, and average work 〈W〉 ≈ 0.274ω /2 versus log 2 (a). Fig. III.4(a) shows the free-energy profiles (solid lines) at different temperatures for Δλ = 0.1λ s=11 , which are bounded above by the average mechanical work 〈W〉 (dashed line). We observe that 〈W〉 at λ 11 is unchanged (≅ 0.274ω /2) over the wide range of temperatures studied here [see Fig. III.4(b)]. The standard deviation 〈(W–〈W〉) 2 〉 1/2 of the work distributions decreases with temperature, and converges to 0.2236ω /2. The unchanged value of 〈W〉 and the convergence of 〈(W–〈W〉) 2 〉 1/2 at low temperatures indicate that the work distribution is practically independent of temperature (at s = 11). This indicates that the ground state (n = 0) at each pulling step significantly contributes to the work distribution at sufficiently low temperatures. At high temperatures (a < 1), the free-energy profiles are not noticeably distinguishable from those at a = 1, but more energy levels and wave functions contribute to the work distributions. As shown in Fig. III.4(b), ΔF(λ 1 ,λ 11 ) at λ 11 = 1×(/mω) 1/2 converges to 0.25 ± 0.01 (ω / 2). The error (≈ 0.01/0.25 = 4%) is due to truncated 55 number n of wave functions. However, ΔF(λ 1 ,λ 11 ) becomes negative when lowering the temperature (a > 1). We recall that ΔF(λ 1 ,λ i ) is the difference between F(λ i ) and F ref (λ 1 ,λ i ), which is not identical to F(λ 1 ) due to the non-zero commutation between the initial Hamiltonian and the applied potential [see Eq. (III.5)], especially at low temperatures. Thus, the negative values of ΔF(λ 1 ,λ i ) indicate the strong effect of the non- zero commutation at a finite number of pulling steps. This means that not only F(λ i ), but also F ref (λ 1 ,λ i ) varies as i runs from 2 to s = 11. Since only the ground state at each pulling step significantly contributes to the work distribution function at low temperatures, the free-energy changes can be estimated from ΔF(λ 1 ,λ s )=kΔλ 2 (s−1) 2 [1−(a−1)/(s−1)]/ 4 (see Appendix F). As a > s, the estimated ΔF(λ 1 ,λ s ) becomes negative, as observed in Figs. III.4(a-b). If s is much larger than a, the estimated ΔF(λ 1 ,λ s ) approaches ΔF Target . As a result, at sufficiently low temperatures (large a) and very small increments Δλ the variation of F ref (λ 1 ,λ i ) with respect to λ i is significant at steps i ~ a, but becomes negligible at steps i >> a. Therefore, we can identify the negative values of ΔF(λ 1 ,λ i ) at steps i ~ a as a quantum effect of the JE at low temperatures. III.3.2 Control Parameter λ as the Spring Constant of a Harmonic Applied Potential To directly compare work distribution functions and free-energy profiles computed by Eqs. (III.6-8) with those derived by using the JE for closed quantum systems (57), we 56 now vary the spring constant following a step-wise protocol. In this case, ˆ H 0 ( ˆ p, ˆ x)= ˆ p 2 /2m+(k 0 −Δk)ˆ x 2 /2, ˆ U( ˆ x,k i )=k i ˆ x 2 / 2, where k i is iΔk, Δk is an increment, and s is the number of pulling steps. The eigenvalues of the coupled Hamiltonian ˆ H( ˆ p, ˆ x;k i )= ˆ H 0 ( ˆ p, ˆ x)+ ˆ U( ˆ x,k i ) are E n (ω i ) = ω i (n+1/2) , where n is an integer, ω i is given by ω 0 [1+(i–1)δ] 1/2 , ω 0 = (k 0 /m) 1/2 , and δ=(ω s 2 −ω 0 2 )/ω 0 2 (s−1) = Δk/k 0 . We choose ω s = 1.3ω 0 as used in Ref. (57). The corresponding eigenstates are ψ n (x,ω i )= (1/2 n n!) mω i /π exp[−mω i x 2 /2]×H n (x mω i /) , using the Hermite polynomials H n (y) = (−1) n exp(y 2 )∂ n exp(−y 2 )/∂y n . The analytical free-energy at each value of ω i is F(ω i ) = sinh(a 0 [1+(i–1)δ] 1/2 /2), where the reduced temperature is again given by a 0 =ω 0 /k B T. Figure III.5. (a) Normalized work distribution at s = 2. (b) Normalized work distribution function at s = 11. (c) Free-energy profiles. The work distributions and free-energy profiles are computed at a 0 =ω 0 /k B T =0.1. 57 First, we examine a case at high temperature, e.g., a 0 = 0.1 (used in Ref.(57)). The work distribution functions ρ i (W) are computed from f i (x i )= ψ j (x i ,ω i ) 2 j=0 n ∑ exp[−βE j (ω i )], where n = 100 is chosen. The expression of work W is computed from 0.5δ / 1+δ(i−1)(x i 2 mω i /) i=1 s−1 ∑ , which is always non- negative. Fig. III.5(a) shows the work distribution function ρ 2 (W) for free-energy difference ΔF(ω 1 =ω 0 ,ω s ) at s = 2. ρ 2 (W) has a sharp peak at W = 0, which resembles the feature for adiabatic processes, ~ exp[–βWω 0 /(ω s –ω 0 )]θ(W), where θ(W) is the Heaviside step function of W. By increasing s to 11, this peak in the work distribution ρ 11 (W) for the same ΔF(ω 1 =ω 0 ,ω s ) smoothens out [see Fig. III.5(b)]. The free-energy profiles shown in Fig. III.5(c) for s = 2 and 11 agree with the targeted ΔF Target = –β –1 ln[F(ω 1 =ω 0 )/F(ω s )]. Note that the work distributions do not have negative tails which were observed in Ref. (57) for non-adiabatic processes. By definition, the values of work are only negative for the cases of reducing the spring constant (δ < 0), which can result into lowering the temperature of the system. Last but not least, we compute free-energy changes at very low temperatures. Analogous to the previous example, we observe that at low temperatures only the ground state at each pulling step significantly contributes to the work distribution functions. This observation allows us to analytically estimate ΔF(ω 1 ,ω s ). Similar to the procedure in Appendix F, we arrive at 58 ΔF(ω 1 ,ω s )= ω 0 2a 0 ln(1+ a 0 δ 2 1+δ(i−1) ) i=1 s−1 ∑ ≅ ω 0 δ 4 1+δ(i−1) i=1 s−1 ∑ +O((a 0 δ) 2 ) ≅ω 0 dy 4 1+y + 0 (ω s 2 −ω 0 2 )/ω 0 2 ∫ O(1/s)+O((a 0 δ) 2 ) = (ω s −ω 0 ) 2 +O(1/s), (16) which is consistent with the low-temperature limit in Ref. (57). III.4 Discussions and Conclusions It is noted that the step-wise pulling protocol and Eqs. (III.6-7) have some similarities with the decomposition scheme applied in the Trotter-Suzuki formula (69, 70), e −β( ˆ H+ ˆ U) = lim q→∞ (e −β ˆ H/q e −β ˆ U/q ) q , where ˆ H and ˆ U are the initial Hamiltonian and applied potential operators, and ˆ U is fixed. The decomposition is useful to derive the path integral representation of the partition function (71, 72). The partition function can be a multi-dimensional integral over q sets of a coordinate variable x, i.e., x j with j being 1 to q. Similarly, Eq. (III.7a) contains a multi-dimensional integral over s – 1 sets of a reaction- coordinate variable x i . In Eqs. (III.6-7), the operators δ ˆ W i = ˆ U( ˆ x,λ i+1 )− ˆ U( ˆ x,λ i ) and the mechanical work W weigh those rare reaction pathways, which have the largest contribution to free-energy changes. A reaction pathway is defined as two series of energy pathways (E 1 , E 2 …E s–1 ) and spatial pathways (x 1 , x 2 …x s–1 ). If s is small, the accuracy of the pathways is poor because the energy and spatial differences between successive pulling steps are large. Equations (III.12-13) (see Fig. III.2) suggest a criterion 59 that certain overlaps of successive wave functions are sufficient to reconstruct reliable rare reaction pathways. Thus, s can be finite at a certain range of temperatures (e.g. larger than Debye temperatures). But at low temperatures (see Sec. III.3), an infinitely large number of s (like for q) will be needed to guarantee the convergence of free-energy changes. This theory of the JE for quantum systems is a generalization from the proof for classical systems. As a result, the same discretized mechanical work can be applied to any classical and quantum systems. Unlike the JE for classical systems, a work distribution function generated from Eqs. (III.7, 9) gives the difference between a final free energy F(λ s ) and the reference free energy F ref (λ 1 ,λ s ). In the classical limit, F ref (λ 1 ,λ s ) becomes F(λ 1 ). For a one-dimensional harmonic oscillator at low temperature, the variation of F ref (λ 1 ,λ s ) due to quantum effects becomes significant at small s, but negligible at very large s (see Sec. III.3). Thus, the definition of F ref (λ 1 ,λ s ) still ensures the convergence of ΔF(λ 1 ,λ s ) by collecting many fluctuations of a reaction coordinate along a given pathway. The fluctuations are affected by the rest of systems and heat baths, which can be arbitrarily coupled in any dynamics. They can be simply computed from wave functions weighted by exponentials of corresponding eigenvalues, and then used for constructing work distribution functions. In the presence of heat baths, the fluctuations can be approximated as exp[−βk(x i −〈x i 〉) 2 /2] to arrive at Eq. (III.10), which is consistent with the Thermodynamic Integral (62), ∂ ˆ H ∂λ λ dλ λ 1 λ s ∫ . Therefore, the convergence and consistency suggest the validity of the quantum expressions of the JE, Eqs. (III.6-8). 60 The introduction of the discretized mechanical work W into Eqs. (III.6-8) is useful to determine which reaction pathways are optimal. Without W, it is unclear to prove that the most optimal reaction pathways follow the principle of detailed balance, P i−1→i /P i−1←i = exp(−βE i−1 )/exp(−βE i ). We also pointed out that the transition probabilities P i−1→i and P i−1←i should contain information of wave functions along any optimal transition pathway, although they can be arbitrarily defined in practice (72). In the paper by Metropolis et al. (73), the idea of using the transition probabilities satisfying the principle of detailed balance is to quickly drive systems into equilibrium states for classical systems; hence sampling of canonical distributions can be done less costly. The sampling based on the transition probabilities might be enhanced or smoothed by using Eqs. (III.12-13), which restrict the overlaps between successive states. Addressing the advantages of using Eqs. (III.12-13) and time evolution of the transitions would be of interest for future research. Since the time evolution has not yet been discussed in the theory and test case, it is also essential to estimate the relaxation time to characterize the limits of non-equilibrium processes, as examined in classical systems (60). On one hand, from the JE we have obtained detailed balance for the most optimal transition pathways. On the other hand, using the principle of detailed balance Jarzynski and Crooks (19, 68) derived the JE for classical stochastic processes. Furthermore, Boltzmann proved that the principle of detailed balance sets a sufficient condition for entropy growth (74). As a result, we infer that the entropy growth follows the most optimal transition pathways for any dynamics. 61 One possible consequence of this theory is to suggest an approach to test the JE in Bose-Einstein condensates (BECs) (59). The major difficulty of testing the JE is to measure work distribution functions in microscopic systems without interfering with the quantum dynamics. Since an applied harmonic potential can be used to trap BECs (75), it is possible to take advantage of that potential to perform work to observe how the condensate particles interact with the rest. The conventional way of constructing work distribution functions is to perform as many pulling trajectories as possible at a certain pulling speed. In other words, one might have to replicate many BECs and examine the effects of the pulling speed like in the case of unfolding RNAs (7). One challenge of the conventional way is to know how many pulling trajectories are sufficient. If the potential is monitored using a step-wise pulling protocol, in which at each pulling step the system is relaxed long enough so that the quantum dynamics is not destroyed in a single step- wise pulling trajectory, then the work distribution functions can be constructed from the distributions of the fluctuating center-of-mass of the condensate particles. Burger et al. (76) showed in an experiment that for small increments (displacements) Δλ ≤ 50 µm, BECs have oscillating frequencies shifted and are undamped on time scales of milliseconds. Motivated by the experiment, we suggest that work distribution functions for Eq. (III.1) can be constructed from the distributions of the center-of-mass of the oscillating BECs deduced from absorption images (76). Ideally, a single harmonic trap is enough to perform step-wise pulling protocols on a Bose-Einstein condensate. Compared to other proposed methods, e.g., trapped ions in a linear Paul trap (77) or heat-transfer 62 measurements (78), our approach provides an alternative perspective and insight into the equilibration dynamics of quantum systems. In summary, we have proposed a generalization of Jarzynski’s Equality [Eqs. (III.6-8)] for quantum systems based on a step-wise pulling protocol. We showed that the mechanical work in Eq. III.3(c) can be used to generate work distribution functions, and evaluate free-energy changes via Eqs. (III.6-8). The work distribution functions can be constructed from (i) eigenstates and eigenvalues of a coupled Hamiltonian operator, or (ii) by collecting the thermal and quantum fluctuation distributions of a reaction coordinate. Using a simple harmonic potential to perform work and based on the variational principle, we derived two sets of equations to identify optimal transition pathways. The optimal transition pathways satisfying both sets of the equations were found to follow the principle of detailed balance. Finally, we tested the theory by explicit analysis of a quantum harmonic oscillator, computing free-energy changes using Eqs. (III.6-8). At temperatures T ~ω /2k B , the convergence of the free-energy changes requires a finite number of many eigenstates and eigenvalues as small as 7, for a step- wise increment along the reaction pathway as small as 0.1×(/mω) 1/2 . By varying the angular frequency, we obtained the same limits derived from the JE for closed systems. At low temperatures, the ground state at each pulling step dominantly contributes to the work distribution function, and a large number s of pulling steps is required to have convergent free-energy profiles. 63 CHAPTER IV Non-equilibrium Approach to Thermodynamics using Jarzynski’s Equality and Diagonal Entropy IV.1 Introduction A formalism using diagonal entropy has recently been developed to account for the thermodynamic entropy in out-of-equilibrium quantum systems (79–82). It is based on the time average of time-dependent density operators, which makes all off-diagonal terms vanish. This time average mimics experimental measurements of physical quantities. Diagonal density operators resulting from the time average can be used to explain experimental observations that initial pure quantum states eventually evolve into mixed states. The diagonal entropy coincides with the equilibrium micro-canonical entropy in a chaotic regime, but does not converge to the entropy of a generalized Gibbs ensemble, whose system is integrable and has conserved quantities (81). Thus, diagonal entropy can be used to evaluate temperature and other thermodynamic quantities in the chaotic regime, but is not satisfactory to do the same for integrable and closed quantum systems (81, 83, 84). To measure temperature in experiments one must perturb systems sufficiently weakly and reach thermodynamic balance between systems and experimental instruments, which read temperature. In thermodynamics, the balance can be maintained when 64 systems are imbedded in very large heat baths, thus the perturbation is negligible and temperature is well defined. In this picture, experimental instruments and heat baths constantly interact with systems. A single quench performed on a closed quantum system, i.e. mimicking the interactions of experimental instruments with the system only once, generally does not induce the conventional thermodynamic Gibbs states (81, 84, 85). Therefore, such a process cannot fully model the balance between quantum systems and experimental instruments. Rather, processes involving multiple quenches are needed to give rise to a reasonable notion of temperature. Furthermore, the use of explicit heat baths in computational studies is numerically expensive due to their exponentially large Fock space, and thus it is practically difficult to examine the emergence of thermodynamic balance and of temperature in such a setting. In contrast, computation using diagonal entropy for integrable and closed systems is numerically feasible. In this paper, we examine a method to extract temperature, its meanings and thermodynamic quantities from diagonal entropy even in systems are far from equilibrium. Here we focus on thermodynamic quantities such as work and free-energy changes computed via Jarzynski’s Equality (JE). Free-energy changes are used to demonstrate energetic properties of chemical reactions along pathways. Their free-energy profiles can be evaluated from out-of-equilibrium processes, whose values of work are exponentially weighted in the JE approach (5, 6, 55–58, 60, 86). Note that it is possible to 65 combine the JE with the formalism of diagonal entropy since both of them are applicable to out-of-equilibrium dynamics (80). This combination offers a way to examine thermodynamic properties based on energy fluctuations in thermally isolated cyclically driven systems. Unlike other developments on the JE (5, 56, 58), which use temperature of initial thermal states to compute work distribution functions and free-energy profiles, the proposed formalism based on the JE requires that temperature is maintained along a reaction pathway of a control parameter coupled to large heat baths inducing canonical ensembles (60, 86). So far, it has not been shown how to maintain temperature along a reaction pathway in integrable and closed quantum systems, thus enabling capture of thermodynamics in non-equilibrium chemical reactions. This paper has three aims: (1) we present an approach mimicking the physics of measuring temperature in experiments via changing a control parameter in closed quantum systems using the formalism of diagonal entropy; (2) we demonstrate protocols of quenches to maintain temperature along a pathway of the control parameter; and (3) we show a way to combine the formalism of diagonal entropy with the JE to estimate free-energy profiles. We organize the paper as follows. Section I discusses formalism of diagonal entropy, temperature combined with Jarzynski’s Equality. Section II presents two examples of applying this approach to a quantum harmonic oscillator and a lattice of hard-core bosons. Finally, in Section III we discuss the notion of temperature and conclude our findings. 66 Figure IV.1 Protocol of multiple quenches to generate a set of diagonal density operators Ω i ≡Ω λ i (Δλ) . The parameter λ denotes an external control, which can tune λ at will. Each circle and square is associated with the Hamiltonian H(λ i ). Here, the ground states are chosen as initial states for Examples 1 and 2, but not restricted for more general cases as long as the characteristic temperature is the same for all quenches. IV.2 Theory A diagonal density operator can be generated in a single quench by changing a control parameter λ along a reaction pathway with an increment Δλ. Given an initial state ψ 0 (λ−Δλ) belonging to a Hamiltonian H(λ–Δλ), at time t ≥ 0 a perturbation is instantaneously turned on, and the system subsequently evolves with Hamiltonian H(λ). The time-dependent density operator is then given by Ω(t,λ,Δλ)=e −iH(λ)t ψ 0 (λ−Δλ) ψ 0 (λ−Δλ) e iH(λ)t . A diagonal density operator is defined as Ω λ (Δλ)= lim τ→∞ 1 τ dtΩ(t,λ,Δλ) 0 τ ∫ = E n E n p n n ∑ , (IV.1) 67 where E n is an eigenstate of H(λ) and p n = E n ψ 0 (λ−Δλ 2 . The diagonal entropy can then be defined asS(λ,Δλ)=−TrΩ λ (Δλ)lnΩ λ (Δλ)=− p n lnp n n ∑ . The inverse temperature β = 1/T (k B = 1) is computed from β= ∂S ∂E " # $ % & ' λ ≡ lim ε→0 S(λ,Δλ+ε)−S(λ,Δλ) E(λ,Δλ+ε)−E(λ,Δλ) , (IV.2) where E(λ,Δλ)= E n p n n ∑ is the averaged energy. If diagonal entropy coincides with micro-canonical entropy (81), T is an equilibrium temperature. If not, T represents an out- of-equilibrium temperature, whose meaning will be discussed through out this paper. T is the so-called a characteristic temperature associated with the diagonal density operator Ω λ (Δλ) and the diagonal entropy S(λ,Δλ). Now suppose that we perform a series of quenches, which yield the same characteristic temperature T = β –1 (see Fig. IV.1), and wish to compute free-energy profiles, ΔF(λ 1 ,λ s ) via Jarzynski’s Equality (JE), exp[−βΔF(λ 1 ,λ s )] = ∫dWρ(W)exp(−βW) ≡ 〈exp(−βW)〉, (IV.3) where ρ(W)= dx i f i (x i )δ W − [U(x i ,λ i+1 )−U(x i ,λ i )] i=1 s−1 ∑ # $ % & ' ( ∫ i=1 s−1 ∏ is the work distribution function, ˆ U( ˆ x,λ) is a potential operator with coupling λ (λ i = (i–1)Δλ) to a reaction coordinate operator ˆ x , and f i (x i )= TrΩ λ i (Δλ)δ(ˆ x− x i ) is the distribution function of eigenvalue x i at the i-th quench. The free-energy profile ΔF(λ 1 ,λ s ) converges to F(λ s ) – F(λ 1 ), where F(λ i ) is −β −1 ln[Tre −βH(λ i ) ], as the number s of discretized equilibration 68 times is taken towards infinity and the central limit theorem holds. One remarkable feature of the JE is that once small rare values of work corresponding to the optimal pathways are sampled, they are exponentially weighed, so collected data in equilibrium do not substantially change ΔF(λ 1 ,λ s ). We find that such small rare values of work can be generated in the protocol illustrated in Fig. IV.1. This protocol is motivated by the fact that small rare values of work are available as long as the probabilities of measuring a reaction coordinate x in successive quenches overlap (86). In this protocol, the external potential U(x,λ) is used to couple a system with an external control represented by λ. By quenching λ, we drive the system along a reaction pathway. The diagonal density operator associated with the ith quench and Hamiltonian H(λ i ) is denoted by Ω i ≡Ω λ i (Δλ) , which generally depends on initial states. For large systems connected to heat baths, Ω i approaches the fully thermalized Gibbs state regardless of the initial states (11, 87), and thus the requirement of the same temperature for all quenches is automatically satisfied. For small closed systems, however, one has to choose appropriate initial states for a series of quenches to have the same temperature for evaluating free-energy profiles via Eq. (IV.3). A series of quenches characterized by a single value of T and the corresponding free-energy profile can be used to quantify the thermodynamic information extracted from the quenches. We now show how to implement this technique of combining diagonal density operators with JE by simply choosing the appropriate initial states for all quenches within a sequence. 69 IV.3 Examples IV.3.1 Example 1: Simple Harmonic Oscillator Let us first consider a simple harmonic oscillator, whose Hamiltonian is given by H(λ)= ˆ p 2 2m + kˆ x 2 2 + k(ˆ x−λ) 2 2 =ω(a λ + a λ + 1 2 )+ kλ 2 4 , (IV.4) where ω=(2k/m) 1/2 , and a λ = mω 2 ˆ x− j mω ˆ p− λ 2 " # $ % & ' ,a λ + = mω 2 ˆ x+ j mω ˆ p− λ 2 " # $ % & ' are the annihilation and creation operators. The initial state for each quench in Fig. IV.1 is the ground state of H(λ−Δλ) . Taking advantage of a λ =a λ−Δλ − mω /2Δλ /2 and a λ−Δλ n λ−Δλ = 0 = 0, one obtains the diagonal density operator Ω λ =e −y n λ n λ y n λ /n λ ! n λ =0 ∑ , with y=mωΔλ 2 /8 . In this case, the energy distribution function p n λ =e −y y n λ /n λ ! is exactly the Poisson distribution function. One can then easily compute the averaged energy,E=ω(y+1/2)+kλ 2 /4, and the entropyS= y−ylny+e −y y n λ lnn λ !/n λ ! n λ =0 ∑ . Since S is a function of y only, varying y (or Δλ) changes the entropy and energy at the same time while keeping λ fixed. As a result, one can use the chain rule to compute the derivative in Eq. (IV.2) as 1 T = ∂S ∂E " # $ % & ' λ = ∂S ∂y / ∂E ∂y " # $ % & ' λ = e −y y l [ln(l+1)−lny]/l! l=1 ∑ ω . (IV.5) Figure IV.2(a) shows how T increases monotonically with respect to the control parameter y, in comparison with T B computed from the boson distribution function n =1/[exp(ω /T B )−1], where 〈n〉 = y if one takes average of the number operator a λ + a λ 70 over the density operator. For sufficiently small quenches with y < 0.5, T is equivalent to T B , and the entropy S is the same as S B [see Fig. IV.2(b)], which is the canonical entropy of the harmonic oscillator at the same temperature T <ω maintained by heat baths. However, for large quenches, while T is increasingly higher than T B at y = 〈n〉, S becomes increasingly smaller than S B at the same temperature. S smaller than S B is expected because large quenches produce non-equilibrium states, whose entropy is always less than the equilibrium entropy. Figure IV.2 (Color online). (a) Temperature T (in units of ω /k B ) versus y=mωΔλ 2 /8 computing via Eq. (IV.5). (b) Diagonal entropy S (in units of k B ) versus T in a quantum harmonic oscillator. T B and S B are the temperature and entropy of a non-interacting boson system computed by ω /ln(1+ n −1 ) and [(1+〈n〉)ln(1+〈n〉) – 〈n〉ln〈n〉], respectively. Here, 〈n〉 is the expectation value of the number operator a λ + a λ . The red (A → B) and black (A → C) arrows present the transitions from an out-of-equilibrium state to the equilibrium states. The transition A → B has 〈n〉 = y, and A → C has T B = T. 71 To elucidate the meaning of the out-of-equilibrium temperature T being higher than T B , we consider transitions from an out-of-equilibrium state A without heat baths to equilibrium states B and C, which are controlled by heat baths with 〈n〉 = y and T B = T, respectively. In the first case [A → B in Fig. IV.2], T ~ 2y higher than T B ~ 〈n〉 at y = 〈n〉 indicates that the system reduces its temperature to T B when it is coupled to a heat bath maintained at T B to transform from the out-of-equilibrium state A to the equilibrium state B. This indicates that for large quenches the out-of-equilibrium state Ω λ is hotter than the equivalent equilibrium canonical ensemble, whose average occupation number 〈n〉 is smaller than 1/[exp(ω /T)−1], and hence T B < T. In contrast, when we consider a transition at constant temperature T B = T [A → C in Fig. IV.2], the system’s entropy is increased, and the expectation value of the number operator grows from y to 〈n〉. This means that in this case the system changes its initial Poisson distribution of the energy to the canonical distribution, and increases its average energy with the entropy while the temperature remains unchanged. Indeed, the system starts absorbing energy of ω( n −y) to thermalize with the heat bath for all final states in the regime between B and C. Finally, for transitions from A to final states in the regime between the origin (y=0,T=0) and B, the system emits energy ω(y− n ) to thermalize with the heat bath, while S either increases or decreases, depending on the particular heat-bath temperature. Now, applying the protocol described in Fig. IV.1, we obtain the same temperature and the same diagonal density operators for all single quenches. Given that we proceed to compute free-energy profiles. The distributions functions of x i are 72 f i (x i )= x i n λ i 2 p n λ i n λ i =0 ∑ , where x i n λ i are the wave functions of the coupled harmonic oscillator at λ i = (i–1)Δλ. At low temperatures or small y, the ground states dominantly contribute to the free-energy change, ΔF(λ 1 ,λ s )≈ k(s−1) 2 Δλ 2 4 ΔF Target       + k(s−1)Δλ 2 4 (1− ω 2T ) ΔF Can      + k(s−1)Δλ 2 4 T ω . (IV.6) The diagonal ensemble introduces a positive shift [the last term in Eq. (IV.6)] to ΔF Can , which is computed from the canonical ensembles, f i (x i )= Trδ(ˆ x−x i )e −H(λ i )/T /exp[−F(λ i )/T] (86). For a quench with T /ω=(−1+ 3)/2≈ 0.37, the last two terms cancel each other, thus ΔF(λ 1 ,λ s ) can be equal to ΔF Target . For large quenches, the last term dominates the second term, thus one expects ΔF(λ 1 ,λ s ) higher than ΔF Target . To illustrate the contributions of excited states, we numerically evaluate the free- energy profiles for a protocol with various Δλ. We find that more than 50 excited states neither change temperature nor the computed free-energy profiles within an uncertainty of 10 -6 for y ∈ [0:10] because of the Poisson distribution of energy. We note that while the distribution functions f i (x i ) look Gaussian [Fig. IV.3(a)] for Δλ = 0.6935 (y = 0.06), they exhibit double peaks at Δλ = 4.0 (y = 2) [Fig. IV.3(c)]. This double-peak feature in f i (x i ) at large y indicates the dominance of excited states in the Poisson distribution, which is in contrast to the always largest contributions of the ground states in canonical 73 ensembles. It indicates that the central limit theorem breaks down in out-of-equilibrium processes where coherent or excited states become dominant (88, 89). These distributions due to large quenches are similar to those of Lohschmidt’s Echo in the small-quench regime of a critical quantum XY chain (85), indicating poor equilibration. It is striking to find that the diagonal density operators can reproduce accurate free-energy profiles [see Figs. IV.3(b)-(d)] even in regimes, where diagonal density operators are different from canonical operators. Figure IV.3 (a) Distribution of reaction coordinate at λ = 0 and (b) free-energy profile at T = 0.35, Δλ = 0.6935. (c) Distribution of reaction coordinate at λ = 0 and (d) free-energy profile at T = 3.52, Δλ = 4.0. The distributions at other values of λ are identical to (a) and (c). Free energy change ΔF, T, and λ are in units of ω /2 ,ω /k B , and  /mω , respectively. DE-JE represents a free-energy profile computed by Diagonal Ensemble and Jarzynski’s Equality. The error bars are the standard deviations of the work distribution functions. 74 IV.3.2 Example 2: One-dimensional Hardcore Boson Lattice Let us now consider a one-dimensional lattice of hard-core bosons trapped in a harmonic potential (90), whose Hamiltonian after the Jordan-Wigner transformation is given by H(λ)=−J (f k + f k+1 +h.c) k=1 N ∑ +V f k + f k (k−a) 2 k=1 N ∑ +V f k + f k (k−λ) 2 k=1 N ∑ , (IV.7) where f k and f k + are fermion annihilation and creation operators at site k, N is the number of lattice sites, J is the hopping parameter, V denotes the spring constant, and a is a constant. We choose V/J = 0.0225 to obtain the superfluid phase for N b = 10 hard-core bosons (equivalently 10 fermions) in a lattice of N = 40 sites, and λ from λ 1 = a = 13 to λ s = 20. We follow the method described in Ref. (90): the initial state ψ 0 (λ i −Δλ) for the i-th quench is constructed by filling the N b lowest energy levels, whose eigenstates are obtained by diagonalizing H(λ i –Δλ). The excited many-body eigenstates of H(λ i ) are constructed by assigning fermions in H(λ i )’s ground states to the other (N–N b ) energy levels. We note that for any Δλ 2 , one must consider all many-body states that have overlaps with the initial ground states to have non-zero values of p n . To estimate an effective temperature, we use Eq. (IV.5) with entropies and energies evaluated from two quenches with Δλ and Δλ + ε at λ = 15 where ε = 0.1Δλ. The total energy E(λ=15) = ψ 0 (λ−Δλ)H(λ)ψ 0 (λ−Δλ) is approximately J(0.112Δλ 2 –0.495). Consistent with the results of Example 1, we observe that for small Δλ 2 the ground and a few excited many- body states overlapping with the initial states are sufficient to estimate the diagonal entropy and effective temperature. Figure IV.4(a) shows that T is approximately 75 proportional to Δλ 2 /8 ≡ y when all other parameters are made unity. This is similar to the dependence of T B on 〈n〉 for large 〈n〉 in Example 1. This result suggests that by increasing the number of particles, T becomes closer to T B , i.e., the system of ten particles appears to be a simple harmonic oscillator coupling to a heat bath. While T increases slower with y than T in Example 1, Figure IV.4(b) shows that the diagonal entropy S increases quicker with T than S in Example 1. This is consistent with fact that the entropy of ten particles should be larger than that of one particle at the same temperature. Figure IV.4 (a) Temperature T (in unit of J /k B ) versus Δλ 2 . (b) Diagonal entropy S (in units of k B ) versus T. (c) Distribution of the center-of-mass x of the superfluid during the time-evolution of ψ(t,λ i ) following a quench with λ i = 14. The distributions for other values of λ are almost identical. (d) Free-energy profile in a lattice of Hard-Core bosons trapped in a harmonic potential versus λ (Example 2). 76 The final test is to compare the free-energy profile with the target, F(λ s )–F(λ 1 ) = VN b (λ s –a) 2 /2. The work distribution functions are computed from the distributions of the center-of-mass x i = f k + f k i ×k k=1 N ∑ / N b and the work W =VN b Δλ (2λ i +Δλ−2x i ) i=1 s ∑ , where Δλ = 1 and 〈...〉 i is an expectation value using time-evolving many-body state ψ(t,λ i ) = exp(−jH(λ i )t)ψ 0 (λ i −Δλ) . Here, we solve the time-evolution for the state instead of evaluating the diagonal density operators, because it is practical to approximate the distributions by f i (x i )= Trδ(ˆ x− x i )Ω λ i ≈ lim τ→∞ 0 τ dt ∫ τ δ(x i (t)− x i ), (IV.8) where x i (t) is ψ(t,λ i ) ˆ xψ(t,λ i ) . This is a good approximation as long as lim τ→∞ 0 τ dt ∫ τ −∞ ∞ dk (−jk) n n! (ˆ x− x i ) n i −[ ˆ x− x i i ] n ( ) n ∑ ∫ is negligible. To obtain the convergent distributions of x i we simulate the evolution following each quench for times larger than N 2 (85). Similar to the distribution in Fig. IV.3(c), the double-peak feature in Fig. IV.4(c) indicates poor thermalization for the quenches in the lattice. The characteristic temperature for these distributions is 0.1953. Using this temperature and work distribution functions in Eq. (IV.3), we obtain the free-energy profile plotted in Fig. IV.4(d), which is approximately 10% higher than the target at λ = 20. 77 IV.3 Discussions and Conclusions We have observed that the temperature computed via Eq. (IV.2) coincides with the equilibrium temperature for sufficiently small quenches on a simple harmonic oscillator, whose diagonal entropy also converges to the canonical entropy in the non-chaotic regime. This suggests that the procedure of extracting an effective temperature T e from 〈H〉 = Tr[Hexp(H/k B T e )]/Tr[exp(H/k B T e )], where H is a many-body Hamiltonian operator of a integrable and closed system, 〈...〉 denotes an average over an initial quantum state, and Tr denotes trace, is problematic (81, 84). While T e can be incorrectly non-zero when initial states are pure, T from Eq. (IV.2) is zero if quenches represented by (Δλ + ε) and Δλ >> ε change energy 〈H〉 but maintain the purity, hence diagonal entropy in systems. Moreover, Example 1 shows that even if T B (similar to T e if there are heat baths) is equal to T, the energies evaluated over a diagonal density operator and the canonical density operator can be very different. Therefore, T e should not be used to compute and compare thermodynamic quantities between diagonal and other canonical entropies. Equation (IV.2) suggests that information from a single quench is not sufficient to examine how thermodynamic quantities emerge in quantum mechanics, even though it offers interesting and useful information of thermalization processes in terms of time scales, system sizes, entanglement, and so on. The limit in Eq. (IV.2) for two quenches (Δλ + ε) and Δλ indicates the convergence of external controls on systems. It implies the thermodynamic balance in the interactions between systems and external controls. It also implies that it is possible to model heat baths via controlling a parameter such as λ. This idea is supported by experiments of using laser-cooling techniques, in which variation of 78 frequencies (i.e. control parameters) coupled to systems of few particles can change temperature (91). Equation (IV.2) offers an efficient way of estimating temperature for closed quantum systems, which might not be restricted in the framework of diagonal entropy. The most expensive computational cost for many-body systems is in evaluating the entropy S, and then expressing S in terms of energy. Equation (IV.2) suggests that the difference of between diagonal entropies in two quenches (Δλ + ε) and Δλ is sufficient to compute temperature. In the two examples we find that they require a number of single- particle or many-body states as small as 50, in contrast to an exponentially large number of many-body states in Fock space to accurately compute a single value of T. Example 1 suggests one interesting effect that even if T computed for the diagonal density operator is equal to the temperature of heat baths, the system of a simple harmonic oscillator can still absorb a net energy to thermalize with heat baths by redistributing occupancy probabilities over energy levels, thus increasing entropy. This effect suggests a way to experimentally verify the temperature by measuring if the fluctuations of the heat-bath temperature before and after coupling with an out-of- equilibrium system are the same, while the system absorbs a significant energy (ΔE > 0). Note that according to the approach, the opposite effect with a net energy emission (ΔE < 0) cannot occur if there is no significant change in the fluctuations of temperature during the system and heat-bath coupling, because the entropy increases (ΔS > 0). In Example 2, we observe that the integrable and closed system of ten particles has temperature computed via Eq. (IV.2) similar to the equilibrium temperature of a 79 harmonic oscillator coupled to heat baths. This further confirms the procedure of computing temperature via Eq. (IV.2), which yields the equilibrium temperature as a number of particles increases (11). This procedure provides a generalized notion of temperature more general than the traditional Gibbs construction, which requires a large number of particles and chaotic collisions for thermodynamic equilibrium. This approach also offers a way to examine thermodynamics in chemical reactions, which are prototypical out-of-equilibrium processes. One is able to compute temperature via Eq. (IV.2) to model and characterize the thermodynamic balance between systems and surroundings, whose direct computation is expensive. Estimates of free-energy profiles from out-of-equilibrium processes of chemical reactions via the diagonal entropy and the JE can be used to generate energetic diagrams describing how chemical reactions may occur. We are currently attempting to apply this approach to examine chemical reactions in testable quantum systems such as hydrochloric acid and water molecules. In conclusion, we have presented an approach combining Jarzynski’s Equality with diagonal entropies to out-of-equilibrium closed quantum systems. We tested this approach on a harmonic oscillator and on a lattice of hard-core bosons verifying that even though diagonal ensembles can be different from canonical ensembles, it is possible to extract meaningful thermodynamic quantities such as temperature, work and free-energy profiles. 80 CHAPTER V Non-Equilibrium Dynamics Determine Ion Selectivity In The KcsA Channel V.1 Introduction Steady state concentration differences for inorganic ions between the cytoplasm and the extracellular bathing solution are essential for cell viability. Ion-selective channels embedded in the cell membranes that separate the cytoplasm from the extracellular solution facilitate the diffusion of specific ions across cell membranes and play important roles in cell physiology and pathophysiology. The ability of ion channels in neurons to discriminate between different monovalent cations, for example, underlies the initiation and propagation of action potentials, and the loss of ion selectivity in a G protein-gated K + -selective ion channel in mice leads to cell death and neurodegeneration (92). The mechanism of ion selectivity by ion channels has been most intensively studied in K + - selective channels because of the availability of several atomic resolution protein structures of these channels. These structures show that the K + ions are dehydrated when they are located within the selectivity filter of the channel, and it is likely that the process of dehydration of the ions is important to the mechanism of ion selectivity (93). It was suggested by Bezanilla and Armstrong in 1972 that the mechanisms of ion selectivity would emphasize either selective binding of the ions in the selectivity filter of the channel 81 or selective exclusion of ions from the selectivity filter (94). Multiple binding sites for ions in the selectivity filter have also been suggested to be necessary for a high throughput flux of K + (95–101), but how they are linked with the ion selectivity is under debate (102–106). In order to investigate the mechanism of ion dehydration and selectivity by potassium selective channels, numerous studies have employed free energy perturbation and umbrella sampling methods in molecular dynamics (MD) simulations (107). MD simulations of ion selectivity in the bacterial KcsA K + -selective channel have identified numerous parameters of the ion-protein interaction such as ligand field strength, coordination geometry or number, or the surrounding protein matrix, that are important factors for determining ion selectivity (100, 103, 106, 108–112). These methods used in MD simulations are powerful approaches that can provide insight into selectivity mechanisms from free-energy barriers; however, they usually employ constraints or algorithms to achieve fast sampling of rare trajectories that are not readily available in straightforward molecular dynamics simulations. To quantify the selectivity of K + over Na + , the methods are often used to compute free energy differences between equilibrium states for different ions bound by the channel. However, these equilibrium states are themselves the consequences of ion selectivity and may not contain information about the mechanisms that led to these specific bound states. Several studies indicate that the structure of the selectivity filter of KcsA may depend on the ionic environment. Zhou et al. showed that the structure of the selectivity filter of KcsA in 200 mM K + is compatible with high-throughput ion permeation whereas 82 the structure of the selectivity filter in 3 mM K + represents a non-conducting state (113). They proposed that ion conduction occurs only when the selectivity filter snaps into a conformation similar to that observed in the protein at high K + concentrations. Grottesi et al. (114) also showed in simulation studies that the conformational flexibility of the selectivity filter leading to specific ion-bound states is not only essential for ion selectivity (103), but is also linked with the gating of potassium channels. Shrivastava et al. simulated the movement after 2 ns of relaxation of both Na + and K + ions that were initially placed in the S1 and S3 sites in the selectivity filter. They found that the ions relaxed to different positions and that the selectivity filter became distorted, with the distortion in the presence of Na + being larger than that in K + (50). Finally, Nimigean and Miller experimentally demonstrated that cytoplasmic Na + blocks the channel for observable times. It is not known, however, how high concentrations of K + trigger formation of the conducting structure of the KcsA selectivity filter, nor what structural features of the selectivity filter promote blockage by Na + and conduction of K + . In addition, although a number of studies using free-energy perturbation MD methods concluded that site S2 in the selectivity filter of KcsA is the most selective binding site for K + over Na + (98, 99, 108, 109, 112, 115, 116), Kim and Allen (117) have suggested that Na + does not bind to site S2 in multiple-ion configurations, and that ion selectivity is based on selective exclusion of Na + from the selectivity filter rather than on selective binding of K + to sites within the selectivity filter. To address these issues, we use a newly developed approach to take into account transient interactions between the ions, water, and the channel. These dynamical 83 interactions underlie the conformation flexibility of the selectivity filter to induce binding sites and must be included in a complete description of ion selectivity (103–106, 118). We have developed a method based on Jarzynski’s Equality (5) that uses step-wise pulling protocols to generate and exponentially weigh trajectories of single K + and Na + ions first entering the selectivity filter. These trajectories are then used to compute free- energy changes in both equilibrium and non-equilibrium dynamics (60, 86, 119). Because the system is relaxed at the end of each pulling step, this method provides information about transient processes between non-equilibrium and full equilibrium (i.e., quasi- equilibrium) states to extract movement trajectories of the ions and the fluctuations of other atoms involved in ion selectivity. It allows direct estimates of conductance while other methods such as steered molecular dynamics simulations (25, 28), adaptive biasing force (32), free-energy perturbation (40) and umbrella sampling methods (36) do not. These simulations of single Na + and K + ions pulled through the KcsA selectivity filter show that the most stable locations for K + and Na + ions in the selectivity filter of KcsA are different and are consistent with previous studies, and that entry of K + into the selectivity filter is favored over Na + by approximately 3.7 kcal/mol. The simulations also show that the selectivity filter of the KcsA selectivity filter undergoes structural rearrangements in response to the ions as they start entering the filter, and that these rearrangements are different for Na + and K + ions. The structural changes that are induced in the selectivity filter in response to the different ions favor entry and permeation of K + ions and rejection of Na + ions at the entrance of the selectivity filter, and reveal a dynamic component in the mechanism of ion selectivity in this ion channel. This is the 84 first step for multiple-ion permeation requiring multiple-binding sites favorable for only potassium ions. The simulation data not only agree with a number of other simulations and experiments, but also suggest a way to resolve the inconsistency between the results from Kim and Allen and other simulation results. The simulation results add a dimension of non-equilibrium dynamics based on Jarzynski’s Equality and step-wise pulling protocols to the studies of the ion selectivity. V.2 Materials and Methods Molecular dynamics simulations were done using NAMD (43, 120) to pull single K + and Na + ions step-wise from the center of mass (z = 0) of the simulation system along the z- axis toward the extracellular surface of the membrane. The simulation system consisted of the tetrameric KcsA channel (PDB accession number 1K4C) embedded in a POPC lipid bilayer with associated water molecules. The 1K4C crystal structure of KcsA was obtained at high K + concentration and is compatible with ion conduction (113). We prepared the two simulation systems with either 0.4M KCl or NaCl, which have no ion in the selectivity filter, and contain 47332 atoms in total. The periodic boundary condition was applied to the x, y and z directions. The systems were first thermalized at 310 °K for 1.5 ns to melt the lipid membrane and exclude water molecules from the hydrophobic region of the lipid membrane while the ion channel is kept fixed (121). This lipid membrane is found sufficiently melted to bind to the ion channel, meaning that water molecules cannot enter the region between the channel and lipid molecules by thermal motion. Then the channel is unconstrained, and an ion was placed at the center of mass of 85 the system at z = 0 Å in vestibule and coupled with a soft harmonic potential (k = 0.6 kcal/mol/Å 2 ≈ 1.0 k B T/Å 2 ) having λ = 0 Å as the center of the potential. The system was further thermalized for 0.5 ns with harmonic constraints (spring constant = 10 kcal/mol/Å 2 ) on the lipid membrane in all directions. The simulations were done in the absence of an applied voltage difference across the membrane. Details of the simulation protocols to vary λ and to compute work distributions and free-energy profiles are given below. Charmm27 force fields (44, 122, 123) were used with the corrections to the Lenard-Jones interactions between potassium sodium ions, and carbonyl oxygen atoms, which were proposed by Noskov and Roux (108). Simulations were run on the High Performance Computing Center at USC and on Ranger at the Texas Advanced Computer Center, through the XSEDE portal. Step-Wise Pulling Protocol to Analyze and Control Non-Equilibrium Dynamics Theory: The free energy profile is one of the most important features of ion channels that can be used to understand ion permeation. Recently, Ngo developed a computational technique based on the Jarzynski’s Equality, Eq. (V.1), and using step-wise pulling protocols that can be used to compute free energy profiles from work distributions in processes approaching equilibrium, without applying any additional algorithms for fast sampling of rare trajectories, such as are used in free energy perturbation and umbrella sampling methods. These step-wise pulling protocols (60, 86, 119) require a finite number of pulling steps, and a relaxation time at each step (Figure V.1A-B), which were determined by comparison with the potential of mean force computed by steered 86 molecular dynamics simulations on stretching deca-alanine (25, 28). The relaxation time is necessary to account for thermal effects (103), to allow the ion to move into stable or transient states, and to generate trajectories in work distributions. These trajectories are then exponentially weighted in the Jarzynski’s Equality. Importantly, this method is suitable to observe the transient processes from non-equilibrium to equilibrium, i.e., quasi-equilibrium. Figure V.1 Schematic diagram for a step-wise pulling protocol. (A) Control parameter λ, and (B) expectation value of an applied potential versus time, U(z, λ) = k(z – λ) 2 /2. Coordinate z is the position of an ion along the z-direction. The pulling in the direction of increasing λ is called forward; the opposite direction of pulling is called backward. (C) Comparison between profiles for PMF (blue) and free energy changes (red) computed using our method. A convergent free energy profile computed by the technique can be the same as the potential of mean force PMF when an applied harmonic potential is strong with large 87 relaxation times and large number of points of λ, which are also the essence of the thermodynamic integration [Eq. (V.2) or see Ref. (60) for more details]. However, when weak applied harmonic potentials are used, PMF and free energy profiles computed by the technique can be different (Figure V.2C), but their free-energy differences between stable states determined by local minima of a PMF should be similar. The rationale is that weak applied harmonic potentials are insufficient to keep ions at transitional states (denoted by local maxima in the PMF). However, such a transition between two local minima should require the same amount of thermodynamic work, i.e., a free-energy change, regardless of applied potentials (60). The advantage of using weak applied potentials in timescales of nanoseconds is that one can observe more subtle transition processes between the local minima than can be observed using strong potentials, which can easily bias the movements of ions. Furthermore, using strong potentials or other conventional techniques to compute PMF, it might not be possible to observe the dynamical adaptation and mutual responses of ions such as positions and velocities with respect to the changes of the selectivity filter. Such adaptation and responses include the fluctuations of reaction coordinates (~ vibrational modes), which can actually provide corrections to PMF profiles (124). The simulations using the step-wise pulling protocols generate the distributions of the z-coordinate g i (z) and averaged positions 〈z〉 I of the ion. From g i (z) and 〈z〉 i , free- energy changes can be computed as 88 ΔF(λ i ,λ 1 )=F(λ i )−F(λ 1 )=−k B T ln[ dW i e −W i /k B T ρ(W i ) ∫ ], (V.1) ΔF(λ i ,λ 1 )≈ kΔλ ( j=1 i−1 ∑ λ j − z j ) Δλ→0 & → && ∂H ∂λ λ dλ λ 1 λ i ∫ , (V.2) ρ(W i )= dz j j=1 i−1 ∏ g j (z j ) j=1 i−1 ∏ δ W i − k 2 [(z j −λ j+1 ) 2 − j=1 i−1 ∑ (z j −λ j ) 2 ] $ % & ' ( ) , ∫ (V.3) where λ i = (i–1)Δλ is the center of the applied potential, Δλ is an increment, H is the Hamiltonian, T is temperature, ρ(W i ) is the distribution function of work W i computed from g i (z j ) with z j being all possible values of z at the j-th pulling step, and the Delta- Dirac function δ is used. Eqs. (1) and (2) hold for finite relaxation times indicating non- equilibrium, and infinite relaxation times indicating equilibrium, respectively. From the overlaps between successive g i (z), important trajectories having small values of work W are weighted by exp(–βW) in Eq. (V.1). We have found (86) that the most optimal reaction pathways having the largest contributions to free-energy profiles satisfy the principle of detailed-balance: ψ E i (x i−1 ) 2 ψ E i−1 (x i ) 2 =e (E i −E i−1 )/k B T , (V.4) where ψ E i−1 (x i ) 2 is a forward transition probability for the system at E i–1 moving into E i , and ψ E i (x i−1 ) 2 is a reverse transition probability for the system at E i returning to E i–1 . The solution of Eq. (V.4) can be represented by the Frank-Condon principle for general chemical reactions to occur, which requires certain overlaps among probability functions of reactants. Therefore, Equation (V.4) is useful to unveil the important transition 89 pathways occurring in quantum systems by computing the probability distributions of reactants and electrons. Pulling protocols: To initiate forward pulling of one ion through the selectivity filter, the coupled system having λ = 0 Å was used as the initial state, and sequential pulling simulations were first performed along the z-axis in which the initial configurations for λ i were successively taken from the final configurations of λ i-1 . An increment λ i – λ i–1 = Δλ = 1.0 Å (60) yielding reliable results in stretching deca-alanine was used here. After each Δλ step, a spatial displacement of the ion that depends on the interaction of the ion with the system as well as on the harmonic pulling potential may occur. Table V.1 shows the resultant positions along the z-axis for both Na + and K + during step-wise pulling through the KcsA channel as described above. Note that for some positions of the ions, small incremental increases in pulling force associated with Δλ do not result in movement of the ions. In each sequential step, the systems characterized by 25 values of λ from 0 to 24 Å are relaxed for τ 1 = 0.5 ns, which is a minimum time found in Ref. (60). Then, the 25 simulations were run in parallel for τ 2 (~10 ns) of further relaxation to ensure the convergence of physical quantities (Figure V.2C). This relaxation time τ 2 is also about the order for the selectivity filter to dehydrate, transfer and re-dehydrate K + ions in experiments (97), thus it is reasonable to have all relevant data at each pulling step to sufficiently describe the ion movement. Moreover, the parallel feature is more computationally efficient to collect such data than the steered MD simulations without using step-wise pulling protocols (60). 90 TABLE V.1: Z-Coordinates (Å) for Stable Positions for Na + and K + Ions After Each Pulling Step (Δλ = 1 Å) λ Stable z-positions (Å) for K + Stable z-positions (Å) for Na + 0 2.1, 4.4 0.4, 2.3, 4.9 1 2, 4.5 2.4, 5 2 4.6 5.2 3 4.7 5.2 4 4.8, 7 5.3 5 5.2, 7.1 5.5 6 7.1 5.5 7 7.2 5.6 8 7.2 5.6, 5.8 9 7.3 5.7, 7 10 7.4 6.7 11 7.5 5.8, 6.8, 8.8 12 7.6 6.8, 9 13 7.7 7, 8.8 14 7.8 9 15 7.8, 10.3 9 16 7.9 9.3 17 7.9, 10.3, 13.9 9.1 18 8, 10.3, 14 9.2 19 10.7, 15.7 9.3 20 10.6, 14.5, 16, 17 9.4 21 10.6, 15.8, 17.7 9.6, 11.8, 21.9 22 17.5, 22.7 9.5, 10.5, 22.4 23 18, 22.7 9.5, 10.5, 11.3, 15.7, 22.5 24 18.1, 23, 24.5 9.7, 11.7, 15.8, 17.5, 22.1, 24 Na + and K + ions were pulled in increments (Δλ) of 1 Å using a soft harmonic potential of 0.6 kcal/mol/Å 2 that has its minimum value at λ. At the start of the pulling sequence Δλ is centered at z = 0. As λ is incremented by Δλ, the spatial coordinates of the Na + and K + ions are unconstrained during the relaxation period τ. The z-coordinate of the ions at the end of τ is shown in the Table. Multiple values in the Table reflect the different z-coordinates observed in the simulations. A histogram of the frequency of the occurrence of the different values for z for each ion is shown in Figure V.2. 91 Estimate of Channel Conductance From Work Distributions Since work applied to an ionic charge q is equivalent to heat emitted by a resistance R = G –1 , we propose a simple expression of conductance G= I 2 (t j −t i +(j−i)τ) / W , (V.5) where I = q[z(t j )− z(t i )]/[(t j −t i +(j−i)τ )(λ j −λ i )] , and z(t i ) is the ion position along the axis of motion at time t i ∈ [0:τ] with j > i being the index of the pulling step. Here, the bracket denotes the average over any possible trajectories from the i-th to j-th pulling steps, and 〈W〉 is the averaged value of work for the trajectories. The physical interpretation of this current is that two electrodes are put at positions λ i < λ j to detect a charge q moving from z(t i ) to z(t j ). If there is a steady flow of a single ion moving with a constant velocity v measured within τ = L/v, then I = qv/L where L = λ j – λ i is the distance between the two applied electrodes and t j – t i = (z(t j ) – z(t i ))/v. Taking into account the fact that we sample z(t j ), t j , z(t j ) and t i independently (i.e., t j – t i can be negative), we collect their values such that t j – t i + (j – i)τ = (z(t j ) – z(t i ))/v to mimic a steady flow having t j – t i + (j – i)τ > 0. In the picture of a steady flow, an electric field caused by the electrodes always performs one value of work W on the charge within a given duration, and R is related to a level of the friction or heat between the charge and filter, which is equal to W. In a non-equilibrium framework, the steady flow belongs to a set of many flows caused by various values of work mimicked by a harmonic applied potential. The heat of these flows is proportional to I 2 (t j −t i +(j−i)τ)=[q 2 /(λ j −λ i ) 2 ][z(t j )− z(t i )] 2 /(t j −t i +(j−i)τ), (V.6) 92 which is equal to a diffusion coefficient D ij (125) multiplied with the factor in the square bracket. Therefore, for infinite relaxation times, one would obtain the convergent values of both numerator and denominator in Eq. (V.5), i.e., conductance G is a characteristic and convergent value. For a finite relaxation time, this conductance represents the averaged response of the selectivity filter toward all possible movements of an ion under non-equilibrium conditions. Note that G can be negative for a pair of pulling steps, which appears encounter-intuitive, but a total conductance is positive as described below. Since work is more biased (60) for a pair having j > i + 1 than a pair with j = i + 1, we compute the conductance G i+1 for a pair of successive steps i-th and (i+1)-th, in which the work 〈W i+1 〉 is averaged from U(z(t i ),λ i + Δλ) – U(z(t i ),λ i ) over all possible values of z(t i ). Note that G i+1 can largely fluctuate along the selectivity filter due to the fact that 〈W i+1 〉 and D ij are not the same for all pairs of successive steps. We compute a non-negative total conductance as follows 1 G total = 1 G i i=1 s−1 ∑ = W i D ij i=1,j=i+1 s−1 ∑ ≤ Wtotal [D ij ] min (V.7) Since the total work W total = ∑ i 〈W i 〉 to enforce ions through the channel and the minimum of D ij are always positive, this total conductance is therefore positive. The first equality in Eq. (V.7) is known for a total resistance of multiple independent resistors in series. This suggests that the Eq. (V.7) is valid when all pairs of successive steps are uncorrelated. 93 V.3 Results V.3.1 Free-energy Profiles To simulate the movement of K + or Na + ions through the KcsA channel, a harmonic pulling potential that couples an ion's z-coordinate with the center of the potential (λ) was used. The protocol uses 24 discrete pulling steps with increments of 1.0 Å from λ = 0, and requires relaxation times τ at each step. The relaxation time permits collection of thermal distributions of the spatial coordinates that are used to identify stable positions of atoms, and to calculate work distributions that are used to compute free-energy profiles. Free energy profiles for the ions were calculated from the work distributions as described in Materials and Methods. In these simulations, each ion was initially hydrated at the center of mass of the simulation system, in the water-filled central cavity (vestibule) of KcsA, and was assigned a reference free energy value of zero. As the K + and Na + ions are pulled through the KcsA channel they encounter different free-energy barriers due to differences in the interactions between the ions and the channel. Although single Na + or K + ions were pulled through the channel in these simulations in order to simulate dynamic interactions between the individual ions and the protein, the structure of the KcsA channel used was obtained at high K + concentration, and represents the structure of the channel when the selectivity filter is occupied by multiple ions. Figure V.2A shows the calculated value of the free energy difference (ΔF) for each ion as it moved from the 94 center of mass of the system (z = 0) in response to sequential pulling steps (Δλ) of 1 Å along the z-axis. Figure V.2 Free energy profiles for Na + and K + ions in the KcsA channel. (A) Free energy difference (ΔF) for Na + and K + ions after each pulling step (λ) compared to λ = 0. (B) Magnitude of the difference between Na + and K + ion free energy differences (ΔΔF) at each position of λ shown in (A). (C) The convergence of ΔΔF at Δλ = 11 Å during relaxation time t. The uncertainty for computing free-energy profiles is about 1 kcal/mol. 95 The free-energy profiles show that relative to the reference position, K + has its free-energy minimum, -7.8 kcal/mol, when λ = 7 Å, and Na + has its free-energy minimum, -7.0 kcal/mol, when λ = 5.5 Å. The large free-energy minima indicate that both ions are strongly attracted to these sites. Figure V.2B-C shows that there is a peak in the convergent free-energy barrier difference (ΔΔF) of approximately 3.7 kcal/mol for the two ions at λ = 11 Å. The ΔΔF profile of the ions decreases at 12 Å < λ ≤ 17 Å, and then increases. Note that a free-energy profile computed from this method has a direct physical meaning: since it is computed from distributions of work, it indicates an amount of thermodynamic energy, including any thermodynamic effects such as entropic effects, required to transform a system from one state to another state denoted by different values of λ (Figure V.1). Thus, a difference in free-energy profiles for the two ions directly indicates an amount of work necessary for a transition between stable positions that are identified from a histogram of positions. Table V.1 shows the corresponding z-coordinate values of the K + and Na + ions after each incremental pulling step Δλ, and histograms of these data showing the frequency of occurrence of each z-value throughout the step-wise pulling simulations is shown in Figure V.3. 96 Figure V.3. Histogram showing the frequency (probability) of finding the Na + and K + ions at each position (z) in the selectivity filter during step-wise pulling. The histogram is construct from 25 pulling-step simulations for each ion. Sites S0-S4 correspond to sites of stable K + binding in the selectivity filter of KcsA. Stable Na + positions are located in the plane of the carbonyl oxygen atoms that separate stable K + binding sites. Note that Na + ions are not stable within the selectivity filter at positions beyond the S2-S3 junction until they emerge from the channel at S0. Z = 0 corresponds to the center of mass of the protein-lipid- water system, within the central water-filled cavity in the transmembrane domain of the channel. V.3.2 Location of Stable Ion Binding Sites The peaks in the histograms shown in Figure V.3 identify stable positions of the K + and Na + ions in the KcsA channel during the 25 pulling simulations. The peaks in the histograms for K + are in the same locations as the crystallographic K + binding sites S4 (z = 7.1 Å), S3 (z = 10.4 Å), S2 (z = 13.6 Å), S1 (z = 17.0 Å), and also the S0 region (z = 23.1 Å) (113). Site S4 coincides with the location of K + at the minimum of ΔF for K + shown in Figure V.2B (see also Table V.1). The histograms for Na + show that there is a 97 very stable site for Na + located in the vestibule just outside of the selectivity filter (z = 5.5 Å), coincident with the location of Na + at the minimum in ΔF for Na + (Figure V.2A), and that within the selectivity filter Na + is stabilized by the coordination of the planar arrangement of carbonyl oxygen atoms found between the K + sites S4/S3 and S3/S2. The simulations indicate that there are no stable positions for Na + within the selectivity filter beyond the S3/S2 junction. Using free energy perturbation MD simulations of ion binding in KcsA, Thompson et al (126) and Kim and Allen (117) also found that the binding sites for Na + and K + in the selectivity filter of KcsA are different, and also suggested that selective permeation may involve barriers that exclude Na + from the selectivity filter. The stable positions of the ions identified in the non-equilibrium simulations are the result of unconstrained motions of the ions in response to the incremental pulling potential. The peak in the ΔΔF profile at λ = 11 Å (Figure V.2B) corresponds to the difference in the amount of work required to move a Na + ion from its stable position in the vestibule (z ≈ 5.5 Å) to the junction between S4 and S3 (z ≈ 9.2 Å) compared to the amount of work needed to dislodge K + from its stable position at S4. As is shown below, the movement of K + into site S4 from the vestibule occurs with a high probability and is associated with little energy cost. The free energy difference of 3.7 kcal/mol derived from the non-equilibrium simulations indicates that the selectivity filter selects K + ions over Na + ions with a selectivity ratio for K + /Na + of approximately 400. This value can be compared to the experimentally determined lower limit for K + /Na + selectivity of 150 (127). 98 V.3.3 Movement of Ions Between Stable Binding Sites The height of the peaks shown in Figure V.3 does not represent relative equilibrium binding affinities at different positions within the channel for the ions but may be interpreted as probabilities for finding the ions at each value of z along the axis of the KcsA channel. These peaks were analyzed in more detail for the occurrence of each value of z at different values of λ and for the magnitude of the force associated with the ion at each value of z. The force on the ions in the z-direction at any value of z, f(z), is a result of the harmonic pulling force in the z-direction sampled at all values of λ. A histogram of frequency versus z-coordinate for each peak in Figure V.3 at all values of λ showed that each peak is associated with a single distribution of z-values that can be approximated by a Gaussian distribution with a mean centered at the position of the maximum of the peak. Means and standard deviations (SD) were calculated for each peak. Histograms of f(z) associated with each peak in Figure V.3, however, were sometimes approximated better by multiple Gaussian distributions with more than a single maximum value for the force. In these instances a Gaussian distribution was also used; however, a larger SD in f(z) resulted from this procedure than for peaks where a single Gaussian distribution was observed. Figure V.4 shows a plot of the force f(z) associated with either Na + or K + ions at each z-coordinate position in the KcsA channel. Each data point corresponds to a different peak in Figure V.3, with associated uncertainty in both z-position and force shown as SD. 99 Figure V.4 Force (kcal/mol/Å) on Na + (circles) or K + (squares) ions in the z- direction as they are pulled step-wise through the KcsA channel. Zero on the z- coordinate axis represents the center of mass of the system. Site S4 in the selectivity filter occurs approximately at z = 7.1 Å, site S3 approximately at z = 10.4 Å, site S2 approximately at z = 13.6 Å, and site S1 approximately at z = 17.0 Å. Histograms of the z positions of each of the two ions were created from all different values of λ (Figure V.2). Univariate normal mixture decomposition implemented in STATA (STATA Corp., State College, TX) was used to determine the number of Gaussians that contribute to the complex distribution of each ion distribution in Figure V.3. The mean of each Gaussian was associated with a stable position for the ions along the z- axis of the KcsA channel. The 95% confidence interval for each Gaussian was used to partition the force, f(z), to an equal number of segments of normal distributions. Means and standard deviations (SD) for the force were calculated for each segment. Mean values of f(z) are shown in the plot with SD estimates in both z- coordinate and f(z). We define the probability of transition from one stable position to another, P(z i - > z i+1 ), as the probability that the force in the current position, f(z i ), is equal to or larger to the f(z i+1 ) in the subsequent position, P(f(z i ) ≥ f(z i+1 )). Thus, we assume that the transitions between to stable positions of the ions can occur by diffusion or by attraction to the subsequent position without any energy cost. For transitions between two stable points with small SD around the mean f(z), we consider that the probability of transition to the subsequent position is 1 if the mean f(z i ) ≥ f(z i+1 ). Subsequently, the probability for the ion of going back once it reaches stable position z i+1 is 0. For transitions between stable position of the ion where either the mean for the force, f(z i ), or the subsequent f(z i+1 ) has a small SD and the other one has a large SD, we calculate the probability P(f(z i ) ≥ f(z i+1 )) by focusing on the f(z) with the large SD. Within this given segment for f(z) we calculate the probability as sum of the observations that satisfy the inequality, f(z i ) ≥ f(z i+1 ), divided by the total number of observations within that segment. Finally, for transitions between stable positions of the ion with coordinate z and a f(z) with very large SD, the probability of transition between two positions, P(f(z i ) ≥ f(z i+1 )), will be represented with P(f(z i ) ≥ f(z i+1 ) ∩ f(z i+1 ) ≤ f(z i )) = P(f(z i ) ≥ f(z i+1 )) * P(f(z i+1 ) ≤ f(z i )). To calculate the two probabilities first find the observations in both segments that satisfy the inequality f(z i ) ≥ f(z i+1 ) and then we divide the respective number by the total number of observations in the given segment. Furthermore, the above formula implies that we consider the two events independent. 100 Figure V.4 shows that the stable positions for the ions in the channel are defined fairly precisely, with SD of less than 1 Å. The force on the ions at each position is also fairly precisely determined when the ions are outside the selectivity filter in the vestibule of the channel (z < 5 Å) or in the selectivity filter beyond the position of the carbonyl oxygen atoms that separate sites S4 and S3 (z > 12 Å). The force on the ions in the region 5 Å < z < 12 Å, however, is highly variable. This observation is significant because the ions are fully hydrated within the vestibule of the channel, whereas in site S4 the K + ions are coordinated by an average of about 0.5 water molecules, and at all positions within the selectivity filter the Na + ions are hydrated with at least two water molecules (Figure V.5). It is likely, therefore, that the variation in force shown for the ions in the region 5 Å < z < 12 Å is related to the mechanism of dehydration of the ions, which is the rate limiting step in ion permeation. For K + ions there is a single stable position at S4 in the region 6.0 Å < z < 8.5 Å and the SD for f(z) for this site is large. Within this region, 386 values of f(z) from a total of 1300 values were found to be < 0, indicating that a K + ion can jump spontaneously from the vestibule where it is hydrated to site S4 where it is dehydrated, with a probability of approximately 0.3. The probability that a K + ion will jump spontaneously from site S4 to site S3 was calculated from the overlap of f(z) distributions at these two positions and is 0.11. These calculations reinforce the conclusion that S4 is a stable location for a single K + ion within the selectivity filter and that K + can enter the selectivity filter with little or no energy cost. For Na + ions in the region 5 Å < z < 12 Å there are two positions where the f(z) values are highly variable, 101 corresponding to locations at the minimum of ΔF just outside the selectivity filter in the vestibule at z ≈ 5.6 Å and in the selectivity filter between sites S4 and S3 at z ≈ 9.1 Å. The probability of Na + moving between these two positions was calculated from the overlap of the two f(z) histograms and was found to be 0.02. Thus, under the conditions of these simulations, Na + is 15 times less likely to move from the vestibule to its stable position between S4 and S3 than K + is to move from the vestibule into site S4. Alternatively, since rates of transitions between states can be considered to be probabilities per unit time, this result indicates that the rate of Na + movement into the selectivity filter is at least 15 times less frequent than movement of K + ions. Note that these simulation conditions are for single ions only and do not incorporate effects of multiple ions. V.3.4 Differential Dehydration Of Na + and K + By The Selectivity Filter Crystal structures of K + ions in the selectivity filter of K + -selective channels indicate that the K + ions are dehydrated, and the higher energy cost to dehydrate Na + ions compared to K + ions may be one of the key reasons that K + -selective channels exclude Na + ions. Numerous physical parameters such as apparent hydration number, x-ray and neutron diffraction data, and the Jones-Dole viscosity B coefficient indicate that Na + is strongly hydrated in aqueous solutions while K + is weakly hydrated (128). Figure V.5 shows that the average number of water molecules within the first hydration shell (3 Å) of K + and Na + in the vestibule of KcsA (z < 5 Å) is 5-6 for both ions, and that the number of bound water molecules fluctuates with different standard deviations for Na + and K + that reflect 102 the shorter lifetime of water attached to K + than to Na + (129). Some water accompanies K + during entry into the selectivity filter, and the number of H 2 O molecules associated with K + in S4 is approximately 0.5 on average. This is consistent with the ratio 0.9 ± 0.2 of K + to H 2 O in the multiple-ion conduction under electric fields as observed in much longer time MD simulations (~µs) (101) than the single-ion pulling simulations. The figure also shows that Na + is always accompanied by at least two water molecules when it moves through the selectivity filter. The absence of data points for Na + in Figure V.5 between site S3 and the extracellular solution is consistent with the histograms of the probability of finding the ions at specific positions within the selectivity filter (Figure V.3), and indicates that there are no stable binding sites for Na + ions between sites S3 and S0. Taking into account the average number of water molecules binding to the Na + , the effective diameter of Na + is approximately 4.6 - 5.0 Å, which is significantly larger than the ionic diameter of K + . The dimensions of S4 in the KcsA crystal structure can easily accommodate a dehydrated K + ion but not a hydrated Na + ion. 103 Figure V.5 The number of water molecules within 3 Å of Na + and K + ions as they are pulled incrementally through the selectivity filter of KcsA from the cytoplasmic side of the membrane (left) to the extracellular surface of the membrane (right). On the x-axis, z = 0 corresponds to the center of mass of the system in the vestibule of the channel. The first data point on the left is a position in the vestibule below site S4. Site S4 is located at z ≈ 7 Å. Values shown are means ± SD. V.3.5 Structural Rearrangement of The Selectivity Filter in Response To Na + and K + Rapid fluctuations in the positions of the carbonyl oxygen atoms of the amino acids in the KcsA selectivity filter were observed in the simulations. These fluctuations were examined in more details when either single Na + or K + was pulled from the vestibule into 104 the selectivity filter since they directly quantify the flexibility of the selectivity filter. Kim et al. have suggested that the thermal fluctuations under external driving forces (from ions) are essential to the mechanism of ion selectivity, but these fluctuations have not been yet received sufficient attention (103). Figure V.6 shows that the positions of the carbonyl oxygen atoms depend on whether a Na + ion (upper panels) or a K + ion (lower panels) moves from the vestibule into the selectivity filter, altering the structure of the selectivity filter of KcsA in an ion-dependent manner. The figure shows histograms of the distribution of the positions of carbonyl oxygen atoms from the same amino acid in the four subunits of the tetrameric KcsA channel when the center of the harmonic pulling potential is located either just below S4 (λ = 0), within S4 (λ = 10), or just beyond S4 (λ = 15). The histograms show results from the last 4 ns of each simulation. 105 Figure V.6 Positions of carbonyl oxygen atoms for amino acids T75-G79 in the KcsA selectivity filter during step-wise pulling of either Na + (top row) or K + (bottom row) from the vestibule toward the extracellular surface of the membrane. The y-axis is the frequency (probability) of finding the oxygen atoms at the position shown on the x-axis. Zero in the middle of the x-axis corresponds to the position of the atoms observed in the crystal structure. The carbonyl atoms corresponding to each amino acid in the tetrameric selectivity filter are shown in a single color, and each amino acid is represented by a different color (T75, black; V76, green; G77, blue; Y78, red; G79, light blue). In this figure, the value z = 0 corresponds to the position of the carbonyl oxygen atoms when they are arranged in a plane separating stable K + binding sites, and deviations from this planar arrangement above and below the plane are shown to the left and right of z = 0. The effect of ion identity on the carbonyl oxygen atoms is observed 106 most dramatically for the carbonyl oxygen atoms of V76 (green traces) that separate sites S3 and S2, although a similar pattern is seen for the carbonyl oxygen atoms of several of the other amino acids in the selectivity filter. When a Na + ion is located in the vestibule below S4 (λ = 0), the four carbonyl oxygen atoms of V76 are all found near the position observed for these atoms in the crystal structure, which defines a plane of oxygen atoms and a potential Na + binding site between S3 and S2 (117, 126). As the Na + ion is pulled into S4 (λ = 10) and beyond S4 (λ = 15), the arrangement of the carbonyl oxygen atoms of V76 becomes disordered, disrupting the planar binding site for the Na + ion between S3 and S2. Disruption of the planar binding site for Na + between S3 and S2 will decrease the stability of Na + binding to this site and hence reduce the probability that Na + will move to this site. When K + is located in the vestibule below S4, the carbonyl oxygen atoms of the four V76 residues are distributed between two positions located 0.5–1 Å above and below the plane. Movement of K + into and beyond site S4 induces a planar arrangement of the carbonyl oxygen atoms of V76 and optimizes the structure of the S3 and S2 binding sites for the coordination of K + by stabilizing the planar arrangement of carbonyl oxygen atoms that is seen in crystal structures. K + entry into the selectivity filter also reduces the dispersion of the other carbonyl oxygen atoms in the selectivity filter around planar positions, creating an ordered structure in the selectivity filter that can accommodate multiple K + ions. This structural rearrangement of the selectivity filter induced by K + represents a possible mechanism for the formation of the four contiguous binding sites for K + in KcsA that Derebe et al. concluded is essential for K + selectivity in this channel (106). 107 The distortion versus order caused by Na + versus K + is also consistent with the simulations of Shrivastava et al. (130), in which two Na + ions were placed near S1 and S3, and cause the distortion during further relaxation of 2 ns. Our simulations show that such distortion occurs even before Na + ions enter the two sites. V.3.6 Estimation of Channel Conductance Because the non-equilibrium simulations calculate work distributions for the movement of ions through the channel, an estimate of channel conductance can be obtained from the equivalence of work and the heat generated by an ionic current through a resistor. For a finite relaxation time, the calculated value of the conductance represents the averaged response of the selectivity filter to all possible movements of an ion in the non- equilibrium pulling processes at 0.4 M of salt concentration. We found that the total conductance values (G) associated with the pulling of single potassium and sodium ions are 2.9 and 1.8 pS at zero applied voltage, respectively. Even though the values of the total conductance are small, we found that the calculated average conductance G ave of a single K + ion in the region of the KcsA channel between sites S4 and S1 in the selectivity filter in the absence of a voltage gradient was 29 pS, and was higher outside of the selectivity filter (Figure V.7). The measured value of the zero-voltage conductance of KcsA depends on KCl concentrations and is 97 pS at 0.1M KCl and approximately 150 pS at 0.4M KCl (127), values that are significantly higher than G ave . Both experimental data and kinetic models of ion occupancy in K + channels indicate that the channel is occupied by 2-3 K + ions during steady state conduction (95, 131). If three K + ions move 108 independently through the selectivity filter within a unit of time, the value of conductance from our approach is estimated to be 29 × 3 = 87 pS, which is only half of the experimental value. Substituting the measured conductance of 150 pS into Eq. (V.5) for a steady ion flow with Δλ = 1 Å and z(t j ) – z(t j ) = 0.5 Å [~ thermal fluctuations (103)] for t j – t i + (j – i)τ = 10 ns [translocation time of K + through the selectivity filter (97)], shows that the work done in this ion flow, 〈W〉 ≈ 0.6 kcal/mol ~ k B T, a value that is consistent with the “knock-on” mechanism of permeation in a barrier-free free-energy landscape for multiple ions (97–99, 132, 133). Therefore, concerted movement of multiple ions incorporated into the diffusional and dissipative dynamical factors (99) that increase ion conductance in a multi-occupancy pore through the “knock-on” mechanism (95) is the likely reason for the deviation of the single conductance from experiments, in agreement with previous observations that ions do not move independently through the pore. 109 Figure V.7 Calculated values of conductance (G) for a K + ion pulled through the KcsA channel using a step-wise pulling protocol. Values of λ are the center of the harmonic pulling potential, and values between λ = 7 Å and λ = 17 Å occur when the center of the pulling potential is within the selectivity filter of KcsA. A negative value of G simply implies negative work (see Materials and Method) corresponding to the attraction (not resistance) from the selectivity filter at a pulling step. But for all pulling steps, the total conductance is always positive because the total work and diffusion coefficients are always positive. V.4 Discussions A mechanism for ion selectivity and K + permeation by KcsA emerges from the results of these simulations. When a K + ion approaches the selectivity filter from the vestibule, it is attracted to and stably bound to site S4 in the selectivity filter. Movement of K + from the vestibule into S4 occurs spontaneously with a high probability and during this process the K + ion is almost completely dehydrated. The dehydration processes of K + and Na + are significantly different in the simulations, in agreement with a number of experimental and simulation studies. Although the average number of water molecules (~ 0.5) binding 110 to K + at S4 is slightly different from unity found in experimental experiments (97, 98) or 0.9 ± 0.2 observed in simulations of K + flow in Kv1.2 voltage-gated channels (101), this difference may be explained by lack of concerted K + flow in our simulations. Elucidation of the mechanistic details of the dehydration process is likely to require quantum mechanical considerations, and the simulations performed here indicate that the organization of water molecules by the channel in the region just outside of the selectivity filter in the vestibule may be important for this mechanism (see below). Even in the absence of an electropositive driving force a K + ion in site S4 will jump spontaneously from S4 to S3 with a modest probability. The structures of the S3 and S2 sites are themselves induced and stabilized by the presence of K + in S4. Crystal structures show that a water molecule is found in the selectivity filter between adjacent K + ions, and Markov chain models and calculations of the potential of mean force are consistent with the concerted movement of multiple K + ions through the selectivity filter of KcsA, with sites S0 through S4 alternately occupied by a K + ion or by a water molecule (99). The simulations of single ion movement into the selectivity filter discussed here show that the region of the vestibule just outside of the selectivity filter is relatively depleted in water molecules when a K + ion is located at its stable position in the vestibule or in site S4. The two or three water molecules in this region of the vestibule may be the source of the water molecules that separate adjacent K + ions in the selectivity filter; however, in preliminary simulations of Na + and K + movement through a non-selective Na/K channel with a selectivity filter amino acid sequence similar to that of KcsA, occupancy of adjacent sites in the selectivity filter by K + or Na + ions is often observed (data not shown). 111 Thus, these simulations suggest the possibility that water molecules may not always be interspersed between ions in the selectivity filter of KcsA. On the other hand, dewetting of the vestibule just outside the selectivity filter has been associated with hydrophobic collapse of the selectivity filter and loss of conduction during gating of K + channels (134), and so the significance of the depletion of water in this region of the vestibule in KcsA ion selectivity is not entirely clear. In contrast to the movement of K + , when a Na + ion approaches the selectivity filter, it is stabilized in a location just outside the selectivity filter. Na + must overcome an energy barrier approximately 3.7 kcal/mole higher than K + to enter the selectivity filter, and the probability of a single Na + ion moving spontaneously from the vestibule to the selectivity filter is only about 0.02 in the absence of a voltage gradient across the membrane. When a Na + ion does enter the selectivity filter, it binds stably in the plane of carbonyl oxygen atoms between sites S4 and S3 together with approximately two water molecules. The movement of Na + into this location induces disorder in the carbonyl oxygen atoms in the selectivity filter, however, and distorts potentially favorable binding sites for Na + distal to this site. Thus, the Na + ion is more likely to remain near the entrance to the selectivity filter than to move into the selectivity filter to the extracellular surface of the membrane unless it is driven by a strong electropositive potential (see below). A similar result was also observed in simulations by Shrivastava et al. (130), in which one Na + prefers positions below the selectivity filter and two Na + ions located near S1 and S3 distort the binding sites made of the carbonyl oxygen atoms during relaxation of 2 ns. Our data suggests that such distortion can be caused by a single Na + , and that Na + 112 is not able to bind to S1 for such a long time, hence explaining the “punch through” relief of cytoplasmic Na + block as discussed below. Our data support the hypothesis originally made by Bezanilla and Armstrong (94) and more recently by Kim and Allen (117) that ion selectivity is based on selective exclusion from the selectivity filter rather than on selective binding to sites within the selectivity filter. It is known that Na + blocks the KcsA channel from an internal site suggested to be located within the vestibule of the channel (94, 134). A stable location for Na + in the vestibule was identified in these simulations (Figure V.3) where the Na + ion is stabilized by interactions with the hydroxyl groups of the four T75 residues of the channel. The simulations also indicate that Na + can bind stably in the plane of the carbonyl oxygen atoms between S4 and S3, although under physiological conditions this site is not likely to be occupied. Current-voltage curves for KcsA show that block of the channel by cytoplasmic Na + occurs at potentials near 100 mV and is relieved by internal positive potentials >200 mV. This effect of voltage on Na + block has been called “punch through”. Na + block and the concept of “punch through” can be understood in energetic and kinetic terms from the profile of f(z) for Na + shown in Figure V.4. The probability of a Na + ion jumping from its stable position in the vestibule into the site between S4 and S3 is small (0.02), indicating that the cytoplasmic potential must be sufficiently positive to overcome the barrier for Na + entry into the vestibule. Furthermore, from the overlap of f(z) profiles for the Na + site between S4 and S3 and the highest energy site for Na + between S3 and S2, the probability of Na + jumping to the latter site is calculated to be > 0.5. Since there are no stable Na + binding sites distal to this site in the selectivity filter, movement of Na + 113 occurs spontaneously from the site between S3 and S2 to the extracellular surface in the S0 region. Our simulation data suggest that at membrane potentials between 100 – 200 mV, Na + would be driven into and stabilized in the site in the vestibule where ΔF is a minimum. At positive potentials >200 mV, Na + ion would be dislodged from this site and would be stabilized in the plane of carbonyl oxygen atoms between sites S4 and S3. The high probability of Na + jumping to the site between S3 and S2 from this location, however, even in the absence of a voltage gradient, and the absence of distal binding sites for Na + in the selectivity filter, would favor movement of Na + directly from the vestibule through the selectivity filter at high positive potentials. This mechanism is supported by our additional conclusion that work done during steady state ion conductance is approximately equal to kT, and by the conclusion of Berneche and Roux that the energy barriers to ion diffusion between adjacent sites in the selectivity filter are close to zero at moderately high positive values of the membrane potential (99). The absence of overlap between histograms of f(z) for K + at positions S1 and S0 indicates that backward movement of K + from S0 to S1 occurs with a low probability, and this observation can explain the mild outward rectification observed for the KcsA channel (127). The location of a stable site for Na + in the vestibule just outside of the selectivity filter agrees with the simulations by Shrivastava et al (130), and a K + ion was observed near this location in crystal structures of KcsA and other K + channels that were crystalized in the presence of potassium. Analysis of the reasons for the stability of Na + at this location in the simulations provides some insight into the differential dehydration 114 of K + as it enters the selectivity filter. Na + is stabilized in the vestibule just outside the selectivity filter because the hydroxyl groups of four T75 residues at the entrance to the selectivity filter can substitute for four of the six water molecules around the hydrated Na + ion, and because the remaining two water molecules are tightly bound to the Na + ions. Because the motions of the hydroxyl groups are more restricted than the water molecules, this arrangement is energetically favored over fully hydrated Na + in the vestibule. The hydroxyl oxygen atoms of T75 also replace four water molecules around the K + ion, but in contrast to Na + , the weakly bound water molecules around the K + ion are easily replaced by surrogate ligands. The restricted movement of the oxygen atoms on the T75 relative to the water oxygen atoms serves to further stabilize the K + ion in site S4. Thus, movement of K + into S4 occurs with no net energy cost for dehydration. This result suggests that further investigation into the role of T75 in ion selectivity in KcsA and in the non-selective NaK channel (118) by simulating Na + and K + movement into the selectivity filter of these channels containing amino acid substitutions at this position may provide valuable information about the difference in selectivity between these two channels. The results reported here also suggest a way to resolve the inconsistency between simulations that support a selective binding mechanism of ion selectivity versus those that favor a selective exclusion mechanism. In simulations where the most selective binding site in the selectivity filter was identified as S2, the potential of mean force of Na + and K + ions and free-energy differences for binding of the ions to each of the four sites S1-S4 in the crystal structure were calculated after alchemical transformation of K + 115 into Na + in the same site. The potential of mean force contains the energetic cost of moving the ions to the crystallographic binding sites in the selectivity filter, and free- energy differences in transforming K + into Na + in these same sites. In a recent simulation, however, also using alchemical transformation, it was suggested that there is no such a special site for ion selectivity found in multiple-ion configurations, and that ion selectivity is based on selective exclusion from the selectivity filter rather than on selective binding to sites within the selectivity filter (117). The simulation data presented here indicate that there are no stable binding sites for Na + near S2 or S1 (Figure V.3) because the selectivity filter is already significantly distorted by the approach of Na + . This result could be interpreted to indicate that S2 is the most selective binding site, as previous simulations have concluded. As discussed above, however, the simulations also indicate that selectivity appears to occur even before Na + enters S4, favoring the selective exclusion hypothesis. One can also perform simulations of long duration (~ µs) (101) to mimic the conduction of K + under electric fields, but this may not be possible for Na + because the large selectivity ratio (~ 400) of KcsA might require prohibitively long simulation times. Although permeation of only single K + and Na + ions was simulated using step-wise pulling protocols, the results of the simulations are consistent with other experimental and simulation data and mimic the realistic movements of ions in the channel. In these protocols, the weak harmonic potential and relaxation times allow both ions and the selectivity filter to respond mutually to each other. No assumptions about stable positions for the ions are assumed in these simulations, but rather the energetics of the system 116 determine where the ions are stable in the channel. In this way, it was found that K + easily enters its first stable position at S4 and induces the formation of the next binding sites, thus enabling subsequent K + ions to follow with much less energy cost, as proposed in the “knock-on” mechanism of ion permeation (95, 115). In contrast, Na + prefers positions below the selectivity filter, but if it is forced into the selectivity filter, the selectivity filter responds by becoming more disordered, raising the energy cost for the entry of subsequent Na + ions, i.e., dynamically rejecting Na + . This result not only agrees with the findings of Shrivastava et al. (130), but also suggests that if the distorted selectivity filter cannot accommodate multiple Na + ions, then imposing multiple Na + ions in simulations at the S1-S4 sites in the selectivity filter may not represent a physiologically realistic situation. An alternative approach may be to prepare a configuration of multiple Na + ions by sequentially pulling two or three Na + ions towards to the filter to observe how the ions and selectivity filter mutually respond to each other. It may then be possible to compare the energy differences between the pulling of multiple K + and Na + ions to determine whether they agree with the results by Kim and Allen (117). These simulations are currently in progress in our laboratory. In conclusion, simulations of non-equilibrium interactions between ions and the KcsA channel have identified an adaptation of the selectivity filter of the channel that adjusts the structure of the selectivity filter to favor entry of K + ions into the selectivity filter with continued permeation of K + through the membrane, and rejection of Na + ions. This feature has been incorporated into a model for selective ion permeation in KcsA that is consistent with experimental measurements of conductance, rectification, and channel 117 block. The mechanism of ion selectivity obtained from this analysis is able to extend results of previously published equilibrium MD simulations and to provide a new perspective on the mechanism of ion selectivity by KcsA. Although single Na + or K + ions were investigated in these simulations, the results obtained from stepwise pulling protocols combined with Jarzynski’s Equality are quantitatively consistent with several experimental measurements of selective ion permeation and indicate that this technique can also be applied to investigate the responses of the selectivity filter in the presence of multiple permeant ions. These results suggest a way to resolve current controversies about the mechanism of selective ion permeation. Although more subtle calculations based on quantum mechanics (110, 111, 135) can be used to obtain additional insight into the mechanism of the ion selectivity, our simulation data suggest that application of non- equilibrium molecular dynamics to ion channels provides an additional perspective on the dynamical adaptation and mutual responses of the ions and the channel. 118 CHAPTER VI Is The G-Quadruplex An Effective Nanoconductor For Ions? VI.1 Introduction Nucleic acid sequences that appear in single-stranded fashion in the chromosomal telomeres, whose length and folding control aging processes and affect genome instabilities (136), are able to form quadruplexes of guanines (Gs), or G-quadruplexes (GQs), or G4-DNA. Evidence for the presence of GQ motifs in the human telomeric region in vivo was gained only very recently (137), but this form of nucleic acid has been known and studied for several years. In particular, GQs can be synthesized in vitro, and structural NMR characterization has demonstrated different folding patterns: monomolecular, bimolecular and tetramolecular (138, 139). Reviews (140–142) on the structure and stability of GQs disclose their outmost importance in various contexts, though their role is still an object of investigation in the medical, biological, biochemical and chemical physics communities. In 1994, the first high-resolution X-ray crystal structure of a parallel-stranded GQ was published (PDB code 244D, resolution 1.2 Å) (143), which was further refined in 1997 (PDB code 352D, resolution 0.95 Å) (144). These works gave an unprecedented precise knowledge of how guanines arrange and form quadruple helices. The unit component of a GQ is the G-quartet, which is a very strong stacking assembly made of 119 four coplanar guanine bases sustained by eight hydrogen bonds. GQs from telomeric sequences are stabilized by monovalent cations such as Na + , K + or NH + 4 , which are coordinated by the carbonyl O6 atoms lining up a central pore or channel. Due to the existence of such a central channel and stable structures, a lively debated issue is whether GQs can be used to select and conduct ions (145). Hud and colleagues (146) argued that since the hydration free-energies of K + and Na + are different, the pore of a GQ is able to distinguish between ions. The group of Davis (147) successfully synthesized a unimolecular GQ, which folds into a conformation to conduct Na + along its symmetry axis across phospholipid bilayer membranes. In potassium-selective KcsA channels, ion selectivity presumably results not only from the channel characteristics, but also from inherent properties of K + (94, 95, 112). This selectivity occurs at the filter region where four identical subunits have the TVGYG amino acid sequence. The crystal structure (113) of this filter region reveals that the amino acids have negatively charged carbonyl oxygen atoms, which line up the channel in a similar way as in the GQ pore. These amino acids allow conducting potassium ions sandwiched by water molecules, while they inhibit flow of sodium ions. The favorable conduction of K + goes together with the evidence found from the crystal structure that the ionic size of K + fits the KcsA pore more favorably than Na + . For GQs, do the guanine bases selectively bind and conduct ions, and how? So far, it has been rather well established that GQs bind more strongly K + than Na + ; however, understanding the mobility of different ions inside GQs is still elusive. Deng and Braunlin (148) showed by experiments using 23 Na NMR that the selectivity of a GQ d(G 4 T 4 G 4 ) favors K + over Na + . 120 The free-energy change measured for the conversion of [d(G 3 T 4 G 3 )] 2 bonded by two Na + to [d(G 3 T 4 G 3 )] 2 bonded by two K + is –1.7 ± 0.2 kcal/mol (146). This free-energy change suggests that K + fits better the GQ pore than Na + , also indicating the preference for K + . The lifetime of bound Na + in GQs, ~ 10 to 250 µs (148), is much shorter than that of NH + 4 , 250 ms (149), which would be similar to that of K + due to the close ionic sizes. This indirect evidence suggests that Na + moves much faster and more easily along the GQ channels than the other ions. Therefore, the concepts of ion selectivity and speed of ion transport in GQs seem to be in conflict, at odds with the behavior of KcsA. Understanding the origin of this discrepancy is a fundamental question that has potential impact for the design of molecular machines. From the experimental point of view, it is problematic to attain a direct comparison between different ionic species in the same GQ. Therefore, simulations assume a particularly relevant role for addressing this problem. Also simulations, however, have their difficulties. The characteristic times of the involved phenomena, namely ions entering the GQ channel from solution and moving through the pore, are too long for standard molecular dynamics (MD) techniques and enhanced techniques must be invoked. As a matter of fact, the sole simulation of the problem was published in 2012 (150). Akhshi et al. investigated ion mobility in a parallel-stranded d[TG 4 T] 4 G- quadruplex by computing the potential of mean force (PMF) profiles for Na + , K + , and NH + 4 ions along the central pore (150). We summarize their findings here for reference and we comment in the Final Remarks how we add relevant knowledge. (i) The analysis of energy profiles gives the following information (150): 121 • the energy barriers between any two subsequent binding sites along the core of the GQ are about 4-5 kcal/mol for Na + , 13-15 kcal/mol for K + and NH 4 + ⇒ Na + ions move faster than the other species; this is rationalized in terms of atomic sizes, because the small Na + ion does not need to perturb the internal size of the channel during it axial motion, at odds with K + and NH 4 + ; • the leakage of internal ions from the sides is blocked by very high energy barriers (>50 kcal/mol); this finding of large energy barriers for sideways escape of ions from the channel into the solution is a clear proof of experimental indications of long binding times of NH 4 + (151). • energy barriers for leakage through the edges are also large, ~20 kcal/mol for K + and NH 4 + , ~14 kcal/mol for Na + ; thus, Na + motion from the channel into the solution is less energetically costly, which is in line with experimental measurements of residence times (152, 153). (ii) The analysis of the motion of water molecules gives the following information: • all ionic species are de-hydrated inside the pore and fully hydrated in solution; • occasionally, water molecules enter the channel and approach the first coordination shell of the ions. These findings shed light on important aspects of ion motion along the GQ axis. However, other important issues related to ion stability and motion in GQs are not tackled (143, 147, 154). No comments are given on different residence sites for the different ionic species. No explicit comments are given on energy barriers for entering 122 the channel from the solution, though their data in Figure VI.1 suggests that K + and NH 4 + experience an energy barrier of approximately 4-5 kcal/mol, while Na + does not experience an entrance energy barrier. Furthermore, the energy profiles are compared to those of KcsA ion channels, but conductivity values are not estimated. To shed direct light on the selectivity and conduction properties of GQs, solve uncertain points from the previous simulation (150) and unveil whether this system can be exploited as a natural or artificial ion channel, we have carried out a computational study using biased molecular dynamics according to an efficient protocol developed by one of us (60). We simulate a condition typical of natural ion channels, e.g. KcsA (101, 113), in which ion motion is accompanied with water. We present the results of such simulations for d[G 9 ] 4 channels that contain K + , Na + or NH + 4 ions in the pore, in explicit water solution (Figure VI.1). While this system is not related to any telomeric sequence, d[G 9 ] 4 is related to long G4-DNA wires that can be produced by enzymatic synthesis, can exist in the absence of inner cations and have been proposed for nanoelectronics applications (155–158). Note that their existence in the absence of stabilizing cations was proved by both experiments (155, 157) and simulations (154). Furthermore, though no X- ray or NMR structure could be resolved so far, d[G 9 ] 4 exhibits a stacking motif that conforms to the three-dimensional structure of tetramolecular GQs (154, 159). We compute the free-energy profiles using Jarzynski’s Equality (JE) (60, 86, 119) to examine how ions of the different species accommodate and move in the d[G 9 ] 4 channel. Our results show that the GQ is selective for K + ions, but can conduct Na + with almost no energy barrier within its channel, whereas the guanine bases are disturbed distinguishably 123 as ions enter the channel. We also observe in the trajectories that the ion movement along the axis of the helix is accompanied by spontaneous leaking of water molecules from the channel. Finally, we estimate the conductance via the equivalent relation between work computed in the pulling protocol and heat generated by a current. Figure VI.1 Snapshot of the G-quadruplex in the presence of K + ions, after 10-ns equilibration. The quadruplex is visualized in a ribbon mode that highlights the backbone; water molecules (O red; H white) and potassium ions (green) are visualized as ball&stick. A planar position is defined as the crossing point of a G- quartet plane and the z-axis, while a cage-like position (S1 to S8) is the center of any two successive G-quartets along the z-axis. The three-dimensional structure is rendered by VMD (160). VI.2 Methodology We use the average structure of a G-quadruplex (GQ) from a 5-ns free MD trajectory at room temperature and standard pressure (154). This structure contains nine G-quartets organized in four parallel strands, having the same total length of 30.6 Å as the GQ 124 synthesized to be a Na + transporter (147). Specifically, we start from the equilibrated 9- plane GQ filled with 8 K + ions, remove the inner ions, solvate the structure with TIP3P water molecules and randomly neutralize it with potassium ions in the solution at the concentration of 0.57 M. These simulation strategies for solvation and neutralization are standard for the vast majority of biological systems that have been addressed by MD (154, 161). We remark, as we mentioned in the Introduction, that long empty G- quadruplexes can exist in the absence of internal cations (154, 155, 157), at odds with the situation in short G-quadruplexes(150). We choose to simulate a condition in which the ions of different species pass through an empty channel because this is similar to the conduction state in natural KcsA ion-channels(101, 113), which is our goal. Keeping all the 8 K + ions at the core during the pulling simulations would not be a typical conduction state, because also water molecules should have access to the conduction path in ion channels, at least in the systems that are nowadays understood (101, 113). The system has 9238 atoms in a supercell of 44×44×48 Å 3 . Its symmetry axis is along the z-direction centered at x 0 = y 0 = 0. To avoid total disruption and translational movement of the GQ during pulling simulations, we apply harmonic constraints to the phosphate groups with a spring constant of 2.0 kcal/mol/Å 2 . These constraints allow us to preserve the equilibrated structure as required for a durable channel, but limit only motion of the external boundary of the channel, yet enabling fluctuations of the bases: the internal guanine atoms that surround the passing ions are completely free to move at the established temperature. Therefore, the bases are free to respond to the passage of ions, in a way that may depend on the ionic species. 125 We use the parm99 AMBER force field (162) and other simulation parameters as assessed before (154). The parm99 force field reproduces canonical DNA structures in simulations of double-stranded DNA on the time scale below ~10 ns (163) and has given good results on G-quadruplexes on the time scale 5-20 ns. Since our simulations at each pulling step are of 5 ns, we are confident that the chosen computational setup is appropriate for a correct description of the structure at hand. For what concerns the energetics, parm99 gives accurate values of H-bonding and stacking energies against experimental data and quantum chemistry calculations (164, 165). Confident in this setup, we equilibrate the system at the temperature T = 300 K and pressure P = 1 atm. We first minimize the energy of the system using conjugate gradients for 5000 steps, after which we perform a MD run for 0.5 ns with a time step of 1 fs at T = 200 K using Langevin dynamics with a damping constant equal to 1 ps –1 . Then the system is further equilibrated for 10 ns at T = 300 K and P = 1 atm using Langevin piston dynamics with a time step of 1 fs, which produce isothermal-isobaric ensembles (166). To prepare the systems with Na + and NH + 4 ions, we simply mutate potassium in the 10ns-equilibration system to sodium and ammonium and then apply the same multi-step minimization- equilibration procedure. Finally, each equilibrated quadruplex is subjected to ion pulling: an ion just outside the 5’ end is dragged into the channel to span its axis via a step-wise protocol (60). We find that in each of the three equilibrated systems there is one ion capping at each edge of the channel (see Figure VI.1), either slightly outside (5’) or slightly inside (3’). The ion at the 3’ end moves into the channel during the equilibration phase simply 126 by thermal motion and attraction of the channel. At the 5’ end, where we note somewhat larger distortions, in line with previous findings, the ions can be internal or external depending on the species (154). The step-wise pulling protocol requires relaxation times for all discretized pulling steps and allows us to estimate free-energy profiles. In each pulling simulation, we constrain one ion at different subsequent positions along the axis of the GQ by applying to it a harmonic potential 0.5k[(x – x 0 ) 2 + (y – y 0 ) 2 + (z – z 0 – λ) 2 ], where x, y, and z are the ion coordinates, k is 0.6 kcal/mol ~ k B T and λ is a control parameter that scans the direction of motion. The same reference coordinates x 0 = y 0 = z 0 = 0 Å are used for potassium, sodium and ammonium ions. Every τ = 5 ns, we instantaneously increase λ by 1.0 Å, starting from λ = –17 Å, so that the external force is increased in a modest manner to pull an ion along the positive z-axis. The relaxation time, τ = 5 ns, is ten times larger than the minimum tested value (60). We perform s = 25 discretized steps of pulling NH + 4 and Na + or K + ions to pass the middle of the longitudinal channel. Every 50 fs during the interval τ = 5 ns at each value of λ, we collect z to construct the distribution g i (z) and compute average positions 〈z〉 i = ∑ z g i (z)z, where i from 1 to s is the index for the i-th step of increasing λ. From the g i (z) we compute free-energy changes ΔF(λ 1 ,λ) along the z axis, which directly indicate amounts of work including entropic effects (60, 86, 119). Since the work applied to an ionic charge q is equivalent to the heat emitted by a resistance R = G –1 , we propose a simple expression of conductance , (VI.1) G= I 2 (t j −t i +(j−i)τ) / W 127 where and z(t i ) is the ion position along the axis of motion at time t i ∈ [0:5] ns, with j > i being the index of the pulling step. Here, the brackets denote the average taken over all possible trajectories between pulling steps i-th and j-th and 〈W〉 is the averaged value of work over such trajectories. The heat of ionic flow is whose average is equal to a diffusion coefficient (125) D ij multiplied by the term in the square bracket. Since work is more biased for a pair having j > i + 1 than for a pair with j = i + 1 (60), we compute the conductance G i+1 for a pair of successive steps i-th and (i+1)-th, in which the work 〈W i+1 〉 is averaged from U(z(t i ),λ i + Δλ) – U(z(t i ),λ i ) over all possible values of z(t i ). Note that G i+1 can largely fluctuate along the selectivity filter due to the fact that 〈W i+1 〉 and D ij are not the same for all pairs of successive steps. We compute a non-negative total conductance as follows 1 G total = 1 G i i=1 s−1 ∑ = W i D ij i=1,j=i+1 s−1 ∑ ≥ Wtotal [D ij ] max (VI.2) Since the total work W total = ∑ i W i to enforce ions through the channel and the minimum of D ij are always positive, this total conductance is therefore positive. The first equality in Eq. (VI.2) is known for a total resistance of multiple resistors in series. The steered MD method adopted in this work, in conjunction with Jarzynski’s Equality, produces a free-energy profile of stretching Deca-alanine in excellent agreement with steered MD simulations evaluating the potential of mean force (PMF) (25, 28, 60). It offers an efficient way to both estimate free-energy profiles and sample I = q[z(t j )− z(t i )]/[(t j −t i +(j−i)τ)(λ j −λ i )] I 2 (t j −t i +(j−i)τ)=[q 2 /(λ j −λ i ) 2 ][z(t j )− z(t i )] 2 /(t j −t i +(j−i)τ), 128 reaction pathways. Importantly, it outdoes PMF for the characterization of thermal fluctuations and estimation of conductance. VI.3 Results and Discussions We remark that the three equilibrated systems contain one ion in the GQ core at the 3’ ter (S8 in Figure VI.1): Na + in a planar site, K + and NH4 + in a cage site. These are indeed binding sites for the ions and consequently they are not abandoned by the unconstrained ions. The pulling protocol is applied only to the ion that is located at the mouth of the channel at the 5’ ter at the end of the equilibration procedure. Therefore, at the 3’ ter each of the three simulated GQs is clogged. The structural portion around the 5’ ter is, however, rather far from the obstructed portion of the pore and can be retained as representative of a range open to free motion. Thus, the results in this part of the systems can be generalized to interpret absolute and relative ion mobility. VI.3.1 Binding sites Figure VI.2 (a) Histograms of ion’s positions along the z-direction in all pulling steps. The histograms are the combination of all separated normalized g i (z) for s = 25 pulling steps for each ion. (b) Averaged position 〈z〉 i = ∑ z g i (z)z of ions along the z-direction. 129 To show how the ions stably bind to the G-quadruplex channel, we plot cumulative distributions of z-coordinates and the averaged positions at each value of λ during the pulling (see Figure VI.2), for each of the three investigated systems. The plot in Figure VI.2(a) reveals most probable values of the z coordinate across the whole pulling protocol in the three cases, while the plot in Figure VI.2(b) gives the average at each value of the control parameter λ. We note that the simulation of the system with Na + does not give a probability peak for z ~ –15 Å below the 5’ entrance of the channel, as the other two simulations do. In fact, inspecting the average positions, for λ = –17 Å (namely for the initial step that corresponds to an ion at the entrance of the channel in 5’) we find that 〈z Na 〉 = –12.7, 〈z K 〉 = –14.5, and 〈z NH4 〉 = –15.3 Å. These results about the position distributions together indicate that while potassium and ammonium ions reside for a finite time outside the channel, sodium ions enter directly. This evidence does not imply that the GQ prefers Na + to the other ionic species; it is in qualitative agreement with the absence of an entrance barrier found for Na + ions by Akhshi et al. (150) The order of these averaged positions for ions close to the entrance, 〈z Na 〉 > 〈z K 〉 > 〈z NH4 〉, is consistent with that of the ionic sizes r Na < r K < r NH4 and it reflects the fact that the smaller the ion size is, the more easily the ion enters the channel. The distinguishable peaks of the histograms (e.g., around z = –12 Å) are consistent with the observations (see Figures VI.2–3) that K + and NH + 4 ions are preferably accommodated at the cages between G- quartets (148, 149, 154), while Na + ions can be metastable both between and within the planes of G-quartets (143, 159). 130 Figure VI.3 Three-dimensional structures of the investigated systems at different stages of the simulations, illustrating the axial motion of Na + (orange, a-e), K + (green, f-j) and NH + 4 (blue&white, k-o) ions in the GQ channel. The snapshots are at 5 ns of each pulling step. Na + also moves along the channel more easily than the other ions: for instance, at λ = –8 Å, Na + is at <z> ~ – 8 Å, farther from the entrance than the other ions [see Figure VI.2(b)]. Figure VI.3 shows snapshots of the ions moving in progression S1 → S2 → S3 → S4 at the end of each pulling step characterized by a value of λ. The series of Figures VI.3(a-o) shows that at the same cage-like positions, λ Na ≤ λ K ≤ λ NH4 . Since the external 131 force f due to the harmonic potential is equal to –k(z – λ), at the same z it requires higher forces to pull larger ions to upper cage-like positions: f Na ≤ f K ≤ f NH4 . When the initial Na + ion moves in the GQ core from 5’ to 3’, it is accompanied by other ions and water molecules. The movement of the Na + ions implies the conduction of multiple ions similar to the “knock-on” mechanism (95), in which the channel attracts extra ions in support of the conduction of the first ion. In the case of potassium ions, most simulation stages are characterized by a conduction configuration of –K + -H 2 O-K + -H 2 O– in which all cage positions (namely, positions between two consecutive G-quartets) are occupied by either ions or water molecules [see Figure VI.3(h)]. Even though exceptional situations occur, in which one or more cage positions are empty [see Figure VI.3(j)], this conduction behavior resembles that observed in potassium-selective channels (101). Note that a spatial arrangement of oxygen atoms (–0.51|e| in CHARMM force fields) (122) in potassium-selective KcsA channels (113), which are decisive to select ions, is similar to the structure of oxygen atoms lining the GQ channel. We show below, based on further analysis of free-energy profiles, that the GQ channel has a higher affinity (selectivity) to K + over Na + ions, because of the optimal fit to the cages of G-quartets. However, it allows the easiest conduction of Na + . VI.3.2 Water Molecules in The Channel We examine if and how water molecules accompany the motion of ions in the channel. Before applying the pulling protocol to each system, seven water molecules fill the channel in the equilibrated structure of the GQ in the presence of Na + and K + ions, while 132 eight water molecules fill the channel in the presence of NH 4 + ions. During the simulations the internal water molecules form two opposite zigzag chains of hydrogen bonds [denoted by arrows in Figure VI.4(a)-(h)-(p)] with variable lengths. These two opposite zigzag chains compete with each other to (i) prevent ions to move farther into the channel, (ii) exert forces on the upper ions, and (iii) knock water molecules out of the channel from the channel sides. This competition is a reason for the ejection of water molecules as the ion motion proceeds (see Figure VI.5). Water molecules are also able to move around the pulled ions to lower positions as shown in Figures VI.4(f-g), VI.4(n) and VI.4(t). 133 Figure VI.4 Positions of water molecules and ions along the axis of motion at various stages of the dynamical simulations, pruned from three-dimensional snapshots of the entire systems. (a-g) Sub-system extracted from the simulation with Na + ions (orange spheres). (h-n) Sub-system extracted from the simulation with K + ions (green spheres). (o-t) Sub-system extracted from the simulation with NH + 4 ions (blue&white spheres). The arrows indicate directions of hydrogen bonds starting from oxygen and ending at hydrogen atoms. The snapshots are at 5 ns of each pulling step. 134 Another interesting behavior that emerges from our simulations is how water molecules escape from the channel as Na + and K + move along the symmetry axis. Figures VI.5(a-c) show a water molecule leaking from the channel after 50 ps when K + is located at a metastable cage-like position (〈z K 〉 ≈ –8.6 Å) and faces a water complex in a triangle configuration. They also show some bases tilted by about 20 degrees: this deformation allows a water molecule to interfere with the H-bond pattern of the guanines, eventually loosening such pattern to find a way out. Figure VI.5(d) shows that Na + located at a metastable position with 〈z Na 〉 ≈ –5 Å also faces a triangle of water molecules. The zigzag water chain in this configuration is under strong compression due to the movement of the pulled ion. This compression causes the top ion [see Figure VI.4(d)] to move up, towards the 3’ ter, and release the stress on the water chain. Then this top ion, due to the strong attraction of the channel, returns to its original metastable position and continues to compress the water chain. At the same time the compression causes the bases especially around the water triangle to be disturbed significantly. One base is tilted by 29 degrees and a water molecule slides into the region-bond pattern of that base, eventually abandoning the channel. These processes occur at 0.17 ns. It then takes 1.52 ns for the second water molecule to escape the channel while Na + is still at the same stable position. 135 Figure VI.5 Three-dimensional structures of portions of the simulated systems at selected snapshots in a ball&stick representation, to visualize the escape of water molecules from the GQ core. Water molecules that abandon the channel are represented in black. Na + and K + ions are at λ = –6 and –4 Å, respectively. Na + and K + ions are represented as orange and green spheres, respectively. Carbon, oxygen, nitrogen and hydrogen atoms are shown as cyan, red, blue and white spheres, respectively. VI.3.3 Fluctuations of Bases around Ions In Figure VI.6 we examine the response of the channel to the ions, by plotting the probability density of the root mean square deviation (RMSD) of the first 16 guanines, with respect to the equilibrated structure (154), for pulling with λ = –11 to –7 Å. Clear differences emerge in how the channel responds to Na + , K + and NH + 4 . The RMSD in the presence of NH + 4 ions is peaked at smaller values than in the presence of the other ionic species, independently of the value of λ at least in the early stages considered here. A reason for this might be that eight water molecules in the channel stabilize the overall base motif better than seven water molecules. Furthermore, the ionic size of NH + 4 constrains the movement of bases more than the other smaller ions: it is interesting to 136 note that, despite the small size difference between NH4 + and K + (0.1 Å, smaller than that between K + and Na + – 0.38 Å), the channel distinguishes between the two ionic species, which points to a possible role of physico-chemical effects. When the NH + 4 ion is at S1 (λ = –11 and λ = –7 Å), the RMSD is larger than when it is still external at the mouth of the channel (λ = –17 Å). The situation is opposite for Na + and K + ionic species, in the sense that the RMSD decreases when an ion enters the GQ channel, relative to the values when it is at the mouth of the tube. The RMSD curves for GQ in the presence of Na + are less broadened at λ = –11 Å, and smaller at λ = –7 Å than those in the presence of K + . This implies that Na + can maintain the initial canonical structure of the bases somewhat better than K + , which may be attributed to the in-plane favorable location of Na + ions. Figure VI.6 Normalized histogram of RMSD evaluated over the four nearest- neighbor G-quartets to a given ion (16 guanine bases), with respect to the equilibrated structure (154). The RMSD is evaluated on the heavy atoms of the bases, excluding the backbone. Since we do not allow shrinking of the channel, the fluctuations do not represent the overall stability of the GQ due to different ions. The RMSDs are collected over 5000 frames during 5 ns for each λ. 137 Cavallari and coworkers (154) reported that the total RMSD computed on guanine heavy atoms for GQs in which the channel is filled with K + and Na + ions, relative to the equilibrated structure, is 0.6 Å and 0.9 Å, respectively, which is in line with the fact that K + is a better stabilizer and therefore a K + -filled pore is less flexible. This is not in contrast to our present RMSD analysis, because we are focusing here on the reaction of the guanines surrounding a certain ion to the passage of the ion. Indeed, we also find that K + does have stronger binding to a cage-like position than Na + to a planar position, meaning that K + has a more powerful stabilization effect. VI.3.4 Free-energy Profiles Figure VI.7 Free-energy profiles for the motion of different ionic species in the GQ channel d[G 9 ] 4 , as a function of the parameter λ that scans the axis of the quadruplex. 138 Figure VI.7 illustrates the free-energy profiles of the three ionic species. These free- energy profiles directly indicate energy or work required to induce transitions along reaction pathways. The curves in Figure VI.7 thus express the free energy changes of the ions binding to the GQ channel with respect to their energies in the solution. These values include entropic effects encompassed by the formalism of Jarzynski’s Equality (60, 86, 119). A comparison with potential-of-mean-force profiles 19 is given in the Supporting Information. The minima of the free-energy profiles are ΔF K min = –7.78 kcal/mol at λ = –11 Å, ΔF Na min = –7.49 kcal/mol at λ = –8 Å, and ΔF NH4 min = –5.71 kcal/mol at λ = –11 Å, in the order ΔF K min < ΔF Na min < ΔF NH4 min . Although the difference ΔF K min – ΔF Na min = –0.3 kcal/mol is very small, the discrimination between the two atomic species is significant, as supported by the order of magnitude of the energy gain found experimentally for changing d[(G 3 T 4 G 3 ] 2 bound to Na + to d[(G 3 T 4 G 3 ] 2 bound to K + , of ~0.8 kcal/mol/ion (146). The order of minima in Figure VI.7 reveals that the GQ binds slightly stronger to K + than to Na + , and more sensibly stronger (by 2 kcal/mol) than to NH + 4 , whose ionic size is marginally larger than K + . These minima are not related to the facility of transporting ions but to the stabilization effects of different ionic species, which agrees with the analysis presented in Figure VI.2 and with previous indications (149, 167) that K + fits to GQs better than Na + . These values are consistent with the minima that we find for the attraction between the selectivity filter of KcsA and K + or Na + ions (168). The attraction is attributed to the negatively charged carbonyl oxygen atoms in the filter, 139 which line the KcsA channel in the same pattern as the O6 atoms of guanine in the G- quadruplex. However, the binding positions of K + and Na + ions at the energy minima are different from those in KcsA. While Na + at the minimum binds in the KcsA filter at a position more external than that of K + , in the GQ channel Na + penetrates farther and faster than K + . This means that the GQ is not only able to choose one species of ions for the optimal fit to its pore, but also capable of choosing a different species of smaller ions for the easiest ion conduction. Figure VI.7 shows that all the ionic species do not experience energy barriers for entering the channel. This is consistent with the fact that during the initial 10-ns equilibration of the system ions get close to the edges and enter the channel (see Figures VI.1 and VI.3). This finding is in apparent discrepancy with the PMF profiles computed via the adaptive biasing force method for a [d(TG 4 T] 4 G-quadruplex (150), which indicate that K + and NH + 4 ions encounter energy barriers of about 3–4 kcal/mol to enter the pore. We discuss this aspect in the final remarks. The opposite values of the free-energy minima in Figure VI.7 represent the free- energy barriers for the ions to move from the channel into the bulk water. These values are in the range of folding free-energy barriers of GQs, which were estimated between –3 and –15 k B T (2–9 kcal/mol) from temperature dependence studies and Kramers’ theory (169). The computed value for NH 4 + is half the experimentally measured free-energy barrier (151). The significant difference may be due to the fact that we impose constraints on phosphate groups to maintain the overall quadruplex symmetry, while experimental GQs are able to shrink for further energy reduction, thus increasing free-energy barriers 140 for ions to move out into the bulk water. Thus, if we take into account the shrinking effect and subtract the estimated “deformation” energy (~2-9 kcal/mol) (169) from the measured free-energy barrier, we find that our data is in fair agreement with experimental outcome. The broad free-energy minimum of Na + indicates that Na + is moving with a small energy cost from λ = –13 (or <z> = –12) to λ = –3 (or <z> = –4) Å. Thus, the GQ can conduct Na + very well in this region of two G-quartet steps, and also through the entire channel if one edge is not capped (see Figure VI.3). This is in qualitative agreement with the PMF-based finding that Na + ions move faster than K + and NH 4 + ions within a [d(TG 4 T] 4 G-quadruplex, due to the small energy barriers between subsequent metastable binding sites (150). Note that the shape of each free-energy profile is affected by the presence of one capping ion at the 3’ end (Figure VI.1): since no constraints are applied to such ion during the simulation, it remains at its metastable binding site S8. Therefore, as we noted before, the channel is obstructed at one side and consequently after some steps of 5’-to-3’ motion the internal ions start to feel the electrostatic repulsion of the capping ion and the free energy becomes positive. The free-energy profile for the Na + - GQ becomes positive at λ = 2 Å after the ion has gone through an axial distance of about 10 Å (or three G-quartet steps). The motion of the ammonium ion is the most limited: ΔF quickly becomes positive at λ = –7 Å after the ion has moved by only one G-quartet step (~3.4 Å). K + can go farther, by about 10 Å, but requires large energy cost because the free-energy profile of K + becomes positive at λ = –4 Å: thus, a K + ion moves 141 exothermically only for a distance of about 3.4 Å, as evident in Figures VI.3(h-j), in which a free ion at S1 is unable to enter farther. VI.3.5 Conductance of G-quadruplex We can finally estimate by Equation (VI.1) the conductance G for the Na + -GQ, K + -GQ and NH 4 + -GQ systems, namely the efficiency of the G-quadruplex for transporting Na + , K + and NH 4 + ions through the core. First we note that the sequential values of G i estimated from Equation (VI.1) fluctuate significantly from negative to positive values, due to the signs and non-equilibrium values of work measured for all instantaneous pairs of steps. Such instantaneous conductance values are therefore trivial. Nonetheless, the total conductance from Equation (VI.2) is physically meaningful because it is always positive in our simulations, reflecting the fact that the total work due to external forces is positive. The total conductance computed from our steered MD trajectories is 7.7, 4.0 and 2.6 pS for conducting Na + , K + and NH 4 + in the d[G 9 ] 4 quadruplex, approximately in the range 1-10 pS. This means that the G-quadruplex conducts Na + better than K + and NH 4 + , which was already stated from the free-energy profiles of Figure VI.7 and from the previous simulation of a shorter quadruplex (150). These values compare unfavorably with the estimated conductance of 97–150 pS (127) for the KcsA potassium channel, by about one order of magnitude. The same order of magnitude distinguishes the GQs for conducting different ionic species. G=1/ G i −1 ∑ 142 VI.4 Final Remarks We have presented the results of biased molecular dynamics simulations to disclose the binding and motion of K + , Na + and NH + 4 ions along the core of a d[G 9 ] 4 G-quadruplex, in the presence of explicit water molecules in the surrounding solution and partially filling the G-quadruplex channel. We have shown results on (i) metastable binding sites for different ionic species, (ii) free-energy barriers for ionic motion in the channel and for entrance into the channel, (iii) hydration patterns and (iv) species-dependent ionic conductivity and performance as ion channels. We point out here the novelty and timeliness of our results and their relation to previous knowledge on the subject. First of all we note that experimental investigations on ion mobility in GQs are essentially restricted to NH 4 + ions and that there exists a single theoretical work on a quadruplex that is different from that considered by us. Our results are in line with experimental data. They are also qualitatively in agreement with the theoretical data, while they add compelling evidence on some aspects and reveal new features. Akhshi’s Communication based on a PMF approach was focused on energy profiles and did not target the dynamics, except for hydration features. A critical summary of those results, pertaining to a steady-state ion motion, is traced in the Introduction. Our analysis here is more elaborated and brings new insights into the mechanism of ion motion through a G-quadruplex, because we have simulated a transient state for ions entering the channel and have also characterized the binding locations and the values 143 of conductance. Furthermore, we have investigated a different system, which allows us to extend our results to longer channels that may be more relevant for practical applications. a) Na + ions can be transported within the quadruplex axis with minimal cost, namely with a vanishing energy cost for displacement between subsequent binding sites, even though there is an energy cost of moving Na + from the inside to the outside of the channel. There is, instead, a finite energy cost of about 4 kcal/mol for the motion of K + from the minimum at λ = –9 Å (<z>=–12 Å) to λ=–6 Å (<z>=–8.5 Å), corresponding to an advancement by one G-quartet step into the channel. These findings are qualitatively in agreement with the simulation of the shorter quadruplex (150), but the energy profiles are different. Our simulation of a transient state gives us access to the energy gain/cost of entering and exiting the pore and of moving through the pore, neglecting the energy barriers for hopping between successive metastable states, due to local maxima. The relation between the computed energy profile and the ionic conductance is direct through Equation (VI.2). b) We have shown that K + ions optimally fit the GQ core at cage positions. Inter-plane positions are also metastable binding sites for NH + 4 ions, while Na + ions are preferentially bound at in-plane sites. This evidence is in line with previous experiments and simulations. It also further rationalizes the energetic behavior: a Na + ion passes through an “empty” cage space for moving from one in-plane binding site to the next, while a K + ion needs to squeeze in G-quartet plane for passing from one cage binding site to the next, which has a deformation energy cost. 144 c) We have described the interference between the motion of the ions and the motion of water molecules that penetrate the channel, with particular reference to triangular configurations of water molecules that perturb the G- quartet H-bonding pattern and thus find a way out. d) Finally, we propose an order of 1–10 pS for the conductance of the synthetic nine-base G-quadruplex. In summary, we have reported on the second simulation of ion mobility through a G-quadruplex, finding qualitative agreement with the previous results (150) that the GQ favors K + ions for binding and favors the motion of Na + ions. We extend this behavior to the longer GQ treated in this work, which is a step towards generalization for what concerns quadruplex length. We have calculated the conductance of this potential artificial ion channel and related it to that of the natural ion channel KcsA. Although the motion of Na + through GQ is somewhat faster than the motion of K + , this unbalance is tiny relative to their noticeable selectivity ratios in the KcsA. Therefore, it is still doubtful whether the motion of Na + ions through GQ can be a better basis for the implementation of an artificial ion channel. 145 CHAPTER VII Supercrystals of DNA-Functionalized Gold Nanoparticles: A Million-Atom Molecular Dynamics Simulation Study VII.1 Introduction The Mirkin group has shown that aggregates of gold nanoparticles (AuNPs) functionalized with DNAs can form body-centered-cubic (bcc) and face-centered-cubic (fcc) supercrystals via DNA hybridization (170). A sequence of two single-stranded DNAs is designed to have (i) 10-base single-stranded segment connected to AuNPs via alkanethiol molecules, (ii) long double-stranded segment to enhance the sequence’s rigidity, and (iii) a single-stranded “sticky” segment having more than 4 bases and 5’-end. The single-stranded DNA of the sequence, which is connected to AuNPs, is called a marker. The other one is called a hybridizing linker. Some sticky bases of the linkers are complementary. At low temperature, the hybridization occurring among those complementary sticky bases induces the formation of the crystallized structures. As a result, the single-component fcc and binary-component bcc structure can be grown if one and two different kinds of linkers are used, respectively. On one hand, the formation of the fcc structure is dominantly driven by entropic effects (171, 172), which depend upon DNA markers (173, 174), linkers (170, 174–176), and ion concentrations (177–179). On the other hand, the formation of the bcc structure is mainly due to enthalpic effects, which maximize the DNA hybridization (170). 146 The supercrystals have a number of interesting properties. The profiles of their melting temperatures are sharper than those of isolated DNAs (178). To account for the melting-temperature profiles, Long et al. (179) proposed the cooperative melting mechanism, in which the increase of melting temperatures is correlated to the enhancement of ion concentrations around DNAs. The melting-temperature properties have been used to precisely probe single-base mismatches, deletions or insertions, and even monitor the DNA denaturization (180, 181). Mirkin and colleagues (182) proposed the six systematic rules of adjusting the supercrystals’ lattice constants to fine-tune the optical properties for a colorimetric detection method. The classification of the optical properties is based on the adsorption frequencies of the surface plasmon of AuNPs, which change in accordance with the lattice constants and assembly sizes (183). Leunissen et al. (184) further demonstrated that the attraction among DNA elements in the assemblies can be turned on and off by introducing folded secondary DNA structures having hairpins and loops. The Gang group (176) also examined the possibility of monitoring the attraction and repulsion in the aggregates by adjusting the ratio of hybridizing linker- DNAs to non-hybridizing DNAs. It is also noted that water can occupy as large as 90% volume of the aggregates (185, 186). The large volume of water provides a plenty of space for carrying drugs and genes for treating diseases (187, 188). The capability for drug delivery using AuNPs functionalized with DNAs has been also intensively exploited (189, 190). Although the properties have been widely investigated, their mechanical properties have not yet been reported. The mechanical properties such as Young’s and 147 bulk moduli determine the mechanical stability of the supercrystals in applications. Are they similar to those of gold, or DNA, or just water? Both gold and DNA are known as remarkable materials. Thus, it is of interest to examine if the supercrystals preserve the characteristics of gold and DNA. For example, Tam et al. (191) showed that when functionalized with oleic acid molecules, lead sulfide nanoparticles also grow into face- centered-cubic supercrystals. They concluded that the mechanical properties resemble those of the functionalized molecules by using nano-indentation techniques. However, it might be elusive to use the same techniques for the supercrystals of DNA-functionalized AuNPs because of the high water volume. Since the structures of DNAs and ion distributions in the aggregates are important (174, 178, 184, 192), Lee and Schatz used all-atom molecular dynamics simulations to investigate the structures of DNAs and ion distributions on small (1.8 nm in diameter) single faceted gold nanoparticles (193) and a flat gold surface (194). The structures of four single-stranded oligonucleotides on the nanoparticles significantly deviate from the canonical B-DNA structure, but the structure of the middle DNA in an assembly of seven DNAs on the flat gold surface is somewhat similar to the canonical B-DNA structure. Nevertheless, the structure of DNAs and ion distributions in modeled bcc and fcc supercrystals at the molecular level are still incomplete. In addition, the competition between enthalpic and entropic effects on the stability of the supercrystals cannot be examined in the simulations. Here we use million-atom molecular dynamics simulations to investigate bcc (BCCS) and fcc (FCCS) supercrystals of DNA-functionalized gold nanoparticles, which 148 are solvated with water and neutralized with sodium ions (see Sec. II Methodology). The two supercrystals contain 2.77 and 5.05 million atoms. Having large numbers of DNAs and hexanethiols attached to 3nm-diameter gold nanoparticles, we observe smooth changes of the DNA structures over simulation time, and are able to construct histograms of hexanethiol lengths, base-base stacking distances and hydrogen bonds (see Sec. III Results and Discussions). We find that within 10 ns the DNA structures are different from the canonical B-DNA structures. We also examine ion distributions around DNAs. From the ion distributions and using the model developed by Jin et al. (178), we estimate the melting-temperature increases, which are ΔT BCCS = 12.9 ± 0.2 K and ΔT FCCS = 8.0 ± 0.1 K. These melting-temperature increases provide evidence for the dominance of entropic over enthalpic effects on the stability of the modeled fcc supercrystal. Finally, we report the Young’s and bulk moduli of the bcc and fcc supercrystals, which resemble those of water. VII.2 Methodology A faceted octahedral gold nanoparticle (AuNP) was cut from a face-centered cubic (fcc) gold lattice. Its diameter is about 3.0 nm. We used function nucgen in AMBER (195) to create one sequence of two stranded canonical B-DNAs [see Figure VII.1(a)], which is 5’-ss(dC) 4 -ds(dC-dG) 6 -ss(dC) 1 -ds(dC-dG) 4 -ss(dG) 1 -ds(dG-dC) 6 -ss(dG) 4 -5’ where dC is Cytosine, dG is Guanine, ss is single-stranded, and ds is double-stranded. In the sequence, the spacers are 5’-ss(dC) 4 and ss(dG) 4 -5’, which are one base shorter than the experimentally used Adenine and Thymine spacers (173). Since a Guanine 149 polynucleotide can form secondary structures through Hoogsteen base pairing (173), using ss(dG) 4 -5’ spacers would be as flexible as 5-base Adenine or Thymine spacers, and reduce final numbers of atoms. Accordingly, 5’-ss(dC) 4 spacers are chosen for symmetry. The linkers are 3’-ss(dG) 11 -5’ and 5’-ss(dC) 11 -3’, which contains 4-base sticky ends. The markers are the 10-base nucleotides connecting the AuNPs, which was specified as a theoretical limit to control the formation of ordered aggregates (174). Figure VII.1 (a) Sequence and atomistic structure of a DNA connecting two neighbor AuNPs via hexanethiol molecules. Here, C is Cytosine, G is Guanine, and hydrogen atoms are not shown. Blue letters represents sticky bases. (b) Body-centered-cubic (bcc) and (c) face-centered-cubic (fcc) supercrystals consisted of AuNP, DNA, hexanethiol, water and Na + . Yellow is AuNP and hexanethiol. Red ribbons are DNA. Green is Na + . Water is not shown. 150 Two neutral hexanethiols are connections between two AuNPs and a DNA given in Figure VII.1(a). A hexanethiol molecule is bonded to a DNA via C-O-P where C is a carbon atom of the hexanethiol, O and P are oxygen and phosphorous of a phosphate (PO 4 ) – at one end of the DNA (193). The charges of the carbon and oxygen atoms are assigned so that the phosphate’s charge is –|e|. To connect hexanethiol molecules with AuNPs we made use of sulfur-gold bonds (196). We distributed 16 and 32 AuNPs to create bcc (BCCS) and fcc (FCCS) supercrystals with two unit cells [see Figure VII.1(b) and VII.1(c)]. Note that the initial distances between the nearest-neighbor AuNPs should be comparable to the length of the hexanethiol-capped DNAs; otherwise too small or large bonds and distances between gold and sulfur atoms would cause simulation instability. Given the initial distances, we arranged 64 and 192 hexanethiol-capped DNA molecules to connect the nearest-neighbor AuNPs in BCCS and FCCS, respectively. Each AuNP in BCCS and FCCS is bonded to 8 and 12 DNA molecules, respectively. Periodic boundary condition is applied to those connections across the system boxes between DNAs and AuNPs. We solvated the two systems with TIP3P (197) water. We randomly distributed 2560 (≈ 0.15 M) and 7680 (≈ 0.25 M) sodium ions to neutralize BCCS and FCCS, respectively. The total numbers of atoms are about 2.77 and 5.05 millions for BCCS and FCCS, respectively. The systems are built in VMD (160). The superlattice constants after relaxation are 15.15 and 18.48 nm for BCCS and FCCS, respectively. 151 We used AMBER force fields for DNA, water and ions (195). The Lenard Jones interaction for gold has σ = 2.569 Å and ε = 0.458 eV (198). Hautman and Klein’s force field (196) is for hexanethiols. We used NAMD package to run simulations (43). Particle Mesh Ewald (PME) (199) method for long-range Coulomb interactions was used. Non- bonded interactions are truncated at 12 Å and smoothly shifted at 10 Å. Shake algorithm (200) was turned on to keep all bonds of hydrogen atoms rigid. For both systems, we used conjugate gradient (201) method for 5000 steps. For the first 0.5 ns the systems were relaxed by using NVT coupling at T = 300 K and a time step of 1 fs. Then we used NPT coupling to thermalize them at P = 1 atm, T = 300 K and a time step of 2 fs. The Langevin dynamics with damping coefficient of 5 ps –1 was used to maintain temperature. Berendsen’s pressure coupling was used with compressibility of 4.57e-5 bar –1 , relaxation time of 0.1 ps and coupling frequency of 10 fs. The production time was 10 ns. Trajectories for analysis were saved very 20 ps. VII.3 Results and Discussions First, we compare the structures of the body-centered-cubic (BCCS) and face-centered- cubic (FCCS) supercrystals with experiments. The supercrystal structures are examined in terms of total numbers of nucleotides per DNA and interparticle distances between AuNPs. According to the findings by Hill et al (175), an interparticle distance D is simply related to a total number x of nucleotides by D = αx + β, (VII.1) 152 where α = 0.255 nm/base, which is a rise per base, is experimentally measured for FCCS, and β is the sum of the diameter of AuNPs (d AuNP ) and twice the average hexanethiol length (l). Since α B-DNA is 0.34 nm/base for B-DNA, the experimental value of α shows the contraction of DNAs in FCCS. Figure VII.2 Normalized histogram of hexanethiol lengths. The histograms are averaged over the last 100 frames (2 ns). Even though the simulated DNA sequence is different from experimental ones(175), the simulated values of α are here only quantities that we can directly compare and show the contraction of DNAs observed in experiments. For the simulated supercrystals, D BCCS is 13.12 nm, D FCCS is 12.83 nm, x is 26, and d AuNP is 3 nm. A hexanethiol length is the distance from the sulfur atom of a hexanethiol to the phosphorous atom of the DNA-phosphate group, to which the hexanethiol is connected. Having the large numbers of hexanethiols (128 and 384), we construct the length histograms to compute l (see Figure VII.2). While there are more hexanethiol lengths < 5 Å in BCCS than FCCS, slightly less hexanethiol lengths are at 8.5 Å in BCCS than FCCS. The averaged hexanethiol lengths are l BCCS = 8.1 Å, and l FCCS = 8.3 Å, hence, β BCCS = 153 4.62 nm, and β FCCS = 4.66 nm. Consequently, the simulated values of α are α BCCS = 0.327 nm/base and α FCCS = 0.314 nm/base. The experimental α is smaller than α BCCS and α FCCS by 22% and 19%, respectively. The value of α B-DNA is larger than α BCCS and α FCCS by 3.8% and 7.6%, respectively. Although the same DNA sequence is used for both supercrystals, α FCCS slightly smaller than α BCCS (or D FCCS < D BCCS ) is consistent with the experimental observations(170) that the close-packed supercrystals (i.e., FCCS) are more entropically favorable than the non-close-packed supercrystals (i.e., BCCS) if the same DNA materials are used. The formation of BCCS is due to more enthalpic than entropic contributions. The enthalpic contribution is to maximize the number of hybridizations. We shall show later that the numbers of hydrogen bonds indicating hybridization levels per DNA are almost the same in BCCS and FCCS. Then, the enthalpic contributions are not distinguishable. Therefore, the stability of BCCS and FCCS is mainly due to entropic effects, which depend on DNA markers (173, 174), linkers (170, 174, 175), and ion concentrations (177–179). To address the entropic effects, Storhoff et al. (173) examined the sequence- dependent stability of DNA-modified AuNPs. They found that 5-Cytosine and 20- Adenine- spacers have almost the same stability on AuNPs. Thus, the use of the 4- Cysosine and 4-Guanine spacers for BCCS and FCCS should not cause any severe instability on AuNPs. Once DNA-modified AuNPs are stable, AuNP curvatures (202, 203) have effects on DNA packing densities (175), which cause the various lengths of hexanethiol-capped DNAs, hence give rise to different values of α. Lee and Schatz found 154 that the simulated values of α for four ss-(dA) 10 and four ss-(dT) 10 on a single faceted AuNP (1.8 nm in diameter) are about α AT = 0.22 nm/base. Their results indicate that non- hybridized ss-DNAs significantly collapse on the AuNP to reduce α. In another study on seven DNAs on a gold flat surface to mimic experimental DNA densities, they observed the extension of the middle DNA (α ~ 0.49 nm/base), whose sequence is similar to those in the experiments. This result suggests that the model of the middle DNA having non- hybridized linkers does not capture the collapse of DNAs on AuNPs in experiments (175). In our simulations, in spite of the high DNA densities (≈ 0.28 and 0.42 chain/nm 2 ) and short spacers [5’-ss(dC) 4 and ss(dG) 4 -5’], the model of BCCS and FCCS shows the reduction of DNA lengths, hence α, and results into some significant structural changes. 155 Figure VII.3 (a) Root mean square deviations (RMSDs) of a single DNA and all DNA molecules with respect to their B-DNA structure. (b) RMSDs of 4bp and 6bp duplexes with respect to their B-DNA structure. (c) Normalized distributions of base-base stacking distance in BCCS (thick line) and FCCS (thin line) during the last 2 ns. Every pair of successive bases in the same DNA strand is collected to measure the distance between the centroids of the 6-membered rings. To determine the structural changes of the DNAs, we measure the root mean square deviations (RMSD) of the DNAs with respect to the canonical B-DNA structure (193, 194), construct histograms for base-base stacking distances, and count the numbers of hydrogen bonds. First, any symmetrical displacements of each individual DNA due to translation and rotation are separately minimized with respect to the B-DNA structure. Then, the RMSDs of all the DNAs are computed as 156 RMSD(  r,  R)=[ (  r j −  R j ) j=1 M ∑ 2 /M] 1/2 , (VII.2) where M is the number of DNA atoms excluding hydrogen atoms,  r j and  R j are the minimized coordinates of the jth atom of DNAs and the initial canonical B-DNAs, respectively. Figure VII.4 Histograms of number of hydrogen bonds per DNA in the BCCS (a) and FCCS (b). 157 Figure VII.5 Number of ions within 15 Å from DNA versus time. Figure VII.3(a) shows the RMSDs of a single DNA and all DNAs in each supercrystal. While the RMSDs of the single DNA has rapid fluctuations of 0.5-1.0 Å, those of all DNAs smoothly change, and at 10 ns reach RMSD BCCS ≈ 5.7 Å, and RMSD FCCS ≈ 5.8 Å (204). These RMSDs are about three times higher than that (~ 2 Å) of the middle DNA in an ensemble of seven DNAs on the flat gold surface (194). Lee and Schatz (194) used the DNA ensemble to mimic tight packing conditions on gold nanoparticles (205), and concluded that the middle-DNA structure is not very different from the canonical B-DNA structure. On the contrary, our RMSDs indicate that the DNA structures in BCCS and FCCS do not preserve the canonical B-DNA structure. They agree with the deviations (206) observed in the transition from A-DNA to B-DNA (~ 6 Å). These RMSDs are about those of ss(dA) 10 (~ 5.6 Å) and ss(dT) 10 (~ 6.7 Å) on 1.8nm- diameter AuNPs (193), which also indicate large deviations from B-DNA. 158 Figure VII.6 Radial distribution functions of (a) ion-ion and (b) ion-phosphorous. Since the sequence has three duplex regions, which are one 4-base-pair (4bp) duplex, and two symmetric 6bp duplexes as seen in Figure VII.1(a), we further examine which duplex region has the largest RMSD. Each duplex region is computed separately by Eq. (VII.2) after minimizing any symmetrical displacements of each individual DNA. The RMSD of the 6bp-duplex region is equal to the average of those of the two symmetric 6bp-duplex regions, and plotted with the RMSD of the 4bp-duplex region in Figure VII.3(b). In each supercrystal, the RMSD of the 4bp-duplex region is the largest as the simulation time is greater than 6 ns. Particularly, the RMSD of the 4bp-duplex region in BCC is almost a constant and higher the others. At 10 ns, while those of the 4bp-duplex region are 6.0 Å in BCCS, and 5.6 Å in FCCS, those of the 6bp-duplex 159 region are 4.8 Å in BCCS, and 5.2 Å in FCCS. The difference of about 1.2 Å between the RMSDs of the 4bp-duplex and 6bp-duplex regions in BCCS indicates that the 4bp-duplex region deviates more significant from the B-DNA structure than the 6bp-duplex one. However, the difference in FCCS is only 0.4 Å, which is about a thermal fluctuation (~ 0.5 Å) of DNA atoms. These results suggest that the structures of the duplexes are no longer the B-DNA structure. Figure VII.3(c) shows that 92% and 86% of base-base stacking distances (between the successive centroids of 6-membered rings in the same strands) are within 4 to 5 Å in BCCS and FCCS, respectively. The base-base (or π–π) stacking is attributed to significantly contribute to the rigidity of DNA backbones(207). Since the base-base stacking distance in B-DNA is 4.01 Å, these results indicate that the base-base B-DNA stacking is still dominant in the supercrystals. The dominance suggests that the rigidity of the simulated DNAs due to the base-base stacking maintains, although the overall structures of the simulated DNAs noticeably deviate from B-DNA. Note that the percentage in FCCS is smaller than that in BCCS. The smaller percentage is consistent with that D FCCS is smaller than D BCCS . We also measure the numbers of hydrogen bonds in the supercrystals to investigate the hybridization properties (see Figure VII.4). A hydrogen bond is counted if a distance between donor and acceptor is less than 3.9 Å and a donor-hydrogen-acceptor angle is between 150 and 180° (194). The histograms of the hydrogen bonds are constructed within 10 ns. They have a width of about one and look similar to Gaussian distributions. The Gaussian similarity indicates that the hybridization is saturated in the 160 supercrystals. The averaged numbers of hydrogen bonds per DNA are hbond BCCS = 42.4 and hbond FCCS = 42.0. Those numbers are about 87% of the initial number (= 48) of hydrogen bonds. They also indicate that not all hydrogen bonds contribute to the hybridization, and that the enthalpic contribution to the stability is almost the same for both supercrystals. This result supports that there are more dominantly entropic effects in FCCS than BCCS. To investigate the roles of ions in stabilizing the hybridization and screening the Coulomb interactions from the DNAs, we count numbers of ions per DNA within 15 Å from all the DNAs (see Figure VII.5), and measure the radial distribution functions g(r) among ions, and between ions and phosphorous atoms (see Figure VII.6). The averaged numbers of ions per DNA in BCCS and FCCS are nion BCCS = 29.6 ± 0.3 and nion FCCS = 33.9 ± 0.2. The numbers having small fluctuations indicate the stability of positive ion clouds around the negatively charged DNAs. The value of nion FCCS higher than nion BCCS is consistent with that more DNA molecules in FCCS than BCCS attract more sodium ions. The local numbers of ions around DNAs helps to maintain the hybridization, therefore increases the melting temperatures for the whole supercrystals (178, 179). So does FCCS have higher melting-temperature increase? To estimate the melting-temperature increases ΔT in BCCS and FCCS, we use the model developed by Jin et al. (178), ΔT = –15.8ln(C 1 /C 2 ), (VII.3) 161 where C 1 is a bulk ion concentrations and C 2 is a local ion concentration in the vicinity of DNA, which is computed from nion. To evaluate ΔT we treat the DNAs as cylinders with the radius of 1 nm and the lengths of 26α BCCS = 8.502 nm for BCCS and 26α FCCS = 8.164 nm for FCCS. Then, ΔT BCCS is 12.9 ± 0.2 K, and ΔT FCCS is 8.0 ± 0.1 K (the fluctuations of nion are used to estimate the uncertainties of ΔT). These values are in agreement with the estimates for the model of DNA clusters (8 K) by Long et al. (179), and for the double helical region of the middle DNA on a flat gold surface (13.5 K) by Lee and Schatz (194). Note that ΔT FCCS is noticeably smaller than ΔT BCCS . This result is consistent with more repulsive interactions among the negatively charged DNAs in FCCS than those in BCCS. Furthermore, the radial distribution functions can be used to explain the higher melting-temperature increase in BCCS. Figure VII.6(a) shows that the ion-ion correlation indicated by the magnitude of peaks in BCCS is higher than those in FCCS, even though the peaks are at the same positions and nion BCCS < nion FCCS (see Figure VII.5). Similarly, the ion-phosphorous correlation in BCCS in Figure VII.6(b) is higher than that in FCCS. The higher correlations indicate that ions bind stronger to DNAs in BCCS than FCCS, thus induce the higher melting temperature increase in BCCS. The lower melting-temperature increase ΔT FCCS strongly indicates that there are more dominantly entropic effects in FCCS than in BCCS. Park et al. (170) described experiments of growing FCCS from the same DNA materials as for growing BCCS. These DNA materials have two different kinds of linkers, which we model as two Cytosine and Guanine polynucleotides in both BCCS and FCCS [see Figure VII.1(a)]. 162 Based on the experiments, they argued that the formation of close-packed FCCS is due to the dominance of entropic over enthalpic effects. The enthalpic effects drive the formation of non-close-packed BCCS, which requires the maximum number of hybridization events. As shown in Figure VII.4, the numbers of hydrogen bonds indicating hybridization levels are almost identical in both supercrystals. Therefore, we arrive at the same conclusion of the more dominantly entropic effects in FCCS as in the experiments. In addition, one can model the competitions among major driving forces for inducing the supercrystal formations, for instance, in the studies by Tkachenko (174), Jin and colleagues (178). Tkachenko investigated the phase transition of DNA-colloidal self- assemblies in terms of concentrations of DNA linkers and markers. Varying these concentrations can change the relative strengths of attraction and repulsion, thus resulting into different supercrystals. However, this model does not consider effects of ion condensation on DNAs. Jin et al. proposed a model, in which ion condensations are attributed to alter DNA melting-entropy via the empirical equation (177) ΔS = ΔS 0 + 0.368N×lnC ion , (VII.4) thus changing melting-temperature of DNAs as described by Eq. (VII.3). Here, ΔS 0 is a predicted entropy at 1 M NaCl, N is the total number of phosphates in duplexes divided by 2, and C ion is an ion concentration (see Ref. (177) for more details). As a result, the decrease or increase of local ion concentrations in the vicinity of DNAs varies the melting-entropy. Moreover, the cooperativity introduced in the model happening across all melting steps sharpens the aggregates’ melting curves. The melting steps identical to a 163 number of linkers have effects on melting enthalpy, but not on entropy. Therefore, the entropic effects on the properties of the supercrystals can be deduced from the cooperative melting model, despite that we use the DNA sequence totally different from those in experiments (170). Figure VII.7 Stress tensor σ xx versus strain ε xx . The error bars are 0.002 GPa for BCCS and 0.0016 GPa for FCCS. Young’s modulus Y is computed by ∂σ xx /∂ε xx . Last but not least, we measure the bulk and Young’s moduli. The bulk moduli of the supercrystals are computed by K = k B T〈V〉/〈ΔV 2 〉 where k B is Boltzmann constant, V is volume of a system, 〈…〉 denotes the average over the production time in NPT coupling and 〈ΔV 2 〉 = 〈(V–<V>) 2 〉. The values of K are K BCCS = 2.29 and K FCCS = 2.26 GPa, which are slightly higher than that of bulk water (2.20 GPa) (208). To compute Young’s modulus, we perform simulations to stretch the supercrystals along the x-direction. The values of strain ε xx are from 1 to 5 percent of the x-dimension. For each value of ε xx , the stress tensor σ xx is computed by σ xx = –[∑ i m i v ix v ix +∑ i<j r ijx f ijx ]/V, (VII.5) 164 where m i and ν i are the mass and velocity of atom i, r ij are vectors from atom j to atom i, f ij are the forces acted on atom i by atom j. To have less fluctuation for σ xx we used the values of the group-pressure along the x-direction specified in NAMD package (43). At each step of stretching, the system is relaxed for 0.5 ns in NVT coupling. The data within the first 0.02 ns are not included to average the stress tensor. The stress tensor σ xx versus the strain ε xx is plotted in Figure VII.7. Young’s moduli, Y = ∂σ xx /∂ε xx , are Y BCCS = 1.49 and Y FCCS = 1.55 GPa. Y FCCS is slightly higher than Y BCCS , but within the uncertainty of about 0.1 GPa the Young’s moduli are indistinguishable. Given the bulk and Young’s moduli, we compute the Poisson ratio by γ = (1 – Y/3K)/2, which is about 0.39 for both supercrystals. The value of γ is close to those of many materials (208), which are 1/3. TABLE VII.1: Quantities computed for BCCS and FCCS a System D (nm) α (nm/base) RMSD (Å) hbond BCCS 13.12 0.327 5.7 87% FCCS 12.83 0.314 5.8 87% System nion/DNA ΔT (K) K (GPa) Y (GPa) BCCS 29.6 12.9 2.29 1.49 FCCS 33.9 8.0 2.26 1.55 a D is interparticle distance, α is base-base spacing, RMSD is root mean square deviation from the canonical B-DNA structure, hbond is fraction of hydrogen bonds per DNA, nion/DNA is average number of ions within 15 Å from a DNA, ΔT is melting-temperature increase, K is bulk modulus, and Y is Young’s modulus. 165 VII.4 Conclusions In this paper, we have modeled body-centered-cubic (BCCS) and face-centered-cubic (FCCS) supercrystals of DNA-functionalized gold nanoparticles by using all-atom molecular dynamics simulations. The two supercrystals contain 2.77 and 5.05 million atoms, which allow us to estimate the important quantities (see Table VII.1). We found that in spite of the large DNA density and short spacers, the base- base spacings α BCCS = 0.327 and α FCCS = 0.314 nm/base reflect the contraction of DNAs on gold nanoparticles observed in experiments. We observed the smooth changes of the DNA structures during simulation time, which noticeably differ from the canonical B-DNA structure. The structure difference is a result of the DNA contraction, the deviations of base-base stacking structure, and the broken hydrogen bonds. The levels of hybridization (numbers of hydrogen bonds) indicate the same enthalpic contribution to the stability of the supercrystals. Thus, the entropic effects play a more important role than the enthalpic ones in FCCS. They result into a lower melting-temperature increase in FCCS than BCCS. Moreover, the radial distribution functions for the correlation between ions and DNA show that ions bind stronger in BCCS than FCCS. This correlation, which agrees with the cooperative melting model proposed by Jin et al. (178), explains the higher melting-temperature increase in BCCS. Having the modeled supercrystals allows us for first time to estimate the bulk and Young’s moduli, which are slightly higher than those of water. 166 CHAPTER VIII Molecular Mechanism of Flip-flop in Triple-layer Oleic-Acid Membrane: Correlation Between Oleic Acid And Water VIII.1 Introduction Oleic acid is one of common fatty acids that are very important in biological processes. Fatty acids not only involve in lipid metabolisms for cells (209), but also play an essential role in mediating stress (210), by trans-membrane movement. The trans-membrane movement of fatty acids without protein transporters is called flip-flop when they migrate from one leaflet into the other leaflet. Fatty acid flip-flop with rates of milliseconds (210– 217) has been considered as a mechanism for transport and stress relaxation across model membranes. Szostak and colleagues suggested that when a membrane is under stress or locally deformed, flip-flop can facilitate the membrane remodeling (210). But there is a noticeable discrepancy between the measurable flip-flop rates and fast absorption rates of fatty acid monomers on lipid bilayers. For example, the absorption rate of palmitic acid to phosphatidylcholine small unilamellar vesicles is 0.024 ns –1 , which is at least thousand times faster than the flip-flop rates, and almost the same for many fatty acid monomers (211, 218, 219). The absorption and transmembrane movement (by either flip-flop or functional proteins) rates should not be discrepant to ensure the equilibrium of fatty acid transport through cellular membranes, which is described by three distinct processes: 167 absorption, trans-membrane movement, and desorption. The discrepancy, however, lies in difficulties of tracking and measuring flip-flop events. It is elusive to directly observe and analyze flip-flop events (211, 214, 220). In advanced fluorescence techniques (210, 217), one first attaches fluorescence molecules to fatty acids, then measures flip-flop rates from the decay of the fluorescence response. The fluorescence techniques are subjected to a time-limited resolution with an order of microseconds. They currently have not been able to identify any flip-flop events occurring in few hundred nanoseconds. The techniques usually provide upper bounds to flip-flop rates. Moreover, the weight of fluorescence molecules might have some effects on the accuracy of the measurements (217). An accurate technique should be able to capture the lowest flip-flop rates. The accuracy of the lowest rates is critically essential because it provides direct evidence for whether the trans-membrane movement by flip- flop is more favorable for cellular membranes than the transport movement by functional proteins (213, 214, 220–223). While the sub-microsecond time-scale to measure the lowest rates has been a challenge, it is achievable in molecular dynamics (MD) simulations. In MD simulations, one can easily identify and analyze the movement of fatty acids across membranes. Although there are MD simulations aimed to investigate the effects of fatty acids on lipid-membrane properties (224, 225), no all-atom MD simulation has been carried out to study the structure, or particularly remodeling of pure oleic acid (OA) membranes. Notman et al. (225) used a coarse-grained model to study the structures of pure OA membranes but did not report a flip-flop mechanism of coarse-grained OAs in membrane 168 remodeling. Gurtovenko et al. (226) examined the correlation between a transient water pore and the molecular flip-flop mechanism of lipid molecules under applied electric fields. They concluded that water molecules instantaneously passing through a lipid membrane promote the flip-flop of lipid molecules. However, the same correlation between the passage of water and flip-flop in pure OA membranes has not been identified. It might suggest a simple molecular mechanism for the membrane remodeling. In this paper, we perform all-atom MD simulations to study the remodeling of a pure OA membrane from a crystalline structure, and investigate the molecular mechanism of flip-flop events. Our resulting OA membrane has the thickness of 46 Å and three layers of hydrophilic COOH head-groups. We measure the pressure profiles to examine the favorable and unfavorable interactions of OAs with water in the membrane, and to compute the surface tension. We observe that the middle layer exists as an intermediate for fast re-arrangement of OAs. Every flip-flopping OA molecule stays most of the time in the middle layer before migrating into the others. We also observe fast passage of water molecules across the membrane (called water events). We hypothesize that the movement of water molecules cooperatively enhances the flip-flop of OAs. We quantify the cooperative movement by counting water events triggering OA flip-flop events. The rates of OA flip-flop and water events are k OA = 0.446 ns –1 and k H2O = 0.673 ns –1 , which are in better agreement with the absorption rates of fatty acid monomers (≈ 0.024 ns –1 ) on lipid vesicles (211) than the measurable values of flip-flop rates (~ ms –1 ) in experiments. 169 VIII.2 Methodology We prepared a bilayer of oleic acid (OA) molecules (Figure VIII.1A). The initial structure of the bilayer is in crystalline γ-phase (227, 228). The density of the crystal membrane is 0.89 g/cc. Each of the top and bottom layers has 441 OAs (Figure VIII.1A). The membrane is then sandwiched by TIP3P water facing the hydrophilic carboxyl groups COOH. The initial dimensions of the system are 10×10×10 nm 3 . The total number of atoms is 95832. Charmm27 force-fields (229, 230) for TIP3P water and OA were used. The force-field parameters of hydrocarbon chains of Dioleoyl-glycero-phosphocholine (DOPC) lipid were applied to OA’s hydrocarbon chain, since they are the same. In Charmm27, since the force-field parameters for COOH of protonated aspartic and glutamic acids based on acetic acid (CH 3 COOH) are indistinguishable, it is reasonable to apply these force-field parameters to the COOH of OA. There is no net charge in the system. The molecular dynamics (MD) simulations were implemented by NAMD package (43). First we used conjugate gradient for 10000 steps to remove any high- energetic contacts in the system; then performed simulations using NVT with Langevin dynamics for 10 ns, and NPT with constant area (NPAT) in the xy-plane to relax the crystalline OA membrane at T = 323 K for 36 ns. The damping coefficient for Langevin dynamics was 5 ps –1 . Time step was 1 fs for NVT, and 2 fs for pressure coupling. The system was coupled with isotropic NPT for production runs. (Constant area was turned off.) The target pressure was set to 1 bar. The barostat oscillation time and barostat damping time were 200 fs and 100 fs, respectively. Particle Mesh Ewald method was 170 used to compute Coulomb interaction with grid-size of about 1 Å. Cutoff for non-bonded interactions was 12 Å. All non-bonded interactions were smoothed at 10 Å. Exclusion 1- 4 interaction was turned on with the scaling being 0.83333. The trajectories were saved every 10 ps for analysis. The total production time was 500 ns. The final dimensions are 10.1×10.1×94.9 nm 3 . The figures were rendered by VMD (160). We adopted the method of computing pressure profiles developed by Lindahl et al. (231) and the Schulten group (232). The local pressure tensor Pαβ(z) within a given slab centered at z is P αβ (z)= 1 ΔV m i  v i ⊗  v i i ∑ −  F ij ⊗  r ij i<j ∑ % & ' ' ( ) * * , (VIII.1) where α ≡ (x,y,z), β ≡ (x,y,z), ΔV is the slab’s volume, and the sums are run over the particles in the slab. The stress profile π(z) is defined as [P xx (z)+P yy (z)]/2 – P zz (z). The surface tension is computed as γ =− π(z) −0.5L z 0.5L z ∫ dz , where L z is the z-dimension normal to the OA membrane. In our simulation, we divided the z-dimension into 94 slabs to compute the diagonal pressures Pαα(z), which are convergent within 335 ns (= 335,000 frames). To observe OA flip-flops, C1 atom of COOH having z C1 with respect to the center-of-mass of the membrane is chosen to detect the starting (t start ) and finishing (t finish ) time of a flip-flop event. When z C1 is either < 22 Å or > –22 Å, t start is marked. When the z C1 becomes either < –22 Å or > 22 Å, t finish is recorded. A complete flip-flop event of OA is accounted if t start < t finish . Flip-flop duration τ is computed as t finish – t start . 171 To observe water traversing the OA membrane, z-coordinate of the oxygen atom of water (z OW ) with respect to the center-of-mass of the membrane is used to detect the starting (t start ) and finishing time (t finish ) of a water event. When z OW is either < 22 Å or > – 22 Å, t start is marked. When the z OW becomes either < –22 Å or > 22 Å, t finish is recorded. A complete event of water is counted if t start < t middle < t finish where t middle is time when a water molecule passes the middle region (–5 Å < z OW < 5 Å). The time for a water molecule traversing the membrane is t finish – t start . VIII.3 Results and Discussions VIII.3.1 Triple-layer Structure First, we describe the remodeling of the initial crystalline bilayer (Figure VIII.1A) to triple-layer structures (Figure VIII.1E-F). Figure VIII.1B shows the oleic acid (OA) bilayer after 10 ns of thermalization with NVT. This bilayer is significantly disordered during 8 ns when pressure coupling (NPAT) was turned on (Figure VIII.1C-D). At the end of this coupling (Figure VIII.1E) the disordered distribution of OAs remodels into a triple-layer structure. The remodeling after being disordered somewhat resembles the re- arrangements after being perturbed in pure fatty acid membranes in experiments. 2 At this point, one can continuously use NPAT for the rest of the simulations, however the availability of area per OA is unknown for keeping the area fixed. Thus, we used NPT for the rest of the simulations to let the system to adjust in all directions. Using isotropic NPT the area and triple-layer structure remain almost the same (Figure VIII.1F). At the middle of this membrane we frequently observe that few water molecules can migrate 172 from both sides and attract OAs as seen Figure VIII.1G. This suggests some interesting properties of this membrane, but before analyzing, we shall provide some arguments for the triple-layer structure. Figure VIII.1 Snapshots of the oleic acid (OA) membrane during simulations. (A) Initial crystalline membrane in γ-phase. (B) OA bilayer in NVT. (C-E) OA membrane in NPAT. (F) OA membrane in isotropic NPT. Red, white and cyan colors are oxygen, hydrogen and carbon atoms, respectively. Water is not shown. (G) Four water molecules surrounded by OAs. Hydrogen atoms bonded to carbon atoms are not shown. Although there is no experimental evidence for the triple-layer structure, let us argue that there is a possibility of the triple-layer OA membrane due to the presence of water. We are not concerned with how the initial bilayer membrane is perturbed, but whether an equilibrium structure of OAs in water can be reached (233). It is noted that without water the crystalline γ-phase has only carboxyl (-COOH) groups forming 173 hydrogen bonds along the normal between two layers in a unit cell (227, 228). This structure exists below -2.2° C, and above 16° C irreversibly transforms into the crystalline β-phase, whose surface structure consists of both methyl (-CH 2 ) and carboxyl groups in two layers in a unit cell. This transformation indicates that OAs flip 180° degree in one layer even without water. Accordingly at 50° C with water, OAs can move drastically in each layer and between the only two layers in Figure VIII.1A-D. Szostak and colleagues observed that such drastic movements like flip-flop help relaxing stress and remodeling in fatty acid membranes (210, 234). In addition, the length of OA is shorter than a Dioleoyl-glycero-phosphocholine (DOPC) lipid, which composes of a dipolar hydrophilic head-group and two hydrocarbon chains with the same length as OA. These hydrocarbon chains form the water-excluded region in lipid bilayers, which it is also observed in the OA membrane (see Figure VIII.2). The weight of the dipolar hydrophilic head-groups on the surface of lipid bilayers is heavier than that of COOH. The lighter COOH with the hydrocarbon chain can diffuse much faster than the dipolar ones, and can be strongly influenced by the diffusion of water (will be discussed). Consequently, the bilayer structure (Figure VIII.1A-B) of OAs with water at 50° C and 1 bar is unstable, and ultimately transforms into the triple-layer structure (Figure VIII.1F). Moreover, the force-field parameters of hydrocarbon chains of DOPC and of protonated aspartic or glutamic acid have been well tested, thus the combination for OAs is reasonable (235). We shall examine the triple-layer structure, and compare with the bilayer structure of lipids to show that this triple-layer structure is physically stable. 174 Figure VIII.2 (A) Snapshot of the OA membrane during simulations. Red (initially at the bottom) and blue (initially at the top) spheres are COOH head- groups. Cyan lines are carbon chains. (B) Snapshot of water in the system. Red and white spheres are oxygen and hydrogen atoms, respectively. (C) Density of water and COOH along the z-direction with a bin width of 2 Å. The data are averaged over 500 ns. Now, we investigate how OA and water molecules distribute in the membrane. Figure VIII.2A shows a snapshot of the membrane evolving from the crystalline structure in Figure VIII.1A. Initially, OAs are at either the top (COOH colored in blue) or bottom (COOH colored in red) layers. During thermalization, OAs form the middle layer and migrate across the membrane, and little water exists at the middle of the membrane (Figure VIII.2B). To characterize the triple-layer structure, we measure water and COOH densities. We observe that the water and COOH densities averaged over 500 ns in NPT are not largely different from those averaged over the last 13.5 ns in NPAT. In Figure VIII.2C, the equilibrium density of water reduces from 0.989 to 0 g/cc from |z| ≈ 38 to |z| ≈ 10 Å. It is 0.012 g/cc at the middle, while it was about 0.02 g/cc during NPAT. The reduction from 0.02 to 0.012 g/cc means that several water molecules in the middle were pushed out the middle during the further thermalization process (NPT). In the COOH 175 density plot, the three peaks have the same order of magnitude (≈ 0.15 g/cc), which does not noticeably deviate from the values, 0.154 to 0.168 g/cc, during the last 13.5 ns in NPAT. These results suggest that thermalization with NPAT, which is usually used for simulating membranes, indeed induces the triple-layer structure. Without a reliable area per OA, the NPT thermalization was then performed. The symmetric distribution of the COOH density confirms that the membrane is in equilibrium. Based on the minima of the COOH density, we define that OAs belong to either the top or bottom leaflet if the z-coordinates of carbon atoms of COOH head- groups with respect to the center-of-mass of the membrane are either greater than 12 Å or less than –12 Å, respectively. The averaged positions of COOH in the top and bottom leaflets along the z-direction are at z = ±23 Å. The thickness is thus 46 Å, whereas the thickness of the initial crystalline membrane is 42 Å. On average, there are 257 OAs in each of the top and bottom leaflets. Thus, the area per OA in the leaflets is 39.7 Å 2 . The middle layer contains the rest, 368 OAs or 41.7 percent of all the OAs. Figure VIII.3 Mean Square Displacements (MSD) of OAs in the xy-plane and z- normal to the membrane. The positions of carbon atoms of COOH head-groups are used to compute the MSDs. 176 The thickness of the OA membrane, ≈ 46 Å, is in a range from 32.8 Å to 48 Å of lipid bilayers (232). Since the area per lipid is typically about 64 Å 2 (236), and each lipid has two hydrocarbon chains, the area per hydrocarbon chain in lipid bilayers is 32 Å 2 , which is about 8 Å 2 smaller than the area per OA, 39.7 Å 2 . If taking into account the size of dipolar lipid head-groups, which is larger than that of COOH, the area per OA is consistently larger than that per hydrocarbon chain in lipid bilayers. The membrane has a small level of hydrophobicity. The level of hydrophobicity can be inferred from the low water-density (c H2O (z=0) = 0.012 g/cc) at the middle of the membrane (see Figure VIII.2C). From the water-density profile, we estimate the potential of mean force (237), which is computed by –RTln[c H2O (0)/c H2O ] = 1.97 kJ/mol, where R is the gas constant, T is 323 K and c H2O = 0.989 g/cc is the bulk density of water outside the membrane. The potential of mean force indicates the free-energy barrier for water to permeate the OA membrane. It is about 13.5 times smaller than the free-energy barrier for water to permeate lipid bilayers (≈ 26 kJ/mol) at T = 350 K (237). To further demonstrate the properties of the OA membrane, we measure the mean square displacements (MSD) in the lateral xy-plane and along the normal to the membrane (see Figure VIII.3). From the slope of the MSD in the xy-plane, the lateral self-diffusion coefficient is 3.5×10 –7 cm 2 /s, which is about nine times higher than the experimental self-diffusion coefficient (238) (≈ 0.45×10 –7 cm 2 /s) of Di-myristoyl- phosphatidyl-choline (DMPC) lipid molecules at 26° C. The DMPC lipid has two little shorter carbon chains but heavier head-group than OA. Thus, due to the larger masses (239) the self-diffusion coefficients of DMPC lipids is consistently nine times smaller 177 than that of OA. In addition, the diffusion coefficient along the normal direction to the membrane, D z = 0.4×10 –7 cm 2 /s, is at least two order-of-magnitude higher than that of lipid molecules (~ 10 –9 cm 2 /s). This diffusion coefficient suggests that the OA membrane is ideal in simulation time-scales to investigate the trans-membrane movement from one (top or bottom) leaflet to the other (bottom or top) leaflet, which is defined as flip-flop. Figure VIII.4 (A) Lateral pressure (P xx + P yy )/2 and normal pressure (–P zz ). (B) Stress profile π(z) = (P xx +P yy )/2 – P zz along the z-direction normal to the membrane with a bin width of 1 Å. To compare the mechanical properties of the OA membrane with those of lipid membranes, we examine the pressure profiles (see Sec. VIII.2). Pressure profiles have been used to describe structures of lipid membranes in gel and liquid phases (232, 240). The structures, favorable and unfavorable interactions are reflected from the modulations, the positive and negative peaks of pressure profiles. The pressure profiles of the OA membrane (see Figure VIII.4) are smaller in magnitude than those of liquid-disordered 178 lipid membranes (240) (Lα), but still preserve the hydrophilic and hydrophobic interfaces of a model membrane. In a direction into a membrane, a hydrophilic or hydrophobic interface is defined as a region where π(z) goes from positive to negative or from negative to positive, respectively. Near the hydrophilic interfaces the small positive peaks of π(z) = 30 ± 3 bar at z ≈ ±33 Å (Figure VIII.4B) are mainly due to the lateral pressure (P xx + P yy )/2 (Figure VIII.4A). These peaks are significantly smaller than those of Lα, which are about 800 bar. The negative peaks (= –405 ± 5 bar) of π(z) at z ≈ ±23 Å near the hydrophobic interfaces indicate that the membrane is not as compressive as Lα. These small negative peaks are consistent with the large diffusion. Interestingly, at z ≈ ±17.5 π(z) is 100 ± 10 bar due to the normal pressure –P zz , that is larger than the first two positive peaks. On the contrary the first two positive peaks of π(z) of Lα are the largest. There is also a small negative peak near z = 0, which indicates slight compression. The peak can be attributed to the presence of water molecules around z = 0 as shown in Figure VIII.1G. Water molecules attract several COOH head-groups, which make the middle layer slightly compressive. The zero values of the stress profile between z = –10 Å and 10 Å indicate that OA and water molecules are also freely diffusive. Those properties of π(z) suggest that water molecules can diffuse more easily into the membrane than lipid bilayers. The resulting surface tension γ(OA), the area under –π(z), is 34.5 mN/m, which falls in a range for those of simulated and experimental lipid bilayers, e.g., γ(POPC) = 35.7 mN/m at 70.56 Å 2 /lipid (232, 241, 242). 179 VIII.3.2 Oleic Acid Flip-Flopping and Water Traversing Membrane The main purpose of this paper is to suggest a strong correlation between oleic acid (OA) flip-flop and the transmembrane movement of water. We find that there are 175 flip-flop events of OAs in 500 ns. Figure VIII.5A shows the histogram of the flip-flop events, where τ denotes flip-flop duration. The shortest duration τ min is 16 ns. Sixty five percent of the events have τ between 50 ns and 200 ns. Figure VIII.5B shows the histogram of the water events, whose τ is also duration for water molecules to traverse the membrane. We find that 305 events of water molecules that finish the migration in 500 ns. The shortest duration τ min is about 1.4 ns. Seventy five percent of events have τ less than 100 ns. As a result, water can move much faster than OA across the membrane. Figure VIII.5 Histograms of flip-flop (A) and traversing (B) durations of OAs and water, respectively. 180 To measure the rates for water and OA flip-flop events, we plot the accumulated numbers (NE H2O and NE OA ) of both events over simulation time (t finish ) in Figure VIII.6. Linearly fitting those numbers as functions of time, we find that NE H2O ~ 0.673t and NE OA ~ 0.446t. The slopes are the rates (k) of the events. The rates of water and OA flip- flop events are k H2O = 0.673 ns –1 and k OA = 0.446 ns –1 . The ratio, k H2O /k OA = 3/2, means that on average there are 3 water and 2 OAs, which finish migrating from one to the other side of the membrane. The trans-membrane movement of water has very high rate, k H2O = 0.673 ns –1 . The rates of water and OA flip-flop events are in the order of the water- permeation rate in bacterial-glycol facilitator proteins (243) (~ 0.5 ns –1 ). They are about 6 times slower than the water-permeation rate in aquaporin-1 proteins(244) (≈ 3.0 ns –1 ), which have a special water-selectivity function (245). The water-selectivity function composes of asparagine, proline and alanine, each of which also possesses a COOH head-group similar to that of oleic acid. The similarity supports the hypothesis that the migrations of water and oleic acid are correlated. Figure VIII.6 Accumulated numbers (NE) of water and OA molecules completely migrating across the membrane versus time (t finish ) (see Sec. VIII.2). 181 Figure VIII.7 (A-F) Snapshots of OA-flip-flopping and water-traversing events. The flip-flopping molecule is highlighted in blue. Its head-group and tail are denoted by balls and sticks, respectively. The surrounding water molecules of the head-group are found within a search of 3.5 Å. The traversing water molecule is highlighted in magenta. Red, cyan and white are oxygen, carbon and hydrogen atoms, respectively. The starting time for these events is 71.4 ns. The numbers denote the durations after the starting time. Figure VIII.7 shows how OA and water molecules migrate across the membrane. These two molecules start moving into the membrane at about 71.4 ns (Figure VIII.7A), which denotes the starting time, t start (see Section VIII.2), and finish the migration in almost the same duration, τ ≈ 129-130 ns. Initially, the OA is surrounded by few water molecules within 3.5 Å from the COOH head-group. The traversing water molecule is quite close to the OA. After only 1.6 ns (Figure VIII.7B), the water molecule reaches the middle of the membrane, and the OA goes about a quarter of the membrane thickness. These molecules are then fluctuating around the middle layer for very long time (~ 100 ns). The COOH head-group is sometimes hydrated and dehydrated, and moves close to the traversing water molecule (Figure VIII.7C-D). It can both make hydrogen bonds with 182 other OAs and with water, which explain the change in the hydration level. Finally at 129 ns, the water molecule completely migrates to the other side (Figure VIII.7E), and the OA delays the migration for 0.6 ns (Figure VIII.7F). Here, the OA does not directly drag the water molecule during migration. One reason is that the hydrogen bonds between OAs and water are not strong enough for water to hold on the COOH head-groups. Since there are water-excluded regions in the membrane, the water molecule moves fast to the middle and vice versa (see Figure VIII.7A-B). Water also has smaller size than OA, on average it takes longer time for OA to reach and move away from the middle layer (see Figure VIII.5). For those reasons, a water and OA molecules cannot together migrate across the membrane in every step. We observe that in many cases water molecules surrounding COOH help triggering the flip-flop events. In Figure VIII.8, we show the number of water molecules, N H2O , surrounding the COOH head-groups of the flip-flopping OAs through out 500 ns. This number is counted in every slap along the z-direction with a bin width of 1 Å, and averaged over 175 flip-flop events. While its variation indicates the relative change of hydration level of COOH across the membrane, its zero value implies no OA located at |z| > 36 Å and no neighboring water molecules. Since there are few water molecules at the middle, the peak of N H2O is significantly smaller than the others. Although the other two peaks at z = ± 22.5 Å are not the same, they simply imply the different residing times of flip-flopping OAs when starting and finishing migration. Thus, these two peaks indicate almost the same and highest hydration level near the hydrophobic interfaces (see Figure VIII.4B). The large values of N H2O at |z| < 22.5 Å suggests that some water 183 molecules accompany the OAs into the interfaces. Importantly, the non-zero minimum values of N H2O , ≈ 90, at z = ± 12 Å indicate that the flip-flopping OAs always contact with water molecules several times as already seen in Figure VIII.7C. As a result, these water molecules help to reduce barriers at the hydrophobic interfaces. Figure VIII.8. Averaged number of water molecules around COOH per flip- flopping OA versus z during migration. The bin width of slaps along the z- direction is 1 Å. Water molecules are counted within 3.5 Å of COOH through out 500 ns in each slap. No OA molecule is found for |z| > 36 Å. To further quantitatively correlate the 305 water events and the 175 OA flip-flop events, we analyze their starting time t start (see Sec. VIII.2), regardless of migration directions. We divide the 500 ns simulation time into 100 bins. WA(i) and OA(i) store the number of the events in the ith-bin. Here index i is floor(t start /5) plus 1, which returns an integer. For example, there are 11 water and 5 OA events in the 3 rd bin, which start migrating between 10 to 15 ns. We define water and OA events as totally correlated if [WA(i) – WA(i–1)]×[OA(i) – OA(i–1)] > 0; partially correlated if [WA(i) – WA(i– 1)]×[OA(i) – OA(i–1)] = 0 and WA(i) ≥ OA(i) > 0; uncorrelated if [WA(i) – WA(i- 1)]×[OA(i) – OA(i–1)] < 0. We find 33 bins (n cor ) having totally correlated events, 16 bins 184 having partially correlated events, and 22 bins (n uncor ) having uncorrelated events. The totally correlated events mean that as a number of water events increases or decreases, so does a number of flip-flop events within an observation interval. In contrast to the totally correlated events, the uncorrelated ones mean that water have no effect on OA flip-flop. The probability of the correlated events is (33+16)/71 = 69%. The probability greater than 50% indicates that the trans-membrane movement of water and OA are likely cooperative. (If the probability less than 50% means that the events of water and OA are totally random.) The ratio of the totally correlated to uncorrelated events is n cor /n uncor = 3/2. It indicates that in every 5 events, there are 3 water events totally correlated with flip-flop events. This ratio n cor /n uncor coincides with k H2O /k OA . This coincidence confirms the quantitative analysis of the correlation. In comparison with the lipid flip-flop mechanism, which is a pore-mediated process (226, 246), our findings bare some similarities and differences. In the lipid mechanism, the migration of lipid molecules across bilayers occurs simultaneously with the formation of transient water pores. Due to the formation, lipid molecules around the pores re-arrange to facilitate equilibrium processes and mediate stress. The stress mediation is also found in experiments by Szostak’s group on fatty acid membranes (210). In our oleic acid membrane, water molecules also help triggering and correlate with the flip-flop events of OAs, however, not as directly as in the lipid flip-flop mechanism, since there is no transient water pore. Once a transient pore appears, a lipid flip-flops within 10 ns, which is about the order of the minimum flip-flopping time of OAs, τ min = 16 ns. Most of the flip-flop times are about 100 ns because OAs can reside in the 185 intermediate middle layer, which can facilitate the migration of both water and OAs without the need of transient water pores like in lipid bilayers. This correlation between the rates of water and flip-flop events might be the simplest mechanism for fatty acid transports. Now in what cases or membranes, can this mechanism take place? First, Szostak and colleagues showed the existence of fatty acid (including oleic acid) membranes, which was thought to be the primitive membranes (210, 234). Second, many researchers (228, 247, 248) observed the crystalline structures of oleic acid without water, whose unit cells have two layers of oleic acids. But the evidence of the bilayer structures does not rule out the possibility of the triple-layer structure. Third, it should be noted that our triple-layer structure was thermally remodeled from the γ-phase crystalline structure, which exists below –2° C. With the presence of water at 50° C, the favorable interactions between the COOH head-groups and water, and the unfavorable interactions between the aliphatic chains and water always result in a water-excluded region. Such a water-excluded region in the OA membrane with the same thickness as lipid membranes is unlikely to have the bilayer structure, since the length of OAs is not as long as those of lipids, and COOH is not as heavy as dipolar lipid head- groups. It is noted that the area per OA, 39.7 Å 2 in the top and bottom leaflets, is consistent with the area per hydrocarbon chains in several bilayers (236) (~ 32 Å 2 ), and the surface tension, 34.5 mN/m, is about the same as those of simulated and experimental lipid bilayers (232, 241, 242). These results suggest that it is possible for pure OA triple- layers to exist within lipid bilayers without changing the elasticity of lipid membranes. 186 Finally, the flip-flop rate is only 20 times faster than absorption rates (≈ 0.024 ns – 1 ) of fatty acid monomers on lipid vesicles in experiments (211, 218, 219). Note that this absorption rate is almost the same for many fatty acid monomers. It is many order-of- magnitude faster than the experimentally measurable values of flip-flop rates (~ ms –1 ). In any cases, one would expect that the transmembrane movement by either flip-flop or proteins and absorption rates of fatty acids should not be widely different. Accordingly, flip-flop mechanism is more favorable than transport proteins, but is currently not well supported by the direct experimental data, i.e., accurate flip-flop rates close to 0.024 ns –1 . If the absorption rates are truly extremely faster than the transport rates, there is a possibility that many OAs fast absorbed on lipid surfaces can form a separated aggregate probably having the triple-layer structure. As shown in our simulations, this triple-layer structure suggests an enhancement in the transport rate without the need of transport proteins. As a result, the triple-layer structure is probably critical for an extremely fast transport mechanism in general cellular membranes. VIII.4 Conclusions We have simulated a pure oleic acid (OA) membrane in water that consists of three layers. The triple-layer structure also has the pressure profiles somewhat similar to those of lipid membranes. The surface tension, 34.5 mN/m, is in a range for those of lipid membranes. It suggests a possible existence of separated pure OA aggregates in lipid membranes. We observed that (i) several water molecules surrounding COOH head-groups help to reduce the barriers at the hydrophobic interfaces to trigger flip-flop events, and (ii) the middle 187 layer serves as an intermediate for water and OAs to migrate from one side to the other side of the membrane (defined as flip-flop). Within 500 ns there are 175 flip-flop events of OAs and 305 events of water traversing the membrane. The ratio of water-traversing rate (k H2O = 0.673 ns –1 ) to OA flip-flop rate (k OA = 0.446 ns –1 ) is 3/2. The ratio of the totally correlated water-OA events to the totally uncorrelated water-OA events, n cor /n uncor , is also 3/2. The probability of having the partially and totally correlated events is 69%. The results indicate that the trans-membrane movement of water and OAs is cooperative and correlated, and agrees with the experimentally measured absorption rate, 0.024 ns –1 . They support the idea that OA flip-flop is more favorable than transport by means of functional proteins. This study might provide further insight into how primitive cell membranes work, and how the interplay and correlation between water and fatty acids may occur. 188 Bibliography 1. Landau, L.D., and E.M. Lifshitz. 1980. Statistical Physics. Third. Butterworth-‐ Heinemann. 2. Baglione, M. 2007. Development of System Analysis Methodologies and Tools for Modeling and Optimizing Vehicle System Efficiency. PhD Thesis, University of Michigan. pp 52-‐54. 3. Yasuda, R., H. Noji, K. Kinosita Jr, and M. Yoshida. 1998. F1-‐ATPase Is a Highly Efficient Molecular Motor that Rotates with Discrete 120° Steps. Cell. 93: 1117–1124. 4. Noji, H., R. Yasuda, M. Yoshida, and K. Kinosita. 1997. Direct observation of the rotation of F1-‐ATPase. Nature. 386: 299–302. 5. Jarzynski, C. 1997. Nonequilibrium Equality for Free Energy Differences. Phys Rev Lett. 78: 2690. 6. Crooks, G.E. 1999. Fluctuation theorems. Phys Rev E. 60: 2721. 7. Liphardt, J.T., S. Dumont, S.B. Smith, I. Tinoco Jr., and C. Bustamante. 2002. Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski’s Equality. Science. 296: 1832. 8. E G D Cohen and David Mauzerall. 2004. A note on the Jarzynski equality. J. Stat. Mech. Theory Exp. 2004: P07006. 9. Vilar, J.M.G., and J.M. Rubi. 2008. Failure of the Work-‐Hamiltonian Connection for Free-‐Energy Calculations. Phys. Rev. Lett. 100: 020601. 10. Rodriguez-‐Gomez, D., E. Darve, and A. Pohorille. 2004. Assessing the efficiency of free energy calculation methods. J. Chem. Phys. 120: 3563–3578. 11. Srednicki, M. 1994. Chaos and quantum thermalization. Phys. Rev. E. 50: 888– 901. 12. Collin, D., F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, et al. 2005. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature. 437: 231–234. 13. Douarche, F., S. Ciliberto, A. Petrosyan, and I. Rabbiosi. 2005. An experimental test of the Jarzynski equality in a mechanical experiment. EPL Europhys. Lett. 70: 593. 189 14. Hendrix, D.A., and C. Jarzynski. 2001. A ``fast growth’’ method of computing free energy differences. J. Chem. Phys. 114: 5974–5981. 15. Jarzynski, C. 2007. Comparison of far-‐from-‐equilibrium work relations. Work Dissipation Fluct. Nonequilibrium Phys. Trav. Dissipation Fluct. En Phys. Non-‐ Équilib. 8: 495–506. 16. Jarzynski, C. 1997. Equilibrium free-‐energy differences from nonequilibrium measurements: A master-‐equation approach. Phys. Rev. E. 56: 5018–5035. 17. Jarzynski, C. 2001. How does a system respond when driven away from thermal equilibrium? Proc. Natl. Acad. Sci. 98: 3636–3638. 18. Jarzynski, C. 2008. Nonequilibrium work relations: foundations and applications. Eur. Phys. J. B -‐ Condens. Matter Complex Sysstem. 64: 331. 19. Crooks, G.E. 1998. Discretized mechanical work and principle of detailed balance. J. Stat. Phys. 90: 1481. 20. Hummer, G., and A. Szabo. 2001. Proc Natl Acad Sci USA. 98: 3658. 21. Hummer, G., and A. Szabo. 2003. Biophys. J. 85: 5. 22. Hummer, G., and A. Szabo. 2004. Abstr. Pap. Am. Chem. Soc. 227: U897. 23. Hummer, G., and A. Szabo. 2005. Acc. Chem. Res. 38: 504. 24. Hummer, G., and A. Szabo. 2010. Proc Natl Acad Sci USA. 107: 21441. 25. Park, S., F. Khalili-‐Araghi, E. Tajkhorshid, and K. Schulten. 2003. Free energy calculation from steered molecular dynamics simulations using Jarzynski’s equality. J. Chem. Phys. 119: 3559–3566. 26. Oberhofer, H., C. Dellago, and P.L. Geissler. 2005. Biased Sampling of Nonequilibrium Trajectories:   Can Fast Switching Simulations Outperform Conventional Free Energy Calculation Methods?†. J. Phys. Chem. B. 109: 6902– 6915. 27. West, D.K., P.D. Olmsted, and E. Paci. 2006. Free energy for protein folding from nonequilibrium simulations using the Jarzynski equality. J. Chem. Phys. 125: 204910–7. 28. Park, S., and K. Schulten. 2004. Calculating potentials of mean force from steered molecular dynamics simulations. J. Chem. Phys. 120: 5946–5961. 190 29. Minh, D.D.L., and A.B. Adib. 2008. Optimized Free Energies from Bidirectional Single-‐Molecule Force Spectroscopy. Phys. Rev. Lett. 100: 180602. 30. Isralewitz, B., M. Gao, and K. Schulten. 2001. Steered molecular dynamics and mechanical functions of proteins. Curr. Opin. Struct. Biol. 11: 224–230. 31. Hummer, G. 2001. Fast-‐growth thermodynamic integration: Error and efficiency analysis. J. Chem. Phys. 114: 7330–7337. 32. Darve, E., D. Rodriguez-‐Gomez, and A. Pohorille. 2008. Adaptive biasing force method for scalar and vector free energy calculations. J. Chem. Phys. 128: 144120–13. 33. Bastug, T., P.-‐C. Chen, S.M. Patra, and S. Kuyucak. 2008. Potential of mean force calculations of ligand binding to ion channels from Jarzynski’s equality and umbrella sampling. J. Chem. Phys. 128: 155104–9. 34. Baştuğ, T., and S. Kuyucak. 2007. Application of Jarzynski’s equality in simple versus complex systems. Chem. Phys. Lett. 436: 383–387. 35. Torrie, G.M., and J.P. Valleau. 1977. Nonphysical sampling distributions in Monte Carlo free-‐energy estimation: Umbrella sampling. J. Comput. Phys. 23: 187–199. 36. Kumar, S., J.M. Rosenberg, D. Bouzida, R.H. Swendsen, and P.A. Kollman. 1992. THE weighted histogram analysis method for free-‐energy calculations on biomolecules. I. The method. J. Comput. Chem. 13: 1011–1021. 37. Souaille, M., and B. Roux. 2001. Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations. Comput. Phys. Commun. 135: 40–57. 38. Kollman, P. 1993. Free energy calculations: Applications to chemical and biochemical phenomena. Chem. Rev. 93: 2395–2417. 39. Ozer, G., E.F. Valeev, S. Quirk, and R. Hernandez. 2010. Adaptive Steered Molecular Dynamics of the Long-‐Distance Unfolding of Neuropeptide Y. J. Chem. Theory Comput. 6: 3026–3038. 40. Zwanzig, R.W. 1954. High-‐Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases. J. Chem. Phys. 22: 1420–1426. 41. Schay, G. 2007. Introduction to Probability with Statistical Applications. Boston: Birkhauser Boston. 191 42. Allen, M.P., and D.J. Tildesley. 2003. Computer Simulation of Liquids. Oxford: Oxford University Press. 43. Kalé, L., R. Skeel, M. Bhandarkar, R. Brunner, A. Gursoy, et al. 1999. NAMD2: Greater Scalability for Parallel Molecular Dynamics. J. Comput. Phys. 151: 283–312. 44. MacKerell, D. Bashford, Bellott, Dunbrack, J.D. Evanseck, et al. 1998. All-‐Atom Empirical Potential for Molecular Modeling and Dynamics Studies of Proteins†. J. Phys. Chem. B. 102: 3586–3616. 45. Zuckerman, D.M., and T.B. Woolf. 2002. Overcoming finite-‐sampling errors in fast-‐switching free-‐energy estimates: extrapolative analysis of a molecular system. Chem. Phys. Lett. 351: 445–453. 46. If the PPP is used in the case of k = 1.0 kcal/mol/Å2, one might have to use larger τ to have good statistics of the reaction coordinate at large λ than τ at small λ. . 47. The range of values of work is enlarged to 450 kcal/mol to cover work distributions for either large k or small τ. . 48. Peliti, L. 2008. Comment on “Failure of the Work-‐Hamiltonian Connection for Free-‐Energy Calculations.”Phys. Rev. Lett. 101: 098903. 49. Horowitz, J., and C. Jarzynski. 2008. Comment on “Failure of the Work-‐ Hamiltonian Connection for Free-‐Energy Calculations.”Phys. Rev. Lett. 101: 098901. 50. Kofke, D.A. 2006. On the sampling requirements for exponential-‐work free-‐ energy calculations. Mol. Phys. 104: 3701–3708. 51. Lu, N., D.A. Kofke, and T.B. Woolf. 2004. Improving the efficiency and reliability of free energy perturbation calculations using overlap sampling methods. J. Comput. Chem. 25: 28–40. 52. Crooks, G.E. 2000. Path-‐ensemble averages in systems driven far from equilibrium. Phys. Rev. E. 61: 2361–2366. 53. Miyamoto, S., and P.A. Kollman. 1993. Absolute and relative binding free energy calculations of the interaction of biotin and its analogs with streptavidin using molecular dynamics/free energy perturbation approaches. Proteins Struct. Funct. Bioinforma. 16: 226–245. 192 54. Straatsma, T.P., H.J.C. Berendsen, and A.J. Stam. 1986. Estimation of statistical errors in molecular simulation calculations. Mol. Phys. 57: 89–95. 55. Andrieux, D., and P. Gaspard. 2008. QuantumWork Relations and Response Theory. Phys Rev Lett. 100: 230404. 56. Talkner, P., E. Lutz, and P. Hanggi. 2007. Fluctuation theorems: Work is not an observable. Phys Rev E. 75: 050102(R). 57. Deffner, S., and E. Lutz. 2008. Nonequilibrium work distribution of a quantum harmonic oscillator. Phys. Rev. E. 77: 021128. 58. Campisi, M., P. Talkner, and P. Hanggi. 2009. Fluctuation Theorem for Arbitrary Open Quantum Systems. Phys Rev Lett. 102: 210401. 59. Ritort, F. 2009. Viewpoint: Fluctuations in open systems. Phys. 2: 43. 60. Ngo, V.A. 2012. Parallel-‐pulling protocol for free-‐energy evaluation. Phys Rev E. 85: 036702. 61. For s = 1, we can make a convention that W = 0 and, the work distribution is identical to unity. Then, Eq. (III.8) still works for any s. . 62. Fetter, A.L., and J.D. Walecka. 2003. Field Theory at Finite Temperature. In: Quantum Theory of Many-‐Particle Systems. New York: Dover Publications, Inc. pp. 227–254. 63. Mathews, J., and R.L. Walker. 1970. Calculus of Variations. In: Mathematical Methods of Physics. New York: W. A. Benjamin, Inc. pp. 322–340. 64. Franck, J. 1926. Trans. Faraday Soc. 21: 0536. 65. Condon, E. 1926. A Theory of Intensity Distribution in Band Systems. Phys. Rev. 28: 1182. 66. Condon, E. 1928. NUCLEAR MOTIONS ASSOCIATED WITH ELECTRON TRANSITIONS IN DIATOMIC MOLECULES. Phys. Rev. 32: 0858. 67. Marcus, R.A. 1956. On the Theory of OxidationReduction Reactions Involving Electron Transfer. J. Chem. Phys. 24: 966. 68. Jarzynski, C. 1997. Equilibrium free-‐energy differences from nonequilibrium measurements: Phys Rev E. 56: 5018. 69. Trotter, H.F. 1959. Proc Amer Math Soc. 10: 545. 193 70. Suzuki, M. 1971. Prog. Theor. Phys. 46: 1054. 71. Feynman, R.P., and A.R. Hibbs. 1965. Quantum Mechanics and Path Integrals. McGraw-‐Hill. 72. Binder, K., and D.W. Heermann. 2002. Monte Carlo Simulation in Statistical Physics, An Introduction. Fourth. Springer. 73. Metropolis, N., A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller. 1953. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys. 21: 1087. 74. Boltzmann, L. 1964. Lectures on the gas. Berkeley, CA, USA: U. of California Press. 75. Grossmann, S., and M. Holthaus. 1995. On Bose-‐Einsteincondensation in harmonictraps. Phys. Lett. A. 208: 188–192. 76. Burger, S., F.S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, et al. 2001. Superfluid and Dissipative Dynamics of a Bose-‐Einstein Condensate in a Periodic Optical Potential. Phys. Rev. Lett. 86: 4447–4450. 77. Huber, G., F. Schmidt-‐Kaler, S. Deffner, and E. Lutz. 2008. Employing Trapped Cold Ions to Verify the Quantum Jarzynski Equality. Phys. Rev. Lett. 101: 070403. 78. Crooks, G.E. 2008. On the Jarzynski relation for dissipative quantum dynamics. J. Stat. Mech. Theory Exp. 2008: P10023. 79. Polkovnikov, A. 2011. Microscopic diagonal entropy and its connection to basic thermodynamic relations. Ann. Phys. 326: 486–499. 80. Bunin, G., L. D’Alessio, Y. Kafri, and A. Polkovnikov. 2011. Universal energy fluctuations in thermally isolated driven systems. Nat Phys. 7: 913–917. 81. Santos, L.F., A. Polkovnikov, and M. Rigol. 2011. Entropy of Isolated Quantum Systems after a Quench. Phys. Rev. Lett. 107: 040601. 82. Yukalov, V.I. 2011. Equilibration and thermalization in finite quantum systems. Laser Phys. Lett. 8: 485–507. 83. Rigol, M., V. Dunjko, V. Yurovsky, and M. Olshanii. 2007. Relaxation in a Completely Integrable Many-‐Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-‐Core Bosons. Phys. Rev. Lett. 98: 050405. 194 84. Rigol, M. 2009. Breakdown of Thermalization in Finite One-‐Dimensional Systems. Phys. Rev. Lett. 103: 100403. 85. Campos Venuti, L., N.T. Jacobson, S. Santra, and P. Zanardi. 2011. Exact Infinite-‐Time Statistics of the Loschmidt Echo for a Quantum Quench. Phys. Rev. Lett. 107: 010403. 86. Ngo, V.A., and S. Haas. 2012. Demonstration of Jarzynski’s equality in open quantum systems using a stepwise pulling protocol. Phys. Rev. E. 86: 031127. 87. Tasaki, H. 1998. From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example. Phys. Rev. Lett. 80: 1373–1376. 88. Kinoshita, T., T. Wenger, and D.S. Weiss. 2006. A quantum Newton’s cradle. Nature. 440: 900–903. 89. Breitenbach, G., S. Schiller, and J. Mlynek. 1997. Measurement of the quantum states of squeezed light. Nature. 387: 471–475. 90. Rigol, M., and A. Muramatsu. 2005. Ground-‐state properties of hard-‐core bosons confined on one-‐dimensional optical lattices. Phys. Rev. A. 72: 013604. 91. Leibfried, D., R. Blatt, C. Monroe, and D. Wineland. 2003. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75: 281–324. 92. Navarro, B., M.E. Kennedy, B. Velimirović, D. Bhat, A.S. Peterson, et al. 1996. Nonselective and Gβγ-‐Insensitive weaver K+ Channels. Science. 272: 1950– 1953. 93. Hille, B. 2001. Ion Channels of Excitable Membranes. Third Edition. Sunderland, MA: Sinauer Associates, Inc. 94. Bezanilla, F., and C.M. Armstrong. 1972. Negative Conductance Caused by Entry of Sodium and Cesium Ions into the Potassium Channels of Squid Axons. J. Gen. Physiol. 60: 588–608. 95. Hodgkin, A.L., and R.D. Keynes. 1955. The potassium permeability of a giant nerve fibre. J. Physiol. 128: 61–88. 96. Doyle, D.A., J.M. Cabral, R.A. Pfuetzner, A. Kuo, J.M. Gulbis, et al. 1998. The Structure of the Potassium Channel: Molecular Basis of K+ Conduction and Selectivity. Science. 280: 69–77. 97. Morais-‐Cabral, J.H., Y. Zhou, and R. MacKinnon. 2001. Energetic optimization of ion conduction rate by the K+ selectivity filter. Nature. 414: 37–42. 195 98. Berneche, S., and B. Roux. 2001. Energetics of ion conduction through the K+ channel. Nature. 414: 73–77. 99. Bernèche, S., and B. Roux. 2003. A microscopic view of ion conduction through the K+ channel. Proc. Natl. Acad. Sci. 100: 8644–8648. 100. Furini, S., and C. Domene. 2011. Selectivity and Permeation of Alkali Metal Ions in K+-‐channels. J. Mol. Biol. 409: 867–878. 101. Jensen, M.Ø., D.W. Borhani, K. Lindorff-‐Larsen, P. Maragakis, V. Jogini, et al. 2010. Principles of conduction and hydrophobic gating in K+ channels. Proc. Natl. Acad. Sci. 107: 5833–5838. 102. Andersen, O.S. 2011. Perspectives on: Ion selectivity. J. Gen. Physiol. 137: 393– 395. 103. Allen, T.W., O.S. Andersen, and B. Roux. 2004. On the Importance of Atomic Fluctuations, Protein Flexibility, and Solvent in Ion Permeation. J. Gen. Physiol. 124: 679–690. 104. Alam, A., and Y. Jiang. 2011. Structural studies of ion selectivity in tetrameric cation channels. J. Gen. Physiol. 137: 397–403. 105. Shi, N., S. Ye, A. Alam, L. Chen, and Y. Jiang. 2006. Atomic structure of a Na+-‐ and K+-‐conducting channel. Nature. 440: 570–574. 106. Derebe, M.G., D.B. Sauer, W. Zeng, A. Alam, N. Shi, et al. 2011. Tuning the ion selectivity of tetrameric cation channels by changing the number of ion binding sites. Proc. Natl. Acad. Sci. 108: 598–602. 107. Domene, C., and S. Furini. 2009. Chapter 7 -‐ Examining Ion Channel Properties Using Free-‐Energy Methods. In: Michael L. Johnson; Gary K. Ackers; Jo M. Holt, editor. Methods in Enzymology. Academic Press. pp. 155–177. 108. Noskov, S.Y., S. Berneche, and B. Roux. 2004. Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands. Nature. 431: 830–834. 109. Noskov, S.Y., and B. Roux. 2007. Importance of Hydration and Dynamics on the Selectivity of the KcsA and NaK Channels. J. Gen. Physiol. 129: 135–143. 110. Varma, S., D.M. Rogers, L.R. Pratt, and S.B. Rempe. 2011. Design principles for K+ selectivity in membrane transport. J. Gen. Physiol. 137: 479–488. 196 111. Varma, S., D. Sabo, and S.B. Rempe. 2008. K+/Na+ Selectivity in K Channels and Valinomycin: Over-‐coordination Versus Cavity-‐size constraints. J. Mol. Biol. 376: 13–22. 112. Roux, B. 2005. ION CONDUCTION AND SELECTIVITY IN K+ CHANNELS. Annu. Rev. Biophys. Biomol. Struct. 34: 153–171. 113. Zhou, Y., J.H. Morais-‐Cabral, A. Kaufman, and R. MacKinnon. 2001. Chemistry of ion coordination and hydration revealed by a K+ channel-‐Fab complex at 2.0[thinsp][angst] resolution. Nature. 414: 43–48. 114. Grottesi, A., C. Domene, S. Haider, and M.S.P. Sansom. 2005. Molecular dynamics Simulation approaches to K channels: conformational flexibility and physiological function. NanoBioscience IEEE Trans. On. 4: 112–120. 115. Egwolf, B., and B. Roux. 2010. Ion Selectivity of the KcsA Channel: A Perspective from Multi-‐Ion Free Energy Landscapes. J. Mol. Biol. 401: 831–842. 116. Noskov, S.Y., and B. Roux. 2006. Ion selectivity in potassium channels. Ion Hydration Spec. Issue. 124: 279–291. 117. Kim, I., and T.W. Allen. 2011. On the selective ion binding hypothesis for potassium channels. Proc. Natl. Acad. Sci. . 118. Alam, A., and Y. Jiang. 2009. Structural analysis of ion selectivity in the NaK channel. Nat Struct Mol Biol. 16: 35–41. 119. Ngo, V., and S. Haas. 2013. Non-‐equilibrium Approach To Thermodynamics Using Jarzynski Equality and Diagonal Entropy. arXiv:1303.2141. . 120. Phillips, J.C., R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, et al. 2005. Scalable molecular dynamics with NAMD. J. Comput. Chem. 26: 1781–1802. 121. Aksimentiev, A., M. Sotomayor, and D. Wells. Protein Tutorial. University of Illinois at Urbana-‐Champaign, Beckman Institute for Advanced Science and Technology, Theoretical and Computational Biophysics Group, Computational Biophysics Workshop. http://www.ks.uiuc.edu/Training/Tutorials/. 122. Mackerell, A.D. 2004. Empirical force fields for biological macromolecules: Overview and issues. J. Comput. Chem. 25: 1584–1604. 123. Mackerell, A.D., M. Feig, and C.L. Brooks. 2004. Extending the treatment of backbone energetics in protein force fields: Limitations of gas-‐phase quantum mechanics in reproducing protein conformational distributions in molecular dynamics simulations. J. Comput. Chem. 25: 1400–1415. 197 124. Garcia-‐Viloca, M., C. Alhambra, D.G. Truhlar, and J. Gao. 2001. Inclusion of quantum-‐mechanical vibrational energy in reactive potentials of mean force. J. Chem. Phys. 114: 9953–9958. 125. Ermak, D.L., and J.A. McCammon. 1978. Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69: 1352–1360. 126. Thompson, A.N., I. Kim, T.D. Panosian, T.M. Iverson, T.W. Allen, et al. 2009. Mechanism of potassium-‐channel selectivity revealed by Na+ and Li+ binding sites within the KcsA pore. Nat Struct Mol Biol. 16: 1317–1324. 127. LeMasurier, M., L. Heginbotham, and C. Miller. 2001. Kcsa: It’s a Potassium Channel. J. Gen. Physiol. 118: 303–314. 128. Collins, K.D. 2012. Why continuum electrostatics theories cannot explain biological structure, polyelectrolytes or ionic strength effects in ion–protein interactions. Biophys. Chem. 167: 43–59. 129. Kiriukhin, M.Y., and K.D. Collins. 2002. Dynamic hydration numbers for biologically important ions. Biophys. Chem. 99: 155–168. 130. Shrivastava, I.H., D. Peter Tieleman, P.C. Biggin, and M.S.P. Sansom. 2002. K+ versus Na+ Ions in a K Channel Selectivity Filter: A Simulation Study. Biophys. J. 83: 633–645. 131. Hille, B., and W. Schwarz. 1978. Potassium channels as multi-‐ion single-‐file pores. J. Gen. Physiol. 72(4): 409–442. 132. Chung, S.-‐H., T.W. Allen, and S. Kuyucak. 2002. Modeling Diverse Range of Potassium Channels with Brownian Dynamics. Biophys. J. 83: 263–277. 133. Aqvist, J., and V. Luzhkov. 2000. Ion permeation mechanism of the potassium channel. Nature. 404: 881–884. 134. Nimigean, C.M., and C. Miller. 2002. Na+ Block and Permeation in a K+ Channel of Known Structure. J. Gen. Physiol. 120: 323–335. 135. Bucher, D., and U. Rothlisberger. 2010. Molecular simulations of ion channels: a quantum chemist’s perspective. J. Gen. Physiol. 135: 549–554. 136. O’Sullivan, R.J., and J. Karlseder. 2010. Telomeres: protecting chromosomes against genome instability. Nat Rev Mol Cell Biol. 11: 171–181. 198 137. Biffi, G., D. Tannahill, J. McCafferty, and S. Balasubramanian. 2013. Quantitative visualization of DNA G-‐quadruplex structures in human cells. Nat Chem. 5: 182–186. 138. Karsisiotis, A.I., N.M. Hessari, E. Novellino, G.P. Spada, A. Randazzo, et al. 2011. Topological Characterization of Nucleic Acid G-‐Quadruplexes by UV Absorption and Circular Dichroism. Angew. Chem. Int. Ed. 50: 10645–10648. 139. Hu, L., K.W. Lim, S. Bouaziz, and A.T. Phan. 2009. Giardia Telomeric Sequence d(TAGGG)4 Forms Two Intramolecular G-‐Quadruplexes in K+ Solution: Effect of Loop Length and Sequence on the Folding Topology. J. Am. Chem. Soc. 131: 16824–16831. 140. Maizels, N. 2006. Dynamic roles for G4 DNA in the biology of eukaryotic cells. Nat Struct Mol Biol. 13: 1055–1059. 141. Davis, J.T. 2004. G-‐Quartets 40 Years Later: From 5′-‐GMP to Molecular Biology and Supramolecular Chemistry. Angew. Chem. Int. Ed. 43: 668–698. 142. Alberti, P., A. Bourdoncle, B. Sacca, L. Lacroix, and J.-‐L. Mergny. 2006. DNA nanomachines and nanostructures involving quadruplexes. Org. Biomol. Chem. 4: 3383–3391. 143. Laughlan, G., A.I.H. Murchie, D.G. Norman, M.H. Moore, P.C.E. Moody, et al. 1994. The High-‐Resolution Crystal Structure of a Parallel-‐Stranded Guanine Tetraplex. Science. 265: 520–524. 144. Phillips, K., Z. Dauter, A.I.H. Murchie, D.M.J. Lilley, and B. Luisi. 1997. The crystal structure of a parallel-‐stranded guanine tetraplex at 0.95Å resolution. J. Mol. Biol. 273: 171–182. 145. Forman, S.L., J.C. Fettinger, S. Pieraccini, G. Gottarelli, and J.T. Davis. 2000. Toward Artificial Ion Channels:   A Lipophilic G-‐Quadruplex. J. Am. Chem. Soc. 122: 4060–4067. 146. Hud, N.V., F.W. Smith, F.A.L. Anet, and J. Feigon. 1996. The Selectivity for K+ versus Na+ in DNA Quadruplexes Is Dominated by Relative Free Energies of Hydration:   A Thermodynamic Analysis by 1H NMR†. Biochemistry (Mosc.). 35: 15383–15390. 147. Kaucher, M.S., W.A. Harrell, and J.T. Davis. 2005. A Unimolecular G-‐Quadruplex that Functions as a Synthetic Transmembrane Na+ Transporter. J. Am. Chem. Soc. 128: 38–39. 199 148. Deng, H., and W.H. Braunlin. 1996. Kinetics of Sodium Ion Binding to DNA Quadruplexes. J. Mol. Biol. 255: 476–483. 149. Hud, N.V., P. Schultze, V. Sklenář, and J. Feigon. 1999. Binding sites and dynamics of ammonium ions in a telomere repeat DNA quadruplex. J. Mol. Biol. 285: 233–243. 150. Akhshi, P., N.J. Mosey, and G. Wu. 2012. Free-‐Energy Landscapes of Ion Movement through a G-‐Quadruplex DNA Channel. Angew. Chem. Int. Ed. 51: 2850–2854. 151. Šket, P., and J. Plavec. 2007. Not All G-‐Quadruplexes Exhibit Ion-‐Channel-‐like Properties:   NMR Study of Ammonium Ion (Non)movement within the d(G3T4G4)2 Quadruplex. J. Am. Chem. Soc. 129: 8794–8800. 152. Wong, A., R. Ida, and G. Wu. 2005. Direct NMR detection of the “invisible” alkali metal cations tightly bound to G-‐quadruplex structures. Biochem. Biophys. Res. Commun. 337: 363–366. 153. Snoussi, K., and B. Halle. 2008. Internal Sodium Ions and Water Molecules in Guanine Quadruplexes: Magnetic Relaxation Dispersion Studies of [d(G3T4G3)]2 and [d(G4T4G4)]2†. Biochemistry (Mosc.). 47: 12219–12229. 154. Cavallari, M., A. Calzolari, A. Garbesi, and R. Di Felice. 2006. Stability and Migration of Metal Ions in G4-‐Wires by Molecular Dynamics Simulations. J. Phys. Chem. B. 110: 26337–26348. 155. Kotlyar, A.B., N. Borovok, T. Molotsky, H. Cohen, E. Shapir, et al. 2005. Long, Monomolecular Guanine-‐Based Nanowires. Adv. Mater. 17: 1901–1905. 156. Cohen, H., T. Sapir, N. Borovok, T. Molotsky, R. Di Felice, et al. 2007. Polarizability of G4-‐DNA Observed by Electrostatic Force Microscopy Measurements. Nano Lett. 7: 981–986. 157. Borovok, N., N. Iram, D. Zikich, J. Ghabboun, G.I. Livshits, et al. 2008. Assembling of G-‐strands into novel tetra-‐molecular parallel G4-‐DNA nanostructures using avidin–biotin recognition. Nucleic Acids Res. 36: 5050– 5060. 158. Shapir, E., L. Sagiv, T. Molotsky, A.B. Kotlyar, R.D. Felice, et al. 2010. Electronic Structure of G4-‐DNA by Scanning Tunneling Spectroscopy. J. Phys. Chem. C. 114: 22079–22084. 159. Calzolari, A., R. Di Felice, E. Molinari, and A. Garbesi. 2002. G-‐quartet biomolecular nanowires. Appl. Phys. Lett. 80: 3331–3333. 200 160. Humphrey, W., A. Dalke, and K. Schulten. 1996. VMD -‐ Visual Molecular Dynamics. J. Molec. Graphics. 14: 33–38. 161. Schwede, T., and M. Peitsch. 2008. Computational Structural Biology: Methods and Applications. Singapore: World Scientific. 162. Cornell, W.D., P. Cieplak, C.I. Bayly, I.R. Gould, K.M. Merz, et al. 1995. A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules. J. Am. Chem. Soc. 117: 5179–5197. 163. Pérez, A., I. Marchán, D. Svozil, J. Sponer, T.E. Cheatham III, et al. 2007. Refinement of the AMBER Force Field for Nucleic Acids: Improving the Description of α/γ Conformers. Biophys. J. 92: 3817–3829. 164. Šponer, J., P. Jurečka, and P. Hobza. 2004. Accurate Interaction Energies of Hydrogen-‐Bonded Nucleic Acid Base Pairs. J. Am. Chem. Soc. 126: 10142– 10151. 165. Šponer, J., P. Jurečka, I. Marchan, F.J. Luque, M. Orozco, et al. 2006. Nature of Base Stacking: Reference Quantum-‐Chemical Stacking Energies in Ten Unique B-‐DNA Base-‐Pair Steps. Chem. – Eur. J. 12: 2854–2865. 166. Feller, S.E., Y. Zhang, R.W. Pastor, and B.R. Brooks. 1995. Constant pressure molecular dynamics simulation: The Langevin piston method. J. Chem. Phys. 103: 4613–4621. 167. Ross, W.S., and C.C. Hardin. 1994. Ion-‐Induced Stabilization of the G-‐DNA Quadruplex: Free Energy Perturbation Studies. J. Am. Chem. Soc. 116: 6070– 6080. 168. Ngo, V.A., D. Stefanovski, S. Haas, and R. Farley. Non-‐Equilibrium Dynamics Determine Ion Selectivity In The KcsA Channel (Submitted to Biophysical Journal). . 169. Ying, L., J.J. Green, H. Li, D. Klenerman, and S. Balasubramanian. 2003. Studies on the structure and dynamics of the human telomeric G quadruplex by single-‐molecule fluorescence resonance energy transfer. Proc. Natl. Acad. Sci. 100: 14629–14634. 170. Park, S.Y., A.K.R. Lytton-‐Jean, B. Lee, S. Weigand, G.C. Schatz, et al. 2008. DNA-‐ programmable nanoparticle crystallization. Nature. 451: 553. 171. Shevchenko, E.V., D.V. Talapin, N.A. Kotov, S. O’Brien, and C.B. Murray. 2006. Structural diversity in binary nanoparticle superlattices. Nature. 439: 55–59. 201 172. Frenkel, D. 2006. Colloidal crystals: Plenty of room at the top. Nat Mater. 5: 85–86. 173. Storhoff, J.J., R. Elghanian, C.A. Mirkin, and R.L. Letsinger. 2002. Sequence-‐ Dependent Stability of DNA-‐Modified Gold Nanoparticles. Langmuir. 18: 6666–6670. 174. Tkachenko, A.V. 2002. Morphological Diversity of DNA-‐Colloidal Self-‐ Assembly. Phys. Rev. Lett. 89: 148303. 175. Hill, H.D., R.J. Macfarlane, A.J. Senesi, B. Lee, S.Y. Park, et al. 2008. Controlling the Lattice Parameters of Gold Nanoparticle FCC Crystals with Duplex DNA Linkers. Nano Lett. 8: 2341–2344. 176. Nykypanchuk, D., M.M. Maye, D. van der Lelie, and O. Gang. 2007. DNA-‐Based Approach for Interparticle Interaction Control. Langmuir. 23: 6305–6314. 177. SantaLucia, J. 1998. A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-‐neighbor  thermodynamics. Proc. Natl. Acad. Sci. 95: 1460 –1465. 178. Jin, R., G. Wu, Z. Li, C.A. Mirkin, and G.C. Schatz. 2003. What Controls the Melting Properties of DNA-‐Linked Gold Nanoparticle Assemblies? J Am Chem Soc. 125: 1643–1654. 179. Long, H., A. Kudlay, and G.C. Schatz. 2006. Molecular Dynamics Studies of Ion Distributions for DNA Duplexes and DNA Clusters:   Salt Effects and Connection to DNA Melting. J Phys Chem B. 110: 2918–2926. 180. Storhoff, J.J., R. Elghanian, R.C. Mucic, C.A. Mirkin, and R.L. Letsinger. 1998. One-‐Pot Colorimetric Differentiation of Polynucleotides with Single Base Imperfections Using Gold Nanoparticle Probes. J Am Chem Soc. 120: 1959– 1964. 181. Elghanian, R., J.J. Storhoff, R.C. Mucic, R.L. Letsinger, and C.A. Mirkin. 1997. Selective Colorimetric Detection of Polynucleotides Based on the Distance-‐ Dependent Optical Properties of Gold Nanoparticles. Science. 277: 1078 – 1081. 182. Macfarlane, R.J., B. Lee, M.R. Jones, N. Harris, G.C. Schatz, et al. 2011. Nanoparticle Superlattice Engineering with DNA. Science. 334: 204 –208. 183. Storhoff, J.J., A.A. Lazarides, R.C. Mucic, C.A. Mirkin, R.L. Letsinger, et al. 2000. What Controls the Optical Properties of DNA-‐Linked Gold Nanoparticle Assemblies? J Am Chem Soc. 122: 4640–4650. 202 184. Leunissen, M.E., R. Dreyfus, F.C. Cheong, D.G. Grier, R. Sha, et al. 2009. Switchable self-‐protected attractions in DNA-‐functionalized colloids. Nat Mater. 8: 590–595. 185. Nykypanchuk, D., M.M. Maye, D. van der Lelie, and O. Gang. 2008. DNA-‐guided crystallization of colloidal nanoparticles. Nature. 451: 549–552. 186. Xiong, H., D. van der Lelie, and O. Gang. 2008. DNA Linker-‐Mediated Crystallization of Nanocolloids. J Am Chem Soc. 130: 2442–2443. 187. Rink, J.S., K.M. McMahon, X. Chen, C.A. Mirkin, C.S. Thaxton, et al. 2010. Transfection of pancreatic islets using polyvalent DNA-‐functionalized gold nanoparticles. Surgery. 148: 335–345. 188. Dhar, S., W.L. Daniel, D.A. Giljohann, C.A. Mirkin, and S.J. Lippard. 2009. Polyvalent Oligonucleotide Gold Nanoparticle Conjugates as Delivery Vehicles for Platinum(IV) Warheads. J Am Chem Soc. 131: 14652–14653. 189. Giljohann, D.A., D.S. Seferos, W.L. Daniel, M.D. Massich, P.C. Patel, et al. 2010. Gold Nanoparticles for Biology and Medicine. Angew. Chem. Int. Ed. 49: 3280– 3294. 190. Maye, M.M., M.T. Kumara, D. Nykypanchuk, W.B. Sherman, and O. Gang. 2010. Switching binary states of nanoparticle superlattices and dimer clusters by DNA strands. Nat Nano. 5: 116–120. 191. Tam, E., P. Podsiadlo, E. Shevchenko, D.F. Ogletree, M.-‐P. Delplancke-‐Ogletree, et al. 2010. Mechanical Properties of Face-‐Centered Cubic Supercrystals of Nanocrystals. Nano Lett. 10: 2363–2367. 192. Lee, O.-‐S., T.R. Prytkova, and G.C. Schatz. 2010. Using DNA to Link Gold Nanoparticles, Polymers, and Molecules: A Theoretical Perspective. J Phys Chem Lett. 1: 1781–1788. 193. Lee, O.-‐S., and G.C. Schatz. 2009. Molecular Dynamics Simulations of DNA-‐ functionalized Gold Nanoparticles. J Phys Chem C. : 2316–2321. 194. Lee, O.-‐S., and G.C. Schatz. 2009. Interaction between DNAs on a Gold Surface. J Phys Chem C. : 15941–15947. 195. Wang, J., R.M. Wolf, J.W. Caldwell, P.A. Kollman, and D.A. Case. 2005. Junmei Wang, Romain M. Wolf, James W. Caldwell, Peter A. Kollman, and David A. Case, “Development and testing of a general amber force field”Journal of Computational Chemistry(2004) 25(9) 1157–1174. J. Comput. Chem. 26: 114. 203 196. Hautman, J., and M.L. Klein. 1989. Simulation of a monolayer of alkyl thiol chains. J. Chem. Phys. 91: 4994–5001. 197. Price, D.J., and C.L. Brooks III. 2004. A modified TIP3P water potential for simulation with Ewald summation. J. Chem. Phys. 121: 10096–10103. 198. Agrawal, P.M., B.M. Rice, and D.L. Thompson. 2002. Predicting trends in rate parameters for self-‐diffusion on FCC metal surfaces. Surf. Sci. 515: 21–35. 199. Darden, T., D. York, and L. Pedersen. 1993. Particle mesh Ewald: An N [center-‐ dot] log(N) method for Ewald sums in large systems. J. Chem. Phys. 98: 10089–10092. 200. Kräutler, V., W.F. van Gunsteren, and P.H. Hünenberger. 2001. A fast SHAKE algorithm to solve distance constraint equations for small molecules in molecular dynamics simulations. J. Comput. Chem. 22: 501–508. 201. Powell, M.J.D. 1977. Restart procedures for the conjugate gradient method. Math. Program. 12: 241–254. 202. Hill, H.D., J.E. Millstone, M.J. Banholzer, and C.A. Mirkin. 2009. The Role Radius of Curvature Plays in Thiolated Oligonucleotide Loading on Gold Nanoparticles. ACS Nano. 3: 418–424. 203. Cederquist, K.B., and C.D. Keating. 2009. Curvature Effects in DNA:Au Nanoparticle Conjugates. ACS Nano. 3: 256–260. 204. If any symmetrical displacements due to translation and rotation are not removed, the RMSDs of DNAs in the supercrystals are about 11-‐12 Å. . 205. Hurst, S.J., A.K.R. Lytton-‐Jean, and C.A. Mirkin. 2006. Maximizing DNA Loading on a Range of Gold Nanoparticle Sizes. Anal Chem. 78: 8313–8318. 206. Cheatham, I., and P.A. Kollma. 1996. Observation of theA-‐DNA toB-‐DNA Transition During Unrestrained Molecular Dynamics in Aqueous Solution. J. Mol. Biol. 259: 434–444. 207. Knotts, T.A., IV, N. Rathore, D.C. Schwartz, and J.J. de Pablo. 2007. A coarse grain model for DNA. J. Chem. Phys. 126: 084901–12. 208. Halliday, D., R. Resnick, and J. Walker. 1997. Fundamental of Physics. Extended. Wiley. 209. Mashek, D.G., and R.A. Coleman. 2006. Cellular fatty acid uptake: the contribution of metabolism. Curr. Opin. Lipidol. 17: 274–278. 204 210. Bruckner, R.J., S.S. Mansy, A. Ricardo, L. Mahadevan, and J.W. Szostak. 2009. Flip-‐Flop-‐Induced Relaxation of Bending Energy: Implications for Membrane Remodeling. Biophys. J. 97: 3113–3122. 211. Hamilton, J.A. 1998. Fatty acid transport: difficult or easy? J. Lipid Res. 39: 467 –481. 212. Hamilton, J.A. 2003. Fast flip-‐flop of cholesterol and fatty acids in membranes: implications for membrane transport proteins. Curr. Opin. Lipidol. 14. 213. Hamilton, J., and K. Brunaldi. 2007. A Model for Fatty Acid Transport into the Brain. J. Mol. Neurosci. 33: 12–17. 214. Hamilton, J.A., and F. Kamp. 1999. How are free fatty acids transported in membranes? Is it by proteins or by free diffusion through the lipids? Diabetes. 48: 2255 –2269. 215. Kamp, F., and J.A. Hamilton. 2006. How fatty acids of different chain length enter and leave cells by free diffusion. Prostaglandins Leukot. Essent. Fatty Acids. 75: 149–159. 216. Kamp, F., D. Zakim, F. Zhang, N. Noy, and J.A. Hamilton. 1995. Fatty Acid Flip-‐ Flop in Phospholipid Bilayers Is Extremely Fast. Biochemistry (Mosc.). 34: 11928–11937. 217. Simard, J.R., B.K. Pillai, and J.A. Hamilton. 2008. Fatty Acid Flip-‐Flop in a Model Membrane Is Faster Than Desorption into the Aqueous Phase. Biochemistry (Mosc.). 47: 9081–9089. 218. Noy, N., and D. Zakim. 1985. Fatty acids bound to unilamellar lipid vesicles as substrates for microsomal acyl-‐CoA ligase. Biochemistry (Mosc.). 24: 3521– 3525. 219. Daniels, C., N. Noy, and D. Zakim. 1985. Rates of hydration of fatty acids bound to unilamellar vesicles of phosphatidylcholine or to albumin. Biochemistry (Mosc.). 24: 3286–3292. 220. Kampf, J.P., D. Cupp, and A.M. Kleinfeld. 2006. Different Mechanisms of Free Fatty Acid Flip-‐Flop and Dissociation Revealed by Temperature and Molecular Species Dependence of Transport across Lipid Vesicles. J. Biol. Chem. 281: 21566 –21574. 221. Hamilton, J., R. Johnson, B. Corkey, and F. Kamp. 2001. Fatty acid transport. J. Mol. Neurosci. 16: 99–108. 205 222. Hirsch, D., A. Stahl, and H.F. Lodish. 1998. A family of fatty acid transporters conserved from mycobacterium to man. Proc. Natl. Acad. Sci. 95: 8625 –8629. 223. Kleinfeld, A.M. 2000. Lipid Phase Fatty Acid Flip-‐Flop, Is It Fast Enough for Cellular Transport? J. Membr. Biol. 175: 79–86. 224. Huber, T., K. Rajamoorthi, V.F. Kurze, K. Beyer, and M.F. Brown. 2001. Structure of Docosahexaenoic Acid-‐Containing Phospholipid Bilayers as Studied by 2H NMR and Molecular Dynamics Simulations. J Am Chem Soc. 124: 298–309. 225. Notman, R., M.G. Noro, and J. Anwar. 2007. Interaction of Oleic Acid with Dipalmitoylphosphatidylcholine (DPPC) Bilayers Simulated by Molecular Dynamics. J Phys Chem B. 111: 12748–12755. 226. Gurtovenko, A.A., and I. Vattulainen. 2007. Molecular Mechanism for Lipid Flip-‐Flops. J Phys Chem B. 111: 13554–13559. 227. Kaneko, F., K. Yamazaki, K. Kitagawa, T. Kikyo, M. Kobayashi, et al. 1997. Structure and Crystallization Behavior of the β Phase of Oleic Acid. J Phys Chem B. 101: 1803–1809. 228. Suzuki, M., T. Ogaki, and K. Sato. 1985. Crystallization and transformation mechanisms of α, β-‐ and γ-‐polymorphs of ultra-‐pure oleic acid. J. Am. Oil Chem. Soc. 62: 1600–1604. 229. Pitman, M.C., F. Suits, A.D. MacKerell, and S.E. Feller. 2004. Molecular-‐Level Organization of Saturated and Polyunsaturated Fatty Acids in a Phosphatidylcholine Bilayer Containing Cholesterol†. Biochemistry (Mosc.). 43: 15318–15328. 230. Feller, S.E., K. Gawrisch, and A.D. MacKerell. 2001. Polyunsaturated Fatty Acids in Lipid Bilayers:   Intrinsic and Environmental Contributions to Their Unique Physical Properties. J Am Chem Soc. 124: 318–326. 231. Lindahl, E., and O. Edholm. 2000. Spatial and energetic-‐entropic decomposition of surface tension in lipid bilayers from molecular dynamics simulations. J. Chem. Phys. 113: 3882–3893. 232. Gullingsrud, J., and K. Schulten. 2004. Lipid Bilayer Pressure Profiles and Mechanosensitive Channel Gating. Biophys. J. 86: 3496–3509. 233. One can prepare a system of randomly distributed oleic acid molecules in water, then relax to see if the triple-‐structure can be obtained. For this system, however, the ratios of oleic acid to water molecules are unknown. Therefore, 206 one reliable set-‐up, as we used, is to start from a well-‐known crystalline structure. . 234. Budin, I., and J.W. Szostak. 2011. Physical effects underlying the transition from primitive to modern cell membranes. Proc. Natl. Acad. Sci. 108: 5249 – 5254. 235. We performed test simulations on the force-‐field parameters of oleic acid by computing elastic moduli of a small orthorhombic crystal in g-‐phase. The bulk modulus is 1.86 GPa. The Young’s moduli are Exx = 3.5 GPa, Eyy = 0.9 GPa, and Ezz = 0.9. Exx is the greatest because of the hydrocarbon chains align in the x-‐ direction. These values agree with those of polystyrene and polyethylene [D. Ciprari, K. Jacob, and R. Tannenbaum, Macromol. 39, 6565-‐6573 (2006); C. Klapperich, K. Komvopoulos, and L. Pruitt, J. Tribol. 123, 624-‐631 (2001)], and the experimentally measured elastic modulus of oleic acid functionalizing lead sulfide nanoparticles (1.7 GPa) [Tam, E.; Podsiadlo, P.; Shevchenko, E.; Ogletree, D. F.; Delplancke-‐Ogletree, M.-‐P.; Ashby, P. D. Nano Lett. 2010, 10, 2363−2367]. . 236. Klauda, J.B., R.M. Venable, J.A. Freites, J.W. O’Connor, D.J. Tobias, et al. 2010. Update of the CHARMM All-‐Atom Additive Force Field for Lipids: Validation on Six Lipid Types. J Phys Chem B. 114: 7830–7843. 237. Marrink, S.-‐J., and H.J.C. Berendsen. 1994. Simulation of water transport through a lipid membrane. J Phys Chem. 98: 4155–4168. 238. Almeida, P.F.F., W.L.C. Vaz, and T.E. Thompson. 1992. Lateral diffusion in the liquid phases of dimyristoylphosphatidylcholine/cholesterol lipid bilayers: a free volume analysis. Biochemistry (Mosc.). 31: 6739–6747. 239. Bearman, R.J., and D.L. Jolly. 1981. Mass dependence of the self diffusion coefficients in two equimolar binary liquid Lennard-‐Jones systems determined through molecular dynamics simulation. Mol. Phys. 44: 665–675. 240. Vanegas, J.M., M.L. Longo, and R. Faller. 2011. Crystalline, Ordered and Disordered Lipid Membranes: Convergence of Stress Profiles due to Ergosterol. J Am Chem Soc. 133: 3720–3723. 241. Baoukina, S., L. Monticelli, S.J. Marrink, and D.P. Tieleman. 2007. Pressure−Area Isotherm of a Lipid Monolayer from Molecular Dynamics Simulations. Langmuir. 23: 12617–12623. 207 242. Crane, J.M., G. Putz, and S.B. Hall. 1999. Persistence of Phase Coexistence in Disaturated Phosphatidylcholine Monolayers at High Surface Pressures. Biophys. J. 77: 3134–3143. 243. Borgnia, M.J., and P. Agre. 2001. Reconstitution and functional comparison of purified GlpF and AqpZ, the glycerol and water channels from Escherichia coli. Proc. Natl. Acad. Sci. 98: 2888 –2893. 244. Zeidel, M.L., S.V. Ambudkar, B.L. Smith, and P. Agre. 1992. Reconstitution of functional water channels in liposomes containing purified red cell CHIP28 protein. Biochemistry (Mosc.). 31: 7436–7440. 245. De Groot, B.L., and H. Grubmüller. 2001. Water Permeation Across Biological Membranes: Mechanism and Dynamics of Aquaporin-‐1 and GlpF. Science. 294: 2353 –2357. 246. Gurtovenko, A.A., J. Anwar, and I. Vattulainen. 2010. Defect-‐Mediated Trafficking across Cell Membranes: Insights from in Silico Modeling. Chem Rev. 110: 6077–6103. 247. Abrahamsson, S., and I. Ryderstedt-‐Nahringbauer. 1962. The crystal structure of the low-‐melting form of oleic acid. Acta Crystallogr. 15: 1261–1268. 248. Chen, S., M.T. Seidel, and A.H. Zewail. 2005. Atomic-‐scale dynamical structures of fatty acid bilayers observed by ultrafast electron crystallography. Proc. Natl. Acad. Sci. U. S. A. 102: 8854–8859. 208 Appendices A Single Pulling Step Here we derive Eqs. (II.5) and (II.6) using canonical ensembles of x 0 and x 1 . Averaging of exp(–βW H ) (β = 1/k B T) over x 1 ’s ensemble is computed as e −βW H x 1 ,λ 1 ,k ( ) = d  r 3N−1 dx 1 e −β W H +H o (  r 3N−1 ,x 1 )+ k 2 (x 1 −λ 1 ) 2 " # $ % & ' ∫ d  r 3N−1 dx 1 e −β H o (  r 3N−1 ,x 1 )+ k 2 (x 1 −λ 1 ) 2 " # $ % & ' ∫ = d  r 3N−1 dx 1 e −β k 2 (x 0 −λ 1 ) 2 +H o (  r 3N−1 ,x 1 ) " # $ % & ' ∫ Z(λ 1 ,k) =e −β k 2 (x 0 −λ 1 ) 2 d  r 3N−1 dx 1 e −βH o (  r 3N−1 ,x 1 ) ∫ Z(λ 1 ,k) , (A.1) which is equivalent to exp −β k 2 (x 0 −λ 1 ) 2 −(x 1 −λ 1 ) 2 " # $ % & ' ( ) * + x 1 ,λ 1 ,k ( ) =exp −β k 2 (x 0 −λ 1 ) 2 " # , $ % - exp βΔF(λ 1 ,k) [ ] . (A.2) Since a single value of x 0 at time t 0 is constant in the average, we cancel both sides the factors related to x 0 . Thus, we arrive at Eq. (II.5): exp β k 2 (x 1 −λ 1 ) 2 " # $ % & ' (x 1 ,λ 1 ,k) =exp βΔF(λ 1 ,k) [ ] . (A.3) A similar procedure can be carried out to derive Eq. (II.6): e −βW H x 0 ,k=0 ( ) = d  r 3N−1 dx 0 e −β W H +H o (  r 3N−1 ,x 0 ) " # $ % ∫ d  r 3N−1 dx 0 e −βH o (  r 3N−1 ,x 0 ) ∫ 209 =e β k 2 (x 1 −λ 1 ) 2 d  r 3N−1 dx 0 e −β k 2 (x 0 −λ 1 ) 2 +H o (  r 3N−1 ,x 0 ) " # $ % & ' ∫ Z(0) , (A.4) which is equivalent to ΔF(λ 1 ,k)=−β −1 ln exp[−β k 2 (x 0 −λ 1 ) 2 ] (x 0 ,k=0) . (A.5) 210 B Series Of Pulling Steps The total work absorbed by the system in a series of pulling steps is given by W total = ∂H(λ 1 ,λ 2 ...λ s ,x) ∂t dt t 0 t s ∫ = ∂ ∂t [H o (  r 3N−1 ,x) t 0 t s ∫ + k 2 (x−λ i ) 2 θ(t−t i−1 ) i=1 s ∑ θ(t i −t)]dt = k 2 (x 0 −λ 1 ) 2 −(x s −λ s ) 2 " # $ % + k 2 (λ i+1 − x i ) 2 −(x i −λ i ) 2 " # $ % i=1 s−1 ∑ = k 2 (x 0 −λ 1 ) 2 −(x s −λ s ) 2 " # $ % + H(λ i+1 ,x i )−H(λ i ,x i ) [ ] i=1 s−1 ∑ . (B.1) Since all x i ’s ensembles are canonical and independent, the average of exp(–βW total ) is computed as e −βW total (x 0 ,x 1 ,...,x s ) = d  r 3N−1 dx 0 e −βH o (  r 3N−1 ,x 0 ) e −β k 2 (x 0 −λ 1 ) 2 ∫ Z(0) × d  r 3N−1 dx i ∫ i=1 s−1 ∏ e −β H o (  r 3N−1 ,x i )+ k 2 (x i −λ i ) 2 % & ' ( ) * e −β k 2 (x i −λ i+1 ) 2 −(x i −λ i ) 2 % & ( ) Z(λ 1 ,k)...Z(λ s−1 ,k) × d  r 3N−1 dx s e −β H o (  r 3N−1 ,x s )+ k 2 (x s −λ s ) 2 # $ % & ' ( e +β k 2 (x s −λ s ) 2 ∫ Z(λ s ,k) =1. (B.2) We first take the average of exp(–βW total ) over x 0 and x 1 to obtain e −βW total x 0 ,x s = d  r 3N−1 dx 0 e −β H o (  r 3N−1 ,x 0 )+ k 2 (x 0 −λ 1 ) 2 " # $ % & ' ∫ Z(0) × d  r 3N−1 dx s e −βH o (  r 3N−1 ,x 0 ) ∫ Z(λ s ,k) e −βW mech 211 =e β F(λ s ,k)−F(λ 1 ,k) [ ] e −βW mech . (B.3) It is noted that based on Eq. (B.1) all possible values of W mech can be constructed from either force-versus-extension (FVE) curves or from the ensembles of x i with i from 1 to s–1. Therefore, averaging the right hand side of Eq. (B3) over all possible FVE curves is equal to averaging the left hand side over the rest of x i . Taking this observation into account and using the identity (B.2), we arrive at a formula identical to JE: ΔF JE (λ s ,λ 1 ,k)=−β −1 ln e −βW mech FVE . (B.4) Furthermore, x i ’s distributions can be approximated as exp[–βk(x i –〈x i 〉) 2 /(2γ i 2 )], where 〈x i 〉 is the averaged position of the reaction coordinate at i th pulling step and γ i 2 is equal to kσ i 2 /k B T with σ i equal to the standard deviation of x i ’s distribution. If all increments λ i+1 – λ i are the same as Δλ = (λ s –λ 1 )/s, we derive an analytical expression for the right hand side of Eq. (B.4): exp[−βW mech ] FVE = dW mech ρ( ∫ W mech )e −βW mech dW mech ρ( ∫ W mech ) (B.5) ≅ dx i exp −βk (x i − x i ) 2 2γ i 2 + Δλ(2λ i +Δλ−2x i ) 2 $ % & & ' ( ) ) * + , - , . / , 0 , ∫ dx i exp[−βk (x i − x i ) 2 /2γ i 2 ( ) ] ∫ i=1 s−1 ∏ = exp − βkΔλ 2 2 [(1−γ i 2 )+2 λ i − x i Δλ ] # $ % & % ' ( % ) % i=1 s−1 ∏ . Then the Gaussian-approximated free energy difference (FED) is given by 212 ΔF G (λ s ,λ 1 ,k)= kΔλ 2 2 (1−γ i 2 ) i=1 s−1 ∑ +kΔλ (λ i − x i ) i=1 s−1 ∑ . (B.6) 213 C Algorithm for Work Distribution Construction Given λ 1 , λ s , Δλ = (λ s –λ 1 )/s and a data set of x i we divide a sufficiently large interval, which covers all values of x i , into K bins with a bin width δx. The distributions ρ i (x i ) of x i are constructed by counting probability of x i falling into each bin. Similarly, we estimate a range having a bin width δW and 2M+1 bins that all mechanical work W mech (W 1 , W 2 … W s–1 ) can fall into. For i = 1, W 1 is zero according to Eq. (II.9). For i = 2, we construct the distribution of W 2 , Ω 2 (W 2 ), as in the following pseudo-code: for (j = 1; j ≤ K; j++) if (ρ 1 (j) ≠ 0) // ρ 1 (x 1 ) are non-zero in small regions around λ 1 . W 2 = kΔλ(2λ 1 – 2jδx + Δλ)/2; w = [W 2 /δW]; // a floor function transforms a real number into an integer. Ω 2 (w) = ρ 1 (j); // all Ω i>2 are initialized to be zero. Endif Endfor For i > 2, the working distributions of these pulling steps are accumulated as the following: for (i = 3; i ≤ s – 1; i++) for (j = 1; j ≤ K; j++) if (ρ i–1 (j) ≠ 0) for (w 1 = –M ; w 1 ≤ M; w 1 ++) 214 if(Ω i–1 (w 1 ) ≠ 0) W i = w 1 ×δW + kΔλ(2λ i–1 – 2jδx + Δλ)/2; w 2 = [W i /δW]; Ω i (w 2 ) += ρ i–1 (j)×Ω i–1 (w 1 ); endif endfor endif endfor endfor We observe that the floor function produces an unwanted spike in work distributions at W i = 0.0. We smooth work distributions at this value by assigning Ω i (0) = [Ω i (–1)+Ω i (1)]/2. From the distributions of work Ω i , it is straightforward to compute FEDs based on Eq. (II.12) or Eq. (B.5). The error analysis of these numerical calculations can be found elsewhere [18, 38]. The variance of FEDs can be estimated by € σ W 2 /Q +β 2 σ W 4 /2(Q−1), where σ W is the standard deviation of work distributions Ω i (W i ) and Q is a number of bins which have non-zero Ω i (W i ). For example, at pulling step i = 35, σ W is about 10 kcal/mol with Q ~ 20000, then the variance is about 0.7 kcal/mol at temperature T = 300 K. Thus the standard deviation of the corresponding FED is about 0.8 kcal/mol. This deviation is about two times smaller than half the difference (~ 2.0 kcal/mol) between ΔF JE (λ s ,λ 1 ,k) and ΔF fluct (λ s ,λ 1 ,k) in Sections IV B and D. Larger Q value (larger K and M) does not make the difference smaller, even though it reduces the 215 deviation. Therefore, we choose [ΔF JE (λ s ,λ 1 ,k) – ΔF fluct (λ s ,λ 1 ,k)]/2 as the major uncertainty of our method. We find that δx = 0.001 Å and δW = 0.01 kcal/mol give reasonable estimates of work distributions and their FEDs investigated in the test case. 216 D The Relation Between ˆ O total And ˆ W Here, we consider the relation between ˆ O total and ˆ W in Eqs. (III.3). Let’s examine the relation in classical systems (60), where the non-operator Hamiltonian is similar to Eq. (III.2). Then, the expression of O total (expectation value) is computed as O total = ∂H(p 3N ,r 3N−1 ,x;λ 1 ,λ 2 ...λ s ) ∂t t 0 t s ∫ dt =U(x 0 ,λ 1 )−U(x s ,λ s )+ [U(x i ,λ i+1 )−U(x i ,λ i )] i=1 s−1 ∑ =U(x 0 ,λ 1 )−U(x s ,λ s )+W, (D.1) where U(x i ,λ i+1 )= U(x,λ i+1 )[∂θ(t−t i )/∂t ∫ ]dt, U(x i ,λ i )=− U(x,λ i )[∂θ(t i −t)/∂t ∫ ]dt, (D.2) W is given by Eq. (III.3c), and x i is a value of the reaction coordinate x at time t i . The relation between O total and W is as clear as in Eq. (D.1). A value of O total indicates absorption energy along a pathway x 0 → x 1 …→ x s and W is mechanical work performed by the applied potential. Since time t i arbitrarily begins from t i–1 , a thermal fluctuation distribution of x i during relaxation time t i – t i–1 exists and plays the same role as the single value of x i at time t i . Thus, the idea of using all possible values of x i during relaxation time (instead of one value at time t i ) is consistent with the ergodic hypothesis of thermodynamics, which implies the equivalence between the averages over time (t i ) and over phase space (x i ) represented by Hamiltonian H i (p 3N ,r 3N−1 ,x;λ i ). Based on this idea, one can construct work distribution functions from the thermal fluctuation distributions of x i (60). 217 However, in quantum mechanics one cannot pass time t i to operator ˆ x as to expectation value x in Eqs. (D.2). As a result, operator ˆ O total is identical to zero and does not have any physical meaning. Strictly speaking, mechanical work operator ˆ W does not indicate useful physical meanings either, but its expectation values defined by Eq. (III.3c). Thus, the relation should be understood in terms of expectation values of ˆ O total and ˆ W over spatial domains along a pathway. 218 E The Recursion Relation To derive Eq. (III.15), we start from Eq. (III.9). First, let W i be [U(x j ,λ j+1 )− j=1 j=i ∑ U(x j ,λ j )] for 1 ≤ i ≤ s – 1, and ρ i+1 (W) be the work distribution of W i . Then we have a simple relation between W i and W i–1 : W i =W i−1 + kΔλ 2 (2λ i +Δλ−2x i ), (E.1) where Δλ = λ s /(s – 1), W 0 = 0, and ρ 1 (W) ≡ 1. As a result of Eq. (III.9) for i = 1, the un- normalized work distribution function ρ 2 (W) is equal to f 1 [λ 1 +Δλ/2 – W/kΔλ]. For i = 2, the un-normalized work distribution function ρ 3 (W) is ρ 3 (W)= dx 1 dx 2 f 1 (x 1 )f 2 (x 2 ) −∞ ∞ ∫ −∞ ∞ ∫ ×δ[W−W 1 − kΔλ 2 (2λ 2 +Δλ−2x 2 )]. (E.2) By changing the variable x 1 = λ 1 + Δλ/2 – W 1 /kΔλ, Eq. (E.2) becomes ρ 3 (W)= dW 1 d(x 2 /kΔλ)ρ 2 (W 1 )f 2 (x 2 ) −∞ ∞ ∫ −∞ ∞ ∫ ×δ[W−W 1 − kΔλ 2 (2λ 2 +Δλ−2x 2 )]. (E.3) Integrating Eq. (E.3) over x 2 , and changing the variable W 1 to w, we obtain ρ 3 (W)=(1/kΔλ) dwρ 2 (w)f 2 (λ 2 + Δλ 2 − W−w kΔλ ) −∞ ∞ ∫ . (E.4) For i > 2, one can easily verify ρ i (W)= dx i−1 −∞ ∞ ∫ × dw −∞ ∞ ∫ {[ dx 1 −∞ ∞ ∫ ... dx i−2 −∞ ∞ ∫ f 1 (x 1 )...f i−2 (x i−2 )δ(w−W i−1 )] ×f i−1 (x i−1 )δ[W−w− kΔλ 2 (2λ i−1 +Δλ−2x i−1 )]} 219 = dx i−1 dw −∞ ∞ ∫ −∞ ∞ ∫ ρ i−1 (w)f i−1 (x i−1 )×δ[W−w− kΔλ 2 (2λ i−1 +Δλ−2x i−1 )] = (1/kΔλ) dwρ i−1 (w)f i−1 (λ i−1 + Δλ 2 − W−w kΔλ ) −∞ ∞ ∫ , (E.5) which is Eq. (III.15), and the normalization factor Q i is computed as Q i =1/ dWρ i (W) −∞ ∞ ∫ . 220 F Contribution of Ground States to Free-Energy Changes Here, we analytically carry out the integrations in Eq. (III.7a) for one-dimensional harmonic oscillator with n = 0, which can be written as e −βΔF(λ 1 ,λ s ) = dx i ∫ ψ 0 (x i ,λ i ) 2 exp[−β( ω 2 + kλ i 2 4 )]exp[−βδW i ) i=1 s−1 ∏ dx i ∫ ψ 0 (x i ,λ i ) 2 exp[−β( ω 2 + kλ i 2 4 )] i=1 s−1 ∏ , (F.1) where δW i =k Δλ 2 (2λ i +Δλ−2x i ), β is 1/k B T, ω = [2k/m] 1/2 , ψ 0 (x,λ)= mω /π 4 exp[−mω(x−λ /2) 2 /2], λ i is (i – 1)Δλ, and Δλ = λ s /(s – 1). Note that the factors exp[−β( ω 2 + kλ i 2 4 )] cancel out in Eq. (F.1), and the integrations in the denominator are equal to unity. By converting x i , λ i , and Δλ into dimensionless variables (in units of  /mω ), we simplify Eq. (F.1) to e −βΔF(λ 1 ,λ s ) = dx i ∫ exp[−(x i − λ i 2 ) 2 − 2x i aΔλ 2 ] i=1 s−1 ∏ ×exp[− aΔλ 2 (2λ i +Δλ)] = dx i ∫ exp[− x i − λ i +aΔλ 2 $ % & ' ( ) 2 ] i=1 s−1 ∏ ×exp[− aΔλ 2 (2λ i +Δλ)+ aΔλ 2 $ % & ' ( ) 2 + aΔλλ i 2 ] =exp − aΔλ 2 (s−2)(s−1) 4 +(s−1) a 2 Δλ 2 4 − aΔλ 2 2 # $ % % & ' ( ( ) * + + , - . . =exp − aΔλ 2 (s−1)(s−a) 4 ) * + + , - . . , (F.2) where a=ω /2k B T. As a = 1, we obtain ΔF(λ 1 ,λ s ) = k(s – 1) 2 Δλ 2 /4, which is kλ s 2 /4 = ΔF Target .

Abstract (if available)

Abstract The fundamental laws of thermodynamics and statistical mechanics, and the deeper understandings of quantum mechanics have been rebuilt in recent years. It is partly because of the increasing power of computing resources nowadays, that allow shedding direct insights into the connections among the thermodynamics laws, statistical nature of our world, and the concepts of quantum mechanics, which have not yet been understood. But mostly, the most important reason, also the ultimate goal, is to understand the mechanisms, statistics and dynamics of biological systems, whose prevailing non-equilibrium processes violate the fundamental laws of thermodynamics, deviate from statistical mechanics, and finally complicate quantum effects. ❧ I believe that investigations of the fundamental laws of non-equilibrium dynamics will be a frontier research for at least several more decades. One of the fundamental laws was first discovered in 1997 by Jarzynski, so-called Jarzynski’s Equality. Since then, different proofs, alternative descriptions of Jarzynski’s Equality, and its further developments and applications have been quickly accumulated. My understandings, developments and applications of an alternative theory on Jarzynski’s Equality form the bulk of this dissertation. The core of my theory is based on stepwise pulling protocols, which provide deeper insight into how fluctuations of reaction coordinates contribute to free‐energy changes along a reaction pathway. We find that the most optimal pathways, having the largest contribution to free‐energy changes, follow the principle of detailed balance. This is a glimpse of why the principle of detailed balance appears so powerful for sampling the most probable statistics of events. In a further development on Jarzynski’s Equality, I have been trying to use it in the formalism of diagonal entropy to propose a way to extract useful thermodynamic quantities such temperature, work and free-energy profiles from far‐from‐equilibrium ensembles, which can be used to characterize non-equilibrium dynamics. ❧ Furthermore, we have been applied the stepwise pulling protocols and Jarzynski’s Equality to investigate the ion selectivity of potassium channels via molecular dynamics simulations. The mechanism of the potassium ion selectivity has remained poorly understood for over fifty years, although a Nobel Prize was awarded to the discovery of the molecular structure of a potassium‐selective channel in 2003. In one year of performing simulations, we were able to reproduce the major results of ion selectivity accumulated in fifty years. We have been even boldly going further to propose a new model for ion selectivity based on the structural rearrangement of the selectivity filter of potassium-selective KcsA channels. This structural rearrangement has never been shown to play such a pivotal role in selecting and conducting potassium ions, but effectively rejecting sodium ions. Using the stepwise pulling protocols, we are also able to estimate conductance for ion channels, which remains elusive by using other methods. In the light of ion channels, we have also investigated how a synthetic channel of telemeric G‐quadruplex conducts different types of ions. These two studies on ion selectivity not only constitute an interesting part of this dissertation, but also will enable us to further explore a new set of ion‐selectivity principles. ❧ Beside the focus of my dissertation, I used million-atom molecular dynamics simulations to investigate the mechanical properties of body-centered-cubic (BCCS) and face‐centered‐cubic (FCCS) supercrystals of DNA-functionalized gold nanoparticles. These properties are valuable for examining whether these supercrystals can be used in gene delivery and gene therapy. The formation of such ordered supercrystals is useful to protect DNAs or RNAs from being attacked and destroyed by enzymes in cells. I also performed all‐atom molecular dynamics simulations to study a pure oleic acid (OA) membrane in water that results into a triple‐layer structure. The simulations show that the trans‐membrane movement of water and OAs is cooperative and correlated, and agrees with experimentally measured absorption rates. The simulation results support the idea that OA flip‐flop is more favorable than transport by means of functional proteins. This study might provide further insight into how primitive cell membranes work, and how the interplay and correlation between water and fatty acids may occur.

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Asset Metadata

Creator Van, Ngo Anh (author)

Core Title Step‐wise pulling protocols for non-equilibrium dynamics

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 01/29/2014

Defense Date 10/18/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag DNA,fatty acid,free‐energy,G-quadruplex,ion channels,ions,lipid,non-equilibrium,OAI-PMH Harvest,protein,supercrystals,temperature,thermodynamics,Work

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haas, Stephan W. (committee chair)

Creator Email huongvan0401@gmail.com,nvan@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-361458

Unique identifier UC11295325

Identifier etd-VanNgoAnh-2237.pdf (filename),usctheses-c3-361458 (legacy record id)

Legacy Identifier etd-VanNgoAnh-2237.pdf

Dmrecord 361458

Document Type Dissertation

Format application/pdf (imt)

Rights Van, Ngo Anh

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA