On stability of spatial patterns for mass-conserved reaction-diffusion systems

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On stability of spatial patterns for mass-conserved reaction-diffusion systems

祐川, 翼

北海道大学. 博士(理学) 甲第15597号

2023-09-25

10.14943/doctoral.k15597

http://hdl.handle.net/2115/90740

theses (doctoral)

Tsubasa_Sukekawa.pdf

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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

博士学位論文

On stability of spatial patterns for mass-conserved reaction-diffusion systems
(保存量をもつ反応拡散系における安定な空間パターンについて)

祐川 翼
北海道大学大学院理学院
数学専攻

2023 年 9 月

Contents
1 Introduction.

2

I Shape of stable steady-state solutions in mass-conserved
linear reaction-diffusion systems
7
2 Introduction.

7

3 Preliminaries.

8

4 Problem setting.

10

5 Main results.

11

6 Numerical simulation.

14

7 Summary and Discussion.

15

II Stability of 2-mode stationary solutions in mass-conserved
reaction-diffusion compartment model
18
8 Introduction.

18

9 Problem setting.
22
9.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9.2 Linearized eigenvalue problem for 2-mode stationary solution . . 23
10 Main results.

24

11 Summary and discussion.

28

A Appendix
31
A.1 An example of Assumption 3, 4. . . . . . . . . . . . . . . . . . . 31
A.2 Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1

1

Introduction.

Reaction-diffusion equations are used as model equations for various pattern
formation phenomena in physics, chemistry, and biology. The existence and
stability of solutions corresponding to characteristic patterns, such as stationary
patterns and traveling waves in actual phenomena are important to understand
the phenomena from the mathematical viewpoint. Recently, mass-conserved
reaction-diffusion systems have been used as mathematical models for various
phenomena such as cell polarity, and phase separation between solid and liquid
phases [5], [21], [26], [13], [10], [27], [16]. This study’s main objective is the
mathematical analysis of mass-conserved reaction-diffusion system for the cell
polarity model.
Cell polarity refers to the spatial localization of some chemical substances
within a cell. For example, in a phenomenon called asymmetric cell division,
it is known that a mother cell localizes a specific protein during cell division
to produce a daughter cell that differs from the mother cell. Cell polarity is
also known to be observed in chemotaxis, in which cells migrate in response
to a concentration gradient of a chemical substance. For these reasons, cell
polarity is considered to be important for cell differentiation and for cells to
acquire anisotropic functions. Various model equations have been introduced to
understand cell polarity from a mathematical point of view [10], [21].
Since cell polarity is an intracellular phenomenon, it is difficult to model and
rigorously analyze all the elements involved in the phenomenon. For example,
cells have a complex shape. In addition, many proteins interact with each other
in a complex manner in the cell polarity, and it is difficult to analyze all of them.
Many previous studies have modeled the cell shape in simple forms, such as a
circle, or examined the models with a reduction for the number of components
under various assumptions [21], [10], [26].
Although we can’t expect that such a conceptual model and the original phenomenon will be in perfect quantitative agreement, the introduction and analysis
of a simple model that reproduces the qualitative properties of cell polarity can
be expected to identify essential elements of the cell polarity phenomenon or to
provide some insight into the phenomenon. For example, previous study [21]
introduced a 6-component reaction-diffusion system under one-dimensional periodic boundary conditions and a 2-component reaction-diffusion system that
simplifies the 6-component system. These model equations reproduce various
features of cell polarity by numerical simulation. In the view of analysis, the 2component equations provide various suggestions for the stability of steady-state
solutions and the dynamics of solutions.
The 2-component model is expressed as

∂2u
∂u



= Du 2 − f (u, v)
∂x
(∗) ∂t
x ∈ (0, K),
2

∂v
∂
v


= Dv 2 + f (u, v)
∂t
∂x
where u and v are the concentrations of proteins U and V in the plasma membrane and cytoplasm, respectively. Although a cell is a three-dimensional object,
the cell membrane and cytoplasm are modeled as intervals (0, K), where periodic boundary conditions are imposed. The parameter K is a positive constant
corresponding to the cell size. Du and Dv are diffusion coefficients of U and
2

V . The reaction term f represents the interconversion between U and V . That
is, f > 0 represents the conversion from U to V , and f < 0 represents the
conversion from V to U . In particular, it can be seen that only the sign of the
reaction term differs in the U and V equations. This means that in the above
model, only the conversion between proteins U and V is considered, ignoring
the effects of synthesis and degradation of proteins. Such an assumption is reasonable because cell polarity phenomena occur on a time scale faster than the
time required for protein synthesis and degradation [5]. From the nature of this
reaction
term and the periodic boundary conditions, the total mass of U and
∫
V , (u + v)dx is shown to be conserved in the system (∗). Due to this property,
equation (∗) is called a mass-conserved reaction-diffusion system.
While the model equation above is 2-component reaction-diffusion system
with periodic boundary condition, various other models have been introduced
as equations with conserved quantities. For example, a cell polarity model with
homogeneous Neumann boundary conditions has been introduced, in which the
mass conservation law holds also. In particular, the dynamics of a front-like
pattern called wave-pinning has been analyzed [13], [14]. A cell polarity model
defined on regions resembling the actual cell shape is called a bulk-surface model.
In the bulk-surface model, the cytoplasm is a region in 2D or 3D space, such as
a unit sphere, and the cell membrane is its boundary. The bulk-surface model
has been the subject of many previous studies, including mathematical analyses
such as the stability of the stationary solutions [23], [24], [3], [17], [18].
We can extend the 2-component model to more multi-component cases. Let
N ≥ 3 be an integer and consider the following N -component equation.
∂ui
∂ 2 ui
= Di 2 + fi (u1 , . . . , uN )
∂t
∂x
N
∑
fi = 0

(i = 1, . . . , N )

i=1

where ui and Di are the concentrations of i-th chemical substance and diffusion
coefficients of these, respectively. fi represents the interaction between the substances. For the above equations, for example,
∫ if periodic boundary conditions
are imposed, the total amount of substance (u1 + · · · + uN )dx is a conserved
quantity. This model equation is derived as a cell polarity model that incorporates more chemicals into the model than the 2-component model [21], [10],
[19], [26].
As mentioned above, many kinds of equations have been introduced as mathematical models of cell polarity. But in general, the more complex the model
equations are, the more complex the mathematical analysis becomes. On the
other hand, the knowledge obtained from the analysis of simple model equations
may be helpful for the analysis of more complex model equations. Based on these
motivations, we limit the study in this thesis to 2-component mass-conserved
reaction-diffusion equations defined on a 1-dimensional interval. Although the
equation is simpler than other model equations, it is known that the model
qualitatively reproduces cell polarity phenomena [5], [21], [13], [10]. Various
mathematical and numerical analyses have been performed on the equations,
including bifurcation analysis, stationary problems, stability of stationary solutions, and dynamics of solutions [16], [14], [15], [6], [7], [25], [12], [2], [8], [9],
[4].
3

One of the crucial issues in mass-conserved reaction-diffusion systems is the
shape of the stable steady-state solutions. Mathematically, the non-constant
stationary solution in (∗) corresponds to the polarity pattern. In particular,
many cell polarity phenomena require that a chemical substance is localized at
only one location in the cell. This situation mathematically corresponds to a
stationary solution with a shape that has only one peak. Therefore, the question
of what shape of the steady-state solution is stable is not only mathematically
but also biologically important [21].
This thesis considers the following equation: a mathematical generalization
of the model equation (∗).
{
∂t u = d1 ∂x2 u + k1 g(x, u, v)
(∗∗)
x ∈ (0, K),
∂t v = d2 ∂x2 v − k2 g(x, u, v)
where u = u(t, x) and v = v(t, x) are unknown functions. The function g
is a real-valued function continuous on I¯ for x and of class C 2 for u and v.
d1 , d2 , k1 , k2 are positive constants. The reaction term depends on the spatial
variable. Such a reaction term can be used, for example, in a model where
chemicals U and V are affected by some other chemical substances in the cell, or
extracellular signals [26], [21], [8]. In particular, if k1 = k2 = 1 and
∫ g(x, u, v) =
−f (u, v), equation (∗∗) is consistent with (∗). For equation (∗∗), I (k2 u+k1 v)dx
is conserved when periodic or homogeneous Neumann boundary conditions are
imposed. In the∫ cell polarity model, u and v represent the concentrations of
chemicals, and I (u + v)dx is conserved if k1 = k2 . This corresponds to the
conservation of mass in the cell. Therefore, when k1 ̸= k2 , model equation
(∗∗) does not correspond to the cell polarity model. On the other hand, the
model equation (∗∗) becomes equivalent to the Fix-Cagnalp model describing
solid-liquid phase separation and each other by choosing the reaction term g
appropriately and applying linear transforming to the unknown functions [16]. ...

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