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Genetic analysis of differentiation of T-helper lymphocytes

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Genetic analysis of differentiation of T-helper lymphocytes

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Content 1

Genetic analysis of differentiation of T-helper lymphocytes

by

Qixin Wang

________________________________________________________________________

A Thesis Presented to
FACULTY OF THE USC GRADUATE SCHOOL
In Partial Fulfillment of the
Requirement for the Degree
Master of Science
(Statistics)

May 2013

2
Acknowledgements

I would like to express my gratitude to all those who helped me during the writing of
this thesis. My deepest gratitude goes first and foremost to Professor Stanley Azen. I am
deeply grateful of his help in the completion of this thesis.
Second, I would like to express my heartfelt gratitude to Professor Michael
Waterman, for his instructive advice and useful suggestions on my research.
Last my thanks would go to my beloved family for their loving considerations and
great confidence in me all through these years. I also owe my sincere gratitude to my
friends and my fellow classmates who gave me their help and time in listening to me and
helping me work out my problems during the difficult course of the thesis.

3
Table of Contents

Acknowledgements 2

Abstract 4

Chapter 1 Literature review 5

Chapter 2 Materials and Methods 8
2.1 Model Building………………………………………………………………….....8
2.1.1 The differentiation of Th1……………………………………………………9
2.1.2 The differentiation of Th2………………………………………………......12
2.2 Model Analysis……………………………………………………………………16

Chapter 3 Results and Discussion 19
3.1 Th1/Th2 differentiation process…………………………………………………..19
3.2 Th1/Th2 re-differentiation process………………………………………………..21
3.3 Th1/Th2 cell without potential of differentiation………………………………....22
3.4 Phase Trajectory………………………………………………………………......24

Chapter 4 Conclusion 26

References 27

4
Abstract

In the human immune system, T-helper cells are able to differentiate into two
lymphocyte subsets: Th1 and Th2. The intracellular signaling pathways of differentiation
form a dynamic regulation network by secreting distinctive types of cytokines, while
differentiation is regulated by two major gene loci: T-bet and GATA-3. We developed a
system dynamics model to simulate the differentiation and re-differentiation process of
T-helper cells, based on gene expression levels of T-bet and GATA-3 during
differentiation of these cells. We arrived at three ultimate states of the model and came to
the conclusion that cell differentiation potential exists as long as the system dynamics is
at an unstable equilibrium point; the T-helper cells will no longer have the potential of
differentiation when the model reaches a stable equilibrium point. In addition, the time
lag caused by expression of transcription factors can lead to oscillations in the secretion
of cytokines during differentiation.

5
Chapter 1 Literature review

CD4 T helper cells play a crucial role in immune reactions and are ubiquitously
known as a significant component of the human immune system. Human T helper cells
are further categorized into two subpopulations: Th1 cells and Th2 cells (Romagnani et
al., 1992). The signaling transduction network that regulates Th1-Th2 differentiation is a
typical dynamic network system.
In their process of differentiation, Th1 cells secrete cytokines such as TNF-beta,
IL-12 and IL-18, as well as IFN-gamma (Viola et al., 1999; Glimcher et al., 2000;
Nakanishi et al., 2001; Gately et al.,1998), while Th2 cells produce IL-4 (massively),
IL-5, IL-6, IL-9 and IL-10 (Romagnani et al., 2000; O’Garra et al., 2000). Through the
secretion of various cytokines, they function correctly as helpers in immune responses of
distinct types.
With ongoing breakthroughs in research on T cells, the mechanism of Th1 and Th2
differentiation has unfolded gradually. CD4+ cells, when remaining undifferentiated, only
secrete a tiny amount of cytokines and they are labeled Th0 cells or naive T helper
precursor cells (Lohning et al., 2002; Murphy et al., 2002; Agnello et al., 2003; Wang et
al., 2007). Of the two differentiation paths, they will finally choose one, and their choice
is decided on the external environment (various cytokines), mainly by IFN-gamma and
IL-4 (Mullen et al., 2002; Szabo et al., 2003; O’Garra et al., 2000).
External IFN-gamma can induce Th0 to differentiate into Th1. First, IFN-gamma
binds to its receptors on the outer surface of the cell membrane, which activates the signal
molecule STAT-1 and triggers its nuclear translocation (Lighvani et al., 2001; Szabo et al.,
2000). Afterwards, STAT-1 activates the transcription factor T-bet, which plays a
dominant role in regulating the differentiation of helper T cells (Chakir et al., 2003).
When T-bet binds to the gene that can be transcribed and translated into IFN-gamma,
histone acetylation and DNA methylation are catalyzed and the gene is modified, hence
6
the alteration of its accessibility, which leads to chromatin remodeling (Lovett-Racke et
al., 2004; Robinson et al., 2002; Mariani et al., 2004; Wang et al., 2007). Generally, the
IFN-gamma gene undergoes morphological changes so that it can be transcribed more
easily. As the production of IFN-gamma is augmented, part of it is transported out of the
cell and once again activates the IFN-gamma/T-bet pathway to establish a positive
feedback loop (Ho et al., 2002; Ouyang et al., 2000; Wang et al., 2008). Moreover, T-bet
can also influence GATA-3’s function in inhibiting the transcription of IL-4 (Szabo et al.,
2000).
Besides, IL-12, mainly secreted by dendritic cells, monocytes and macrophages, is
another crucial cytokine that induces Th1 differentiation. In the early differentiation stage,
IL-12 activates STAT4 via the receptor complex on the cell membrane (Gately et al.,
1998). STAT4 can simultaneously affect the IFN-gamma and T-bet genes (Lovett-Racke
et al., 2004; Robinson et al., 2002), leading to more combination of IFN-gamma and
T-bet, which boosts the translation of IFN-gamma and its production. Following a certain
period of time after the onset of differentiation, differentiation can proceed smoothly even
without IL-12, because the necessity of its promoting effect decreases with time (Lohning
et al., 2002; R Mariani et al., 2004). Note that IL-12 is not an essential cytokine: when the
IL-12 gene of T helper cells is knocked out, these cells can still secrete IFN-gamma, but
at much lower levels than normal (Jankovic et al., 2002).
External IL-4 is able to induce Th0 to differentiate into Th2 (Lametschwandtner et
al., 2004; Ranganath et al., 2001). Initially, IL-4 binds to its receptors on the outer surface
of cell membrane and activates STAT-6 and finally GATA-3 (Lametschwandtner et al.,
2004; Ranganath et al., 2001; Zhou et al., 2001; Smits et al., 2001). Activated GATA-3
then changes the accessibility of IL-4 and thereby stimulates the transcription of IL-4,
forming a positive feedback loop. GATA-3 can also inhibit the function of T-bet in the
same way as T-bet does (Ranganath et al., 2001; Zhou et al., 2001; Smits et al., 2001).
The process of differentiation can be divided into two stages: polarization and
7
differentiation. Polarization is defined as the structural transformation of Th0 towards
either subpopulation of Th1 and Th2 prior to the completion of differentiation. The
structural change of Th0 at this stage is reversible, which means the cell is still able to
switch from Th1 to Th2 or backwards, if its culture environment is properly altered.
However, a Th2 cell with thorough differentiation cannot possibly transform into a Th1
cell, even when a large quantity of IFN-gamma is added to the culture environment or a
high level of T-bet is expressed in cells. Nevertheless, the latter method does generate a
special type of cell that produces both IL-4 and IFN-gamma, which is presumably due to
the subtle change where the IL-4 gene with chromatin remodeling is no longer inhibited
by T-bet (Smits et al., 2001; Zhou et al., 2003; Yates et al., 2004).
Th1 related factor
Th2 related factor
Th1 related gene
Th2 related gene
IFN-r gene
IL-4 gene
STAT-1
STAT-6
IFN-r
receptor
IL-4
receptor
IFN-r
IL-4
IL-4
IFN-r
promote
inhibit
STAT-4
IL-12
IL-12
receptor
T-bet gene
GATA-3 gene
GATA-3
T-bet
Figure 1: The differentiation-related intracellular signaling pathway for Th1 and Th2
cells.

8
Chapter 2 Materials and Methods

2.1 Model Building
System dynamic models have a wide range of applications in the field of biological
regulatory networks. We developed a lag-system dynamical model to simulate the
differentiation process of Th0 cell to Th1/Th2 cells.

Table 1. Symbols and definition of Th1-Th2 differentiation model
Symbols Definition

Production rate coefficient of IL-4

Production rate coefficient of IFN-gamma

binding capacity of T-bet and the IFN-gamma gene

maximum production rate of IFN-gamma

Binding ability coefficient of T-bet to genes with the presence of GATA-3

Binding ability coefficient of GATA-3 to genes without the presence of T-bet

Binding ability coefficient of T-bet to genes without the presence of GATA-3

Binding capacity coefficient of GATA-3 to the IL-4 gene

maximum production rate of IL-4

Binding ability coefficient of GATA-3 to IL-4 gene with presence of T-bet

Binding ability coefficient of T-bet to IL-4 gene without the presence of
GATA-3

Non-negligible time delay that IL-4 needs to stimulate GATA-3

Non-negligible time delay that IFN-gamma needs to stimulate T-bet

Delay operator, which mean concentration of GATA-3 is actually determined
by the concentration of IL-4 time
1
 ago

Delay operator, which mean the concentration of T-bet is actually determined
by the concentration of IFN-gamma time
2
 ago

Concentration of IL-12 at early differentiation stage
9

Length of time that G(i) remains approximately the same value

Length of time that T(i) remains approximately the same value

2.1.1 The differentiation of Th1
Firstly, if we define the concentration of T bet  as () Ti at the timei , the production
rate of IFN   can be modeled as

1
()
1
()
()
v
IFN i produce
m
k T i
v
k T i
 



(1)
Where
1 v
k and
1m
k are meaningful coefficients here. Eq. (1) apparently shows that
() IFN i produce
v
 
increases non-linearly as the concentration of T bet  augments, and have
1
()
1
()
v
IFN i produce
m
k T i
v
k
 
 when ( ) 0 Ti  , which means
() IFN i produce
v
 
is almost directly
proportional to () Ti , when the concentration of T bet  is extremely low.
1m
k denotes
the binding capacity of T bet  and the IFN   gene (the smaller
1m
k is, the stronger
the binding capacity is, and vice versa). We get
( ) 1 IFN i produce v
vk
 
 , when () Ti  . In
the condition that the concentration of T bet  is elevated to an exceptionally high level,
() IFN i produce
v
 
approaches to a fixed value. In another word,
1 v
k represents the production
rate of IFN   , when the IFN   gene is being transcribed continuously. Due to the fact
that the transduction pathway is comparatively long, the concentration of T bet  is
actually determined by the concentration of IFN   time
2
 ago. Here
2
 is in fact the
non-negligible period of time that IFN   needs to stimulate T bet  . Hence the
production rate of T bet  is expressed by:
10

2
2
()
()
()
1
IF IFN i produce
T i produce
IF
IFN i produce
IFT
kv
v
k
v
k










(2)
Further analysis of Eq.(2) reveals the following information:
when
()
0
IFN i produce
v
 
 and
( ) ( ) IFN i produce T i produce
vv
 
,
() T i produce
v approaches to
2
() IF IF i produce
kv




. That is to say, given a low production rate of IFN   , the production
rate of T bet  is to a very large extent decided by
() IF i produce
v
 
and
IF
k , which are the
production rate of IFN   and the amplification factor of the signal cascade,
respectively.
When
( ) 1 IFN i produce v
vk
 
 and
( ) ( ) IFN i produce T i produce
vv
 
 , we have
1
()
1
IFT IF v
T i produce
IFT IF v
k k k
v
k k k


,
which means that the production rate of T bet  approaches to a fixed value
1
1
IFT IF v
IFT IF v
k k k
k k k 
, when
() IFN i produce
v
 
is close to
1 v
k and at the same time not far from
() G i produce
v .
When
( ) 1 IFN i produce v
vk
 
 and
( ) ( ) IFN i produce T i produce
vv
 
, we will expect to have
() T i produce IFT
vk  . When the production rate of IFN   is much higher than that of
T bet  , the T bet  production rate will approach to, but is always lower than
IFT
k ,
due to the speed limitation of T bet  transcription.
If we substitute Eq.(1) into Eq.(2), we will arrive at:
2
2
2
2
1
1
()
1
1
()
()
(3)
()
1
()
v
IF
m
T i produce
v IF
IFT m
k T i
k
k T i
v
k T i k
k k T i













or its simplified form:
11

2
2
1
()
11
()
( ) ( )
IF v IFT
T i produce
m IFT IFT IF v
k k k T i
v
k k k k k T i






(4)
Since 3 GATA  can affect the binding ability of T bet  to the IFN   gene, the
binding rate of T bet  and IFN   is given by the following formulas:
()
()
()
vT
T i bind
mT
k T i
v
k T i


(5)

12
()
(1 )(1 )
c IL B
mT Tm
Gm
kC G i I
kk
ki

  
,
1,
0,
B
iB
I
iB









1
1
( ) ( )
B
i
in
B
B i T i T i
n












(6)
where
mT
k represents the binding ability of T bet  to genes with the presence of
3 GATA  ;
Gm
k represents the binding ability of 3 GATA  to genes without the presence
of T bet  ;
Tm
k represents the binding ability of T bet  to genes without the presence of
3 GATA  ;
12 IL
C

is the concentration of
12 IL 
at early differentiation stage.
As () Gi augments, it inhibits T bet  more significantly, and when less 3 GATA 
produce, more T bet  will binds to the IFN   gene, hence
Gm
k and
mT
k are
negatively correlated. If () Ti remains approximately the same value for a rather long
period of time, it means that the Th cells have reached the stage of differentiation
completion. Under such circumstances the process of differentiation is irreversible even
when a large quantity of 4 IL  is added to the culture environment or a high level of
3 GATA  is expressed in cells, in another word, the change of () Gi cannot affect the
binding ability of T bet  to the IFN   gene. At the early stage of Th1
differentiation, 12 IL  is able to simultaneously stimulate both IFN   and T-bet gene
through 12 4 IL STAT    transduction pathway and accelerates the IFN   gene
12
transcription. As the process of differentiation proceeds, the promoting effect of 12 IL  diminishes,
till differentiation progresses self-reliantly even without the presence of 12 IL  (when
i  ,
12
11
c IL
kC
i


).
We substitute Eq. (6) into Eq. (5) to get:

()
12
()
()
(1 )(1 ) ( )
vT
T i bind
c IL B
Tm
Gm
k T i
v
kC G i I
k T i
ki


  

(7)
Then we assume thatT bet  won’t degrade before binding to the IFN   gene, based on
which we have:
( ) ( )
()
T i produce T i bind
dT i
vv
dt

(8)
We substitute Eq. (4), Eq.(7) into Eq.(8), so we have:
2
2
1
12
11
( ) ( ) ()
()
( ) ( )
(1 )(1 ) ( )
IF v IFT vT
c IL B
m IFT IFT IF v
Tm
Gm
k k k T i k T i dT i
kC G i I
dt k k k k k T i
k T i
ki







  
(9)

2.1.2 The differentiation of Th2
We define () Gi as the concentration of 3 GATA  in cells at the moment i , and the
production rate of 4 IL  is modeled as:

6
4 ( )
6
()
()
v
IL i produce
m
k G i
v
k G i



(10)

where
1 v
k and
1m
k are meaningful coefficients here. Eq. (10) shows that
4( ) IL i produce
v

augments non-linearly with the increase of concentration of 3 GATA  , and:
When ( ) 0 Gi  , we have
6
4( )
6
()
v
IL i produce
m
k G i
v
k

 , which means
4( ) IL i produce
v

is almost
directly proportional to () Gi , when the concentration of 3 GATA  is extremely low.
13
6m
k denotes the binding capacity of 3 GATA  to the 4 IL  gene （the smaller
6m
k is,
the stronger the binding capacity is, and vice versa ）
When () Gi  , we have
4( ) 6 IL i produce v
vk

 . In the condition that the
concentration of 3 GATA  is elevated to an exceptionally high level,
4( ) IL i produce
v

approaches to a fixed value. In another word,
6 v
k represents the maximum
production rate of 4 IL  , when the 4 IL  gene is being transcribed continuously. Due to
the fact that the transduction pathway is comparatively long, the concentration of
3 GATA  is actually determined by the concentration of 4 IL  time
1
 ago. Here
1
 is
in the non-negligible time delay that 4 IL  needs to stimulate 3 GATA  . Hence the
production rate of 3 GATA  is expressed by:
1
1
4( )
()
4( )
1
IL IL i produce
G i produce
IL
IL i produce
ILG
kv
v
k
v
k









(11)

Further analysis of Eq.(11) reveals the following information:
when
4( )
0
IL i produce
v

 and
4( ) ( ) IL i produce G i produce
vv

,
() G i produce
v approaches to
1
4( ) IL IL i produce
kv



, which means, given a low production rate of 4 IL  , the production
rate of 3 GATA  is to a very large extent decided by
4( ) IL i produce
v

and
IL
k , which are the
production rate of 4 IL  and the amplification factor of the signal cascade, respectively.
When
4( ) 6 IL i produce v
vk

 and
4( ) ( ) IL i produce G i produce
vv

 , we have
6
()
6
ILG IL v
G i produce
ILG IL v
k k k
v
k k k


,
which means that the production rate of 3 GATA  approaches to a fixed value
6
6
ILG IL v
ILG IL v
k k k
k k k 
, when
4( ) IL i produce
v

is close to
6 v
k and at the same time not far from
() T i produce
v .
14
When
4( ) 6 IL i produce v
vk

 and
4( ) ( ) IL i produce G i produce
vv

, we will expect to have
() G i produce ILG
vk  . When the production rate of 4 IL  is much higher than that
of 3 GATA  , the 3 GATA  production rate will approach to, but is always lower
than
ILG
k , due to the speed limitation of T bet  transcription. We substitute Eq.(10) into
Eq.(11), then arrive at:
1
1
1
1
6
6
()
6
6
()
()
()
1
()
v
IL
m
G i produce
v IL
ILG m
k G i
k
k G i
v
k G i k
k k G i













(12)

And its simplified form is:

1
1
6
()
66
()
( ) ( )
IL v ILG
G i produce
m ILG ILG IL v
k k k G i
v
k k k k k G i






. (13)
Since T bet  can affect the binding ability of 3 GATA  to the 4 IL  gene, the binding
rate of 3 GATA  and 4 IL  is given by the following formulas:
()
()
()
vG
G i bind
mG
k G i
v
k G i


(14)
()
(1 )
A
mG Gm
Tm
T i I
kk
k

，
1,
0,
A
iA
I
iA









1
1
( ) ( )
A
i
in
A
A i G i G i
n












(15)
Where
mG
k represents the binding ability of 3 GATA  to 4 IL  gene with the presence
of T bet  ;
Gm
k represents the binding ability of 3 GATA  to 4 IL  gene without the
presence of T bet  ;
Tm
k represents the binding ability of T bet  to 4 IL  gene without
the presence of 3 GATA  .
As () Ti augments, it inhibits 3 GATA  more significantly, and when less T bet 
produce, more 3 GATA  will binds to the 4 IL  gene, hence
Tm
k and
mG
k are
negatively correlated. If () Gi remains approximately the same value for a rather long
15
period of time, it shows that the Th cells have reached the stage of differentiation
completion. Under such circumstances the process of differentiation is irreversible even
when a large quantity of IFN   is added to the culture environment or a high level of
T bet  is expressed in cells, in another word, the change of () Ti cannot affect the
binding ability of 3 GATA  to the 4 IL  gene any more. We substitute Eq.(15) into
Eq.(14) to get:

()
()
()
(1 ) ( )
vG
G i bind
A
Gm
Tm
k G i
v
T i I
k G i
k



(16)

Then we assume that 3 GATA  won’t degrade before binding to the 4 IL  gene, based
on which we have:
( ) ( )
()
G i produce G i bind
dG i
vv
dt
 (17)
We substitute Eq.(13), Eq.(16) into Eq.(17), so we have:
1
1
6
66
( ) ( ) ()
()
( ) ( )
(1 ) ( )
IL v ILG vG
A
m ILG ILG IL v
Gm
Tm
k k k G i k G i dG i
T i I
dt k k k k k G i
k G i
k







(18)
Note that Th cells is almost unable to spontaneously secrete cytokines except
IFN   and 4 IL  without the stimulation of other types of cytokines. Since this
experiment is designed to culture Th cells in the environment where no cytokines exist
but, IFN   , 12 IL  and 4 IL  , the possible existence and influence of other cytokines
can be theoretically ignored. Thus, the differentiation model of Th cells is modeled as:
12
1 1 2 12
61
6 6 1
( ) ( ) ()
( ) ( ) (1 ( ) / )(1 / ) ( )
( ) ( ) ()
( ) ( ) (1 ( ) / ) ( )
IF v IFT vT
m IFT IFT IF v Tm B Gm c IL
IL v ILG vG
m ILG ILG IL v Gm A Tm
k k k T i k T i dT i
dt k k k k k T i k G i I k k C i T i
k k k G i k G i dG i
dt k k k k k G i k T i I k G i









     






    

(19)

16
2.2 Model Analysis
The completion of the differentiation process is reached when:
12
1 1 2
61
6 6 1
( ) ( )
0
( ) ( ) (1 ( ) / ) ( )
( ) ( )
0
( ) ( ) (1 ( ) / ) ( )
IF v IFT vT
m IFT IFT IF v Tm B Gm
IL v ILG vG
m ILG ILG IL v Gm A Tm
k k k T i k T i
k k k k k T i k G i I k T i
k k k G i k G i
k k k k k G i k T i I k G i








    






    

(20)
At this moment, limit points for the equations exist:
When the cells are at Th0 stage, we have ( ) 0 Ti  and ( ) 0 Gi  . According to Eq.(20):
12
1 1 2 12
61
6 6 1
( ) ( )
0
( ) ( ) (1 ( ) / )(1 / ) ( )
(21)
( ) ( )
0
( ) ( ) (1 ( ) / ) ( )
IF v IFT vT
m IFT IFT IF v Tm B Gm c IL
IL v ILG vG
m ILG ILG IL v Gm A Tm
k k k T i k T i
k k k k k T i k G i I k k C i T i
k k k G i k G i
k k k k k G i k T i I k G i





 


     






    


Thus, the ordinary differential equations (Eq.(19)) have a incipient
state:
00
0 0 0
( , ) (0,0) P G T 
The differentiation of Th1 is considered to be completed, when
i  , ( ) 0 Gi  and 0
B
I  . According to Eq.(19),
12
( ) 0
1 1 2 12
1
11
( ) ( )
lim
( ) ( ) (1 ( ) / )(1 / ) ( )
( ) ( )
( ) ( ) ( )
IF v IFT vT
Gi
m IFT IFT IF v Tm B Gm c IL
IF v IFT vT
m IFT IFT IF v Tm
k k k T i k T i
k k k k k T i k G i I k k C i T i
k k k T i k T i
k k k k k T i k T i






     

  
,
upon which we have:

1
11
( ) ( )
0
( ) ( ) ( )
IF v IFT vT
m IFT IFT IF v Tm
k k k T i k T i
k k k k k T i k T i

  

(22)

The solution is:
11
11
()
IF v IFT Tm m IFT vT
IFT vT IF v vT IF v IFT
k k k k k k k
Ti
k k k k k k k k



.
(23)
17
On the other hand:
61
( ) 0
6 6 1
( ) ( )
lim 0
( ) ( ) (1 ( ) / ) ( )
IL v ILG vG
Gi
m ILG ILG IL v Gm A Tm
k k k G i k G i
k k k k k G i k T i I k G i





    

(24)
So one limit point for Eq.(19) is ：
00 11
0 1 1
11
( , ) (0, )
IF v IFT Tm m IFT vT
IFT vT IF v vT IF v IFT
k k k k k k k
P G T
k k k k k k k k




The process of Th2 differentiation is over when i  , ( ) 0 Ti  and 0
A
I  .According
to Eq.(19):
61
( ) 0
6 6 1
6
66
( ) ( )
lim
( ) ( ) (1 ( ) / ) ( )
( ) ( )
( ) ( ) ( )
IL v ILG vG
Ti
m ILG ILG IL v Gm A Tm
IL v ILG vG
m ILG ILG IL v Gm
k k k G i k G i
k k k k k G i k T i I k G i
k k k G i k G i
k k k k k G i k G i





    

  
,
upon which we have:

6
66
( ) ( )
0
( ) ( ) ( )
IL v ILG vG
m ILG ILG IL v Gm
k k k G i k G i
k k k k k G i k G i

  

(25)
The solution is:

66
66
()
IL v ILG Gm m ILG vG
ILG vG IL v vG IL v ILG
k k k k k k k
Gi
k k k k k k k k




(26)
On the other hand:
12
( ) 0
1 1 2
( ) ( )
lim 0
( ) ( ) (1 ( ) / ) ( )
IF v IFT vT
Ti
m IFT IFT IF v Tm B Gm
k k k T i k T i
k k k k k T i k G i I k T i





    

(27)
So another limit point for Eq. (19) is
00 66
0 2 2
66
( , ) ( ,0)
IL v ILG Gm m ILG vG
ILG vG IL v vG IL v ILG
k k k k k k k
P G T
k k k k k k k k




The incipient point for Eq.(19) is
00
0 0 0
( , ) (0,0) P G T 
(28)
18
The limit points for Eq.(19) are

00 11
0 1 1
11
( , ) (0, )
IF v IFT Tm m IFT vT
IFT vT IF v vT IF v IFT
k k k k k k k
P G T
k k k k k k k k



(29)
and

00 66
0 2 2
66
( , ) ( ,0)
IL v ILG Gm m ILG vG
ILG vG IL v vG IL v ILG
k k k k k k k
P G T
k k k k k k k k



(30)
Three equilibrium points in the system dynamics model are shown in Table 2.

Table 2. Equilibrium points of system dynamic model
Differentiation state Results Stable status
Th0 state

Unstable equilibrium point
Th1-differentiation
state

Stable equilibrium point
Th2-differentiation
state

Stable equilibrium point

19
Chapter 3 Results and Discussion

3.1 Th1/Th2 differentiation process
Mathematical modeling plays an important role in analysis of a great deal of
complicated experimental phenomena of biological systems. In this paragraph, we would
like to discuss the relationship between differentiation potential and equilibrium points of
the Th1-Th2 system dynamical model. Mathematical modeling plays an important role in
the analysis of a great deal of complicated experimental phenomena of biological systems
(Xia et al., 2007; Xia et al., 2011; Zu et al., 2012; Zhou et al., 2011). Here, we discuss the
relationship between differentiation potential and equilibrium points of the Th1-Th2
system dynamics model.
Figures 2 and 3 show the differentiation process of both Th1 and Th2 cells. When
there is no stimulation of cytokines or other signals, Th0 cells may differentiate into Th1
or Th2 cells with equal probability. Cytokines (IFN-gamma, IL-4) can stimulate
transcription factors (T-bet, GATA-3), which results in a certain period of time delay,
which in turn brings about oscillation in the expression of T-bet and GATA-3 (the
oscillation becomes more evident when the graph is enlarged), but they cannot influence
the equilibrium point of the differential equation (i.e., the concentration when a steady
state is reached). Therefore, the oscillation can explain the abnormity of the curves
representing T-bet and GATA-3 (Chakir et al., 2003). Moreover, it will take the
differentiation process a longer time to reach the stable point when more considerable
oscillation occurs.
20
0 200 400 600 800 1000 1200 1400 1600 1800
0
0.2
0.4
0.6
0.8
1
1.2
1.4

GATA-3
T-bet

Figure 2: Expression level of major regulation gene in Th1 differentiation process with
time lag.
0 200 400 600 800 1000 1200 1400 1600 1800
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

GATA-3
T-bet

Figure 3: Expression level of major regulation gene in Th2 differentiation process with
time lag.

21
3.2 Th1/Th2 re-differentiation process
In the middle of differentiation (Figure 4), when the expression level of T-bet has
not yet reached a stable equilibrium point, increased expression of GATA-3 will lead to a
decrease in T-bet expression, which means re-differentiation occurs.
In the middle of differentiation (Figure 5), when the expression level of GATA-3 has
not yet reached its stable equilibrium point, an increase in T-bet expression will lead to a
decrease in GATA-3 expression, which means re-differentiation occurs.
0 200 400 600 800 1000 1200 1400 1600 1800
0
0.2
0.4
0.6
0.8
1
1.2
1.4

GATA-3
T-bet

Figure 4: Expression level of major regulation gene in Th1

Th2 re-differentiation
process with time lag.
22
0 200 400 600 800 1000 1200 1400 1600 1800
-0.2
0
0.2
0.4
0.6
0.8
1
1.2

GATA-3
T-bet

Figure 5: Expression level of major regulation gene in Th2

Th1 re-differentiation
process with time lag.

3.3 Th1/Th2 cell without potential of differentiation
After differentiation is complete (Figure 6), the expression level of T-bet cannot be
affected even when a high level of GATA-3 is expressed in cells. However, by transiently
increasing the expression level of GATA-3, it is possible to obtain re-differentiated cells
that can transiently secrete IFN-gamma and IL-4. The mechanism is: after G(i) and T(i)
remain stable for a while, Equation 19 transforms into Equation 26 (G(i) and T(i) cannot
affect each other) ：

12
1 1 2
61
6 6 1
( ) ( ) ()
( ) ( ) ( )
( ) ( ) ()
( ) ( ) ( )
IF v IFT vT
m IFT IFT IF v Tm
IL v ILG vG
m ILG ILG IL v Gm
k k k T i k T i dT i
dt k k k k k T i k T i
k k k G i k G i dG i
dt k k k k k G i k G i








   






   

(31)
Due to the reconstruction of the chromosome, the binding of T-bet to the IFN-gamma
gene is not influenced by GATA-3, and therefore, the production rates of IFN-gamma and
23
T-bet remain constant. Under these circumstances, the fluctuation of G(i) cannot affect
T(i), even when there is a high level of GATA-3 in cells.
Similarly, after differentiation is complete (Figure 7), the expression level of
GATA-3 cannot be affected even when a high level of T-bet is expressed in cells, but it is
possible to obtain re-differentiated cells that can transiently secrete IFN-gamma and IL-4.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
0.2
0.4
0.6
0.8
1
1.2
1.4

GATA-3
T-bet

Figure 6: Th1 cell without potential of differentiation
24
0 500 1000 1500 2000 2500
-0.2
0
0.2
0.4
0.6
0.8
1
1.2

GATA-3
T-bet

Figure 7: Th2 cell without potential of differentiation
Similarly, after through differentiation is completes (Figure 11), the expression level
of 3 GATA  cannot be affected even when a high level of T bet  is expressed in cells,
but it is possible to obtain re- differentiation cells that can transiently secrete
IFN   and 4 IL  .

3.4 Phase Trajectory
We can see from the figure that when the lag increases, commotion is enhanced.
Figures 8 and 9 show that oscillation does not exist when either one of GATA-3 and
T-bet is obviously superior to the other in quantity (comparatively stable state of
differentiation), whereas in an unstable state, the phase trajectory oscillates more
considerably as the delay order increases. In addition, the curve oscillates dramatically
when the level of expression of T-bet is low.

25
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25

commotio lag=5
commotio lag=4
commotio lag=3
commotio lag=2
commotio lag=1
no-commotio

Figure 8: The phase trajectory curve of Th2 differentiation when. We can find from the
figure when the  increase ，the commotion will enhance.
-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4

commotio lag=5
commotio lag=4
commotio lag=3
commotio lag=2
commotio lag=1
no-commotio

Figure 9: The phase trajectory curve of Th1 differentiation Note: the yellow curve in this
figure is the phase trajectory of differentiation ignoring the time lag in the transduction
pathway

26
Chapter 4 Conclusion

Helper T lymphocytes have the potential to differentiate, when a system dynamics
model shows the expression of major control genes at an unstable equilibrium point. The
helper T lymphocyte will no longer have differentiation potential when the model
evolution reaches a stable equilibrium point. In addition, the time lag, caused by the
expression of transcription factors, can lead to an oscillation in the secretion of cytokines
during the differentiation process.

27
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Asset Metadata

Creator Wang, Qixin (author)

Core Title Genetic analysis of differentiation of T-helper lymphocytes

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Statistics

Publication Date 04/23/2013

Defense Date 04/22/2013

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

Tag equilibrium point,genetic analysis,OAI-PMH Harvest,system dynamics model,Th1-Th2

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Azen, Stanley P. (committee chair), Waterman, Michael S. (committee member), Zhang, Jianfeng (committee member)

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