M.tech Thesis

Effect of allelic differences on stochastic gene
expression
A thesis submitted in partial fulﬁllment of
the requirements for the degree of
Master of Technology
by
Kaushalesh Gupta
(Roll No. 144106035)
Under the guidance of
Dr. Biplab Bose
DEPARTMENT OF BIOSCIENCES & BIOENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI
May 2016

CERTIFICATE
This is to certify that the work contained in this thesis entitled
Effect of allelic differences on stochastic gene expression
is the work of
Kaushalesh Gupta
(Roll No. 144106035)
for the award of the degree of Master of Technology, carried out in the
Department of Biosciences and Bioengineering, Indian Institute of Technology
Guwahati under my supervision and that it has not been submitted elsewhere
for a degree.
Supervisor
Signature of HOD
Date:
Place:

DECLARATION
The work contained in this thesis is my own work under the supervision of
the Dr.Biplab Bose. We have read and understood the “M. Tech Ordinances
and Regulations” of IIT Guwahati and the “FAQ Document on Academic Mal-
practice and Plagiarism” of Biosciences and Bioengineering Department of IIT
Guwahati. To the Best of our knowledge, this thesis is an honest representation
of my work.
Kaushalesh Gupta
Date:
Place:

Acknowledgments
The writing of this project work has been one of the most signiﬁcant academic challenges I have
ever had to face. Without the support, patience and guidance of the following people, this study
would not have been completed .It is to them I owe my deepest gratitude.
First of all, I would like to give my sincere thanks to my honoriﬁc supervisor Dr. Biplab
Bose, who accepted me as a MTP student .I appreciate all his contributions of time, ideas and
funding to make my project a productive and stimulating experience. I am also thankful for the
excellent example he has provided as a successful scientist.
I also want to express my profound sense of gratitude and respect to our H.O.D, Prof.Kannan
Pakshirajan , for his help in academic matters. I would like to thank all the members of Cen-
ter for Excellence who have created all necessary conditions conductive to the initiation and
completion of project.
All the Lab members contributed immensely to my personal and professional time. They
have been a source of friendship as well as good advice. I am especially grateful to Vimal,
Thiyagarajan, Sandipan, Poulami, Mahesh, Satendra, poulomi, Neha, Nischal and Namami who
have supported me and encouraged me throughout the project work. I very much appreciate
their patience with me. Last but not the least; I would like to thank my family and friends for
their love and understanding. They form the backbone and origin of my happiness.

Abstract
It is widely known that stochastic gene expression gives rise to ﬂexibility in the biological sys-
tem and makes it adaptable to changing environment. We have used an analytical model to
visualize the effect of stochasticity in a biological system which is previously supposed to be
deterministic as it shows the average behavior of cell population. Simulation are performed
in a simple model of biallelic gene expression to understand how protein behavior changes by
changing transcriptional rate kinetics of an allele. It is observed that in an increasing tran-
scriptional rate kinetics of an allele decreased the variation arise in stochastic gene expression.
Similarly, analytical model is created for protein oscillation and simulation performed to under-
stand how allelic differences in one gene affect the expression of other protein.
iv

Nomenclature
MDM2 Mouse Double Minute 2
SNP Single Nucleotide polymorphism
mRNA messenger RNA
ATM Ataxia Telangiectasia Mutated
Wip1 Wild-type p53-induced phosphatase 1
casp3 Caspase 3
ode Ordinary differential equation
ka Rate of transition of promoter from off to on
kd Rate of transition of promoter from on to off
v

Contents
Abstract iv
Nomenclature v
List of Figures viii
1 Introduction 1
1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Review of Literature 3
2.1 Stochasticity in gene expression . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Allele differences in gene expression . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 P53 and its regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Materials and Methods 6
3.1 Formulation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Model Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 p53 and its regulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Results and Discussion 11
4.1 Network map of p53 regulation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Bi-allele model of gene expression . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.1 Model of bi-allele gene expression . . . . . . . . . . . . . . . . . . . . 12
4.2.2 Model simulation of bi-allelic gene expression Network. . . . . . . . . 12
4.2.3 Stochastic system with changing rate kinetics. . . . . . . . . . . . . . . 13
4.3 Simple oscillatory Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3.1 Model Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
vi

4.3.2 Deterministic and stochastic simulation of oscillatory Network using
random number to check promoter state. . . . . . . . . . . . . . . . . . 15
4.3.3 Deterministic and stochastic simulation of oscillatory Network using
Poisson process for promoter transition between on/off state. . . . . . . 16
4.3.4 Fourier Analysis of Oscillatory Network. . . . . . . . . . . . . . . . . 17
4.3.5 Single cell and Average population behavior of the network with stochas-
tic gene expression. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3.6 Effect of promoter transition rate on homozygous allele system. . . . . 19
4.3.7 Effect of promoter transition rate on heterozygous allele system. . . . 20
4.3.8 Effect of allele differences on stochastic gene expression . . . . . . . . 21
4.3.9 Analyses of protein noises of X1, X2, X3 . . . . . . . . . . . . . . . . 22
5 Conclusion 24
6 Appendix 25
vii

List of Figures
3.1 Schematics representation of deterministic simulation . . . . . . . . . . . . . . 8
3.2 Schematics representation of stochastic simulation . . . . . . . . . . . . . . . 9
4.1 Deterministic and Stochastic schematics of bi-allele gene expression. . . . . . 12
4.2 Dynamics of bi-allelic gene expression for deterministic and stochastic model. . 13
4.3 Dynamics of slow and fast stochastic system. . . . . . . . . . . . . . . . . . . 13
4.4 Histogram plot for stochastic system. . . . . . . . . . . . . . . . . . . . . . . . 14
4.5 A deterministic model of simple oscillatory network. . . . . . . . . . . . . . . 14
4.6 A Stochastic model of simple oscillatory network. . . . . . . . . . . . . . . . . 15
4.7 Deterministic and stochastic simulation result of oscillatory Network using ran-
dom number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.8 Deterministic and stochastic simulation result of oscillatory Network using Pois-
son process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.9 Result of Deterministic and stochastic Network Fourier Analysis. . . . . . . . . 18
4.10 Single cell and Average population behavior of stochastic gene expression . . . 19
4.11 Homozygous allele system behavior with different promoter transition kinetics 20
4.12 Heterozygous allele system behavior with different promoter transition kinetics. 21
4.13 Effect of allele differences on stochastic gene expression. . . . . . . . . . . . . 22
4.14 Results of protein noise analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 23
viii

Chapter 1
Introduction
A biological process in an individual cell is stochastic in nature. In a single cell, stochasticity
can arise at many stages for example molecule diffusion in a cell, conformation rearrangements,
and randomness in a process of transcription and degradation of mRNA and proteins. Stochas-
ticity in gene expression can also arise due to SNPs in the promoter region of an allele and give
rise differences in its expression dynamics which can consequently affect dynamics of other
system depend on it.
We have created an analytical model to visualize how transitions of promoter between
different transcriptional states lead to noisy gene expression among clonal cell population [5,
20].These noises in gene expression is essential for diverse cellular processes, such as the gen-
eration of phenotypic heterogeneity and the regulation of probabilistic cell-fate decisions across
isogenic cell lines .Tight regulation of gene expression noises is also vital for the functioning of
proteins, and breakdown in its regulation can lead to a diverse disease state.
One such system is p53 and MDM2. p53 is a 393 amino acid nuclear phosphoprotein,
acts as a tetrameric sequence-speciﬁc DNA-binding protein to regulate the transcription of a
growing range of genes whose protein products can regulate cell-cycle progression and apop-
tosis [11]. MDM2 protein also known as E3 ubiquitin-protein ligase. it is a 491 amino acid
nuclear phosphoprotein that contains a p53 binding domain at its N-terminus protein. mdm2
bind p53 mediate its rapid degradation through the ubiquitin proteolysis pathway keeps its level
under tight control [6, 8, 12]. It is found that SNPs in the promoter of mdm2 (SNP 309) gene
increases the afﬁnity of sp1 transcriptional factor and lead to a high expression of MDM2 which
consequently alter dynamics of p53 and lead to a diseased state of tumor progression [3].
1

1.1 Objective
• To make Network map of p53 Regulation.
• To study the effect of allelic differences in the production of protein in a single cell.
• To study how allelic differences in promoter region effects oscillation of protein in a
single cell.
2

Chapter 2
Review of Literature
2.1 Stochasticity in gene expression
Stochasticity is a property of a system which lacks any predictable order and outcomes. It com-
pletely different from a deterministic system whose future behavior is fully determined by their
initial conditions and follow a predictable pattern. A biological process in an individual cell is
stochastic in nature. In a single cell, stochasticity can arise at many stages for example molecule
diffusion in a cell, conformation rearrangements and randomness in a process of transcription
and degradation of mRNA and proteins.
Novick and Weinert showed that the beta-galactosidase production in single cells was
highly variable,and on induction the proportion of cells expressing the enzyme rather increased
rather than increasing every cell’s expression level equally. Later studies were performed on the
expression of a glucocorticoid-responsive transgene encoding beta-galactosidase with different
doses of glucocorticoid expression reporter in a single-cells and observed large variability in
the expression of the transgene from cell to cell. It is also observed that with an increase in
dose result in increased frequency of cells displaying a high level of expression instead of the
uniform increase in expression in every cell. [7]. The same observation is observed in the p53
system. They found that p53 was expressed in a series of discrete pulses after DNA damage.
Genetically identical cells had different numbers of pulses: zero, one, two or more. The mean
height and duration of each pulse were ﬁxed and did not depend on the amount of DNA damage.
The mean number of pulses, however, increased with DNA damage [9]. Later there are several
theoretical work has been purposed to describe stochastic in gene expression especially in a P53
system such as A plausible model for the digital response of p53 to DNA damage to describe
3

the variability in p53 system [21], Modelling the role of p53 pulses in DNA damage [2].
Stochastic gene expression has important consequences for cellular function, being ben-
eficial in some contexts and harmful in others. These situations include the stress response,
metabolism, development, the cell cycle, circadian rhythms, and aging [17]. Heterogeneity
arises in isogenic cell population due stochastic gene expression give rise to flexibility in the
system and also play a crucial role in embryonic development and cancer. It is observed that a
positive feedback increases heterogeneity in gene expression and give rise to bistability in the
system. A bistable system has two stable steady states, with lower and higher expression of the
target gene. In such a system, for a given inducing signal, cells can have either higher expression
or lower expression of the gene .This gives rise to a mixed population of cells with bimodal dis-
tribution in expression. Due to the inherent stochasticity in transcriptional processes, a positive
feedback can lead to bimodal gene expression even without bistability Bimodal gene expres-
sion, due to positive feedback in transcription, has been observed for genes involved cellular
differentiation and embryonic development [14]
2.2 Allele differences in gene expression
An allele is one of a pair of genes that appear at a particular location on a particular chromosome
and control the same characteristic, such as blood type or colour-blindness. Some organism
contain only one allele at a particular locus on a chromosome and called as monoallelic such
as bacteria while species like human are contains two alleles at a particular locus called as
biallelic, they inherit one from each parent. If the two alleles are the same, the individual is
homozygous for that gene. If the alleles are different, the individual is heterozygous. When the
alleles of a pair are heterozygous, the phenotype of one trait may be dominant and the other
recessive. The dominant allele is expressed and the recessive allele is masked. This is known
as complete dominance. In heterozygous relationships where neither allele is dominant but
both are completely expressed, the alleles are considered to be co-dominant. Co-dominance is
exemplified in AB blood type inheritance. When one allele in not completely dominant over
the other, the alleles are said to express incomplete dominance.
Promoter regions are an essential component of transcription initiation and the assembly
of RNA polymerase and associated regulators. SNPs in the promoter region of a gene would
be expected to modify the binding affinity of the promoter to transcription factors thereby reg-
4

ulating gene expression. The change in expression due to SNP in promoter can lead to diseased
condition one such example is the development of Alzheimers disease due to a mutation in the
promoter sequence of 4 allele of the Apolipoprotein E gene in human [10].
2.3 P53 and its regulation
p53 is a tumor suppressor protein which regulates the expression of multiple proteins involve
in cell cycle regulation and forms many feedback loops. An outstanding one is p53-MDM2
negative feedback loop.p53 can transcriptionally activate MDM2 expression and in turn MDM2
can targets p53 for proteasome degradation through the ubiquitin proteolysis pathway [6]. P53-
MDM2 feedback loop together with another characterized negative feedback loop involving
WIP1/ATM forms the basis of p53 pulses and previous experiments have conﬁrmed sustained
p53 oscillation under DNA damage conditions [15,18]. MDM2 contained SNP309 its promoter
region which lead to increased afﬁnity of sp1 transcription factor hence show increased level of
MDM2 expression level, hence result in attenuation of p53 expression and its control to regulate
cell cycle which leads to tumor progression [3].
5

Chapter 3
Materials and Methods
3.1 Formulation of the model
A. We consider the three-variable model shown in Fig. 4.5 and described by the following
differential equations for Deterministic Model of gene expression:
.
dX(1)
dt
= k(1) − k(2) ×
x(1)
x(1) + k(3)
× x(3)
.
.
dx(2)
dt
= k(4) × x(1) − k(5) × x(2)
.
.
dx(3)
dt
= k(6) × x(2) − k(7) × x(3)
.
Here x(1), x(2) , x(3) are three variable which production depend on rate constant k(1),
k(4), k(6) respectively, and degradation depend on rate constant k(2), k(5), k(7) respectively.
Model consist of one feedback loop in which x(3) is negatively regulating the expression of
x(1) which is the main cause in the origin of oscillatory behavior in the system.
6

B. We consider the three-variable model shown in Fig. 4.6 and described by the following
differential equations for Stochastic Model of gene expression:
.
dx(1)
dt
= k(1) − k(2) ×
x(1)
x(1) + k(3)
× x(3)
.
.
dx(2)
dt
=
k(4)
2
× (status(1) + status(2)) × x(1) − k(5) × x(2)
.
.
dx(3)
dt
= k(6) × x(2) − k(7) × x(3)
.
Case 1 : Here x(1), x(2) , x(3) are three variable which production depend on rate constant
k(1), k(4), k(6) respectively, and degradation depend on rate constant k(2), k(5), k(7) respec-
tively. In this case promoter state will depend on random number for example, we have set the
initial status of promoter in off state. Then it will generate a random number and compare the
random number with the probability of promoter off → on state. If random number is less the
probability, promoter will turned on else remain in off state. Similarly, when promoter is in
on state it will check the probability of promoter on → off state. If random number is less the
probability, promoter will turned off else remain in on state. When promoter is on, it will result
in protein production and in off state it will not produce protein.
Case 2 : In second case of stochastic simulation we have used interarrival time according
to Poisson process to determine promoter state. Initially we consider promoter is in off state
then according to Poisson process(time = - 1/kon * log(u)), it will check when promoter will
going to be turned on. Similarly, when promoter is in on state It will check when promoter is
going to be turned off (time = - 1/koff*log(u)) .When promoter is on, it will result in protein
production and in off state it will not produce protein.(u is random number)
7

3.2 Model Simulation
All the simulation are performed in Matlab 2014b.
Figure 3.1: Schematics representation of deterministic simulation
In this schematics, showing the working of our code in Matlab for Deterministic simulation.
As shown in the ﬁgure ordinary differential equation (ode) solver call all the equation and input
requires for solving odes. After simulation, it writes data to a text ﬁle.
8

Figure 3.2: Schematics representation of stochastic simulation
It this schematics, showing the working of our code in Matlab for Stochastic simulation. As
shown in the ﬁgure ordinary differential equation (ode) solver call all the equation and input
requires in solving odes. Stochasticity is generated by ﬂuctuating the promoter status according
to Poisson distribution.
9

In the Schematics shown in fig. 3.1 and fig. 3.2 describes the working Matlab code.
Matlab code of four functions: Ode solver, parameter value, Equation and text. The first step in
this process is to convert a biological network into the system of ordinary differential equation
and define all the ordinary differential equation in Equation function. The second step is to find
the Parameter value (for example. the rate of transcription, the rate of translation, the rate of
degradation of protein in a gene regulatory network) and initial condition (for example. Basal
level of protein in a cell) by reviewing it in literature and then define it in parameter function.
The third step is to use Ode solver which is a Matlab inbuilt function, we have used it to solve the
ode defined in Equation function. After simulation data is saved in a text file by text function.
In Matlab code for the stochastic simulation, an addition function is added promoter status to
generate randomness into the system. This function helps Ode solver to check whether the
promoter is switch on (then status will be updated to 1) or switch off condition (then status will
be updated to 0) and accordingly protein will produce.
3.3 p53 and its regulation.
p53 is a tumour suppressor protein which regulates the transcription of a growing range of
genes whose protein products can regulate cell-cycle progression. The intracellular activity of
the p53 is regulated by MDM2. MDM2 mediated degradation of p53 under non-stressed. It is
also reported in the paper that MDM2 inhibit the translation of p53 [16]. In stressed condition
(DNA damage) ATM gets activated through intermolecular phosphorylation. Activated ATM
(i.e. phosphorylated form) then phosphorylates and stabilizes p53 (p53p) ATM also phospho-
rylate MDM2 and strongly diminished its ability to degrade [19]. Phosphorylated p53 species
can activate mdm2 and wip1 transcription. WIP1 (wildtype p53induced phosphatase 1) result
in dephosphorylation of p53p, MDM2p and ATM [15]. ATM protein keeps a check over DNA
damage and gets re-activated again and again till the damage is repaired [1] and this gives rise
oscillatory behaviour of p53. Oscillatory behaviour of p53 has physical significance in biolog-
ical context At low damage levels, few p53 pulses evoke cell cycle arrest by inducing p21 and
promote cell survival, whereas at high damage levels, sustained p53 pulses trigger apoptosis
by inducing Casp3 [13].On the basis of all these information, a network map P53 regulation is
created.
10

Chapter 4
Results and Discussion
4.1 Network map of p53 regulation
p53 is a tumour suppressor protein. The intracellular activity of the p53 is regulated by
MDM2. MDM2 mediated degradation of p53 under non-stressed. In stressed condition (DNA
damage) ATM gets activated through intermolecular phosphorylation. Activated ATM then
phosphorylates and stabilizes p53. ATM also phosphorylate MDM2 and strongly diminished its
ability to degrade p53. Phosphorylated p53 species can activate mdm2 and wip1 transcription.
WIP1 result in dephosphorylation of p53p, MDM2p and ATM.
11

4.2 Bi-allele model of gene expression
4.2.1 Model of bi-allele gene expression
This model consists of only one variable.In deterministic model, it is assumed that allele is
continuously producing protein with rate ’jp’ and protein is continuously degraded at a rate
’kp’. In stochastic model, it is consider that gene is switch randomly between inactive (G)
and active (G*) states according to ﬁrst-order reaction kinetics. ka and kd are activation and
deactivation rate that determines how long a allele remain in the active state. When active, each
allele expresses a protein (G).
Figure 4.1: Deterministic and Stochastic schematics of bi-allele gene expression.
Each active allele (g1*, g2*) allele express protein (G) at rate of jp (nM). G is degraded with a
rate of kp. ka and kd are activation and deactivation rate of allele respectively.
4.2.2 Model simulation of bi-allelic gene expression Network.
As shown in ﬁgure. 4.2 deterministic system, protein ’G’ is showing stable expression of allele
and it is reaching steady state level of 0.5 nM, whereas in stochastic system product G is varying
around a mean value of 0.5 nM with excursions to higher(.75 nM) and lower(.25 nM) protein
levels.
12

Figure 4.2: Dynamics of bi-allelic gene expression for deterministic and stochastic model.
In figure at the right,it is assumed that promoter is always turned on and in figure at the left has
promoter transition kinetics (ka = 2 × 10−3
and kd = 2 × 10−3
).
4.2.3 Stochastic system with changing rate kinetics.
In a stochastic system with rate of promoter activation and deactivation result in variability in
protein expression. Protein levels is varying around a mean value of 0.5 nM with excursions
to higher and lower product levels (fig 4.3). The range of protein level varying from 0.25 nM
to 0.75 nM for slow kinetics to 0.40 nM to 0.60 nM. in fast kinetics as shown in histogram
plot (fig. 4.4). Faster expression kinetics is producing extremely narrow variations in protein
level. Our result matches with the paper [4] which validate that the code we have designed for
simulation is working fine.
Figure 4.3: Dynamics of slow and fast stochastic system.
Product G is showing fluctuation around the mean value of 0.5. In figure at the right has pro-
moter transition rate (ka= 2 × 10−3
and kd= 2 × 10−3
) and in figure at the left has promoter
transition rate (ka= 2 × 10−4
and kd= 2 × 10−4
) than .It is observed that increase in rate kinetics
reduces the variation in the protein expression.
13

Figure 4.4: Histogram plot for stochastic system.
A bell-shaped histogram is observed around a mean value of 0.5 nM. a) Stochastic system with
slow kinetics showing protein product in a range of 0.25 nM to 0.75 nM (ka=kd= 2 × 10−4
) .
b) Stochastic system with fast kinetics showing protein product in a range of 0.4 nM to 0.6 nM
(ka=kd = 2 × 10−3
).
4.3 Simple oscillatory Network.
4.3.1 Model Diagram
Figure 4.5: A deterministic model of simple oscillatory network.
g1* and g2* are allele of X2 and it is transcriptionally activated by X1 protein and resulting in
production of X2. X2 protein is resulting in production of X3 protein which in turn mediating
degradation of X1 to regulate it’s level in a cell.
14

Figure 4.6: A Stochastic model of simple oscillatory network.
g1* and g2* are the allele of X2 and it is transcriptionally activated by X1 protein and resulting
in the production of X3 protein from X2 protein, X3 protein in turn mediating degradation of
the X1 protein. Allele differences arise in the system due switching of allele between on and
off with a rate of ka and kd. ka is rate of activation of allele, kd is rate of deactivation of allele,
k1 is rate of production of X1, k2 is rate of degradation of X1, k3 is Michael’s Melton constant,
k4 is rate of transcription of X2, k5 is rate of degradation of X2, k6 rate of Production of X3, k7
is rate of degradation of X3.
4.3.2 Deterministic and stochastic simulation of oscillatory Network us-
ing random number to check promoter state.
The network shown in fig.4.5 and fig 4.6 is converted into a system of ordinary differential equa-
tion and simulation is performed using Matlab 2014b. In our simulation of deterministic model
stable oscillation of X1, X2, X3 is observed with fixed mean height (fig. 4.7). In stochastic
simulation, we observed decay in oscillatory pulses due to allele differences in X2 gene.
15

Figure 4.7: Deterministic and stochastic simulation result of oscillatory Network using random
number.
X1, X2, X3 showing stable oscillation in deterministic simulation shown whereas in stochastic
simulation oscillation observed . In stochastic system random number is generated to check
promoter state at every step of simulation. In this case, we have set the intial status of promoter
to off state, then it will generate a random number to check that whether random number is less
than the probability we have set for promoter off to on state. If condition is justify promoter
will become active and protein will production otherwise it will remain in off state. Similarly,
when promoter is in on state it will check the probability for promoter on to off state.
4.3.3 Deterministic and stochastic simulation of oscillatory Network us-
ing Poisson process for promoter transition between on/off state.
The network shown in fig.4.5 and fig 4.6 is converted into a system of ordinary differential equa-
tion and simulation is performed using Matlab 2014b. In our simulation of deterministic model
stable oscillation of X1, X2, X3 is observed with fixed mean height (fig. 4.8). In stochastic
simulation, we observed decay in oscillatory pulses due to allele differences in X2 gene.
16

Figure 4.8: Deterministic and stochastic simulation result of oscillatory Network using Poisson
process.
X1, X2, X3 showing stable oscillation in deterministic simulation shown whereas in stochastic
simulation(ka=kd= 2 × 10−2
) decay of oscillation observed (Data shown is average of 1000
simulation). In this simulation promoter status is varied according to Poisson process. Initially
we consider promoter is in off state then according to Poisson process(time = - 1/kon * log(u)),
it will check when promoter will going to be turned on. Similarly, when promoter is in on state
It will check when promoter is going to be turned off (time = - 1/koff*log(u)) .When promoter
is on, it will result in protein production and in off state it will not produce protein.(u is random
number)
4.3.4 Fourier Analysis of Oscillatory Network.
Fourier analysis of oscillatory network is performed for deterministic and stochastic system and
result are shown in ﬁg. 4.9.
17

Figure 4.9: Result of Deterministic and stochastic Network Fourier Analysis.
Fourier analysis of oscillatory network shows that amplitude of X1, X2, X3 is decreased in
stochastic system as compare to deterministic system but the frequency of oscillation is ap-
proximately same in all system i.e 3.1 × 10−5
. Error bar is showing that their is variation in
amplitude in stochastic simulation with promoter transition rate ( ka=kd= 2 × 10−2
).
4.3.5 Single cell and Average population behavior of the network with
stochastic gene expression.
In biological system protein expression usually studied by experiments that are averaged over
cell populations, potentially masking the dynamic behavior in individual cells.We have sim-
ulated the model with different promoter transition rate kinetics and it is observed that the
average population behavior is same for different promoter transition rate as shown in ﬁg.4.10,
whereas there are huge differences in gene expression noises for different rate kinetics within
an individual cell.
18

Figure 4.10: Single cell and Average population behavior of stochastic gene expression
In a single cell with changing rate kinetics, huge variation is observed. Fig shown at left slow
kinetics(ka=kd=2 × 10−4
for both allele) protein is varying from 15 nM to 60 nM for X1, and
60 nM to 200 nM for X2, X3 and at right fast kinetics(ka=kd= 2 × 10−3
for both allele. protein
is varying from 23 nM to 48 nM for X1 and 80 nM to 120 nM for X2, X3.
4.3.6 Effect of promoter transition rate on homozygous allele system.
In this simulation promoter transition rate of both allele is kept same, hence represent a homozy-
gous system. To understand how promoter transition rate effect protein expression stochastic
simulation are performed with following rate constant(a. ka=kd= 2 × 10−2
, b. ka=kd= 2 × 10−3
,
c. ka=kd= 2 × 10−4
, d. ka=kd= 2 × 10−5
). With each rate constant 1000 simulation are per-
formed and noise (deﬁned as the standard deviation divided by the mean) of X1, X2, X3 is
calculated and it is observed that by increasing promoter transition rate reduces the noise in
protein expression(ﬁg.4.11) .
19

Figure 4.11: Homozygous allele system behavior with different promoter transition kinetics
Analysis of protein noises in homozygous allele system shows that by increasing promoter
transition rate we can reduces the noises in protein expression. Noise is deﬁned as the standard
deviation divided by the mean protein. Noise is calculated from simulation result of stochastic
system with following promoter rate transition a. ka=kd= 2 × 10−2
, b. ka=kd= 2 × 10−3
, c.
ka=kd= 2 × 10−4
, d. ka=kd= 2 × 10−5
and calculated noise is plotted against time.
4.3.7 Effect of promoter transition rate on heterozygous allele system.
In this simulation promoter transition rate of both allele is kept different for other allele of a
gene, hence represent a heterozygous system.To understand how promoter transition rate effect
protein expression simulation are performed with following rate constant(a. ka=kd= 2 × 10−3
for allele one and for other is ka=kd= 2 × 10−4
, b. ka=kd= 2 × 10−4
for one allele and for other
ka=kd= 2×10−2
), c. ka=kd= 2×10−3
for one allele and for other ka=kd= 2×10−2
). With each
rate constant 1000 simulation are performed and noise in protein expression is calculated.It is
observed that faster the promoter transition rate lesser will be the noise in protein expression as
shown in ﬁg. 4.12.
20

Figure 4.12: Heterozygous allele system behavior with different promoter transition kinetics.
Analysis of protein noises in heterozygous allele system shows that by increasing promoter
transition rate we can reduces the noises in protein expression. Noise is defined as the standard
deviation divided by the mean protein. Noise is calculated from simulation result of stochastic
system with following promoter rate transition a. ka=kd= 2×10−3
for allele one and for other is
ka=kd= 2 × 10−4
, b. ka=kd= 2 × 10−4
for one allele and for other ka=kd= 2 × 10−2
), c. ka=kd=
2 × 10−3
). and calculated noise is plotted against
time.
4.3.8 Effect of allele differences on stochastic gene expression
Allelic differences in the promoter region can arise due to various reason ex. mutation in single
allele which may modify the binding affinity of the transcription factor at the promoter region
results in the change of promoter activation kinetics which can lead to variation in protein
expression pattern. In an average population, the same amount of proteins is observed for
different promoter rate kinetics, although their huge difference in variation within clonal cell
population as shown in Noise vs Time plot shown in fig 4.13.
21

Figure 4.13: Effect of allele differences on stochastic gene expression.
Allelic differences leads to variation in gene expression in a single cell shown in Noise vs Time
plot whereas average population behaviur of protein is same shown in G vs Time plot. The
average population behavior mask the behavior and variation that arise in single cell. Noise
is deﬁned as the standard deviation divided by the mean protein. Stochastic simulation are
performed with following promoter transition kinetics a. ka=kd= 2 × 10−2
for both allele , b.
ka=kd= 2 × 10−3
), c. ka=kd= 2 × 10−3
for both
allele.
4.3.9 Analyses of protein noises of X1, X2, X3 .
Stochastic network of biallelic gene expression is simulated with with promoter transition rate
ka=kd=2×10−2
using Matlab 2014b. Simulated result of stochastic gene expression is analysed.
Noise in protein expression of X1, X2, X3 is calculated for 1000 simulation. Noise of one
protein is plotted against noise of another protein and it is observed that their is correlation in
their noise as shown in ﬁgure 4.14. It seems that when variation arises in the expression of one
protein it will consequently affect protein expression of other protein. A similar result for other
promoter transition rate.
22

Figure 4.14: Results of protein noise analysis.
Analysis of protein noises showing the correlation between X1, X2, X3 noises in protein expres-
sion. It is evident that when their is variation in expression of one protein it will consequently
effects the expression of other protein. Noise is deﬁned as the standard deviation divided by
the mean protein. Noise is calculated from simulation result of stochastic system with promoter
rate transition ka=kd=2 × 10−2
.
23

Chapter 5
Conclusion
I have created stochastic and deterministic oscillatory network of biallelic gene expression. All
network is converted to system of ordinary differential equation’s and simulation are performed
with different promoter transition rate(1000 simulation for each promoter transition rate) using
Matlab 2014b. On analysis of simulated data following result is observed.
• Stochasticity in gene expression lead to damping of oscillation of average population
protein expression.
• In stochastic system, noises in gene expression is reduced on increasing the promoter
transition rate in homozygous and heterozygous allele system.
• Allelic differences lead to variation at single cell level. Heterozygous allele expression
showing large variation at single cell level as compare to homozygous allele expression
which is not observed in average population behavior of gene expression. It is evident
that average population behavior of protein is masking the variation arise at single cell
level.
24

Chapter 6
Appendix
1 / / Author : Kaushalesh Gupta , Email : kaushalesh . gupta@gmail . com ;
2 Dr . Biplab Bose , Email : biplabbose@gmail . com .
3 / / Date : september 2015.
4 / / This s e t of f u n c t i o n used f o r d e t e r m i n i s t i c s i m u l a t i o n .
5 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
6
7 f u n c t i o n Main
8 c l o s e a l l ; c l e a r a l l ; c l c ;
9 t r y
10 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
11 %The main f u n c t i o n reads the i n p u t data from the model
12 [ model , t , x , k ] = Parameter ;
13 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
14 %The main f u n c t i o n c a l l s the ode s o l v e r to solve the ODE
15 o p t i o n s = odeset ( ’ RelTol ’ ,10ˆ −8 , ’ AbsTol ’ ,10ˆ −8 , ’ MaxStep ’ ,100) ;
16 [T , X] = ode15s (@ ( t , x ) Equation ( t , x , k ) , t , x , o p t i o n s ) ;
17 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
18 %The s i m u l a t e d r e s u l t s are s t o r e d in an a r r a y ’ data ’
19 data = [T , X] ;
20 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
21 %The Text f u n c t i o n w r i t e s the data in a t e x t f i l e with unique id f o r each
22 %run
23 Text ( model , x , k , data ) ;
24 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
25 %The below catch f u n c t i o n r e c o r d s the e r r o r thrown by matlab
26 catch e r r o r
25

27 f i d = fopen ( ’ E r r o r F i l e . t x t ’ , ’ a+ ’ ) ;
28 f p r i n t f ( fid , ’%s n ’ , e r r o r . message ) ;
29 f o r i = 1 : l e n g t h ( e r r o r . s t a c k )
30 f p r i n t f ( fid , ’%s a t %dn ’ , e r r o r . s t a c k ( i ) . name , e r r o r . s t a c k ( i ) . l i n e ) ;
31 end
32 f c l o s e ( f i d )
33 end
34 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
35
36 f u n c t i o n [ model , t , x , k ] = Parameter
37 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
38 %Enter the name of the model
39 model = ’ Reaction ’ ;
40 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
41 %Enter the i n i t i a l values of the s p e c i e s t h a t does not r e q u i r e random
42 %number g e n e r a t i o n
43 x ( 1 ) = 45;
44 x ( 2 ) = 140;
45 x ( 3 ) = 125;
46 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
47 %Enter the i n i t i a l values of the s p e c i e s t h a t r e q u i r e random number
48 %g e n e r a t i o n
49 %To get s h u f f l e d random number every time , use the below two l i n e s
50 s = RandStream ( ’ mt19937ar ’ , ’ Seed ’ , ’ s h u f f l e ’ ) ;
51 RandStream . setGlobalStream ( s ) ;
52 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
53 %Enter the values of the parameters
54 k ( 1 ) = 1∗10ˆ −2; k ( 2 ) = 1∗10ˆ −4; k ( 3 ) = 1∗10ˆ −6; k ( 4 ) =1∗10ˆ −3;
55 k ( 5 ) =2∗10ˆ −4; k ( 6 ) =2∗10ˆ −4; k ( 7 ) =2∗10ˆ −4;
56 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
57 t ( 1 ) =0; t ( 2 ) =250000;
58 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
59 end
60 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
61
62 f u n c t i o n dx = Equation ( t , x , k )
63 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
64 % Enter the e q ua t io n below
65 dx ( 1 ) = k ( 1 )−k ( 2 ) ∗ ( x ( 1 ) / ( x ( 1 ) +k ( 3 ) ) ) ∗x ( 3 ) ;
26

66 dx ( 2 ) = k ( 4 ) ∗ x ( 1 ) − k ( 5 ) ∗ x ( 2 ) ;
67 dx ( 3 ) = k ( 6 ) ∗ x ( 2 ) − k ( 7 ) ∗ x ( 3 ) ;
68 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
69 %j u s t above the l i n e dx = dx ’ ;
70 dx = dx ’ ;%Do not change
71 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
72 end
73 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
74
75 f u n c t i o n Text ( model , x , k , data )
76 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
77 %The below s t a n z a c r e a t e s a new f o l d e r with curremt system time
78 time = d a t e s t r (now , ’yyyymmdd HHMMSS ’ ) ;
79 c u r d i r = pwd ;
80 newdir = f u l l f i l e ( c u r d i r , s p r i n t f ( ’%s %s ’ , model , time ) ) ;
81 mkdir ( newdir ) ;
82 cd ( newdir ) ;
83 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
84 %The below s t a n z a c r e a t e s a t e x t f i l e f o r data
85 filename1 = s p r i n t f ( ’ Data %s . t x t ’ , time ) ;
86 f i d = fopen ( filename1 , ’ wt ’ ) ;
87 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
88 %The below loop p r i n t s l a b e l to the data
89 f p r i n t f ( fid , ’%s t ’ , ’ t ’ ) ;
90 num = l e n g t h ( x ) ;
91 f o r i = 1 : num
92 f p r i n t f ( fid , ’%s%d t ’ , ’x ’ , i ) ;
93 end
94 f p r i n t f ( fid , ’n ’ ) ;
95 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
96 %The below s t a t e m e n t appends the data as tab d e l i m i t e d with t h r e e decimal
97 %p o i n t p r e c i s i o n
98 dlmwrite ( filename1 , data , ’ d e l i m i t e r ’ , ’ t ’ , ’ p r e c i s i o n ’ , ’%.8 f ’ , ’−append
’ ) ;
99 f c l o s e ( f i d ) ;
100 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
101 %The below s t a n z a c r e a t e s a t e x t f i l e with header
102 filename2 = s p r i n t f ( ’ Header %s . t x t ’ , time ) ;
27

104 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
105 %The below loop p r i n t s dashed l i n e to mark the beginning
106 f o r i = 1 : 40
107 f p r i n t f ( fid , ’%c ’ , ’−’ ) ;
108 end
109 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
110 %The below s t a t e m e n t p r i n t s the model name
111 f p r i n t f ( fid , ’nnModel name : %s nn ’ , model ) ;
112 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
113 %The below loop p r i n t s the i n i t i a l c o n d i t i o n s of the s p e c i e s
114 f p r i n t f ( fid , ’ I n i t i a l c o n d i t i o n of the s p e c i e s : n ’ ) ;
115 num = l e n g t h ( x ) ;
116 f o r i = 1 : num
117 f p r i n t f ( fid , ’%s%d :%.8 f ; ’ , ’x ’ , i , x ( i ) ) ;
118 end
119 f p r i n t f ( fid , ’nn ’ ) ;
120 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
121 %The below loop p r i n t s the parameter values
122 f p r i n t f ( fid , ’ Values of the parameters : n ’ ) ;
123 num = l e n g t h ( k ) ;
124 f o r i = 1 : num
125 f p r i n t f ( fid , ’%s%d :%.8 f ; ’ , ’k ’ , i , k ( i ) ) ;
126 end
127 f p r i n t f ( fid , ’nn ’ ) ;
128 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
129 %The below loop p r i n t s dashed l i n e to mark the end
130 f o r i = 1 : 40
131 f p r i n t f ( fid , ’%c ’ , ’−’ ) ;
132 end
133 f p r i n t f ( fid , ’n ’ ) ;
134 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
135 %The l o c a t i o n i s r e v e r t e d to the p a r t e n t d i r e c t o r y f o r the next run
136 f c l o s e ( f i d ) ;
137 cd ( c u r d i r ) ;
138 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
139 end
140 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Listing 6.1: Matlab Script for Deterministic Simulation
28

3 / / Date : November 2015.
4 / / This s e t of f u n c t i o n used f o r s t o c h a s t i c s i m u l a t i o n using random number .
5 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
6
9 t r y
10 %
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−
12 [ model , t , x , k , p ] = Parameter ;
13 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
16 [T , X] = ode15s (@ ( t , x ) Equation ( t , x , k , p ) , t , x , o p t i o n s ) ;
17 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
19 data = [T , X] ;
20 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
22 %run
24 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
25 %Binning f u c t i o n w i l l bin the data and give average value in each bin
26 %a f t e r the binning data w i l l s t o r e in BinMean matrix
27 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
28 %To save the data in t e x t format a f t e r the end of loop
29 %To summed matrix divided by no of loop run .
30 clock
31 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
33 catch e r r o r
29

38 end
39 f c l o s e ( f i d )
40 end
41 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
42
43 f u n c t i o n [ model , t , x , k , p ] = Parameter
44 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
47 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
50 x ( 1 ) = 0;
51 x ( 2 ) = 0;
52 x ( 3 ) = 0;
53 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
59 %For normal d i s t r i b u t i o n follow the below syntax
60 % x ( 1 ) = normrnd (mu, sigma , m, n )
61 % mu i s the mean , sigma i s the s t a n d a r d d e v i a t i o n
62 % m, n i s the s i z e of the a r r a y . I t should be 1 ,1
63 %For log normal d i s t r i b u t i o n follow the below syntax
64 % x ( 1 ) = lognrnd (mu, sigma , m, n )
65 % mu and sigma are the mean and s t a n d a r d d e v i a t i o n , r e s p e c t i v e l y ,
66 % of the a s s o c i a t e d normal d i s t r i b u t i o n
67 % m, n i s the s i z e of the a r r a y . I t should be 1 ,1
68 %x ( 1 ) = normrnd (2 , 0.1 , 1 , 1) ;
69 %x ( 2 ) = lognrnd (2 , 0.1 , 1 , 1) ;
70 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
72 k ( 1 ) = 48∗10ˆ −6; k ( 2 ) = 4.8 ∗10ˆ −5; %k ( 3 ) = 10; k ( 4 ) = 10;
73 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
74 %Enter the i n i t i a l and f i n a l l i m i t of time f o r the s i m u l a t i o n
75 t ( 1 ) = 0; t ( 2 ) = 3600000;
30

76 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
77 %Enter the i n i t i a l c o n d i t i o n of the promoter ( on / o f f )
78 %Enter t r u e i f ON; f a l s e i f OFF
79 %I f t h e r e are more than one promoter , l i s t them in an a r r a y
80 %For e . g . , i n i t p m t r ( 1 ) = t r u e ; i n i t p m t r ( 2 ) = f a l s e ;
81 g l o b a l i n i t p m t r ;
82 i n i t p m t r ( 1 ) = f a l s e ;
83 i n i t p m t r ( 2 ) = f a l s e ;
84 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
85 %Enter the p r o b a b i l i t y of promoter s w i t c h i n g from ON to OFF
86 %p ( 1 , 1 ) p r o b a b i l i t y t h a t the promoter ( 1 ) w i l l switch from OFF to ON
87 %p ( 1 , 2 ) p r o b a b i l i t y t h a t the promoter ( 1 ) w i l l switch from ON to OFF
88 %I f t h e r e are more than one promoter , append them in the a r r a y
89 %For e . g . , f o r promoter 1 −−> p ( 1 , 1 ) = 0 . 1 ; p ( 1 , 2 ) = 0 . 1 ;
90 %f o r promoter 2 −−> p ( 2 , 1 ) = 0 . 5 ; p ( 2 , 2 ) = 0 . 2 ;
91 p ( 1 , 1 ) = 2∗10ˆ −4; p ( 1 , 2 ) = 2∗10ˆ −4;
92 p ( 2 , 1 ) = 2∗10ˆ −4; p ( 2 , 2 ) = 2∗10ˆ −4;
93 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
94 end
95 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
96
97 f u n c t i o n dx = Equation ( t , x , k , p )
98 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
99 %The s t a t u s of the promoter i s s t o r e d in the g l o b a l v a r i a b l e ’ i n i t p m t r ’
100 %The s t a t u s i s updated f o r every run by c a l l i n g teh f u n c t i o n ’ Promoter ’
102 s t a t u s = Promoter ( p ) ;
103 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
104 %To i n t r o d u c e s t o c h a s t i c i t y use the below code .
105 %Each promoter should be t r a e t e d as a s e p a r a t e v a r i a b l e i . e . , x ( 1 ) , x ( 2 )
106 %Place the ode ’ s t h a t w i l l be a f f e c t e d by prmoter ’ s s t a t u s ,
107 %under ’ if ’ and ’ else ’ s t a t e m e n t .
108 %For more than one promoter , copy the below i f and e l s e s t a t e m e n t
109 %a c c o r d i n g l y .
110 %Do not change the o t h e r p a r t of the code .
111 dx ( 1 ) =( k ( 1 ) / 2 ) ∗ ( s t a t u s ( 1 ) + s t a t u s ( 2 ) )−k ( 2 ) ∗x ( 1 )
112 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
113 %Place the ode ’ s t h a t are not a f f e c t e d by the promoter ’ s s t a t u s ,
31

116 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
117 i n i t p m t r = s t a t u s ;
118 end
119 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
120
121 f u n c t i o n p r o m o t e r s t a t u s = Promoter ( p )
122 %Do not make any changes to the t h i s f u n c t i o n .
123 %This f u n c t i o n g e n e r a t e s uniformly d i s t r i b u t e d random number and updates
124 %the promoter s t a t u s f o r each and every run of the s i m u l a t i o n
125 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
127 promoter count = l e n g t h ( i n i t p m t r ) ;
128 p r o m o t e r s t a t u s = zeros ( promoter count , 1) ;
129 random = rand ( promoter count , 1) ;
130 f o r i = 1 : promoter count
131 p r o m o t e r s t a t u s ( i ) = i n i t p m t r ( i ) ;
132 i f p r o m o t e r s t a t u s ( i ) == t r u e
133 i f random ( i ) <= p ( i , 2)
134 p r o m o t e r s t a t u s ( i ) = f a l s e ;
135 end
136 e l s e
137 i f random ( i ) <= p ( i , 1)
138 p r o m o t e r s t a t u s ( i ) = t r u e ;
139 end
140 end
141 end
142 end
143 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
144
146 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
149 c u r d i r = pwd ;
152 cd ( newdir ) ;
153 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
32

157 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
159 f p r i n t f ( fid , ’%s t ’ , ’ t ’ ) ;
160 num = l e n g t h ( x ) ;
161 f o r i = 1 : num
163 end
164 f p r i n t f ( fid , ’n ’ ) ;
165 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
168 dlmwrite ( filename1 , data , ’ d e l i m i t e r ’ , ’ t ’ , ’ p e c i s i o n ’ , ’%.8 f ’ , ’−append ’
) ;
169 f c l o s e ( f i d ) ;
170 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
174 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
176 f o r i = 1 : 40
177 f p r i n t f ( fid , ’%c ’ , ’−’ ) ;
178 end
179 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
182 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
185 num = l e n g t h ( x ) ;
186 f o r i = 1 : num
188 end
189 f p r i n t f ( fid , ’nn ’ ) ;
190 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
33

193 num = l e n g t h ( k ) ;
194 f o r i = 1 : num
196 end
197 f p r i n t f ( fid , ’nn ’ ) ;
198 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
200 f o r i = 1 : 40
201 f p r i n t f ( fid , ’%c ’ , ’−’ ) ;
202 end
203 f p r i n t f ( fid , ’n ’ ) ;
204 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
206 f c l o s e ( f i d ) ;
207 cd ( c u r d i r ) ;
208 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
209 end
210 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Listing 6.2: Matlab Script for stochastic Simulation with promoter transition using random
number.
Listing 6.3: Python Script for binning and extraction of data
3 / / Date : december 2016.
4 / / This program does binning and e x t r a c t i o n of data ( Sampling of Data )
5
6 from o p e r a t o r import add
7 import sys
8 import os
9
10 def bin ( t1 , t2 , n , no var , f i n ) :
11 s t e p s i z e = ( t2−t1 ) / ( n∗ 1 . 0 )
12 c u r b i n = t1 + ( s t e p s i z e / 2 )
13 n e x t b i n = t1 + s t e p s i z e
14 b i n d i c ={} # I n t i l i a z e a empty dic t h a t w i l l c o n t a i n bin as key as
i t s corresponding avg as i t s values
15 b i n d i c [ c u r b i n ] = [ 0 . 0 ] ∗ no var
34

16 c u r b i n c o u n t =0
17 f =open ( f i n + ’ . t x t ’ , ’ r ’ )
18 l i n e l i s t = f . r e a d l i n e s ( ) # s t o r e a l l data in one go
19 f . c l o s e ( )
20 fout name = f i n . r e p l a c e ( ’ Data ’ , ’ bin Data ’ ) + ’ . t x t ’
21 f =open ( fout name , ’w’ )
22 f o r i , item in enumerate ( l i n e l i s t [ 1 : ] ) :
23 c u r l i n e l i s t = item . s t r i p ( ) . s p l i t ( ’ t ’ )
24 # p r i n t ’ c u r l i n e l i s t : ’ , c u r l i n e l i s t
25 ## s c o n v e r t i n g s t r i n g to f l o a t
26 f o r j , item2 in enumerate ( c u r l i n e l i s t ) :
27 c u r l i n e l i s t [ j ] = f l o a t ( c u r l i n e l i s t [ j ] )
28 i f c u r l i n e l i s t [ 0 ] < n e x t b i n :
29 b i n d i c [ c u r b i n ] =map ( sum , zip ( b i n d i c [ c u r b i n ] , c u r l i n e l i s t
[ 1 : ] ) )
30 c u r b i n c o u n t +=1
31 e l s e :
32 # R e i n t i a l i z e s a l l needed temproray v a r i a b l e s
33 i f c u r b i n c o u n t ==0:
34 p r i n t i , cur bin , c u r b i n c o u n t
35 b i n d i c [ c u r b i n ] = map ( lambda x : x / c u r b i n c o u n t , b i n d i c [
c u r b i n ] )
36 s t = s t r ( c u r b i n ) + ’ t ’
37 f o r j , item2 in enumerate ( b i n d i c [ c u r b i n ] ) :
38 s t += ’ %0.8 f t ’ %(item2 )
39 f . w r i t e ( s t + ’n ’ )
40 # update bin value
41 c u r b i n = n e x t b i n + ( s t e p s i z e / 2 )
42 n e x t b i n = n e x t b i n + s t e p s i z e
43 # c u r b i n c o u n t =0
44 b i n d i c [ c u r b i n ] = c u r l i n e l i s t [ 1 : ]
45 c u r b i n c o u n t =1
46 f . c l o s e ( )
47
48 def copy column ( no of bin , no var , b a s e d i r ) :
49 ’ ’ ’ Assumes t h a t each d i r e c t o r y c o n t a i n s bin f i l e and
50 put corresponding in new d i r e c t o r y ’ ’ ’
51 o u t d i r = ’ s e p v a r i a b l e ’ + b a s e d i r
52 t r y :
35

53 os . mkdir ( o u t d i r )
54 except :
55 p r i n t ’ Dir %s a l r e a d y e x i s t s ’ %( o u t d i r )
56 #open l i s t of f i l e s f o r w r i t i n g
57 o u t f i l e p o i n t e r l i s t =[]
58 f o r i in range ( no var ) :
59 c u r f i l e n a m e = ’%s / column %d . t x t ’ %( o u t d i r , i )
60 o u t f i l e p o i n t e r l i s t . append ( open ( c u r f i l e n a m e , ’w’ ) )
61 d i r l i s t = os . l i s t d i r ( b a s e d i r )
62 #open l i s t of f i l e s f o r re ading
63 i n f i l e p o i n t e r l i s t =[]
64 f o r item in d i r l i s t :
65 cur dir name = ’%s/%s ’ %( b a s e d i r , item )
66 c u r d i r l i s t = os . l i s t d i r ( cur dir name )
67 f o r item2 in c u r d i r l i s t :
68 i f item2 . s t a r t s w i t h ( ’ bin Data ’ ) :
69 i n f i l e n a m e = ’%s/%s/%s ’ %( b a s e d i r , item , item2 )
70 i n f i l e p o i n t e r l i s t . append ( open ( i n f i l e n a m e , ’ r ’ ) )
71 # Reading and w r i t i n g s i m u l t a n e o u s l y by each l i n e by l i n e
72 f o r i in range ( n o o f b i n ) :
73 t e m p s t r =[ ’ ’ ] ∗ no var
74 c u r t i m e = ’ ’
75 f o r item in i n f i l e p o i n t e r l i s t :
76 c u r l i n e l i s t = item . r e a d l i n e ( ) . s t r i p ( ) . s p l i t ( ’ t ’ )
77 c u r t i m e = c u r l i n e l i s t [ 0 ]
78 f o r j , item2 in enumerate ( c u r l i n e l i s t [ 1 : ] ) :
79 t e m p s t r [ j ] += item2 + ’ t ’
80 f o r k , item in enumerate ( o u t f i l e p o i n t e r l i s t ) :
81 item . w r i t e ( c u r t i m e + ’ t ’+ t e m p s t r [ k ] + ’n ’ )
82 # c l o s e the f i l e p o i n t e r
83 f o r item in o u t f i l e p o i n t e r l i s t :
84 item . c l o s e ( )
86 item . c l o s e ( )
87
88 def sum row ( no of bin , no var , b a s e d i r ) :
89 o u t d i r = ’ Avg ’ + b a s e d i r
90 t r y :
91 os . mkdir ( o u t d i r )
36

92 except :
93 p r i n t ’ Dir %s a l r e a d y e x i s t s ’ %( o u t d i r )
94 #open l i s t of f i l e s f o r w r i t i n g
95 c u r f i l e n a m e = ’%s / sum row . t x t ’ %( o u t d i r )
96 o u t f i l e p o i n t e r =open ( c u r f i l e n a m e , ’w’ )
98 n o d i r = len ( d i r l i s t )
99 #open l i s t of f i l e s f o r re ading
100 i n f i l e p o i n t e r l i s t =[]
102 cur dir name = ’%s/%s ’ %( b a s e d i r , item )
103 c u r d i r l i s t = os . l i s t d i r ( cur dir name )
104 f o r item2 in c u r d i r l i s t :
105 i f item2 . s t a r t s w i t h ( ’ bin Data ’ ) :
106 i n f i l e n a m e = ’%s/%s/%s ’ %( b a s e d i r , item , item2 )
107 i n f i l e p o i n t e r l i s t . append ( open ( i n f i l e n a m e , ’ r ’ ) )
108 # Reading and w r i t i n g s i m u l t a n e o u s l y by each l i n e by l i n e
109 f o r i in range ( n o o f b i n ) :
110 temp sum = [ 0 . 0 ] ∗ no var
111 c u r t i m e = ’ ’
113 c u r l i n e l i s t = item . r e a d l i n e ( ) . s t r i p ( ) . s p l i t ( ’ t ’ )
114 c u r t i m e = c u r l i n e l i s t [ 0 ]
115 f o r j , item2 in enumerate ( c u r l i n e l i s t [ 1 : ] ) :
116 temp sum [ j ] += f l o a t ( item2 )
117 s t = c u r t i m e
118 f o r item in temp sum :
119 s t += ’ t %0.8 f ’ %(item / n o d i r )
120 o u t f i l e p o i n t e r . w r i t e ( s t + ’n ’ )
121 # c l o s e the f i l e p o i n t e r
122 o u t f i l e p o i n t e r . c l o s e ( )
124 item . c l o s e ( )
125
126 def main ( ) :
127 #Make changes here
128 b a s e d i r = ’ data ’ # Enter base d i r e c t o r y name
129 no var =3 # Enter no of v a r i a b l e s in s i m u l a t i o n
130 n=1000 # Enter no of Bin
37

131 t1 =0 # I n t i a l Time of Simulation
132 t2 =250000 # F i n a l Time of Simulation
135 f i l e l i s t i n c u r d i r = os . l i s t d i r ( b a s e d i r + ’ / ’+ item )
136 f o r item2 in f i l e l i s t i n c u r d i r :
137 i f item2 . s t a r t s w i t h ( ’ Data ’ ) :
138 f i n = ’%s/%s/%s ’ %( b a s e d i r , item , item2 . s t r i p ( ’ . t x t ’ ) )
139 bin ( t1 , t2 , n , no var , f i n )
140 p r i n t ’ Processed f i l e ’ , f i n
141 copy column ( n , no var , b a s e d i r )
142 sum row ( n , no var , b a s e d i r )
143
144
145 i f name == ’ m a i n ’ :
146 main ( )
3 / / Date : January 2016.
4 / / This s e t of f u n c t i o n used f o r s t o c h a s t i c using poisson proce ss .
5 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
6
9 t r y
10 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
12 [ model , t , x , k ] = Parameter ;
13 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
16 [T , X] = ode15s (@ ( t , x ) Equation ( t , x , k ) , t , x , o p t i o n s ) ;
17 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
19 data = [T , X] ;
20 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
22 %run
38

24 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
26 catch e r r o r
31 end
32 f c l o s e ( f i d )
33 end
34 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
35
36
37 f u n c t i o n [ model , t , x , k ] = Parameter
38 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
41 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
44 x ( 1 ) = 45;
45 x ( 2 ) = 140;
46 x ( 3 ) = 125;
47 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
53 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
54 % This f u n c t i o n c a l l s the i n t e r a r r i v a l t i m e f u n c t i o n and g e n e r a t e and s t o r e
55 % promoter s t a t u s f o r a l l e l e 1 in t e x t f i l e .
56 [w, p ] = i n t e r a r r i v a l t i m e a l l e l e 1 ;
57 % i t w i l l import the t e x t f i l e f o r promoter1 f i l e f o r ALLELE 1
58 N = dlmread ( ’ i n t e r a r r i v a l t i m e a l l e l e 1 . t x t ’ ) ;
59 g l o b a l tn ;
60 t l = N( : , 1 ) ;
61 tn = [ t l ; max ( t l ) +1];
39

62 g l o b a l r ;
63 tk = N( : , 2 ) ;
64 r = [ tk ; 1 ] ;
65 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
67 k ( 1 ) = 1∗10ˆ −2; k ( 2 ) = 1∗10ˆ −4; k ( 3 ) = 1∗10ˆ −6; k ( 4 ) =1∗10ˆ −3;
68 k ( 5 ) =2∗10ˆ −4; k ( 6 ) =2∗10ˆ −4; k ( 7 ) =2∗10ˆ −4;
69 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
70 t ( 1 ) =0; t ( 2 ) =250000;
71 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
72 end
73 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
74
75 f u n c t i o n [w, p ] = i n t e r a r r i v a l t i m e a l l e l e
76 c l o s e a l l ; c l c ;
77 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
78 % Make changes here
79 i =1; % i n t i a l i s a t i o n of i ( increment o p e r a t o r )
80 w( i ) =0; % i n t i a l i s a t i o n of timr
81 p ( i ) =0; % i n t i a l i s a t i o n of promoter s t a t u s
82 T= 260000; % Time up to which promoter s t a t u s i s r e q u i r e d
83 kon= 2∗10ˆ −3; % Rate of t r a n s i t i o n of a l l e l e 1 from o f f to on
84 koff =2∗10ˆ −3; % Rtae of t r a n s i t i o n of a l l e l e 1 from on to o f f
85 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
86 while w( i ) <T
87 i f p ( i ) ==0;
88 u =rand ( 1 ) ;
89 time = − 1/ kon∗ log ( u ) ;
90 i f (w( i ) +time ) <= T ;
91 w( i +1)=w( i ) +time ;
92 p ( i +1) =1;
93 e l s e
94 break
95 end
96 i = i +1;
97 e l s e u =rand ( 1 ) ;
98 time= − 1/ koff ∗ log ( u ) ;
99 i f (w( i ) +time ) <T ;
100 w( i +1)=w( i ) +time ;
40

101 p ( i +1) =0;
102 e l s e
103 break
104 end
105 i = i +1;
106 end
107 end
108 data1 = [w; p ] ;
109 data = data1 . ’ ;
110 %time = d a t e s t r (now , ’yyyymmdd HHMMSS’ ) ;
111 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
113 %filename1 = s p r i n t f ( ’ i n t e r a r r i v a l t i m e %s . txt ’ , time ) ;
114 filename1 = s p r i n t f ( ’ i n t e r a r r i v a l t i m e a l l e l e 1 . t x t ’ ) ;
116 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ ) ;
120 f c l o s e ( f i d ) ;
121 end
122 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
123
124 f u n c t i o n dx = Equation ( t , x , k )
125 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
126 %The s t a t u s of the promoter i s s t o r e d in the g l o b a l v a r i a b l e ’ i n i t p m t r ’
127 %The s t a t u s i s updated f o r every run by c a l l i n g teh f u n c t i o n ’ Promoter ’
128 s t a t u s = P r o m o t e r a l l e l e 1 ( t ) ;
129 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
130 % Enter the e q ua t io n below
131 dx ( 1 ) = k ( 1 )−k ( 2 ) ∗ ( x ( 1 ) / ( x ( 1 ) +k ( 3 ) ) ) ∗x ( 3 ) ;
132 dx ( 2 ) = ( k ( 4 ) / 2 ) ∗ ( s t a t u s 1 + s t a t u s 2 ) ∗ x ( 1 ) − k ( 5 ) ∗ x ( 2 ) ;
133 dx ( 3 ) = k ( 6 ) ∗ x ( 2 ) − k ( 7 ) ∗ x ( 3 ) ;
134 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
137 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
138 end
41

139 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
140
141 f u n c t i o n p r o m o t e r 1 s t a t u s = P r o m o t e r a l l e l e ( t )
142 %Do not make any changes to the t h i s f u n c t i o n .
143
144 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
145 % Below f u n c t i o n w i l l check the time and l o c a t e the p o s i i t i o n of time
146 % and a c c o r d i n g l y check the promoter s t a t u s a t t h a t p o s i i t i o n
147 g l o b a l posi ;
148 g l o b a l tn ;
149 g l o b a l r ;
150 i f ( any ( t == tn ) ) ;
151 posi = f i n d ( tn == t ) ;
152 p r o m o t e r 1 s t a t u s 1 = r ( posi ) ;
153 e l s e
154 i f ( tn ( posi )<= t && t <= tn ( posi +1) )
156 e l s e i f ( t <= tn ( posi ) ) ;
157 posi = posi −1;
159 e l s e
160 posi = posi + 1 ;
162 end
163 end
164 end
165 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
166
168 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
171 c u r d i r = pwd ;
174 cd ( newdir ) ;
175 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
42

179 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
181
182 f p r i n t f ( fid , ’%s t ’ , ’ t ’ ) ;
183 num = l e n g t h ( x ) ;
184 f o r i = 1 : num
186 end
187 f p r i n t f ( fid , ’n ’ ) ;
188 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ ) ;
192 f c l o s e ( f i d ) ;
193 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
197 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
199 f o r i = 1 : 40
200 f p r i n t f ( fid , ’%c ’ , ’−’ ) ;
201 end
202 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
205 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
208 num = l e n g t h ( x ) ;
209 f o r i = 1 : num
211 end
212 f p r i n t f ( fid , ’nn ’ ) ;
213 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
43

216 num = l e n g t h ( k ) ;
217 f o r i = 1 : num
219 end
220 f p r i n t f ( fid , ’nn ’ ) ;
221 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
223 f o r i = 1 : 40
224 f p r i n t f ( fid , ’%c ’ , ’−’ ) ;
225 end
226 f p r i n t f ( fid , ’n ’ ) ;
227 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
229 f c l o s e ( f i d ) ;
230 cd ( c u r d i r ) ;
231 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
232 end
233 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Listing 6.4: Matlab Script for stochastic Simulation with promoter transition using Poisson
process.
3 / / Date : March 2016.
4 / / This f u n c t i o n used f o r F o u r i e r Transformation .
5 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
6 f u n c t i o n F o u r i e r T r a n s f o r m a t i o n
7 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
8 % to read t e x t f i l e f o r which we want to do f o u r i e r t r a n s f o r m a t i o n
9 % Make changes here
10 M = dlmread ( ’ x1 . t x t ’ ) ;
11 T = M( 2 , 1 )−M( 1 , 1 ) ;
12 Fs = 1/T ; % Sampling frequency
13 [L ,N] = s i z e (M) ;
14 f = Fs∗ ( 0 : ( L / 2 ) ) / L ;
15 Data ( : , 1 ) = f ;
16 % to c a l c u l a t e f o u r i e r t r a n s f o r m of a l l row one by one
17 f o r j =2:N;
18 X=M( : , j ) ;
44

19 % to c a l c u l a t e amplitude
20 Y = f f t (X) ;
21 P2 = abs (Y/ L) ;
22 P1 = P2 ( 1 : L/2+1) ;
23 P1 ( 2 : end −1) = 2∗P1 ( 2 : end −1) ;
24 Data ( : , j ) =P1 ;
25 end
26 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
27 %To save data in t e x t f i l e
28 dlmwrite ( ’ f o u r i e r d a t a . t x t ’ , Data , ’ d e l i m i t e r ’ , ’ t ’ , ’ p r e c i s i o n ’ , ’%.8 f ’ ) ;
29 end
Listing 6.5: Matlab Script for Fourier Transformation
45

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48

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