UNIVERSITY OF CALIFORNIA, SAN DIEGO

Partial Differential Equation Models and Numerical Simulations ofRNA Interactions and Gene Expression

A dissertation submitted in partial satisfaction of the

requirements for the degree

Doctor of Philosophy

in

Mathematics

by

Maryann Elisabeth Hohn

Committee in charge:

Professor Bo Li, ChairProfessor Gaurav AryaProfessor Li-Tien ChengProfessor Jiawang NieProfessor Shyni Varghese

2013

Copyright

Maryann Elisabeth Hohn, 2013

All rights reserved.

The dissertation of Maryann Elisabeth Hohn is approved,

and it is acceptable in quality and form for publication

on microfilm and electronically:

Chair

University of California, San Diego

2013

iii

DEDICATION

To those who think their work is futile.

iv

EPIGRAPH

Multiplication is vexation;

Division is as bad;

The Rule of Three perplexes me,

And fractions drive me mad!

— Nursery rhyme

v

TABLE OF CONTENTS

Signature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Epigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Abstract of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 RNA and Gene Expression . . . . . . . . . . . . . . . . . 11.2 Existing Models and Studies . . . . . . . . . . . . . . . . 41.3 Summary of Thesis Work . . . . . . . . . . . . . . . . . . 6

Chapter 2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . 102.1 Derivation of the Mean-field Model of Two Species . . . . 102.2 Two-species Models in Multiple Dimensions . . . . . . . 182.3 Multiple-species Models in Multiple Dimensions . . . . . 20

Chapter 3 Mathematical Analysis of the Models . . . . . . . . . . . . . . 223.1 Ordinary Differential Equations Analysis . . . . . . . . . 22

3.1.1 The System of ODEs for Reaction and Its Lin-earization . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Linear Stability . . . . . . . . . . . . . . . . . . . 273.2 Partial Differential Equations Analysis . . . . . . . . . . 29

3.2.1 Well-posedness of Two-species Model I . . . . . . 293.2.2 Well-posedness of Multiple-species Model I . . . . 313.2.3 Behavior Analysis of Multiple-species Models . . . 32

Chapter 4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 374.1 Methods for Multiple-species Models in 1-D . . . . . . . 37

4.1.1 Finite Difference and The Neumann BoundaryCondition . . . . . . . . . . . . . . . . . . . . . . 38

vi

4.1.2 Alternating Iteration . . . . . . . . . . . . . . . . 394.1.3 The Crank–Nicolson Method . . . . . . . . . . . . 40

4.2 Methods for Two Species in 2-D . . . . . . . . . . . . . . 414.2.1 Finite Difference Discretization . . . . . . . . . . 424.2.2 Newton’s Method and Gauss–Seidel Iteration . . . 424.2.3 Alternating Iteration . . . . . . . . . . . . . . . . 444.2.4 Explicit vs. Implicit Scheme . . . . . . . . . . . . 44

Chapter 5 Computational Results . . . . . . . . . . . . . . . . . . . . . . 485.1 Multiple-species Models in 1-D . . . . . . . . . . . . . . . 48

5.1.1 Multiple-species Model I . . . . . . . . . . . . . . 505.1.2 Multiple-species Model II . . . . . . . . . . . . . . 535.1.3 Multiple-species Model III . . . . . . . . . . . . . 57

5.2 Two-species Models in 2-D . . . . . . . . . . . . . . . . . 575.2.1 Two-species Model I . . . . . . . . . . . . . . . . 605.2.2 Two-species Model II . . . . . . . . . . . . . . . . 645.2.3 Two-species Model III . . . . . . . . . . . . . . . 64

Chapter 6 Conclusions and Discussions . . . . . . . . . . . . . . . . . . . 686.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . 686.2 Parameters of Interest . . . . . . . . . . . . . . . . . . . 696.3 Accuracy of the Model . . . . . . . . . . . . . . . . . . . 716.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

vii

LIST OF FIGURES

Figure 1.1: DNA-RNA-Proteins . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2: MiRNA and target mRNA binding . . . . . . . . . . . . . . . . 3

Figure 2.1: Kinetic scheme of mRNA and sRNA concentrations . . . . . . . 11

Figure 4.1: Stencils of 1-D modified finite difference method . . . . . . . . . 38Figure 4.2: Multiple-species Model I, test of numerical methods . . . . . . . 39Figure 4.3: Multiple-species Model II, test of Alternating Iteration . . . . . 40Figure 4.4: Multiple-species Model III, test of Crank–Nicolson Method . . . 41Figure 4.5: Stencils of 2-D modified finite difference method . . . . . . . . . 42Figure 4.6: Two-species Model I, Newton’s Method and Gauss–Seidel Iter-

ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 4.7: Two-species Model II, Alternating Iteration . . . . . . . . . . . 44Figure 4.8: Two-species Model III, explicit scheme . . . . . . . . . . . . . . 45Figure 4.9: Two-species Model III, implicit scheme . . . . . . . . . . . . . . 46Figure 4.10: Two-species Model III, explicit vs. implicit scheme . . . . . . . 47

Figure 5.1: Transcription profiles of mRNA and sRNA . . . . . . . . . . . . 49Figure 5.2: Replicated results . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 5.3: Change in mRNA and sRNA concentrations in Multiple-species

Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 5.4: Change in mRNA and sRNA concentrations in Multiple-species

Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 5.5: Change in mRNA and sRNA concentrations in Multiple-species

Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 5.6: Change in mRNA and sRNA concentrations in Multiple-species

Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 5.7: Change in mRNA and sRNA concentrations in Multiple-species

Model III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 5.8: Transcription rates and steady state concentrations of Gene 1 . 59Figure 5.9: Transcription rates and steady state concentrations of Gene 2 . 60Figure 5.10: Sharpening of the interface, Two-species Model I, Gene 1 . . . . 61Figure 5.11: Sharpening of the interface, Two-species Model I, Gene 2 . . . . 62Figure 5.12: Range of diffusion coefficient values for Gene 1 . . . . . . . . . 63Figure 5.13: Range of diffusion coefficient values for Gene 2 . . . . . . . . . 63Figure 5.14: Sharpening of the interface, Two-species Model II, Gene 1 . . . 65Figure 5.15: Sharpening of the interface, Two-species Model II, Gene 2 . . . 66Figure 5.16: Stability analysis of Two-species Model . . . . . . . . . . . . . . 67

viii

LIST OF TABLES

Table 4.1.1:Test functions for 1-D models . . . . . . . . . . . . . . . . . . . 38Table 4.2.1:Test functions for 2-D models . . . . . . . . . . . . . . . . . . . 41

Table 5.1.1:Production rates of mRNA and sRNA . . . . . . . . . . . . . . . 49

ix

ACKNOWLEDGEMENTS

I would like to offer my special thanks to Dr. Bo Li for his enthusiasm,

motivation, and valuable support during the last five years. You helped me discover

a way to meld several disciplines that I love and galvanized me to pursue research

in mathematical biology. Often, our weekly meetings were the most inspiring part

of my week; I will miss them dearly.

I would like to express my great appreciation to all of my fellow doctoral

students for their encouragement, understanding, and friendship that kept the

isolation and loneliness that accompanies doctoral students at bay. Commiserating

with you kept me in the game.

I would like to thank Casey for her understanding and support – even when

she had no idea what I was talking about. I would also like to thank Jeremy who

cajoled me into continuing with my program when I seriously thought of leaving

it.

I am eternally grateful to Tom for all of the love, support, and patience he

has shown me. Your insight and humor has kept me going more times that I can

recall.

And last, but not least, I would like to thank my family whose advice and

encouragement helped me follow my dreams one step at a time. I am eternally

grateful for your endless love and support. Thank you for wanting to talk to me

even when, throughout one week, my personality resembled each of the Seven

Dwarfs. I devoured the elephant!

x

VITA

2005 B.S. in Mathematics with Honors and B.A. in Italian Studieswith Honors, University of California, Santa Barbara

2010 M.A. in Mathematics, University of California, San Diego

2010 Graduate Student Researcher for the Mathematics DiagnosticTesting Project, Univeristy of California, San Diego

2010 Graduate Student Researcher for Bo Li, University of Cali-fornia, San Diego

2010 Ph.D. Candidate in Mathematics, University of California,San Diego

2010-2012 San Diego Fellowship recipient, University of California, SanDiego

2010-2012 Graduate Teaching Assistant, University of California, SanDiego

2010-2013 Junior Research Fellow for the Center for Theoretical Biolog-ical Physics, Univeristy of California, San Diego

2011 Associate Instructor, University of California, San Diego

2013 Ph.D. in Mathematics, University of California, San Diego

xi

ABSTRACT OF THE DISSERTATION

Partial Differential Equation Models and Numerical Simulations ofRNA Interactions and Gene Expression

by

Maryann Elisabeth Hohn

Doctor of Philosophy in Mathematics

University of California, San Diego, 2013

Professor Bo Li, Chair

Our genetic information is stored in the nucleus of our cells via a double he-

lical chain of nucleotides called deoxyribonucleic acid (DNA). DNA is transcribed

into a single chain of nucleotides called ribonucleic acid (RNA) which is then trans-

lated into proteins. New discoveries of other non-coding macromolecules and their

functions along with a new understanding of post-transcriptional protein regula-

tion have influenced the study of these processes. For example, small, non-coding

RNAs (sRNA) such as microRNA (miRNA) or small interfering RNA (siRNA)

regulate developmental events through certain interactions with messenger RNA

(mRNA). By binding to specific sites on a strand of mRNA, sRNA may cause a

gene to be activated or suppressed which may turn a gene “on” or “off.” To un-

xii

derstand these interactions, we developed a mathematical model which consists of

N+1 coupled partial differential equations that describe mRNA and sRNA interac-

tions across cells and tissue. These equations illustrate how one small, non-coding

RNA segment and N target mRNA segments interact with each other depending

on transcription rates, independent and dependent degradation rates, and the rate

of intercellular mobility of each species. By varying diffusion coefficients (mobility

of each species) and time dependence (creating a steady state), the system of N+1

coupled PDEs can be studied as three separate well-posed systems of equations:

a single, nonlinear diffusion equation; coupled diffusion equations at steady state;

and coupled diffusion equations with time dependence. This dissertation analyzes

the mathematical models created and shows the implementation of consistent, ef-

ficient numerical methods such as modified finite difference methods and a form

of alternating iteration to solve these equations. The numerical simulations show

that when sRNA has mobility across tissue, the concentration profiles of mRNA

display a sharp interface between tissue with high mRNA concentration and tis-

sue with low mRNA concentration. If mRNA mobility across tissue is added, the

concentration profile of mRNA is smoothed across the tissue. These simulations

suggest that the mobilities of sRNA and mRNA contribute to the behavior of

mRNA concentrations across tissue. In addition, this model may be utilized to

illustrate similar types of interactions between multiple chemical species.

xiii

Chapter 1

Introduction

This dissertation concerns the development of mathematical models and

numerical methods to understand interactions between different types of RNA

molecules in cells and tissues and their consequences in the expression of a gene.

1.1 RNA and Gene Expression

Each of us has genetic information encoded within our cells via a double

helical molecule called deoxyribonucleic acid (DNA) which looks like a twisted

ladder. Each rung of the ladder consists of two bases, either adenine and thymine

or cytosine and guanine, held together by a bond. The vertical sides of the ladder

are called the backbone of the DNA molecule and consist of alternating groups of

sugar (deoxyribose) and phosphate. If we combine a base, sugar, and phosphate

together, we call it a nucleotide.

To keep the genetic information safe from being tampered with or changed

inadvertently, DNA replicates itself guardedly and uses a delegate to tell the rest

of the body what to make. The main representative the DNA uses is ribonucleic

acid or RNA. Messenger RNA (mRNA) is a single stranded copy of one side of

the double stranded DNA with a small change in the four bases; the base thymine

is replaced by uracil. mRNA transports the genetic information from the DNA

to the cytoplasm of the cell where the mRNAs are translated into proteins. Fig-

ure 1.1 displays the general transfers of DNA information including transcription

1

2

Figure 1.1: DNA is transcribed into RNA and then translated into proteins.

Ribosomes assemble proteins using amino acids delivered by transfer RNAs (tR-

NAs) [22].

and translation.

In addition to mRNA, we have small, non-coding RNAs that do not code

for proteins like mRNA, but do effect the proteins that become translated. These

small RNAs bind to target sites along the mRNA strand in the cytoplasm, resulting

in proteins that were previously translated being suppressed. This suppression

changes the expression of that gene. Particularly, we are interested in how certain

changes in the spatial concentrations of small, non-coding RNAs and mRNAs effect

the expression of a gene across tissue.

Small, non-coding regulatory RNAs (sRNAs) are now known to be one

of the primary regulators of gene expression. These sRNA include micro RNA

(miRNA) and short-interfering RNA (siRNA), both short RNA molecules about

3

21 nucleotides long. Each type base-pairs with a mRNA target, resulting in either

inhibiting translation or causing degradation. Humans may express about one

thousand miRNAs, most of them occurring during embryonic development and

after birth. In addition, many miRNAs can target more than one mRNA.

Micro RNA gene regulation appears in all multicellular plants and animals,

although the type of base-pair binding may differ. A miRNA may bind to a target

mRNA site at a place of base-pair complementarity, either near perfectly or imper-

fectly. In near perfect complementarily binding, the base-pairs are in a formation

of a near perfect duplex and leads to mRNA cleavage and degradation. In imper-

fect complementarity binding, miRNA regulates a gene by binding to multiple sites

that code for specific proteins, negatively regulating expression. Figure 1.2 shows

biogenesis of miRNA including the different base-pair bindings and the resulting

inhibition or degradation. In both cases, miRNA regulates gene expression on a

translational level.

Figure 1.2: Simplified diagram of miRNA biogenesis pathway. Notice miRNA

binding to its target mRNA through imperfect complimentarity causing transla-

tional repression or perfect complimentarity causing RNA interference (reprinted

from Cuellar and McManus [3]).

4

By base-pairing to a mRNA, a miRNA may turn a gene on or off. Some of

these mRNA and miRNA bindings effect major cell processes such as cell growth,

tissue differentiation, or programed cell death. Since cancer occurs as a result of a

disruption in the balance of cell growth and cell death, miRNAs may play a role

in certain types of cancers. Over-expression of cell growth or under-expression of

cell death results in cell overgrowth — a characteristic of cancer. For example,

specific miRNAs called miR-15a and miR-16-1 negatively regulate the BCL2 gene

which creates a family of regulator proteins Bcl-2 that regulate cell death. Damage

to the BCL2 gene has been identified in a number of cancers such as melanoma,

breast, prostate, chronic lymphocytic leukemia, and lung cancer.

Because diseases such as cancer are related to mRNA and sRNA interac-

tions, there is a great interest in modeling these molecular interactions. Addition-

ally, what influences the boundary between cells that become one particular type

and those that do not may give insight into gene expression across tissue. We are

particularly interested in a generalized model where one can input quantitative

date of several sRNAs and mRNAs and get results of possible gene expression

across tissue.

1.2 Existing Models and Studies

Several studies of mRNA-sRNA interaction via modeling already exist,

mostly comprising of one of two types of models: stochastic or mean-field. Both

models create similar equations that describe mRNA and sRNA concentrations at

some time t. A major difference between the stochastic approach and the mean-

field approach is in the scale and quantity of mRNA and sRNA that will be mod-

eled. The stochastic models tend to depict mRNA and sRNA interactions in the

cell through master equations derived from rate diagrams [17, 25]. The stochastic

models focus on knowing the quantity of mRNA and sRNA molecules inside the

cell and tend to take into account protein bursting and other fluctuations. By do-

ing so, these stochastic models do not include the spatial variation and influential

movement of mRNA and sRNA between cells.

5

The mean-field models tend to be variations of the stochastic models, sim-

plified in order to find the concentrations of mRNA and sRNA across many cells.

By disregarding some information about mRNA and sRNA interactions in each

individual cell, the mean-field models are able to model the overall expected popu-

lation of mRNA and sRNA at both the cell and tissue level. Several models using

the mean-field approach show the concentrations of one mRNA and one sRNA

across tissue in one-dimension [13–15]. One of these models includes the diffusion

of sRNA across tissue [14], allowing sRNA movement from cell to cell [28, 31].

However, research has indicated that not only do sRNA move between cells, but

certain mRNA do as well [30]. Hence, these models lack some of the mRNA’s and

sRNA’s interactions across tissue.

In addition, some models bring into consideration interactions with other

proteins such as Argonaute [17] in the cell. Some models measure the protein

concentrations that are created (or not created) by mRNA and sRNA interactions

[27]. Although this information is useful in that it gives information about other

types of interactions within the cell involving proteins, these interactions will not

effect the measurement of the quantity of mRNA and sRNA across tissue.

Some recent studies influenced how we designed our models. We considered

the study of bacterial gene expression that showed that small RNAs provided a

safety mechanism against random fluctuations and transient signals within the

cell by establishing a threshold level for the expression of their target [13,15]. For

example, if sRNA had a single mRNA target and its growth (transcription) rate is

more than the mRNA target, then the mRNA target will not be expressed at steady

state since sRNA would bind to the mRNA and repress expression. However, if

sRNA’s growth rate is less than the mRNA target, the unpaired mRNA would

continue transcribing, and the mRNA target would be expressed. The expressed

protein level would be linearly proportional to the difference between the two

growth rates [13]. In our model, we take into account this threshold level for the

expression of the target mRNA.

Because of new research that indicates that certain mRNAs and sRNAs

move between cells [28, 30, 31], we are interested in a mean-field model of mRNA

6

and sRNA concentrations in which each RNA has the ability to diffuse across

tissue. Furthermore, since some genes that are effected by sRNA movement are

suggested to be dose dependent [1], we want our model to also allow us to choose

specific doses of sRNA that may turn a gene on or off. Since sRNAs can regulate

dozens of other genes [26], our model should allow one sRNA to regulate N mRNA

target genes.

1.3 Summary of Thesis Work

The concentrations of both mRNA and sRNA in cells are linked to the ex-

pression of a gene. To gain insight into how these concentration levels turn a gene

on or off, we created a mathematical model depicting how the generation rates of

each species, the independent death rate of each species, the couple death rate of

each species, and the increase or decrease in the diffusion coefficient of each species

effects the gene’s spatial concentrations across tissue. Predicting the expression of

genes across tissue could then be analyzed by a simple change of parameters in the

model.

Mathematical Model

One sRNA species may regulate several different genes by binding to several

mRNA targets creating different effects in the expression of a gene. To model this

effect, we allow one sRNA to couple with N different segments of a mRNA strand.

This interaction is described in the following N +1 system of equations named the

Multiple-species Model:

Let Ω be a bounded domain in R3, where the mRNA and sRNA interact.

Our model is a system of N + 1 reaction-diffusion partial differential equations

7

(PDEs):

∂ui∂t

= Di∆ui − βiui − kiuiv + αi (i = 1, . . . , N)

∂v

∂t= D∆v − βv −

N∑i=1

kiuiv + αin Ω× [0,∞), (1.3.1)

∂ui∂n

=∂v

∂n= 0 (i = 1, . . . , N) on ∂Ω× [0,∞), (1.3.2)

ui(·, 0) = (ui)0 (i = 1, . . . , N) and v(·, 0) = v0 in Ω. (1.3.3)

Here, ui is the nonnegative function which represents the concentration of the ith

mRNA target, and v is the nonnegative function which represents the concentration

of the sRNA. For i = 1, . . . , N , Di, D, βi, β, and ki are positive constants. Di and

D are diffusion coefficients of the ith mRNA and sRNA, respectively. The terms

βiui and βv describe the self-degradation of the ith mRNA and sRNA species. The

nonlinear terms, kiuiv and∑N

i=1 kiuiv, describe the coupled reaction between the

ith mRNA species and sRNA species in which ki is the reaction rate. αi, α are

nonnegative given functions bounded on Ω which describe the transcription profiles

or production rates for each species. The initial data, (ui)0 and v0, are two given

functions on Ω. n denotes the unit exterior normal at ∂Ω, the boundary of Ω, and

∂∂n

denotes the partial derivative along the normal n.

To better understand the behavior of these partial differential equations

(PDEs), we divided the Multiple-species Model into several individual models by

varying the diffusion coefficients (mobility of each species) and time dependence

(creating a steady state). The three types of equation models constructed are

a nonlinear diffusion equation, coupled diffusion equations at steady state, and

coupled diffusion equations with time dependence as in Equation (1.3.3). We also

considered separately the case in which N = 1 where one sRNA interacts with one

target mRNA and named it the Two-species Model.

The Multiple-species Model is derived from a two species stochastic model

of mRNA and sRNA interactions within the cell. In order to describe mRNA and

sRNA interactions across tissue, we created a mean-field model of their interac-

tions, resulting in the concentrations of mRNA and sRNA being both spatially

and time dependent. This mean-field model may be viewed as a generalized model

8

of chemical species across tissue that allows the parameters that are specific to

particular sRNA and mRNA interactions to be inputted easily.

Analysis of the Models

The analysis of our Multiple-species Model is divided into two types of sys-

tems: ordinary differential equation (ODE) systems and PDE systems. Analysis

of the ODE systems that are derived from our Multiple-Species Model shows that

the equilibrium solutions are stable subject to perturbations. The section on PDE

systems analysis illustrates the well-posedness of the three types of equations: a

single, nonlinear equation, coupled steady-state diffusion equations, and the time

dependent system of coupled equations. We also show using calculus of varia-

tions that the nonlinear equations associated with the Two-species Model and the

Multiple-species Model are unique.

Numerical Methods

In order to understand the behavior of our models numerically, we em-

ployed several numerical schemes. First, we discretized the space in which the

equations would lie by modifying the Finite Difference Method to account for our

Neumann boundary conditions. In addition, we created an alternating iteration

method to solve for our coupled PDEs. To approximate the time step, we em-

ployed the Forward Euler method and the Backward Euler method. Although the

Backward Euler scheme required more computations, we found that in our system,

the scheme converged faster than the Forward Euler scheme.

Computational Results

The numerical simulations for both the Two-species Model and the Multiple-

species Model show that when sRNA has a certain diffusion coefficient tolerance,

the concentration profile of mRNA displays a sharp interface between tissue with

9

high mRNA concentration and tissue with low mRNA concentration. That is, if

sRNA has some mobility across tissue, there is a sharp distinction between tissue

with high numbers of mRNA and tissue with low numbers of mRNA. When a

diffusion coefficient is added for the mRNA, the concentration profile of mRNA is

smoothed across the tissue. In other words, if mRNA has the ability to move across

tissue, the distinction recognized between tissue with high levels of mRNA and low

levels of mRNA is diminished. Another interest was how stable the interface be-

tween two species would be if small perturbations were introduced. We created a

perturbed sharp interface, and at steady state, the interface became sharp again.

That is, small fluctuations on a sharp interface will not change the shape of the

interface.

A sharpening of the concentration profile of mRNA may suggest a biological

regulation mechanism to minimize the number of cells where a gene is not strongly

expressed as on or off. Our model may give insight into how much mRNA and

sRNA molecules are needed for this type of regulation and how much diffusion (if

any) of each species leads to a working regulation mechanism. In some cancers for

which large concentrations of miRNA congregate in cells, the model may help us

predict how the concentrations effect gene expression across tissue.

Additionally, this numerical model also allows a simple input of specific

mRNA and sRNA characteristics such as production rates, independent degrada-

tion rates, and coupled degradation rates to see coupled behavior in tissue. In

addition, this model may be utilized to illustrate similar types of interactions be-

tween multiple chemical species.

Chapter 2

Mathematical Models

To construct the model of mRNA and sRNA interactions across tissue,

we first model the mRNA and sRNA interactions within the cell using a rate

diagram showing a network of states. From the diagram, we will create an ordinary

differential equation, the so-called master equation, that describes the change in

the mRNA and sRNA states at time t. This model of interactions within the cell

will be the basis for our partial differential equation (PDE) model of mRNA and

sRNA interactions across tissue.

2.1 Derivation of the Mean-field Model of Two

Species

A rate diagram of sRNA-mediated gene silencing within the cell presents a

nice visualization of the states and connections between mRNA and sRNA popula-

tions. Figure 2.1 depicts the different rates in which mRNA and sRNA populations

may change at time t. In the figure, ασ and αµ describe the sRNA and mRNA gen-

eration or production rates, βσ and βµ describe the sRNA and mRNA independent

death or degradation rates, and γ describes the coupled death or degradation rate

at time t. Notice in the rate diagram that when mRNA and sRNA bind together,

the process is irreversible.

The range of parameters used for each one of the rates is determined via

10

11

experimental data [15]. For example, to find the coupled degradation rate, one

can inhibit transcription of mRNA and monitor its decay rate (total decay =

independent decay + coupled decay). If one knows the mRNA degradation rate

alone, then the coupled degradation rate is found. The parameters may change

based upon types of sRNA and the types of reactions occurring. Therefore, a range

of parameters will be explored to determine interaction responses.

sRNAmRNAαμ

ασ

βσ

γ

βμ

ø

ø

ø

Figure 2.1: Kinetic scheme of mRNA and sRNA concentrations. ασ and αµ rep-

resent the production rates of sRNA and mRNA, respectively. βσ and βµ represent

the independent degradation rates of sRNA and mRNA, respectively. γ represents

the couple degradation rate of sRNA and mRNA.

To construct our quantitative mathematical model from the diagram in

Figure 2.1, we first need to define some terms. Let (Mt)t≥0 and (St)t≥0 be two

continuous time processes with t ∈ [0,∞), where Mt represents the number of

mRNA in the cell at time t and St represents the number of sRNA in the cell at

time t. For notational convenience, let Nt = (Mt, St) represent the pair of RNA

populations in which the first number depicts the mRNA population at time t

and the second number depicts the sRNA population at time t. We will make the

following assumption regarding this time process:

Assumption. We assume that (Nt)t≥0 is a time-homogeneous continuous-time

Markov process with state space S with the following ordering:

S = ((0, 0), . . . , (m− 1, s− 1), (m− 1, s), (m, s− 1), (m, s), (m, s+ 1), . . .) .

12

Note that the number of mRNA and sRNA that can be created is bounded

above (by, say, the number of atoms in the universe). Because the number of both

mRNA and sRNA that can be created has a limit, the state space is finite.

Let P (t) be the time dependent (row) vector where

Pm,s(t) = P (Nt = (m, s)) = P (Mt = m,St = s).

Then,

P (t) = aP(t)

where a be the row vector indexed by S that represents the initial distribution of

the process and P(t) is the transition matrix.

Assumption. Let Pm,s(t) represent the (m, s) element of the vector P (t). The vec-

tor P (t) is assumed to satisfy the following master equation (Kolmogorov forward

equation),

Pm,s(t) = ασPm−1,s(t) + αµPm,s−1(t)− θm,sPm,s(t)

+ βσ(s+ 1)Pm,s+1(t) + βµ(m+ 1)Pm+1,s(t)

+ γ(m+ 1)(s+ 1)Pm+1,s+1(t)

(2.1.1)

where

Pm,s(t) =∂

∂tPm,s(t);

ασ and αµ represent the generation rates or production rates of sRNA and mRNA,

respectively; βσ and βµ represent the independent death or degradation rates of

sRNA and mRNA, respectively; and γ represents the coupled death or degradation

rate of mRNA and sRNA. The term

θm,s := αµ + ασ + βµm+ βσs+ γms

represents the rate in which the system leaves the state (m, s).

Since (Nt)t≥0 is a time-homogeneous continuous-time Markov chain with

transition matrix P(t), the master equation defines the infinitesimal generator Q

of the process, and P(t) = etQ. Because P (t) = aP(t) = aP(t)Q = P (t)Q, the

13

master equation tells us that the column of Q corresponding to the state (m, s)

must be

column (m, s) of Q =

0...

0

ασ

αµ

−θm,sβσ(s+ 1)

βµ(m+ 1)

γ(m+ 1)(s+ 1)

0...

0

.

Notice that the diagonal elements of Q consist of −θ terms, and that the rows of

Q sum to 0. This means that Q is a Markov infinitesimal generator.

Proposition 2.1.1. Let ν ∈ RS = f | f : S → R. Then,

E [ν(Nt)] = P (t)ν.

On the left hand side, we view ν(Nt) as the composition of ν : S → R with the

random variable Nt. On the right hand side, we view P (t)ν as vector multiplication

between the row vector P (t) = aP(t) and ν which is identified as a column vector

indexed by S.

Proof. We have

E [ν(Nt)] =∑

(m,s)∈S

ν(m, s)P (Nt = (m, s)) =∑

(m,s)∈S

v(m, s)Pm,s(t) = P (t)ν.

Now, we have all of the tools to prove the following lemma that describes

the change in average mRNA and sRNA concentrations over time. This lemma

will also help us derive a mean-field model illustrating mRNA and sRNA molecule

movement within the cell and across tissue.

14

Lemma 2.1.1. For a random variable X, we denote the expectation of X by

〈X〉 = E [X]. Then, we have

∂

∂t〈Mt〉 = αµ − βµ 〈Mt〉 − γ 〈MtSt〉 (2.1.2)

∂

∂t〈St〉 = ασ − βσ 〈St〉 − γ 〈MtSt〉 . (2.1.3)

Proof. Define [A]m,s to be the (m, s) entry of the matrix A. Let πµ and πσ be the

projection maps from S → R given by πµ(m, s) = m and πσ(m, s) = s. Then,

〈Mt〉 = E [πµ(Nt)] and 〈St〉 = E [πσ(Nt)]. Using Proposition 2.1.1 with ν = πµ and

using the properties of Q, we get

∂

∂tE [πµ(Nt)] =

∂

∂tP (t)πµ = [P (t)Q] πµ

=∑

(m,s)∈S

πµ(m, s) [P (t)Q]m,s

=∑

(m,s)∈S

m [P (t)Q]m,s

By recalling that [P (t)Q]m,s = Pm,s(t) and using our master equation (Equa-

tion (2.1.1)), the last equality becomes∑(m,s)∈S

m [P (t)Q]m,s =∑

(m,s)∈S

m

(αµPm−1,s(t) + ασPm,s−1(t)− θm,sPm,s(t)

+ βσ(s+ 1)Pm,s+1(t) + βµ(m+ 1)Pm+1,s(t)

+ γ(m+ 1)(s+ 1)Pm+1,s+1(t)

).

Since we are summing over a finite set S, we interchange the sum and the differ-

ential operator to get the following:

15

∂

∂t

∑(m,s)∈S

mPm,s(t) = αµ

∑(m,s)∈S

mPm−1,s(t)−∑

(m,s)∈S

mPm,s(t)

+ ασ

∑(m,s)∈S

mPm,s−1(t)−∑

(m,s)∈S

mPm,s(t)

+ βµ

∑(m,s)∈S

m(m+ 1)Pm+1,s(t)−∑

(m,s)∈S

m2Pm,s(t)

+ βσ

∑(m,s)∈S

m(s+ 1)Pm,s+1(t)−∑

(m,s)∈S

msPm,s(t)

+ γ

∑(m,s)∈S

m(m+ 1)(s+ 1)Pm+1,s+1(t)

− γ∑

(m,s)∈S

m2sPm,s(t)

∂

∂t

∑(m,s)∈S

mPm,s(t) = αµ

∑(m−1,s)∈S

(m+ 1)Pm,s(t)−∑

(m,s)∈S

mPm,s(t)

+ ασ

∑(m,s−1)∈S

mPm,s(t)−∑

(m,s)∈S

mPm,s(t)

+ βµ

∑(m+1,s)∈S

(m− 1)mPm,s(t)−∑

(m,s)∈S

m2Pm,s(t)

+ βσ

∑(m,s+1)∈S

msPm,s(t)−∑

(m,s)∈S

msPm,s(t)

+ γ

∑(m+1,s+1)∈S

(m− 1)msPm,s(t)−∑

(m,s)∈S

m2sPm,s(t)

16

∂

∂t

∑(m,s)∈S

mPm,s(t) = αµ

∑(m,s)∈S

(m+ 1)Pm,s(t)−∑

(m,s)∈S

mPm,s(t)

+ ασ

∑(m,s)∈S

mPm,s(t)−∑

(m,s)∈S

mPm,s(t)

+ βµ

∑(m,s)∈S

(m− 1)mPm,s(t)−∑

(m,s)∈S

m2Pm,s(t)

+ βσ

∑(m,s)∈S

msPm,s(t)−∑

(m,s)∈S

msPm,s(t)

+ γ

∑(m,s)∈S

(m− 1)msPm,s(t)−∑

(m,s)∈S

m2sPm,s(t)

∂

∂t

∑(m,s)∈S

mPm,s(t) = αµ∑

(m,s)∈S

Pm,s(t)

+ βµ

∑(m,s)∈S

(m2 −m)Pm,s(t)−∑

(m,s)∈S

m2Pm,s(t)

+ γ

∑(m,s)∈S

(m2s−ms)Pm,s(t)−∑

(m,s)∈S

m2sPm,s(t)

And,

∂

∂t

∑(m,s)∈S

mPm,s(t) = αµ − βµ∑

(m,s)∈S

mPm,s(t)− γ∑

(m,s)∈S

msPm,s(t).

Since∑

(m,s)∈SmPm,s(t) = 〈Mt〉,

∂

∂t〈Mt〉 = αµ − βµ 〈Mt〉 − γ 〈MtSt〉 .

Mimicking the above process for 〈St〉 = E [πσ(Nt)] = P (t)πσ using the master

equation (Equation (2.1.1)), we have

∂

∂t〈St〉 = ασ − βσ 〈St〉 − γ 〈MtSt〉 .

17

Because we are looking for insight into the concentrations of mRNA and

sRNA molecules not just inside cells, but concentrations at the tissue level, we

modify Equations (2.1.2) and (2.1.3) using mean-field theory to examine the be-

havior of many mRNA molecules at once. The mean-field assumption is that

〈MtSt〉 = 〈Mt〉 〈St〉 . (2.1.4)

Therefore, by integrating this assumption into Equations (2.1.2) and (2.1.3), we

have the following coupled differential equations:

Let u(t) = 〈Mt〉 and v(t) = 〈St〉.∂u

∂t= αµ − βµu− γuv,

∂v

∂t= ασ − βσv − γuv .

Although the above equations describe binding of mRNA and sRNA within

a cell across tissue, they do not describe the process of sRNA movement between

cells documented in cell differentiation and development [28, 31]. To describe this

movement, we must add a diffusion process with an appropriate diffusion coefficient

Dσ to the sRNA equation [14]. We have the following equations:

∂u

∂t= αµ − βµu− γuv, (2.1.5)

∂v

∂t= ασ − βσv − γuv +Dσ∆v . (2.1.6)

Furthermore, we account for the intercellular macromolecule movement of mRNA

such as with exosome-mediated transfer [30] by adding mRNA movement across

tissue. We model this behavior by adding a diffusion processes with an appropriate

diffusion coefficient Dµ to Equation (2.1.5) to form the following coupled equations:

∂u

∂t= αµ − βµu− γuv +Dµ∆u, (2.1.7)

∂v

∂t= ασ − βσv − γuv +Dσ∆v. (2.1.8)

The coupled reaction-diffusion equations above illustrate mRNA and sRNA

concentration levels across tissue developed from the stochastic master equation

(2.1.1). These equations will transform into two types of models: two-species

models and multiple-species models.

18

2.2 Two-species Models in Multiple Dimensions

A dynamic model allowing for both mRNA and sRNA to move inside and

outside of the cell as well as across tissue in a multiple-dimension model would

bring accuracy to the behavior of RNA molecules across tissue. By adapting

Equations (2.1.7) and (2.1.8) and adding some reasonable boundary conditions, we

create coupled reaction-diffusion equations for a multiple dimensional environment.

Let Ω be a bounded domain in R3 where the mRNA and sRNA interact.

Let u = u(x, t) and v = v(x, t) denote the concentrations of mRNA and sRNA,

respectively, at a spatial point x = (x1, x2, x3) ∈ Ω at time t. The following

reaction-diffusion equations model the concentrations with specified boundary and

initial conditions:

∂u

∂t= D1∆u− β1u− k1uv + α1

∂v

∂t= D∆v − βv − k1uv + α

in Ω× [0,∞), (2.2.1)

∂u

∂n=∂v

∂n= 0 on ∂Ω× [0,∞), (2.2.2)

u(·, 0) = u0 and v(·, 0) = v0 in Ω. (2.2.3)

Here, u is the nonnegative function which represents the concentration of the

mRNA, and v is the nonnegative function which represents the concentration of

the sRNA. D, D1, β, β1, k1, and k1 are positive constants. D1 and D are diffusion

coefficients of mRNA and sRNA, respectively. The terms β1u and βv describe the

self-degradation of the two species mRNA and sRNA, respectively. The nonlinear

terms, k1uv and k1uv, describe the coupled reaction between the two species in

which k1 and k1 are the reaction rates. α1 and α are nonnegative given functions

defined on Ω which describe the transcription profiles or production rates for each

species. The initial data, u0 and v0, are two given functions on Ω. n denotes

the unit exterior normal at ∂Ω, the boundary of Ω, and ∂∂n

denotes the partial

derivative along the normal n.

We consider three different cases of the above model, and name the cor-

responding models Two-species Model I, Two-species Model, II, and Two-species

Model III.

19

Two-species Model I: A single, nonlinear PDE

First, we let mRNA diffuse very slowly with D1 = 0 and D > 0. Moreover, we as-

sume the concentration of mRNA reaches a steady-state in which the time deriva-

tive of u vanishes. The time independent equation for u from Equation (2.2.1)

becomes

u =α1

β1 + k1vin Ω.

Thus, the steady-state concentration v = v(x) of sRNA is governed by the following

nonlinear, diffusion equation and boundary condition:

D∆v − βv − k1α1v

β1 + k1v+ α = 0 in Ω, (2.2.4)

∂v

∂n= 0 on ∂Ω. (2.2.5)

Two-species Model II: Coupled, steady-state diffusion equations

We consider the steady-state concentrations of both species with D1, D > 0 and

study the following coupled steady-state diffusion equations:

D1∆u− β1u− k1uv + α1 = 0

D∆v − βv − k1uv + α = 0in Ω, (2.2.6)

∂u

∂n=∂v

∂n= 0 on ∂Ω. (2.2.7)

Two-species Model III: Coupled, time-dependent diffusion equations

This model is our original coupled reaction-diffusion Equations (2.2.1)–(2.2.3):

∂u

∂t= D1∆u− β1u− k1uv + α1

∂v

∂t= D∆v − βv − k1uv + α

in Ω× [0,∞),

∂u

∂n=∂v

∂n= 0 on ∂Ω× [0,∞),

u(·, 0) = u0 and v(·, 0) = v0 in Ω.

20

2.3 Multiple-species Models in Multiple Dimen-

sions

We consider the situation in which one sRNA may regulate more than one

gene. That is, one sRNA species may bind to several different parts of a mRNA

strand creating different effects in the expression of several different genes. To

model this effect, we allow one sRNA to couple with N different segments of a

mRNA strand. This interaction is described in the following N + 1 system of

equations:

Let Ω be a bounded domain in R3 where the mRNA and sRNA interact.

Our model is the system of reaction-diffusion equations:

∂ui∂t

= Di∆ui − βiui − kiuiv + αi (i = 1, . . . , N)

∂v

∂t= D∆v − βv −

N∑i=1

kiuiv + αin Ω× [0,∞), (2.3.1)

∂ui∂n

=∂v

∂n= 0 (i = 1, . . . , N) on ∂Ω× [0,∞), (2.3.2)

ui(·, 0) = (ui)0 (i = 1, . . . , N) and v(·, 0) = v0 in Ω. (2.3.3)

Here, ui is the nonnegative function which represents the concentration of the ith

mRNA target, and v is the nonnegative function which represents the concentration

of the sRNA. For i = 1, . . . , N , Di, D, βi, β, and ki are positive constants. Di and

D are diffusion coefficients of the ith mRNA target and sRNA, respectively. The

terms βiui and βv describe the self-degradation of the ith mRNA target species and

sRNA species. The nonlinear terms, kiuiv and∑N

i=1 kiuiv, describe the coupled

reaction between the ith mRNA target species and sRNA species in which ki is the

reaction rate. αi, α are nonnegative given functions bounded on Ω which describe

the transcription profiles or production rates for each species. The initial data,

(ui)0 and v0, are two given functions on Ω. n denotes the unit exterior normal at

∂Ω, the boundary of Ω, and ∂∂n

denotes the partial derivative along the normal n.

Similar to our two species models above, we divide our multiple-species

model into three separate cases.

Multiple-species Model I: A single, nonlinear PDE

21

As with Model I, we suppose ith mRNA target diffuses very slowly. We let Di = 0

and D > 0 to have the following nonlinear PDE at steady state:

D∆v − βv −N∑i=1

αikiv

βi + kiv+ α = 0 in Ω, (2.3.4)

∂v

∂n= 0 on ∂Ω. (2.3.5)

Multiple-species Model II: Coupled, steady-state diffusion equations

We consider the steady-state concentrations of all species with Di, D > 0 and

study the following coupled steady-state diffusion equations:

Di∆ui − βiui − kiuiv + αi = 0 (i = 1, . . . , N)

D∆v − βv −N∑i=1

kiuiv + α = 0in Ω, (2.3.6)

∂ui∂n

=∂v

∂n= 0 (i = 1, . . . , N) on ∂Ω. (2.3.7)

Multiple-species Model III: Coupled, time-dependent diffusion equa-

tions

This model is the coupled reaction-diffusion Equations (2.3.1)–(2.3.3):

∂ui∂t

= Di∆ui − βiui − kiuiv + αi (i = 1, . . . , N)

∂v

∂t= D∆v − βv −

N∑i=1

kiuiv + αin Ω× [0,∞),

∂ui∂n

=∂v

∂n= 0 (i = 1, . . . , N) on ∂Ω× [0,∞),

ui(·, 0) = (ui)0 (i = 1, . . . , N) and v(·, 0) = v0 in Ω.

Chapter 3

Mathematical Analysis of the

Models

The mathematical analysis of the Two-species Model and the Multiple-

species Model is divided into two sections: analysis dealing with ordinary differen-

tial equations (ODEs) derived from our models and analysis of PDEs associated

with our models. The ODE section asks if the equilibrium solutions are stable

subject to perturbations. The PDE section addresses existence and uniqueness of

the models derived from the Two-species Model and the Multiple-species Model.

3.1 Ordinary Differential Equations Analysis

First, we neglect the spatial dependence of the Multiple-species Model sys-

tem of equations and study the resulting ODEs. We will then solve for steady

state solutions of the ODE system and linearize a system around such a steady

state solution. The eigenvalues of the associated matrix to the linearized system

determines stability.

22

23

3.1.1 The System of ODEs for Reaction and Its Lineariza-

tion

The ODE system created by Two-species Model III with D1 = 0 and D = 0

is a special case of the system of ODEs from the Multiple-species Model III in

which N = 1, D1 = 0, and D = 0. Because of this special case, the following

analysis focuses on Multiple-species Model III. The eigenvalues of the stability

matrix created by the ODE system play a prominent role in the behavior of the

ODE. Recall that if the real part of the eigenvalues of the stability matrix which

represents the ODE system is negative, the system is considered linearly stable.

Consider a spatial homogenous system where our Multiple-species Model

III (Equations (2.3.1)–(2.3.3)) depends solely on time t. Then, since we are no

longer relying on spatial coordinates, the terms Di∆ui = 0 for i = 1, . . . , N and

D∆v = 0 creating the following ODE system:

duidt

= −βiui − kiuiv + αi (i = 1, . . . , N)

dv

dt= −βv −

N∑i=1

kiuiv + α.(3.1.1)

First, we need to show a steady state solution exists.

Theorem 3.1.1. For the system of equations

−βiui − kiuiv + αi = 0 (i = 1, . . . , N) (3.1.2)

−βv −N∑i=1

kiuiv + α = 0, (3.1.3)

there exists a unique equilibrium solution for u1, u2, . . . , uN , v with ui ≥ 0 and

v ≥ 0 for all i = 1, . . . , N .

Proof. From Equation (3.1.3),

v =α

β + U

where U =∑N

i=1 kiui. By substituting v into Equation (3.1.2), we have the equa-

tions

−βiui −kiuiα

β + U+ αi = 0 (i = 1, . . . , N).

24

Then,

ui =αi

βi + kiα (β + U)−1 (i = 1, . . . , N),

and

U =N∑i=1

kiui =N∑i=1

kiαi

βi + kiα (β + U)−1 .

Now, we want to show that the equation

U −N∑i=1

kiαi

βi + kiα (β + U)−1 = 0

has a unique solution. Let the function g : [0,∞)→ R be defined by

g(s) = s−N∑i=1

kiαi

βi + kiα (β + s)−1 , (3.1.4)

and recall that βi, β, and ki are positive constants and αi and α are nonnegative

functions. g is smooth in our domain [0,∞). Since

g(0) = −N∑i=1

kiαi

βi + kiα (β)−1 ≤ 0

and the lims→∞ g(s) = +∞, there exists a s ∈ [0,∞) such that g(s) = 0. That is,

there exists a U ∈ [0,∞) such that g(U) = 0. Then,

v =α

β + U,

and v has a solution. Similarly,

ui =αi

βi + kiv(i = 1, . . . , N),

and ui has a solution for all i. To find the behavior of the possible root(s), we first

look at the derivatives of g. The first derivative of g is

g′(s) = 1−N∑i=1

k2i αiα

(βi (β + s) + kiα)2 ,

25

and the second derivative of g is

g′′(s) = 2αN∑i=1

k2i αiβi

(βi (β + s) + kiα)3 ≥ 0.

Notice that g′′(s) = 0 on [0,∞) only when α = 0 or αi = 0 for all i. In both

cases, g(s) is then a linear equation with only one solution, and g(s) has a unique

solution U such that g(U) = 0.

Suppose g′(s) ≥ 0 on [0,∞). Since g is convex and concave up (g′′(s) > 0),

g(U) = 0 is the unique minimizer. Suppose, on the other hand, g′(s) < 0. Then, g

attains its minimum in [0, U ] at some s0 ∈ (0, U) where g′(s0) = 0. In the interval

[0, s0), g′(s) < 0 and g(s) < 0. In the interval (s0,∞), g′(s) > 0 and g′′(s) > 0.

Hence, g(s) = 0 in only one place. That is, g(s) has at most one root. This proves

uniqueness.

Define u0i and v0 to be the solutions to the coupled equations (3.1.1). That

is,

−βiu0i − kiu0iv0 + αi = 0 (i = 1, . . . , N)

−βv0 −N∑i=1

kiu0iv0 + α = 0.

For ε > 0, define the equations ui(t) and v(t) by

ui(t) = u0i + εu1i(t) +O(ε2)

(i = 1, . . . , N) (3.1.5)

v(t) = v0 + εv1(t) +O(ε2). (3.1.6)

By rewriting Equations (3.1.5) and (3.1.6), we have the following equations de-

scribing our ODE:

εu1i = ui − u0i +O(ε2) (i = 1, . . . , N) (3.1.7)

εv1 = v − v0 +O(ε2). (3.1.8)

For each i = 1, . . . , N , we take the derivative of both sides of Equation (3.1.7) with

26

respect to t,

εdu1i

dt=duidt− du0i

dt+O(ε2)

= (−βiui − kiuiv + α1)− (−βiu0i − kiu0iv0 + α1) +O(ε2)

= −βi(ui − u0i)− ki(uiv − u0iv0) +O(ε2)

= −βi(ui − u0i)− ki((ui − u0i)(v − v0) + uiv0 + u0iv − 2u0iv0) +O(ε2)

= −βi(ui − u0i)− ki((ui − u0i)(v − v0) + v0(ui − u0i) + u0i(v − v0))

+O(ε2)

= −βi(ui − u0i)− kiv0(ui − u0i)− kiu0i(v − v0)− ki(ui − u0i)(v − v0)

+O(ε2)

= −(βi + kiv0)(ui − u0i)− kiu0i(v − v0)− ki(ui − u0i)(v − v0) +O(ε2)

= −ε(βi + kiv0)u1i − εkiu0iv1 +O(ε2).

Then, we havedu1i

dt= −(βi + kiv0)u1i − kiu0iv1 +O(ε).

Similarly, by taking the derivative of Equation (3.1.8),

εdv1

dt=dv

dt− dv0

dt+O(ε2)

= (−βv −N∑i=1

kiuiv + α)− (−βv0 −N∑i=1

kiu0iv0 + α) +O(ε2)

= −β(v − v0)−N∑i=1

ki(uiv − u0iv0) +O(ε2)

= −β(v − v0)−N∑i=1

ki((ui − u0i)(v − v0) + uiv0 + u0iv − 2u0iv0) +O(ε2)

= −β(v − v0)−N∑i=1

ki [u0i(v − v0) + (ui − u0i)v0 + (ui − u0i)(v − v0)]

+O(ε2)

= −(β +N∑i=1

kiu0i)(v − v0)−N∑i=1

ki [(ui − u0i)v0 + (ui − u0i)(v − v0)]

+O(ε2)

27

= −ε(β +N∑i=1

kiu0i)v1 − εN∑i=1

kiv0u1i +O(ε2).

And, we have

dv1

dt= −

(β +

N∑i=1

kiu0i

)v1 −

N∑i=1

kiv0u1i +O(ε).

For i = 1, . . . , N , we have the coupled system of N + 1 equations

du1i

dt= −(βi + kiv0)u1i − kiu0iv1 +O(ε)

dv1

dt= −

(β +

N∑i=1

kiu0i

)v1 −

N∑i=1

kiv0u1i +O(ε).

Define w =

u11

u12

...

u1N

v1

. We obtain the linearized system

dw

dt= Mw,

where

M =

−(β1 + k1v0) 0 . . . 0 −k1u01

0 −(β2 + k2v0) . . . 0 −k2u02

......

. . ....

0 0 −(βN + kNv0) −kNu0N

−k1v0 −k2v0 . . . −kNv0 −(β +∑N

i=1 kiu0i)

(3.1.9)

We call this matrix the stability matrix. If the real parts of the eigenvalues of M

are negative, the system is considered to be linearly stable.

3.1.2 Linear Stability

To show the real parts of the eigenvalues of M are negative, we will show

that the diagonal dominance and the negative diagonal elements of M cause each

28

eigenvalue of M to be contained in a Gershgorin disc lying in the left-half plane.

Since <(λ(M)) < 0, the ODE is stable. We start by showing the column diagonal

dominance of M .

Proposition 3.1.1. Assume β, βi, ki > 0 and u0i , v0 ≥ 0 for all i = 1, . . . , N .

Then, the matrix M from (3.1.9) is strictly column diagonally dominant.

Proof. M = [mij] is said to be strictly column diagonally dominant if for all

j = 1, . . . , N + 1,

|mjj| >N+1∑i=1i 6=j

|mij| .

For columns j = 1, . . . , N ,

|mjj| = |βj + kjv0| > |kjv0| =N+1∑i=1i 6=j

|mij| .

For j = N + 1,

|mN+1,N+1| =

∣∣∣∣∣β +N∑i=1

kiu0i

∣∣∣∣∣ >∣∣∣∣∣N∑i=1

kiu0i

∣∣∣∣∣ =N∑i=1

|mi,N+1| .

Hence, M is strictly column diagonally dominant.

Since M is strictly column diagonally dominant, MH is row diagonally

dominant with corresponding eigenvalues λ(M) = λ(MH). We will now show that

the eigenvalues of MH have negative real part by the following theorem.

Theorem 3.1.2. Let A = [aij] ∈ Rn×n be strictly row diagonally dominant. Then,

if all main diagonal entries of A are negative, then all the eignenvalues of A have

negative real part.

Proof. Let A = [aij] be strictly row diagonally dominant with all negative main

diagonal entries. Let λ be an eigenvalue of A with the corresponding eigenvector

x. For x = [xi] 6= 0, there exists a p such that |xp| ≥ |xi| for all i = 1, . . . , n, and

|xp| 6= 0. Since Ax = λx,

λxp = [λx]p = [Ax]p =n∑i=1

apixi = appxp +n∑i=1i 6=j

apixi,

29

and

(λ− app)xp =n∑i=1i 6=j

apixi.

Hence,

|λ− app| |xp| = |(λ− app)xp| =

∣∣∣∣∣∣∣n∑i=1i 6=j

apixi

∣∣∣∣∣∣∣ ≤n∑i=1i 6=j

|api| |xi| ≤ |xp|n∑i=1i 6=j

|api| .

Since A is row diagonally dominant,

|λ− app| ≤n∑i=1i 6=j

|api| < |app| .

Therefore, λ ∈ z : |z − app| < |app|. That is, λ exists in a disk in the complex

plane centered at app ∈ R− of radius less than |app|. Thus, <(λ) exists in the

left-half plane.

Since M is strictly column diagonally dominant, MH is strictly row diago-

nally dominant. By Theorem 3.1.2, the eigenvalues of MH have all negative real

part. Since λ(MH) = λ(M), the real part of the eigenvalues of M all have negative

real part, and the system is asymptotically stable.

3.2 Partial Differential Equations Analysis

The following section focuses on the analysis of the PDEs derived from the

Two-species Model and the Multiple-species Model. In particular, we find that

the solutions to the Two-species Model I and the solutions to the Multiple-species

Model I exist and are unique. We also find that a solution to Two-species Model

II exists, but may not necessarily be unique.

3.2.1 Well-posedness of Two-species Model I

We show that Two-species Model I has a unique solution in Ω.

30

Theorem 3.2.1. Assume Ω has a Lipschitz-continuous boundary. The boundary-

value problem

D∆v − βv − k1α1v

β1 + k1v+ α = 0 in Ω, (3.2.1)

∂v

∂n= 0 on ∂Ω. (3.2.2)

(Two-species Model I, Section 2.2, Equations 2.2.4 and 2.2.5) has a unique solution

v ∈ H1(Ω) such that v ≥ 0, and v is smooth in Ω.

Proof. Define J : H1(Ω)→ R ∪ ±∞ by

J [v] =

∫Ω

D

2|∇v|2 +

β

2v2 +

k1α1β1

k21

[k1

β1

v − ln

(1 +

k1

β1

v

)]− αv

dx.

Define ln s = −∞ for any s ≤ 0. Notice that the function

g(s) = s− ln(1 + s) (−1 < s <∞)

is strictly convex and reaches its minimum g(0) = 0. Thus, there exists constants

C1 > 0 and C2 ≥ 0 such that

J [v] ≥ C1‖v‖2H1(Ω) − C2 ∀v ∈ H1(Ω). (3.2.3)

If we denote θ = infv∈H1(Ω) J [v], then θ is finite. Let vj ∈ H1(Ω) (j = 1, 2, . . . ) be

such that J [vj]→ 0 as j →∞. It follows from Equation (3.2.3) that, passing to a

subsequence if necessary, vj∞j=1 converges to some v ∈ H1(Ω), weakly in H1(Ω),

strongly in L2(Ω), and a.e. in Ω. Since the function g(s) ≥ 0 for all s ∈ (−1,∞),

Fatou’s Lemma implies that

lim infj→∞

∫Ω

[k1

β1

vj − ln

(1 +

k1

β1

vj

)]dx ≥

∫Ω

[k1

β1

v − ln

(1 +

k1

β1

v

)]dx.

The weak convergence in H1(Ω) and strong convergence in L2(Ω) of vj∞j=1 to v

now imply that

lim infj→∞

J [vj] ≥ J [v].

Hence, J [v] = θ. The strict convexity of J implies that v ∈ H1(Ω) is the unique

minimizer of J : H1(Ω)→ R ∪ +∞.

31

Now we will show that v ≥ 0 a.e. in Ω. By the uniqueness of the minimizer

of J over H1(Ω), it suffices to show that |v| ∈ H1(Ω) is also a minimizer. Notice

that |∇|v| | ≤ |∇v| in Ω. Since α is nonnegative in Ω, we have −α|v| ≤ −αv a.e.

in Ω. Notice that for any s ∈ (−1, 0]

g(|s|)− g(s) = |s| − ln(1 + |s|)− [s− ln(1 + s)] = −2s+ ln

(1 + s

1− s

).

It is easy to verify that this continuous function of s has a positive derivative in

(−1, 0) and is equal to 0 at s = 0. Therefore, g(|s|) ≥ g(s) for s ∈ (−1, 0). Applying

this to the case, s = k1v(x)/β1 ∈ (−1, 0] for some some x ∈ Ω. Therefore, we obtain

that J [|v|] ≥ J [v]. Hence v ≥ 0.

It now follows that

d

dt

∣∣∣∣t=0

∫Ω

ln

(1 +

k1

β1

(v + tw)

)dx =

(k1/β1)w

1 + (k1/β1)v∀w ∈ H1(Ω).

Since v minimizes J over H1(Ω), we have

d

dt

∣∣∣∣t=0

J [v + tw] = 0 ∀w ∈ H1(Ω).

Standard calculations then imply that v ∈ H1(Ω) is a weak solution to Equa-

tion (2.2.4) and Equation (2.2.5). The smoothness of v inside Ω follows from a

standard bootstrapping technique.

3.2.2 Well-posedness of Multiple-species Model I

We will show that Multiple-species Model I is unique.

Theorem 3.2.2. Assume Ω has a Lipschitz-continuous boundary. The equation

D∆v − βv −N∑i=1

kiαiv

βi + kiv+ α = 0 in Ω, (3.2.4)

∂v

∂n= 0 on ∂Ω,

has a unique solution v ∈ H1(Ω) such that v ≥ 0, and v is smooth in Ω.

Proof. Follow the proof of Theorem 3.2.1 with the functional

J [v] =

∫Ω

D

2|∇v|2 +

β

2v2 +

N∑i=1

αiβiki

[kiβiv − ln

(1 +

kiβiv

)]− αv

dx.

32

3.2.3 Behavior Analysis of Multiple-species Models

Multiple-species Model I Behavior

Recall that Multiple-species Model I has the form:

D∆v − βv −N∑i=1

kiαiv

βi + kiv+ α = 0 in Ω, (3.2.5)

∂v

∂n= 0 on ∂Ω.

We want to know the behavior of solutions to this boundary-value problem. One

way to do so is to look at the behavior of Equation (3.2.5) as some parameters get

very small. First, we will look at the independent death rate of mRNA as it gets

very small (βi → 0).

Suppose βi → 0. Then, we have the following equation:

D∆v − βv −N∑i=1

αi + α = 0 in Ω, (3.2.6)

∂v

∂n= 0 on ∂Ω.

We now consider the one-dimensional system for which we can find some solution

for v. To solve for v, we will find Green’s function to

∆v − β

Dv =

α(x)−∑N

i=1 αi(x)

D.

Let λ =√

βD

and f(x) =α−

∑Ni=1 αiD

. In one dimension, we are solving

v′′ − λ2v = f(x)

with the boundary conditions v′(0) = v′(1) = 0. First, we will find the solution to

homogeneous equation v′′−λ2v = 0. Using the Undetermined Coefficients Method,

Green’s function will be of the form

G(x, s) =

A(s)(c1eλx + c2e

−λx) for x < s

B(s)(c3eλx + c4e

−λx) for x > s

33

for constants c1, c2, c3, c4 and functions A(s) 6= 0 and B(s) 6= 0. We can solve for

these variables using the conditions which G(x, s) must satisfy such as boundary

conditions, continuity at x and s, and the derivative “jump” when x → s. For

x < s,

G′(x, s) = A(s)(c1λeλx − c2λe

−λx)

G′(0, s) = A(s)(c1λ− c2λ)

c2 = c1.

For x > s,

G′(x, s) = B(s)(c3λeλx − c4λe

−λx)

G′(1, s) = B(s)(c3λeλ − c4λe

−λ)

c4e−λ = c3e

λ

c4 = c3e2λ.

Then,

G(x, s) =

A(s)(c1eλx + c1e

−λx) for x < s

B(s)(c3eλx + c3e

2λ−λx) for x > s.

From continuity at x = s,

A(s)(c1eλs + c1e

−λs) = B(s)(c3eλs + c3e

2λ−λs).

From the derivative “jump” where G′(s+, s)−G′(s−, s) = 1,

B(s)(λc3eλs − λc3e

2λ−λs)− A(s)(c1λeλs − c1λe

−λs) = 1.

Together,A(s)(c1λeλs + c1λe

−λs)−B(s)(c3λeλs + c3λe

2λ−λs) = 0

−A(s)(c1λeλs − c1λe

−λs) +B(s)(c3λeλs − c3λe

2λ−λs) = 1,

and so,

2A(s)c1λe−λs − 2B(s)c3λe

2λ−λs = 1.

34

Then,

A(s)c1 =1

2λeλs +B(s)c3e

2λ.

By substitution,(1

2eλs + λB(s)c3e

2λ

)(eλs + e−λs

)−B(s)c3λe

λs(1 + e2λ−2λs

)= 0

1

2

(e2λs + 1

)+ λB(s)c3e

2λ+λs + λB(s)c3e2λ−λs − λB(s)c3e

λs(1 + e2λ−2λs

)= 0

1

2

(e2λs + 1

)+ λB(s)c3e

2λ+λs − λB(s)c3eλs = 0

1

2

(e2λs + 1

)+ λB(s)c3e

λs(e2λ − 1

)= 0

Solving for c3,

λB(s)c3eλs(e2λ − 1

)= −1

2

(e2λs + 1

)B(s)c3 = −1

λ

(e2λs + 1

2eλs (e2λ − 1)

)Substituting A(s)c1 and B(s)c3 into G(x, s), we have

G(x, s) =

(

12λeλs − 1

λ

(e2λs+1

2eλs(e2λ−1)

)e2λ

)(eλx + e−λx) for x < s

− 1λ

(e2λs+1

2eλs(e2λ−1)

)(eλx + e2λ−λx) for x > s.

By collecting terms, we have

G(x, s) =

(e2λs(e2λ−1)−(e2λs+1)e2λ

2λeλs(e2λ−1)

)(eλx + e−λx

)for x < s

−(

e2λs+1

2λeλs(e2λ−1)

)(eλx + e2λ−λx) for x > s.

Further consolidation,

G(x, s) =

(e2λ+2λs−e2λs−e2λs+2λ−e2λ

2λeλs(e2λ−1)

)(eλx + e−λx

)for x < s

−(

e2λs+1

2λeλs(e2λ−1)

)(eλx + e2λ−λx) for x > s.

35

G(x, s) =

−(

e2λs+e2λ

2λeλs(e2λ−1)

)(eλx + e−λx

)for x < s

−(

e2λs+1

2λeλs(e2λ−1)

)(eλx + e2λ−λx) for x > s.

G(x, s) =

−(e2λs(1+e2λ−2λs)

2λeλs(e2λ−1)

)(eλx + e−λx

)for x < s

−(

e2λs+1

2λeλs(e2λ−1)

)(1+e2λ−2λx

e−λx

)for x > s.

G(x, s) =

−(eλs−λ(1+e2λ−2λs)

2λe−λ(e2λ−1)

)(eλx + e−λx

)for x < s

−(

e2λs+1

λeλs−λ(e2λ−1)

)(1+e2λ−2λx

2eλ−λx

)for x > s.

G(x, s) =

− 1λ

(e2λ−2λs+1

2eλ−λs

)(eλ

e2λ−1

) (eλx + e−λx

)for x < s

− 1λ

(e−λs(e2λs+1)e−λ(e2λ−1)

)(1+e2λ−2λx

2eλ−λx

)for x > s.

G(x, s) =

−cosh (λ−λs) cosh (λx)

λ sinh (λ)for x < s

− cosh (λ−λx) cosh (λs)λ sinh (λ)

for x > s.

Notice that G(x, s) is symmetric about x and s.

To find v(x), we must solve

v(x) =

∫ 1

0

G(x, s)f(s)ds

where f(s) =α(s)−

∑Ni=1 αi(s)

D.

v(x) =

∫ 1

0

G(x, s)

(α(s)−

∑Ni=1 αi(s)

D

)ds

= −∫ x

0

cosh (λ− λs) cosh (λx)

λ sinh (λ)

(α(s)−

∑Ni=1 αi(s)

D

)ds

−∫ 1

x

cosh (λ− λx) cosh (λs)

λ sinh (λ)

(α(s)−

∑Ni=1 αi(s)

D

)ds

= − cosh (λx)

λ sinh (λ)

∫ x

0

cosh (λ− λs)

(α(s)−

∑Ni=1 αi(s)

D

)ds

− cosh (λ− λx)

λ sinh (λ)

∫ 1

x

cosh (λs)

(α(s)−

∑Ni=1 αi(s)

D

)ds

36

For special choices of α(s) and αi(s), let

αi(s) = 0.5Aui

(tanh

(xtsxi − xξtsx

)+ 1

)(i = 1, . . . , N),

α(s) = 0.5Av

(tanh

(x− xtsxξtsx

)+ 1

),

where Av, Aui , xtsx, xtsxi , and ξtsx are constants. Then,

v(x) = − cosh (λx)

Dλ sinh (λ)

∫ x

0

cosh (λ− λs)

(α(s)−

N∑i=1

αi(s)

)ds

− cosh (λ− λx)

Dλ sinh (λ)

∫ 1

x

cosh (λs)

(α(s)−

N∑i=1

αi(s)

)ds

= −0.5Av cosh (λx)

Dλ sinh (λ)

∫ x

0

cosh (λ− λs)(

tanh

(s− xtsxξtsx

)+ 1

)ds

+0.5 cosh (λx)

Dλ sinh (λ)

∫ x

0

cosh (λ− λs)N∑i=1

(Aui

[tanh

(xtsxi − sξtsx

)+ 1

])ds

− 0.5Av cosh (λ− λx)

Dλ sinh (λ)

∫ 1

x

cosh (λs)

(tanh

(s− xtsxξtsx

)+ 1

)ds

+0.5 cosh (λ− λx)

Dλ sinh (λ)

∫ 1

x

cosh (λs)N∑i=1

(Aui

[tanh

(xtsxi − sξtsx

)+ 1

])ds

Solutions to Equation (3.2.6) under varying parameters are found by inserting

information about αi and α. These solutions give us information on the behavior

of our nonlinear PDE as βi → 0 for all i = 1, . . . , N .

Chapter 4

Numerical Methods

To solve our coupled partial differential equations numerically, we chose to

use the finite difference method (FDM). With our varying diffusion coefficients and

varying production rates, using FDM allowed easy implementation and changes to

variables. Benefits of using the finite element method such as the ability to create

complex geometries were not needed since our domains are simple geometries.

Although the two-species models are a special case of the multiple-species

models with N = 1, the dimension in which the numerical simulations occurred

changed the numerical methods used. Hence, the 1-D methods for Multiple-species

Model I-III will be divided from the multiple dimensional methods used to repre-

sent Two-species Model I-III.

4.1 Methods for Multiple-species Models in 1-D

The following subsections describe the methods employed to model Multiple-

species Model I, II, and III. The first subsection describes how each multiple-species

model requires a modified FDM to account for the Neumann boundary conditions.

The second subsection describes a numerical scheme for Multiple-species Model II

and III in which we created an alternating scheme resembling a Gauss–Seidel like

iteration. The the last subsection explains the use of the Crank–Nicolson Method

to discretize the time element for Multiple-species Model III. Table 4.1.1 displays

the test functions used in the following numerical results.

37

38

Table 4.1.1: Test functions for 1-D models

1-D Test Functions Function Color in Graphs

u1(x, t) = cos(πx) blue

u2(x, t) = x2(12− 4x− 3x2) green

u3(x, t) = 5x2 − 53x3 − 5

4x4 black

v(x, t) = x2(1− x)2 red

4.1.1 Finite Difference and The Neumann Boundary Con-

dition

The Neumann boundary condition in all of the Multiple-species Models

requires a modified finite difference scheme to discretize ∆ui and ∆v. The scheme

consists of two cases, each based upon the grid point location on a uniformly spaced

grid (divided into p sections) in Ω = (0, 1).

The grid points are divided into interior points (points inside Ω = (0, 1))

and boundary points of Ω. Interior points follow the traditional central finite

difference scheme, while the boundary points follow a scheme created by Taylor

series approximations with an error on the order of O(h2) with h equal to the

distance between each grid point. Figure 4.5 displays the weights for the two

different types of points in Ω.

-0.54-3.5

(a) Boundary point

1-21

(b) Interior point

Figure 4.1: Stencils of 1-D finite difference method. The point being evaluated

is colored blue with weights distributed as shown.

Figure 4.2 displays the error created when testing Multiple-species Model

39

I for N = 3. The error decreases as iterative steps increase and as the grid size

increases. The test functions are shown in Table 4.1.1.

1 2 3 4 5 6 70

1

2

3

4

5

Iterative steps

Err

or

Student Version of MATLAB

(a) Error vs iterative steps, grid size p = 40

100 101 10210−4

10−3

10−2

10−1

100

Grid size

Err

or

Student Version of MATLAB

(b) Log plot of error vs grid size

Figure 4.2: Multiple-species Model I numerical methods test for N = 3 and

Di = 0. Note that the color blue denotes the function u1, the color green denotes

the function u2, the color black denotes the function u3, and the color red denotes

the function v.

4.1.2 Alternating Iteration

For Multiple-species Model I, the nonlinear PDE could be solved using

Gaussian Elimination. However, for Multiple-species Model II and III, this tactic

required too many calculations. Therefore, we fabricated an iterative method to

solve for ui and v by successive Gauss–Seidel type iterations that we call Alternat-

ing Iteration (AI). AI consists of solving for each ui via Gauss–Seidel iteration and

then, using the updated ui, solving for v via Gauss–Seidel iteration. For example,

let uqi be the qth iteration of ui, and let vq be the qth iteration of v. For each

i = 1, . . . , N , we solve for uqi using Gauss–Seidel:

Di∆uqi − βiu

qi − kiu

qivq−1 + αi = 0 .

40

Then, we solve the following equation for vq with the just calculated uqi ,

D∆vq − βvq −N∑i=1

kiuqivq + α = 0 ,

leaving an alternating Gauss–Seidel-like scheme. Figure 4.3 shows the error from

this scheme.

2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

1.4

Iterative steps

Err

or

Student Version of MATLAB

(a) Error vs time steps, grid size p = 40

100 101 10210−4

10−2

100

102

Grid size

Err

or

Student Version of MATLAB

(b) Log plot of error vs grid size

Figure 4.3: Multiple-species Model II numerical methods test for N = 3 and

Di 6= 0. Note that the color blue denotes the function u1, the color green denotes

the function u2, the color black denotes the function u3, and the color red denotes

the function v.

4.1.3 The Crank–Nicolson Method

Based on the central difference method in space and the trapezoidal rule

in time, the Crank–Nicolson method gives a second order convergence in time.

Figure 4.4 displays the numerical error from this method which describes Multiple-

species Model III.

41

0 1000 2000 3000 4000 5000 60000

1

2

3

4

Iterative steps

Err

or

Student Version of MATLAB

(a) Error vs. time steps, grid size p = 40

100 101 10210−4

10−2

100

102

Grid size

Err

or

Student Version of MATLAB

(b) Log plot of error vs. grid size

Figure 4.4: Multiple-species Model III numerical methods test for N = 3. Note

that the color blue denotes the function u1, the color green denotes the function

u2, the color black denotes the function u3, and the color red denotes the function

v.

4.2 Methods for Two Species in 2-D

To solve Two-species Model I, II and III in two dimensions, we needed a

different modified FDM than the one used previously to account for the Neumann

boundary conditions in two dimensions. In addition, we used Newton’s method

and Gauss–Seidel iteration to numerically solve the nonlinear PDE in Two-species

Model I. We use Alternating Iteration for Two-species Model II, and the implicit

method for Two-species Model III. Table 4.2.1 displays the test functions used in

all of the following computations.

Table 4.2.1: Test functions for 2-D models

2-D Test Functions

u(x, y, t) = cos2(πx) cos2(πy)

v(x, y, t) = x2(1− x)2y2(1− y)2

w(x, y, t) = sin2(πx) sin2(πy)

42

4.2.1 Finite Difference Discretization

The Neumann boundary condition in all three of the models derived from

the Two-species Model requires a modified finite difference scheme to discretize

∆u and ∆v on the boundary. The scheme consists of three cases, each based upon

the grid point location on a uniformly spaced p× p grid in Ω = [0, 1]× [0, 1].

The grid points are divided into interior points (points in Ω), boundary

points (points on ∂Ω), and corner points ((0, 0), (0, 1), (1, 0), (1, 1)). Interior

points follow the traditional central finite difference scheme, while the boundary

and corner points follow a scheme created by Taylor series approximations. The

boundary point and corner point schemes have an error on the order of O(h2)

where h is equal to the distance between each grid point on the uniformly spaced

p× p grid. Figure 4.5 displays the numerical weight for each type of point in Ω.

-4

1

1

1

1

(a) Center point

-0.5

-0.5

4

4-3.5

(b) Corner boundary point

-0.5

1

4

-5.51

(c) Edge boundary point

Figure 4.5: Stencils of 2-D modified finite difference method. The points being

evaluated are colored blue with weights distributed as shown.

4.2.2 Newton’s Method and Gauss–Seidel Iteration

For Two-species Model I, our numerical scheme required a linearization of

the nonlinear term to decrease computing time and increase the effectiveness of

computing large sparse matrices. Implementing Newton’s method to the nonlin-

ear part of Equation (2.2.4) will approximate the nonlinear term linearly at each

iterative step.

43

Let F (v) be defined by

F (v) = α− βv − k1α1v

β1 + k1v.

Two-species Model I can be written as

D∆v + F (v) = 0 . (4.2.1)

Applying Newton’s method to F (v), the m+ 1st iterative step of F (v) becomes

F (vm+1) = F (vm) + F ′(vm)(vm+1 − vm)

and Equation (4.2.1) becomes

D∆vm+1 + vm+1F ′(vm) = vmF ′(vm)− F (vm).

To compute our relatively sparse, diagonally dominate matrix created by

Two-species Model I, we used Gauss–Seidel iteration, updating entries as com-

puted. Figure 4.6 shows the computational error using Newton’s method and

Gauss–Seidel iteration simultaneously.

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

Student Version of MATLAB

(a) Error vs iterative steps, grid size p = 64

100 101 10210−3

10−2

10−1

100

Grid size

Err

or

Student Version of MATLAB

(b) Log plot of error vs grid size

Figure 4.6: Test of numerical methods for Two-species Model I with Newton’s

method and Gauss–Seidel iteration. Note that the color red denotes the function

v.

44

4.2.3 Alternating Iteration

For the coupled time-independent PDE, Model II, we used the AI method

from Multiple-species Model II (see Section 4.1.2). Here, AI consists of solving

for u via Gauss–Seidel iteration and then, using an updated u, solving for v via

Gauss–Seidel iteration. Figure 4.7 shows the numerical computational error of

Two-species Model II using AI.

0 200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

Err

or

Iterative steps

Student Version of MATLAB

(a) Error vs iterative steps, grid size p = 64

100 101 10210−4

10−2

100

102

Grid size

Err

or

Student Version of MATLAB

(b) Log plot of error vs grid size

Figure 4.7: Test of numerical methods for Two-species Model II with Alternating

Iteration. Note that the color blue denotes the function u and the color red denotes

the function v.

4.2.4 Explicit vs. Implicit Scheme

We selected the forward Euler method to discretize the time element for

Two-species Model III and found results using both the explicit and implicit spatial

discretization methods. Define L1(u, v) and L2(u, v) by the following:

L1(u, v) = −β1u− k1uv + α1

L2(u, v) = −βv − k1uv + α .

45

With forward Euler and explicit spatial discretization, Two-species Model III’s

discretization for both u and v becomes

uk+1i,j − uki,j

∆t= L1(uki,j, v

ki,j),

vk+1i,j − vki,j

∆t= L2(uki,j, v

ki,j) .

where uki,j = u(xi, xj, tk) and vki,j = v(xi, xj, tk).

0 0.5 1 1.5 2 2.5x 105

0

0.2

0.4

0.6

0.8

1

Err

or

Iterative steps

Student Version of MATLAB

(a) Error vs time steps, grid size p = 64, time

steps T = 100

100 101 10210−3

10−2

10−1

100

101

Grid size

Err

or

Student Version of MATLAB

(b) Log plot of error vs. grid size

Figure 4.8: Numerical error of Two-species Model III using the explicit scheme.

Note that the color blue denotes the function u and the color red denotes the

function v.

The scheme for solving the coupled time dependent equations involves al-

ternating between solving for u at time t and then for v at time t in AI fashion. To

ensure convergence of the explicit scheme, a CFL condition must be met, which

requires that the time step ∆t be limited by the following equation

maxD1, D∆t

(∆x)2≤ 1

2. (4.2.2)

This restriction may create an undesirable number of increased iterations from the

decrease in time step size. Figure 4.8 represents our numerical error of our explicit

scheme using test functions u and v defined above.

46

The implicit scheme for Two-species Model III is not harbored down by the

CFL restriction, but instead requires a new equation to be solved at each iterative

step. The discretization for the implicit method is

uk+1i,j − uki,j

∆t= L1(uk+1

i,j , vki,j) (4.2.3)

vk+1i,j − vki,j

∆t= L2(uk+1

i,j , vk+1i,j ) . (4.2.4)

where uki,j = u(xi, xj, tk) and vki,j = v(xi, xj, tk). Figure 4.9 depicts the numerical

error created by the implicit scheme using a test function.

0 2 4 6 8 10 12x 105

0

0.2

0.4

0.6

0.8

1

Err

or

Iterative steps

Student Version of MATLAB

(a) Error vs time steps, grid size p = 64, time

steps T = 100

100 101 10210−4

10−2

100

102

Grid size

Err

or

Student Version of MATLAB

(b) Log plot of error vs. grid size

Figure 4.9: Numerical error of Two-species Model III using the implicit scheme.

Note that the color blue denotes the function u and the color red denotes the

function v.

Comparing these two schemes shows that the implicit scheme convergences

more quickly by less step iterations than the explicit scheme with our test functions

(Figure 4.10).

47

0 0.5 1 1.5 2 2.5x 105

0

0.2

0.4

0.6

0.8

1

Err

or

Iterative steps

Student Version of MATLAB

(a) Explicit scheme, error vs time steps

0 0.5 1 1.5 2 2.5x 105

0

0.2

0.4

0.6

0.8

1

Err

or

Iterative steps

Student Version of MATLAB

(b) Implicit scheme, error vs time steps

Figure 4.10: Numerical error of Two-species Model III of both the explicit and

implicit scheme. Note that the color blue denotes the function u and the color red

denotes the function v.

Chapter 5

Computational Results

Now we present our computational results showing mRNA and sRNA in-

teractions in a one-dimensional environment and two-dimensional environment.

Parameters for both mRNA and sRNA were picked based on some experimental

data and varied in some instances to show how changes may effect interactions

between the two species. The first section depicts interactions of multiple mRNA

species interaction with sRNA in a one-dimensional environment. The second

section shows interactions of one mRNA and one sRNA in two dimensions and

addresses numerical stability of the interface between the species.

5.1 Multiple-species Models in 1-D

For the following numerical simulations, unless noted otherwise, we set

N = 3, βi = β = 10−2, and Ω = [0, 1]. We define the production rates or

transcription rates of the multiple mRNA strands and the sRNA strand by the

functions displayed in Table 5.1.1. Two of the functions, α1 and α, shown in

Table 5.1.1 and Figure 5.1 describe mRNA and sRNA production rates derived

from experimental data [14]. The other functions are variations of α1, showing

how other similar mRNA target concentrations react with the concentration of

sRNA under the same conditions.

48

49

Table 5.1.1: Production rates of mRNA and sRNA

Transcription

profileAm As xtsx ltsx Function

α1 2 1 0.5 0.2 α1 = 0.5Am

(tanh

(xtsx − xltsx

)+ 1

)α2 2 1 0.65 0.15 α2 = 0.5Am

(tanh

(xtsx − xltsx

)+ 1

)α3 2 1 0.25 0.1 α3 = 0.5Am

(tanh

(xtsx − xltsx

)+ 1

)α 2 1 0.5 0.2 α = 0.5As

(tanh

(x− xtsxltsx

)+ 1

)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tissue lengh

Tra

nscr

iptio

n ra

te

Student Version of MATLAB

(a) Transcription profiles (production rates) of

mRNA and sRNA.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(b) Steady state concentrations of mRNA and

sRNA with no diffusion.

Figure 5.1: Transcription profiles (production rates) and steady state concen-

trations of mRNA targets and sRNA with no diffusion. Note that the color blue

denotes the concentration of mRNA target 1 (u1), the color green denotes the

concentration of mRNA target 2 (u2), the color black denotes the concentration of

mRNA target 3 (u3), and the color red denotes the sRNA concentration (v).

50

5.1.1 Multiple-species Model I

Recall that the Multiple-species Model I is characterized by the equations:

D∆v − βv −3∑i=1

αikiv

βi + kiv+ α = 0 in Ω, (5.1.1)

∂v

∂n= 0 on ∂Ω. (5.1.2)

Biologically, Multiple-species Model I shows how mRNA and sRNA concentrations

interact with each other at steady state when mRNA has no diffusion coefficient

and sRNA moves across tissue.

Before simulating Multiple-species Model I with N = 3, we recreated a

numerical simulation of Multiple-species Model I with N = 1 and compared it

to a previous simulation that correlated with collected experimental data [14]. In

Figure 5.2, the results from our simulation of Multiple-species Model I seen in

Figure 5.2a closely resemble the experimental data simulation in Figure 5.2b. Our

results, in accordance with the previous numerical results, show that the mRNA

interface between high concentrations and low concentrations sharpens over time

and the sRNA interface smooths over time. The sharpening occurrs when D, the

diffusion coefficient of sRNA, had magnitude 10−2. The greatest movement of the

mRNA interface between the curve with no diffusion and D = 10−2 was 0.212 units

where the curve moving from right to left on our standardized Ω = [0, 1] grid.

Because of the successful outcome, in the following simulations of Multiple-

species Model I, we let N = 3. Recall that Di = 0 for all i. Since we are interested

in seeing a sharp concentration interface change of mRNA, we let the order of

magnitude of D range such that 10−4 ≤ D ≤ 102. Figure 5.3 shows the trend

in concentration boundary movement based on the diffusion coefficient D. Notice

that a sharp interface for u1 and u2 only occurs when D ≤ 10−2 and u3 is relatively

unchanged until D = 102.

The order of magnitude of ki may change the concentration interface of

both mRNA and sRNA. To understand their connection to the movement of the

interface, we let ki = 10 and performed the same simulations seen in Figure 5.3.

In Figure 5.4 which depicts the results with k1 = 10 and 10−4 ≤ D ≤ 102, the

51

0 0.2 0.4 0.6 0.8 1

0

1

2

1.5

0.5

mRNA

miRNA

Fraction of Tissue Length

Tra

nscription R

ate

0 0.2 0.4 0.6 0.8 1

0

100

200

150

50

Fraction of Tissue Length

Concentr

ation

mRNA

miRNA

0 0.2 0.4 0.6 0.8 1

0

100

200

150

50

Fraction of Tissue Length

Concentr

ation mRNA

miRNA

(a) 1-D numerical results from left to right: transcription profile of mRNA (red) and sRNA

(green), steady state concentration of mRNA (u0) and sRNA (v0) with D1 = D = 0, steady

state concentration of mRNA (u) and sRNA (v) with D1 = 0, D = 10−2.

(b) Results [14]: A) transcription profile of mRNA (red) and sRNA (green), B) steady state

concentration of mRNA and sRNA with D1 = D = 0, C) steady state concentration of mRNA

and sRNA with D1 = 0, D = 10−2.

Figure 5.2: Replicated results from Levine et al. [14].

52

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(a) Tissue length vs. concentration, D = 10−4.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(b) Tissue length vs. concentration, D = 10−2.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(c) Tissue length vs. concentration, D = 1.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(d) Tissue length vs. concentration, D = 102.

Figure 5.3: Change in mRNA and sRNA concentrations in Multiple-species

Model I where k1 = k2 = k3 = 1 and D1 = D2 = D3 = 0. Solid lines depict

concentrations where both mRNA and sRNA have no diffusion (D = 0) and dot-

ted lines indicate the concentrations of mRNA and sRNA at respective diffusion

coefficients. Note that the color blue denotes the concentration of mRNA target 1

(u1), the color green denotes the concentration of mRNA target 2 (u2), the color

black denotes the concentration of mRNA target 3 (u3), and the color red denotes

the sRNA concentration (v).

53

higher ki value resulted in a sharp interface for a higher value of D. That is, a

higher coupled degradation coefficient resulted in the ability for a higher diffusion

coefficient of sRNA.

5.1.2 Multiple-species Model II

Recall that for Multiple-species Model II, we are solving the equation:

D1∆u1 − β1u1 − k1u1v + α1 = 0 in Ω,

D2∆u2 − β2u2 − k2u2v + α2 = 0 in Ω,

D3∆u3 − β3u3 − k3u3v + α3 = 0 in Ω,

D∆v − βv −3∑i=1

kiuiv + α = 0 in Ω,

∂u1

∂n=∂u2

∂n=∂u3

∂n=∂v

∂n= 0 on ∂Ω.

Multiple-species Model II differs from Multiple-species Model I in that we let both

D and Di be greater than 0. This means that both the mRNA and the sRNA

diffuse within tissue.

In the previous Multiple-species Model I simulation, the sharp interface

occurred when D = 10−2 and when ki = 1. Hence, we fixed D at that order of

magnitude and looked for interface changes as Di ranged from 10−4 to 102. We also

wanted to see if D = 10−3 changed the interface significantly. Figure 5.5 depicts

the changes in the concentration boundaries for ki = 1, D = 10−2 or D = 10−3,

and Di varying between 10−4 to 102. Notice that a sharp interface only occurs for

u1 and u2 when Di = 10−4. This may suggest that mRNA may not diffuse much

in tissue if a sharp interface is desired.

As with Multiple-species Model I, ki = 10 may change the range of values

where a sharp interface for mRNA may occur. In Figure 5.6, for all i, we let

ki = 10, D = 10−2 or D = 10−3, and varied Di.

Notice that ki = 10 did not change the range of values that create a sharp

interface for mRNA. That is, a sharp interface only occurs for u1 and u2 when

Di = 10−4.

54

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(a) Tissue length vs. concentration, D = 10−4.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(b) Tissue length vs. concentration, D = 10−2.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(c) Tissue length vs. concentration, D = 1.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(d) Tissue length vs. concentration, D = 102.

Figure 5.4: Change in mRNA and sRNA concentrations in Multiple-species

Model I where k1 = k2 = k3 = 10 and D1 = D2 = D3 = 0. Solid lines de-

pict concentrations where mRNA and sRNA have no diffusion (D = 0) and dotted

lines indicate the concentrations of mRNA and sRNA at respective diffusion co-

efficients. Note that the color blue denotes the concentration of mRNA target 1

(u1), the color green denotes the concentration of mRNA target 2 (u2), the color

black denotes the concentration of mRNA target 3 (u3), and the color red denotes

the sRNA concentration (v).

55

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(a) Tissue length vs. concentration,

Di = 10−4, D = 10−2.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(b) Tissue length vs. concentration,

Di = 10−3, D = 10−2.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(c) Tissue length vs. concentration,

Di = 10−4, D = 10−3.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(d) Tissue length vs. concentration,

Di = 10−3, D = 10−3.

Figure 5.5: Change in mRNA and sRNA concentrations in Multiple-species

Model II where ki = 1 for all i and D = 10−2 or D = 10−3 respectively. Solid

lines depict concentrations where mRNA and sRNA have no diffusion (for i=1,2,3,

Di = D = 0) and dotted lines indicate the concentrations of mRNA and sRNA at

respective diffusion coefficients. Note that the color blue denotes the concentration

of mRNA target 1 (u1), the color green denotes the concentration of mRNA target

2 (u2), the color black denotes the concentration of mRNA target 3 (u3), and the

color red denotes the sRNA concentration (v).

56

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(a) Tissue length vs. concentration,

Di = 10−4, D = 10−2.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(b) Tissue length vs. concentration,

Di = 10−3, D = 10−2.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(c) Tissue length vs. concentration,

Di = 10−4, D = 10−3.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(d) Tissue length vs. concentration,

Di = 10−3, D = 10−3.

Figure 5.6: Change in mRNA and sRNA concentrations in Multiple-species

Model II where ki = 10 for all i and D = 10−2 or D = 10−3 respectively. Solid

lines depict concentrations where mRNA and sRNA have no diffusion (for i=1,2,3,

Di = D = 0) and dotted lines indicate the concentrations of mRNA and sRNA at

respective diffusion coefficients. Note that the color blue denotes the concentration

of mRNA target 1 (u1), the color green denotes the concentration of mRNA target

2 (u2), the color black denotes the concentration of mRNA target 3 (u3), and the

color red denotes the sRNA concentration (v).

57

5.1.3 Multiple-species Model III

Multiple-species Model III is our dynamic PDE that is time dependent.

Recall that for N = 3 the equations are

∂u1

∂t= D1∆u1 − β1u1 − k1u1v + α1 in Ω× [0,∞),

∂u2

∂t= D2∆u2 − β2u2 − k2u2v + α2 in Ω× [0,∞),

∂u3

∂t= D3∆u3 − β3u3 − k3u3v + α3 in Ω× [0,∞),

∂v

∂t= D∆v − βv −

3∑i=1

kiuiv + α in Ω× [0,∞),

∂u1

∂n=∂u2

∂n=∂u3

∂n=∂v

∂n= 0 on ∂Ω× [0,∞),

u1(·, 0) = (u1)0, u2(·, 0) = (u2)0

u3(·, 0) = (u3)0, v(·, 0) = v0

in Ω.

Because Multiple-species Model I and Multiple-species Model II are steady state

versions of Multiple-species Model III, the concentrations of mRNA targets and

sRNA found using the Crank-Nicolson method are the same as those found in Sec-

tions 5.1.1 and 5.1.2. Figure 5.7 shows simulations matching those in Figures 5.3b,

5.4b, 5.5a, and 5.6a.

5.2 Two-species Models in 2-D

After replicating the 1-D numerical results in Figure 5.2a, we created analo-

gous 2-D transcription rate equations for two different gene types. Gene 1, similar

to the equations associated with Figure 5.2, follows the transcription rates

α1 = 1 + tanh(2.5− 5xy),

α = 0.5 + 0.5 tanh(5xy − 2.5) .

Figure 5.8 displays the newly acquired 2-D production rates of Gene 1 as well as

steady state concentrations of both mRNA and sRNA with diffusion coefficients

of 0. Gene 2, a striped gene, follows the equations

58

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(a) Tissue length vs. concentration,

Di = 0, D = 10−2, ki = 1.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(b) Tissue length vs. concentration,

Di = 0, D = 10−2, ki = 10.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(c) Tissue length vs. concentration,

Di = 10−4, D = 10−2, ki = 1.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

Tissue Length

Con

cent

ratio

n

Student Version of MATLAB

(d) Tissue length vs. concentration,

Di = 10−4, D = 10−2, ki = 10.

Figure 5.7: Change in mRNA and sRNA concentrations in Multiple-species

Model III where D = 10−2. Solid lines depict concentrations where mRNA and

sRNA have no diffusion (for i=1, 2, 3, Di = D = 0) and dotted lines indicate

the concentrations of mRNA and sRNA at respective diffusion coefficients. Note

that the color blue denotes the concentration of mRNA target 1 (u1), the color

green denotes the concentration of mRNA target 2 (u2), the color black denotes

the concentration of mRNA target 3 (u3), and the color red denotes the sRNA

concentration (v).

59

1'1 101 '"• )

&S

L,§0,F C C

I IC

"Fractkln «TissL>e Lengh C C

1'1 1'1 10

~ 200 _ 200 _ 200 _0

"S ICC 100 , ICC••,•8 c c C

I I IC

"C I Fractkln «TisSL>e Lengh C I C I

Figure 5.8: (a) Transcription rates of mRNA (α1) and sRNA (α) on Ω, (b)

transcription rate of mRNA (α1), (c) transcription rate of sRNA (α), (d) steady

state concentrations of mRNA (u) and sRNA (v) with no diffusion where

D1 = D = 0, (e) steady state concentration of mRNA with no diffusion, (f) steady

state concentration of sRNA with no diffusion.

60

1'1 101

1~ ~O.5~5·····0

C I

1'1 1'1

~ICC ICC0

"s" "••,

•8 c C

I IC

"'C I Fractkln «TisSL>e Lengh C I

Figure 5.9: (a) Transcription rates of mRNA (α1) and miRNA (α2) on Ω, (b)

transcription rate of mRNA (α1), (c) transcription rate of miRNA (α2), (d) steady

state concentrations of mRNA (u) and sRNA (v) with no diffusion where

D1 = D = 0, (e) steady state concentration of mRNA with no diffusion, (f) steady

state concentration of sRNA with no diffusion.

α1 = tanh

(0.7− xy

0.2

)+ 1,

α = 0.6 + tanh

(0.3− xy

0.2

)+ 1.

to show production rates of mRNA and sRNA as well as steady state concentrations

shown in Figure 5.9.

5.2.1 Two-species Model I

The 2-D version of Multiple-species Model I with N = 1, similar to the 1-D,

has a sharpening of the interface occurring along the surface describing u and a

61

Figure 5.10: (a) Steady state concentrations of mRNA with no diffusion where

D1 = D = 0, (b) steady state concentration of mRNA with D1 = 0, D = 10−2,

(c) steady state concentrations of sRNA where D1 = D = 0 , (d) steady state

concentration of sRNA with D1 = 0, D = 10−2.

decrease in the concentration of v seen in the flattening of the curve. Figure 5.10

shows the steady state concentrations of Gene 1 in which sharpening occurs. In

Figure 5.11, the initial concentration profiles for mRNA and sRNA and the sharp-

ening of the mRNA concentration profile of Gene 2 are illustrated.

The interface sharpening occurred only after a specific order of magnitude

of D was reached for each gene type with D values starting at 10−5. For Gene

1, when the diffusion coefficient reached D = 1, the sharpening effect became a

smoothing effect on the interface (Figure 5.12). For Gene 2, the smoothing effect

on the interface came as the diffusion coefficient reached D = 10−2 (Figure 5.13).

62

Figure 5.11: (a) Steady state concentrations of mRNA where D1 = D = 0,

(b) steady state concentration of mRNA with D1 = 0, D = 10−3, (c) steady state

concentrations of sRNA where D1 = D = 0 , (d) steady state concentration of

sRNA with D1 = 0, D = 10−3.

63

Student Version of MATLAB

Figure 5.12: Range of diffusion coefficient values and respective interface behav-

ior for Gene 1. Top row depicts mRNA concentrations and bottom row sRNA

concentrations.

u,02_0

c ,Y,02_0

u, 02 _ 0.0001

c ,Y, 02 _ 0.0001

u, 02 _ 0.001

c ,Y, 02 _ 0.001

u, 02 _ 0.01

Y, 02 _ 0.01

u,02_0.1

,co

"~~.O

C ,

Y,02_0.1

Figure 5.13: Range of diffusion coefficient values and respective interface behav-

ior for Gene 2. Top row depicts mRNA concentrations and bottom row sRNA

concentrations.

64

5.2.2 Two-species Model II

The sharpening of the interface in Two-species Model II of u depends on

ranging values of both D1 and D, as opposed to the adjustment of only the one

parameter D in Two-species Model I. To determine when a sharpening may occur,

we let either D1 or D be constant and looked for changes or patterns. For Gene

1, we chose D2 to be a stationary value and ranged the values of D1 by orders

of magnitude from 10−4 to 1 (Figure 5.14). For Gene 2, values of D1 = 10−4 to

D1 = 1 with D = 0.001 yielded a smoothing of the interface or little to no interface

movement. Figure 5.15 depicts the slight changes in D1 resulted in a decrease in

the maximal concentration level of u and a smoothing of the surface.

5.2.3 Two-species Model III

The implicit and explicit scheme numerical results closely follow those

shown for Two-species Model I and Two-species Model II with the respective dif-

fusion coefficients, showing that indeed steady state concentrations are reached

using the implicit and explicit schemes. The implicit and explicit schemes provide

additional information through observation of the flow and progression of concen-

trations into their respective steady state positions, thus displaying how the change

in interface is achieved.

Stability to Fluctuations: To test the 2-D numerical solutions for stability along

the interface, we created an artificial interface containing oscillations between the

concentrations of Species 1 and Species 2. We let u = 1 and v = 0 on one side of

an oscillatory region and u = 0 and v = 1 on the other. By running our implicit

scheme, the oscillations shown in Figure 5.16a dissipate, and we get a sharp straight

interface as seen in Figure 5.16b.

65

I') (0) '"~ ICC 200, 200 _0

"S ICC 100 _ 100 _••,0

:~8 , c) ) 0.5'- - -.-.------- 05 0C C

"C Fractkln «TissL>e Lengh C ) C )

I') I') 10

:~ ~O_5~50C )

o -._ .-.-.- _.-.---.' 0 -._ ...------.----_

1~01~Oo 1 Fractkln«TissooLengh 0 1

~ 200_

~sE!8

Figure 5.14: (a) Steady state concentration of mRNA with no diffusion where

D1 = D = 0, (b) steady state concentration of mRNA with diffusion coefficients

D1 = 0.0001 and D = 0.01, (c) steady state concentration of mRNA with diffusion

coefficients D1 = 0.001 and D = 0.01, (d) steady state concentration of mRNA

with diffusion coefficients D1 = 0.01 and D = 0.01, (e) steady state concentration

of mRNA with diffusion coefficients D1 = 0.1 and D = 0.01, (f) steady state

concentration of mRNA with diffusion coefficients D1 = 1 and D = 0.01.

66

I') (0) '"~ ICC 100 , 100 _0

"s" "••,

0

8 , c c) ) )

c c c

"C Fractkln «TissL>e Lengh C )

I') I') 10

~ 100 _ 100 _ 100 _0

"S" " "••,

0

8

c )

Figure 5.15: (a) Steady state concentration of mRNA with no diffusion where

D1 = D = 0, (b) steady state concentration of mRNA with diffusion coefficients

D1 = 0.0001 and D = 0.001, (c) steady state concentration of mRNA with diffusion

coefficients D1 = 0.001 and D = 0.001, (d) steady state concentration of mRNA

with diffusion coefficients D1 = 0.01 and D = 0.001, (e) steady state concentration

of mRNA with diffusion coefficients D1 = 0.1 and D = 0.001, (f) steady state

concentration of mRNA with diffusion coefficients D1 = 1 and D = 0.001.

67

(a) Top view of stability analysis

,,

,,

1'1

1111 "1111 "I

"11111

IIIII1,1 II11' 'illll

,,

,,

1111

1'1

(b) Side view of stability analysis

Figure 5.16: Numerical simulations of stability for the Two-species Model.

Chapter 6

Conclusions and Discussions

We have created two mathematical models that simulate interactions be-

tween mRNA and sRNA across tissue. Multiple-species Model I, II, and III de-

scribes one sRNA interacting with several different target mRNA, while Two-

species Model I, II, and III models one sRNA interacting with one mRNA.

6.1 Summary of Results

To construct the Multiple-species Model, we first created a stochastic model

of mRNA and sRNA interactions within the cell from a kinetic diagram. By using

the mean-field assumption, we created PDEs that modeled mRNA and sRNA in-

teractions across tissue with parameters that described production rates, indepen-

dent degradation rates, and coupled degradation rates of mRNA and sRNA. We di-

vided these PDEs into three separate equations: a single, nonlinear equation called

Multiple-species Model I; a coupled PDE at steady state called Multiple-species

Model II; and a time dependent, coupled PDE called Multiple-species Model III.

The Two-species Model is similar to the Multiple-species Model with N = 1 and

was divided into three separate models in a similar manner.

Because the Two-species Model is very similar to the Multiple-species Model

with N = 1, the analysis of the models focused mainly on the Multiple-species

Model. For the Multiple-species Model, we found that the solution to the model’s

associated ODE system is linearly stable. We also found using calculus of variations

68

69

that Multiple-species Model I is well-posed and the solution is unique.

For Multiple-species Model I, II, and III, we let N = 3 and numerically

solved for target mRNA and sRNA concentrations. We were interested in values

that indicated the conditions for which a gene may be turned on or off. We

varied certain D values of our multiple-species models to find conditions in which

a sharpening of the concentration profile of target mRNA may occur. In addition,

we altered the couple degradation coefficient to see how that effected the range in D

values. We found that a increase in the coupled degradation coefficient allowed an

extension of the D values. For Multiple-species Model II, a change in the coupled

degradation coefficient did not change the range of D values. We will comment

more on the parameters later in this chapter.

With 2-D numerical simulations of Two-species Model I, II, and III, we

found a range of values for which a sharpening of the concentration profile at the

interface occurred. For Two-species Model I, sharpening of the interface for Gene

1 began with D ≈ 10−5 and a smoothing of the interface occurred at 1. For

Two-species Model II, a fixed D and a ranging D1 provided mostly a smoothing

of the surface of u1 (mRNA concentration profile) for both Gene 1 and Gene 2.

Using Two-species Model III, we showed the stability of solutions for our coupled

time dependent PDE by starting with oscillatory conditions at the interface and

showing that the surface u converged to a straight interface.

6.2 Parameters of Interest

Notice that βi and β were not changed in the numerical simulations. Equal

changes in both βi and β simultaneously seem to only scale the concentrations of

each species and do not seem to effect the interactions between the species.

Some parameters give insight into when and how the interface will change.

In particular, the ratio of βi, β, αi, α,Di and D are of interest. We look at some

parameter relationships in Two-species Model I, II and III. At steady state with

70

Di = 0 and D = 0, we have the following differential equation:

−β1u− k1uv + α1 = 0

−βv − k1uv + α = 0

Solving for u, we have

−β1u(β + k1u

)− k1uα + α1

(β + k1u

)= 0

−k1β1u2 − β1βu− k1uα + α1k1u+ α1β = 0

u2 +1

k1β1

(β1β + k1α− α1k1

)u− α1β

k1β1

= 0.

Then, we can solve for u using the quadratic formula.

u =− 1k1β1

(β1β + k1α− α1k1

)±√(

1k1β1

(β1β + k1α− α1k1

))2

+ 4α1β

k1β1

2

=1

2β1

β1β

k1

+k1

k1

α− α1 ±

√(β1β

k1

+k1

k1

α− α1

)2

+4α1β1β

k1

.

These u and v were used in the numerical simulations as initial conditions for the

Two-species Model. Let λ = β1β

k1. Then,

u =1

2β1

λ+k1

k1

α− α1 ±

√(λ+

k1

k1

α− α1

)2

+ 4α1λ

.

Notice that if λ→ 0,

u =1

2β1

k1

k1

α− α1 ±

√(k1

k1

α− α1

)2 (6.2.1)

=1

2β1

(k1

k1

α− α1 ±(k1

k1

α− α1

))(6.2.2)

=

1β1

(k1k1α− α1

)0

. (6.2.3)

That is, as λ gets closer to 0, the equation u will get closer to the behavior found

in Equation (6.2.3).

71

6.3 Accuracy of the Model

A disadvantage of the mean-field model may lie in the assumption that

〈MtSt〉 = 〈Mt〉 〈St〉 (Equation (2.1.4)). This means that the covariance, a measure

of how much two random variables change together, is zero. Under certain circum-

stances, this may not be true. However, it is argued that accurate estimates maybe

be found by allowing some parameter k such that 〈MtSt〉 = k 〈Mt〉 〈St〉 [25]. Hence,

a mean-field solution may be found with this modification. In addition, Platini

discusses the circumstances in which the mean-field approximation is accurate.

The data found using the Gillespie algorithm shows that the mean-field model is

not a good approximation for αiβi<< 1 and α

β<< 1 [25].

6.4 Future Work

With some experimental results, our generalized multiple-species model of

mRNA and sRNA interactions may be able to represent specific types of cells and

sRNA interactions. In particular, there is interest in finding the production rates,

independent degradation rates, and couple degradation rates for particular cell

types. Computationally, an increase in the accuracy of modeling mRNA and sRNA

interactions may form from using the Level Set Method or a Phase-field model.

The Multiple-species Model may further be modified by adding an advection term

to mRNA movement as has been suggested by some biophysicists.

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