UNIVERSITY OF CALIFORNIA, SAN DIEGO Partial Differential Equation Models and Numerical Simulations of RNA 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, Chair Professor Gaurav Arya Professor Li-Tien Cheng Professor Jiawang Nie Professor Shyni Varghese 2013
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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.
The dissertation of Maryann Elisabeth Hohn is approved,
and it is acceptable in quality and form for publication
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
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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