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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.
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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
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DEDICATION
To those who think their work is futile.
iv
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EPIGRAPH
Multiplication is vexation;
Division is as bad;
The Rule of Three perplexes me,
And fractions drive me mad!
Nursery rhyme
v
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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
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4.1.2 Alternating Iteration . . . . . . . . . . . . . . . .
394.1.3 The CrankNicolson Method . . . . . . . . . . . . 40
4.2 Methods for Two Species in 2-D . . . . . . . . . . . . . .
414.2.1 Finite Difference Discretization . . . . . . . . . .
424.2.2 Newtons Method and GaussSeidel 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
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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 CrankNicolson Method . . .
41Figure 4.5: Stencils of 2-D modified finite difference method . .
. . . . . . . 42Figure 4.6: Two-species Model I, Newtons Method and
GaussSeidel 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
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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
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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
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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
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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
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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
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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
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2Figure 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
-
321 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]).
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4By 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.
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5The 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 [1315]. 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 mRNAs and
sRNAs 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
sRNAs 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
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6and 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 genes 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 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):
uit
= Diui iui kiuiv + i (i = 1, . . . , N)v
t= Dv v
Ni=1
kiuiv + in [0,), (1.3.1)
uin
=v
n= 0 (i = 1, . . . , N) on [0,), (1.3.2)
ui(, 0) = (ui)0 (i = 1, . . . , N) and v(, 0) = v0 in .
(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, i, , and ki are
positive constants. Di and
D are diffusion coefficients of the ith mRNA and sRNA,
respectively. The terms
iui and v describe the self-degradation of the ith mRNA and sRNA
species. The
nonlinear terms, kiuiv andN
i=1 kiuiv, describe the coupled reaction between the
ith mRNA species and sRNA species in which ki is the reaction
rate. i, are
nonnegative given functions bounded on which describe the
transcription profiles
or production rates for each species. The initial data, (ui)0
and v0, are two given
functions on . n denotes the unit exterior normal at , the
boundary of , 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
-
8of 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
-
9high 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, and describe the sRNA and
mRNA gen-
eration or production rates, and describe the sRNA and mRNA
independent
death or degradation rates, and 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.
sRNAmRNA
Figure 2.1: Kinetic scheme of mRNA and sRNA concentrations. and
rep-
resent the production rates of sRNA and mRNA, respectively. and
represent
the independent degradation rates of sRNA and mRNA,
respectively. 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)t0 and
(St)t0 be two
continuous time processes with t [0,), where Mt represents the
number ofmRNA 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)t0 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) = Pm1,s(t) + Pm,s1(t) m,sPm,s(t)+ (s+ 1)Pm,s+1(t) + (m+
1)Pm+1,s(t)
+ (m+ 1)(s+ 1)Pm+1,s+1(t)
(2.1.1)
where
Pm,s(t) =
tPm,s(t);
and represent the generation rates or production rates of sRNA
and mRNA,
respectively; and represent the independent death or degradation
rates of
sRNA and mRNA, respectively; and represents the coupled death or
degradation
rate of mRNA and sRNA. The term
m,s := + + m+ s+ ms
represents the rate in which the system leaves the state (m,
s).
Since (Nt)t0 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
m,s(s+ 1)
(m+ 1)
(m+ 1)(s+ 1)
0...
0
.
Notice that the diagonal elements of Q consist of terms, and
that the rows ofQ sum to 0. This means that Q is a Markov
infinitesimal generator.
Proposition 2.1.1. Let RS = {f | f : S R}. Then,
E [(Nt)] = P (t).
On the left hand side, we view (Nt) as the composition of : S R
with therandom variable Nt. On the right hand side, we view P (t)
as vector multiplication
between the row vector P (t) = aP(t) and which is identified as
a column vector
indexed by S.
Proof. We have
E [(Nt)] =
(m,s)S(m, s)P (Nt = (m, s)) =
(m,s)S
v(m, s)Pm,s(t) = P (t).
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
tMt = Mt MtSt (2.1.2)
tSt = St MtSt . (2.1.3)
Proof. Define [A]m,s to be the (m, s) entry of the matrix A. Let
pi and pi be the
projection maps from S R given by pi(m, s) = m and pi(m, s) = s.
Then,Mt = E [pi(Nt)] and St = E [pi(Nt)]. Using Proposition 2.1.1
with = pi andusing the properties of Q, we get
tE [pi(Nt)] =
tP (t)pi = [P (t)Q] pi
=
(m,s)Spi(m, s) [P (t)Q]m,s
=
(m,s)Sm [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)Sm
(Pm1,s(t) + Pm,s1(t) m,sPm,s(t)
+ (s+ 1)Pm,s+1(t) + (m+ 1)Pm+1,s(t)
+ (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) =
(m,s)S
mPm1,s(t)
(m,s)SmPm,s(t)
+
(m,s)S
mPm,s1(t)
(m,s)SmPm,s(t)
+
(m,s)S
m(m+ 1)Pm+1,s(t)
(m,s)Sm2Pm,s(t)
+
(m,s)S
m(s+ 1)Pm,s+1(t)
(m,s)SmsPm,s(t)
+
(m,s)S
m(m+ 1)(s+ 1)Pm+1,s+1(t)
(m,s)Sm2sPm,s(t)
t
(m,s)S
mPm,s(t) =
(m1,s)S
(m+ 1)Pm,s(t)
(m,s)SmPm,s(t)
+
(m,s1)S
mPm,s(t)
(m,s)SmPm,s(t)
+
(m+1,s)S
(m 1)mPm,s(t)
(m,s)Sm2Pm,s(t)
+
(m,s+1)S
msPm,s(t)
(m,s)SmsPm,s(t)
+
(m+1,s+1)S
(m 1)msPm,s(t)
(m,s)Sm2sPm,s(t)
-
16
t
(m,s)S
mPm,s(t) =
(m,s)S
(m+ 1)Pm,s(t)
(m,s)SmPm,s(t)
+
(m,s)S
mPm,s(t)
(m,s)SmPm,s(t)
+
(m,s)S
(m 1)mPm,s(t)
(m,s)Sm2Pm,s(t)
+
(m,s)S
msPm,s(t)
(m,s)SmsPm,s(t)
+
(m,s)S
(m 1)msPm,s(t)
(m,s)Sm2sPm,s(t)
t
(m,s)S
mPm,s(t) =
(m,s)SPm,s(t)
+
(m,s)S
(m2 m)Pm,s(t)
(m,s)Sm2Pm,s(t)
+
(m,s)S
(m2sms)Pm,s(t)
(m,s)Sm2sPm,s(t)
And,
t
(m,s)S
mPm,s(t) =
(m,s)SmPm,s(t)
(m,s)S
msPm,s(t).
Since
(m,s)SmPm,s(t) = Mt,
tMt = Mt MtSt .
Mimicking the above process for St = E [pi(Nt)] = P (t)pi using
the masterequation (Equation (2.1.1)), we have
tSt = St 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= u uv,
v
t= v uv .
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
D to the sRNA equation [14]. We have the following
equations:
u
t= u uv, (2.1.5)
v
t= v uv +Dv . (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 D to Equation (2.1.5) to form the
following coupled equations:
u
t= u uv +Du, (2.1.7)
v
t= v uv +Dv. (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 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) at time t. The
followingreaction-diffusion equations model the concentrations with
specified boundary and
initial conditions:
u
t= D1u 1u k1uv + 1
v
t= Dv v k1uv +
in [0,), (2.2.1)
u
n=v
n= 0 on [0,), (2.2.2)
u(, 0) = u0 and v(, 0) = v0 in . (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, , 1, k1, and k1 are positive constants. D1 and
D are diffusion
coefficients of mRNA and sRNA, respectively. The terms 1u and v
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. 1 and are nonnegative
given functions
defined on which describe the transcription profiles or
production rates for each
species. The initial data, u0 and v0, are two given functions on
. n denotes
the unit exterior normal at , the boundary of , 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 =1
1 + k1vin .
Thus, the steady-state concentration v = v(x) of sRNA is
governed by the following
nonlinear, diffusion equation and boundary condition:
Dv v k11v1 + k1v
+ = 0 in , (2.2.4)
v
n= 0 on . (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:
D1u 1u k1uv + 1 = 0Dv v k1uv + = 0
in , (2.2.6)
u
n=v
n= 0 on . (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= D1u 1u k1uv + 1
v
t= Dv v k1uv +
in [0,),
u
n=v
n= 0 on [0,),
u(, 0) = u0 and v(, 0) = v0 in .
-
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 be a bounded domain in R3 where the mRNA and sRNA
interact.Our model is the system of reaction-diffusion
equations:
uit
= Diui iui kiuiv + i (i = 1, . . . , N)v
t= Dv v
Ni=1
kiuiv + in [0,), (2.3.1)
uin
=v
n= 0 (i = 1, . . . , N) on [0,), (2.3.2)
ui(, 0) = (ui)0 (i = 1, . . . , N) and v(, 0) = v0 in .
(2.3.3)Here, ui is the nonnegative function which represents the
concentration of the i
th
mRNA target, and v is the nonnegative function which represents
the concentration
of the sRNA. For i = 1, . . . , N , Di, D, i, , and ki are
positive constants. Di and
D are diffusion coefficients of the ith mRNA target and sRNA,
respectively. The
terms iui and v describe the self-degradation of the ith mRNA
target species and
sRNA species. The nonlinear terms, kiuiv andN
i=1 kiuiv, describe the coupled
reaction between the ith mRNA target species and sRNA species in
which ki is the
reaction rate. i, are nonnegative given functions bounded on
which describe
the transcription profiles or production rates for each species.
The initial data,
(ui)0 and v0, are two given functions on . n denotes the unit
exterior normal at
, the boundary of , 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:
Dv v Ni=1
ikiv
i + kiv+ = 0 in , (2.3.4)
v
n= 0 on . (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:
Diui iui kiuiv + i = 0 (i = 1, . . . , N)
Dv v Ni=1
kiuiv + = 0in , (2.3.6)
uin
=v
n= 0 (i = 1, . . . , N) on . (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):
uit
= Diui iui kiuiv + i (i = 1, . . . , N)v
t= Dv v
Ni=1
kiuiv + in [0,),
uin
=v
n= 0 (i = 1, . . . , N) on [0,),
ui(, 0) = (ui)0 (i = 1, . . . , N) and v(, 0) = v0 in .
-
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 Diui = 0 for i
= 1, . . . , N and
Dv = 0 creating the following ODE system:
duidt
= iui kiuiv + i (i = 1, . . . , N)dv
dt= v
Ni=1
kiuiv + .(3.1.1)
First, we need to show a steady state solution exists.
Theorem 3.1.1. For the system of equations
iui kiuiv + i = 0 (i = 1, . . . , N) (3.1.2)
v Ni=1
kiuiv + = 0, (3.1.3)
there exists a unique equilibrium solution for u1, u2, . . . ,
uN , v with ui 0 andv 0 for all i = 1, . . . , N .
Proof. From Equation (3.1.3),
v =
+ U
where U =N
i=1 kiui. By substituting v into Equation (3.1.2), we have the
equa-
tions
iui kiui + U
+ i = 0 (i = 1, . . . , N).
-
24
Then,
ui =i
i + ki ( + U)1 (i = 1, . . . , N),
and
U =Ni=1
kiui =Ni=1
kii
i + ki ( + U)1 .
Now, we want to show that the equation
U Ni=1
kii
i + ki ( + U)1 = 0
has a unique solution. Let the function g : [0,) R be defined
by
g(s) = sNi=1
kii
i + ki ( + s)1 , (3.1.4)
and recall that i, , and ki are positive constants and i and are
nonnegative
functions. g is smooth in our domain [0,). Since
g(0) = Ni=1
kii
i + ki ()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 =
+ U,
and v has a solution. Similarly,
ui =i
i + 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) = 1Ni=1
k2i i
(i ( + s) + ki)2 ,
-
25
and the second derivative of g is
g(s) = 2Ni=1
k2i ii
(i ( + s) + ki)3 0.
Notice that g(s) = 0 on [0,) only when = 0 or i = 0 for all i.
In bothcases, 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,
iu0i kiu0iv0 + i = 0 (i = 1, . . . , N)
v0 Ni=1
kiu0iv0 + = 0.
For > 0, define the equations ui(t) and v(t) by
ui(t) = u0i + u1i(t) +O(2)
(i = 1, . . . , N) (3.1.5)
v(t) = v0 + v1(t) +O(2). (3.1.6)
By rewriting Equations (3.1.5) and (3.1.6), we have the
following equations de-
scribing our ODE:
u1i = ui u0i +O(2) (i = 1, . . . , N) (3.1.7)v1 = v v0 +O(2).
(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,
du1idt
=duidt du0i
dt+O(2)
= (iui kiuiv + 1) (iu0i kiu0iv0 + 1) +O(2)= i(ui u0i) ki(uiv
u0iv0) +O(2)= i(ui u0i) ki((ui u0i)(v v0) + uiv0 + u0iv 2u0iv0)
+O(2)= i(ui u0i) ki((ui u0i)(v v0) + v0(ui u0i) + u0i(v v0))
+O(2)
= i(ui u0i) kiv0(ui u0i) kiu0i(v v0) ki(ui u0i)(v v0)+O(2)
= (i + kiv0)(ui u0i) kiu0i(v v0) ki(ui u0i)(v v0) +O(2)= (i +
kiv0)u1i kiu0iv1 +O(2).
Then, we havedu1idt
= (i + kiv0)u1i kiu0iv1 +O().
Similarly, by taking the derivative of Equation (3.1.8),
dv1dt
=dv
dt dv0
dt+O(2)
= (v Ni=1
kiuiv + ) (v0 Ni=1
kiu0iv0 + ) +O(2)
= (v v0)Ni=1
ki(uiv u0iv0) +O(2)
= (v v0)Ni=1
ki((ui u0i)(v v0) + uiv0 + u0iv 2u0iv0) +O(2)
= (v v0)Ni=1
ki [u0i(v v0) + (ui u0i)v0 + (ui u0i)(v v0)]
+O(2)
= ( +Ni=1
kiu0i)(v v0)Ni=1
ki [(ui u0i)v0 + (ui u0i)(v v0)]
+O(2)
-
27
= ( +Ni=1
kiu0i)v1 Ni=1
kiv0u1i +O(2).
And, we have
dv1dt
= ( +
Ni=1
kiu0i
)v1
Ni=1
kiv0u1i +O().
For i = 1, . . . , N , we have the coupled system of N + 1
equations
du1idt
= (i + kiv0)u1i kiu0iv1 +O()dv1dt
= ( +
Ni=1
kiu0i
)v1
Ni=1
kiv0u1i +O().
Define w =
u11
u12...
u1N
v1
. We obtain the linearized system
dw
dt= Mw,
where
M =
(1 + k1v0) 0 . . . 0 k1u010 (2 + k2v0) . . . 0 k2u02...
.... . .
...
0 0 (N + kNv0) kNu0Nk1v0 k2v0 . . . kNv0 ( +
Ni=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 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+1i=1i 6=j
|mij| .
For columns j = 1, . . . , N ,
|mjj| = |j + kjv0| > |kjv0| =N+1i=1i 6=j
|mij| .
For j = N + 1,
|mN+1,N+1| = +
Ni=1
kiu0i
>Ni=1
kiu0i
=Ni=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 (M) = (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] Rnn 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 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 = x,
xp = [x]p = [Ax]p =ni=1
apixi = appxp +ni=1i 6=j
apixi,
-
29
and
( app)xp =ni=1i 6=j
apixi.
Hence,
| app| |xp| = |( app)xp| =
ni=1i 6=j
apixi
ni=1i 6=j
|api| |xi| |xp|ni=1i 6=j
|api| .
Since A is row diagonally dominant,
| app| ni=1i 6=j
|api| < |app| .
Therefore, {z : |z app| < |app|}. That is, exists in a disk
in the complexplane centered at app R of radius less than |app|.
Thus,
-
30
Theorem 3.2.1. Assume has a Lipschitz-continuous boundary. The
boundary-
value problem
Dv v k11v1 + k1v
+ = 0 in , (3.2.1)
v
n= 0 on . (3.2.2)
(Two-species Model I, Section 2.2, Equations 2.2.4 and 2.2.5)
has a unique solution
v H1() such that v 0, and v is smooth in .
Proof. Define J : H1() R {} by
J [v] =
{D
2|v|2 +
2v2 +
k111k21
[k11v ln
(1 +
k11v
)] v
}dx.
Define ln s = for any s 0. Notice that the function
g(s) = s ln(1 + s) (1 < s 0 and C2 0 such that
J [v] C1v2H1() C2 v H1(). (3.2.3)
If we denote = infvH1() J [v], then is finite. Let vj H1() (j =
1, 2, . . . ) besuch that J [vj] 0 as j . It follows from Equation
(3.2.3) that, passing to asubsequence if necessary, {vj}j=1
converges to some v H1(), weakly in H1(),strongly in L2(), and a.e.
in . Since the function g(s) 0 for all s (1,),Fatous Lemma implies
that
lim infj
[k11vj ln
(1 +
k11vj
)]dx
[k11v ln
(1 +
k11v
)]dx.
The weak convergence in H1() and strong convergence in L2() of
{vj}j=1 to vnow imply that
lim infj
J [vj] J [v].
Hence, J [v] = . The strict convexity of J implies that v H1()
is the uniqueminimizer of J : H1() R {+}.
-
31
Now we will show that v 0 a.e. in . By the uniqueness of the
minimizerof J over H1(), it suffices to show that |v| H1() is also
a minimizer. Noticethat ||v| | |v| in . Since is nonnegative in ,
we have |v| v a.e.in . 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). Applyingthis to the case, s = k1v(x)/1 (1, 0] for some some
x . Therefore, we obtainthat J [|v|] J [v]. Hence v 0.
It now follows that
d
dt
t=0
ln
(1 +
k11
(v + tw)
)dx =
(k1/1)w
1 + (k1/1)vw H1().
Since v minimizes J over H1(), we have
d
dt
t=0
J [v + tw] = 0 w H1().
Standard calculations then imply that v H1() is a weak solution
to Equa-tion (2.2.4) and Equation (2.2.5). The smoothness of v
inside 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 has a Lipschitz-continuous boundary. The
equation
Dv v Ni=1
kiiv
i + kiv+ = 0 in , (3.2.4)
v
n= 0 on ,
has a unique solution v H1() such that v 0, and v is smooth in
.Proof. Follow the proof of Theorem 3.2.1 with the functional
J [v] =
{D
2|v|2 +
2v2 +
Ni=1
iiki
[kiiv ln
(1 +
kiiv
)] v
}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:
Dv v Ni=1
kiiv
i + kiv+ = 0 in , (3.2.5)
v
n= 0 on .
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 (i 0).Suppose i 0. Then, we have the following
equation:
Dv v Ni=1
i + = 0 in , (3.2.6)
v
n= 0 on .
We now consider the one-dimensional system for which we can find
some solution
for v. To solve for v, we will find Greens function to
v Dv =
(x)Ni=1 i(x)D
.
Let =
D
and f(x) =Ni=1 i
D. In one dimension, we are solving
v 2v = f(x)
with the boundary conditions v(0) = v(1) = 0. First, we will
find the solution to
homogeneous equation v2v = 0. Using the Undetermined
Coefficients Method,Greens function will be of the form
G(x, s) =
A(s)(c1ex + c2ex) for x < sB(s)(c3ex + c4ex) for x > s
-
33
for constants c1, c2, c3, c4 and functions A(s) 6= 0 and B(s) 6=
0. We can solve forthese 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. Forx < s,
G(x, s) = A(s)(c1ex c2ex)G(0, s) = A(s)(c1 c2)
c2 = c1.
For x > s,
G(x, s) = B(s)(c3ex c4ex)G(1, s) = B(s)(c3e c4e)c4e = c3e
c4 = c3e2.
Then,
G(x, s) =
A(s)(c1ex + c1ex) for x < sB(s)(c3ex + c3e2x) for x >
s.From continuity at x = s,
A(s)(c1es + c1e
s) = B(s)(c3es + c3e2s).
From the derivative jump where G(s+, s)G(s, s) = 1,
B(s)(c3es c3e2s) A(s)(c1es c1es) = 1.
Together,A(s)(c1es + c1es)B(s)(c3es + c3e2s) = 0A(s)(c1es c1es)
+B(s)(c3es c3e2s) = 1,and so,
2A(s)c1es 2B(s)c3e2s = 1.
-
34
Then,
A(s)c1 =1
2es +B(s)c3e
2.
By substitution,(1
2es + B(s)c3e
2
)(es + es
)B(s)c3es (1 + e22s) = 01
2
(e2s + 1
)+ B(s)c3e
2+s + B(s)c3e2s B(s)c3es
(1 + e22s
)= 0
1
2
(e2s + 1
)+ B(s)c3e
2+s B(s)c3es = 01
2
(e2s + 1
)+ B(s)c3e
s(e2 1) = 0
Solving for c3,
B(s)c3es(e2 1) = 1
2
(e2s + 1
)B(s)c3 = 1
(e2s + 1
2es (e2 1))
Substituting A(s)c1 and B(s)c3 into G(x, s), we have
G(x, s) =
(
12es 1
(e2s+1
2es(e21)
)e2)
(ex + ex) for x < s
1
(e2s+1
2es(e21)
)(ex + e2x) for x > s.
By collecting terms, we have
G(x, s) =
(e2s(e21)(e2s+1)e2
2es(e21)
)(ex + ex
)for x < s
(
e2s+1
2es(e21)
)(ex + e2x) for x > s.
Further consolidation,
G(x, s) =
(e2+2se2se2s+2e2
2es(e21)
)(ex + ex
)for x < s
(
e2s+1
2es(e21)
)(ex + e2x) for x > s.
-
35
G(x, s) =
(
e2s+e2
2es(e21)
)(ex + ex
)for x < s
(
e2s+1
2es(e21)
)(ex + e2x) for x > s.
G(x, s) =
(e2s(1+e22s)
2es(e21)
)(ex + ex
)for x < s
(
e2s+1
2es(e21)
)(1+e22x
ex
)for x > s.
G(x, s) =
(es(1+e22s)
2e(e21)
)(ex + ex
)for x < s
(
e2s+1
es(e21)
)(1+e22x
2ex
)for x > s.
G(x, s) =
1
(e22s+1
2es
)(e
e21
) (ex + ex
)for x < s
1
(es(e2s+1)e(e21)
)(1+e22x
2ex
)for x > s.
G(x, s) =
cosh (s) cosh (x)
sinh ()for x < s
cosh (x) cosh (s) sinh ()
for x > s.
Notice that G(x, s) is symmetric about x and s.
To find v(x), we must solve
v(x) =
10
G(x, s)f(s)ds
where f(s) =(s)Ni=1 i(s)
D.
v(x) =
10
G(x, s)
((s)Ni=1 i(s)
D
)ds
= x
0
cosh ( s) cosh (x) sinh ()
((s)Ni=1 i(s)
D
)ds
1x
cosh ( x) cosh (s) sinh ()
((s)Ni=1 i(s)
D
)ds
= cosh (x) sinh ()
x0
cosh ( s)((s)Ni=1 i(s)
D
)ds
cosh ( x) sinh ()
1x
cosh (s)
((s)Ni=1 i(s)
D
)ds
-
36
For special choices of (s) and i(s), let
i(s) = 0.5Aui
(tanh
(xtsxi xtsx
)+ 1
)(i = 1, . . . , N),
(s) = 0.5Av
(tanh
(x xtsxtsx
)+ 1
),
where Av, Aui , xtsx, xtsxi , and tsx are constants. Then,
v(x) = cosh (x)D sinh ()
x0
cosh ( s)((s)
Ni=1
i(s)
)ds
cosh ( x)D sinh ()
1x
cosh (s)
((s)
Ni=1
i(s)
)ds
= 0.5Av cosh (x)D sinh ()
x0
cosh ( s)(
tanh
(s xtsxtsx
)+ 1
)ds
+0.5 cosh (x)
D sinh ()
x0
cosh ( s)Ni=1
(Aui
[tanh
(xtsxi stsx
)+ 1
])ds
0.5Av cosh ( x)D sinh ()
1x
cosh (s)
(tanh
(s xtsxtsx
)+ 1
)ds
+0.5 cosh ( x)D sinh ()
1x
cosh (s)Ni=1
(Aui
[tanh
(xtsxi stsx
)+ 1
])ds
Solutions to Equation (3.2.6) under varying parameters are found
by inserting
information about i and . These solutions give us information on
the behavior
of our nonlinear PDE as i 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
GaussSeidel like
iteration. The the last subsection explains the use of the
CrankNicolson 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(pix) blue
u2(x, t) = x2(12 4x 3x2) green
u3(x, t) = 5x2 5
3x3 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 = (0, 1).
The grid points are divided into interior points (points inside
= (0, 1))
and boundary points of . 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 .
-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
Erro
r
Student Version of MATLAB
(a) Error vs iterative steps, grid size p = 40
100 101 102104
103
102
101
100
Grid size
Erro
r
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 GaussSeidel type iterations
that we call Alternat-
ing Iteration (AI). AI consists of solving for each ui via
GaussSeidel iteration and
then, using the updated ui, solving for v via GaussSeidel
iteration. For example,
let uqi be the qth iteration of ui, and let v
q be the qth iteration of v. For each
i = 1, . . . , N , we solve for uqi using GaussSeidel:
Diuqi iuqi kiuqivq1 + i = 0 .
-
40
Then, we solve the following equation for vq with the just
calculated uqi ,
Dvq vq Ni=1
kiuqivq + = 0 ,
leaving an alternating GaussSeidel-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
Erro
r
Student Version of MATLAB
(a) Error vs time steps, grid size p = 40
100 101 102104
102
100
102
Grid size
Erro
r
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 denotesthe function u2, the color black denotes the
function u3, and the color red denotes
the function v.
4.1.3 The CrankNicolson Method
Based on the central difference method in space and the
trapezoidal rule
in time, the CrankNicolson 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
Erro
r
Student Version of MATLAB
(a) Error vs. time steps, grid size p = 40
100 101 102104
102
100
102
Grid size
Erro
rStudent 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
Newtons method
and GaussSeidel 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(pix) cos2(piy)
v(x, y, t) = x2(1 x)2y2(1 y)2w(x, y, t) = sin2(pix)
sin2(piy)
-
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 = [0,
1] [0, 1].The grid points are divided into interior points (points
in ), boundary
points (points on ), 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 .
-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 Newtons Method and GaussSeidel 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 Newtons 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) = v k11v1 + k1v
.
Two-species Model I can be written as
Dv + F (v) = 0 . (4.2.1)
Applying Newtons 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
Dvm+1 + vm+1F (vm) = vmF (vm) F (vm).
To compute our relatively sparse, diagonally dominate matrix
created by
Two-species Model I, we used GaussSeidel iteration, updating
entries as com-
puted. Figure 4.6 shows the computational error using Newtons
method and
GaussSeidel 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 102103
102
101
100
Grid size
Erro
r
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 Newtons
method and GaussSeidel 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 GaussSeidel iteration and then, using an updated u,
solving for v via
GaussSeidel 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
Erro
r
Iterative steps
Student Version of MATLAB
(a) Error vs iterative steps, grid size p = 64
100 101 102104
102
100
102
Grid size
Erro
r
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) = 1u k1uv + 1L2(u, v) = v k1uv + .
-
45
With forward Euler and explicit spatial discretization,
Two-species Model IIIs
discretization for both u and v becomes
uk+1i,j uki,jt
= L1(uki,j, v
ki,j),
vk+1i,j vki,jt
= 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
Erro
r
Iterative steps
Student Version of MATLAB
(a) Error vs time steps, grid size p = 64, time
steps T = 100
100 101 102103
102
101
100
101
Grid size
Erro
r
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
max{D1, D} t(x)2
12. (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,jt
= L1(uk+1i,j , v
ki,j) (4.2.3)
vk+1i,j vki,jt
= L2(uk+1i,j , v
k+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
Erro
r
Iterative steps
Student Version of MATLAB
(a) Error vs time steps, grid size p = 64, time
steps T = 100
100 101 102104
102
100
102
Grid size
Erro
r
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).
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47
0 0.5 1 1.5 2 2.5x 105
0
0.2
0.4
0.6
0.8
1
Erro
r
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
Erro
r
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, i = = 102, and = [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, 1 and
, 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 1, 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
1 2 1 0.5 0.2 1 = 0.5Am
(tanh
(xtsx xltsx
)+ 1
)2 2 1 0.65 0.15 2 = 0.5Am
(tanh
(xtsx xltsx
)+ 1
)3 2 1 0.25 0.1 3 = 0.5Am
(tanh
(xtsx xltsx
)+ 1
) 2 1 0.5 0.2 = 0.5As
(tanh
(x xtsxltsx
)+ 1
)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Tissue lengh
Tran
scrip
tion
rate
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
Conc
entra
tion
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).
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50
5.1.1 Multiple-species Model I
Recall that the Multiple-species Model I is characterized by the
equations:
Dv v 3i=1
ikiv
i + kiv+ = 0 in , (5.1.1)
v
n= 0 on . (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 102. The greatest
movement of the
mRNA interface between the curve with no diffusion and D = 102
was 0.212 units
where the curve moving from right to left on our standardized =
[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 104 D 102. Figure 5.3 shows the
trendin concentration boundary movement based on the diffusion
coefficient D. Notice
that a sharp interface for u1 and u2 only occurs when D 102 and
u3 is relativelyunchanged 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 104 D
102, the
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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
Co
nce
ntr
atio
n
mRNA
miRNA
0 0.2 0.4 0.6 0.8 1
0
100
200
150
50
Fraction of Tissue Length
Co
nce
ntr
atio
n 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 =
102.
(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 = 102.
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
Conc
entra
tion
Student Version of MATLAB
(a) Tissue length vs. concentration, D = 104.
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Tissue Length
Conc
entra
tion
Student Version of MATLAB
(b) Tissue length vs. concentration, D = 102.
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Tissue Length
Conc
entra
tion
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
Conc
entra
tion
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).
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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:
D1u1 1u1 k1u1v + 1 = 0 in ,D2u2 2u2 k2u2v + 2 = 0 in ,D3u3 3u3
k3u3v + 3 = 0 in ,
Dv v 3i=1
kiuiv + = 0 in ,
u1n
=u2n
=u3n
=v
n= 0 on .
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 = 102 and when ki = 1. Hence, we fixed D at that
order of
magnitude and looked for interface changes as Di ranged from 104
to 102. We also
wanted to see if D = 103 changed the interface significantly.
Figure 5.5 depicts
the changes in the concentration boundaries for ki = 1, D = 102
or D = 103,
and Di varying between 104 to 102. Notice that a sharp interface
only occurs for
u1 and u2 when Di = 104. 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 = 102 or D = 103, 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 = 104.
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54
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Tissue Length
Conc
entra
tion
Student Version of MATLAB
(a) Tissue length vs. concentration, D = 104.
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Tissue Length
Conc
entra
tion
Student Version of MATLAB
(b) Tissue length vs. concentration, D = 102.
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Tissue Length
Conc
entra
tion
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
Conc
entra
tion
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).
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55
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Tissue Length
Conc
entra
tion
Student Version of MATLAB
(a) Tissue length vs. concentration,
Di = 104, D = 102.
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Tissue Length
Conc