Generation and motion of interfaces in a mass-conserving reaction-diffusion system Pearson W. Miller1Daniel Fortunato2 Matteo Novaga3

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Generation and motion of interfaces in a mass-conserving reaction-diffusion

system

Pearson W. Miller,1Daniel Fortunato,2, Matteo Novaga3,

Stanislav Y. Shvartsman1,4,5, Cyrill B. Muratov3,6,†

1Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA

2Center for Computational Mathematics, Flatiron Institute, New York, NY 10010, USA

3Dipartimento di Matematica, Universit`

a di Pisa, Largo B. Pontecorvo 5,

56127 Pisa, Italy

4Department of Molecular Biology, Princeton University, Princeton, NJ

08540, USA

5Lewis-Sigler Institute for Integrative Genomics, Princeton University,

Princeton, NJ 08540, USA

6Department of Mathematical Sciences, New Jersey Institute of

Technology, Newark, NJ 07102, USA

†To whom the correspondence should be addressed

E-mail: muratov@njit.edu.

Reaction-diffusion models with nonlocal constraints naturally arise as limiting cases

of coupled bulk-surface models of intracellular signalling. In this paper, a minimal,

mass-conserving model of cell-polarization on a curved membrane is analyzed in

the limit of slow surface diffusion. Using the tools of formal asymptotics and calcu-

lus of variations, we study the characteristic wave-pinning behavior of this system

on three dynamical timescales. On the short timescale, generation of an interface

separating high- and low-concentration domains is established under suitable con-

ditions. Intermediate timescale dynamics is shown to lead to a uniform growth or

shrinking of these domains to sizes which are ﬁxed by global parameters. Finally,

the long time dynamics reduces to area-preserving geodesic curvature ﬂow that

may lead to multi-interface steady state solutions. These results provide a founda-

tion for studying cell polarization and related phenomena in biologically relevant

geometries.

Key words. pattern formation, reaction-diffusion, singular perturbations, Laplace-Beltrami operator, long-time behavior

AMS subject classiﬁcations. 35Q92, 35K57, 92C37

1. Introduction.

Reaction-diffusion models are essential tools for understanding the spatial self-

organization of chemical patterns inside the cell. Over the past decade, research interest has coalesced

around two key features which set intracellular dynamics apart from other classes of models. First, in

contrast to traditional models which exist on a single domain, these models are often bulk-surface models,

in that they feature distinct diffusion processes within the 3D cytosolic volume of the cell and on the

2D cell membrane coupled by a nonlinear boundary condition [46]. Second, many recent studies have

emphasized systems subject to mass-conservation [24,42]. These properties are particularly prevalent in

arXiv:2210.00585v1 [nlin.PS] 2 Oct 2022

2 P. W. MILLER ET AL.

models of cell polarization, a crucial process by which the spatial distribution of proteins within a cell

becomes highly localized to a region of the membrane as a result of spontaneous symmetry breaking or an

external guiding cue [16]. Polarization is essential for a great variety of biological phenomena, including

guiding developmental outcomes, establishing axes for cell division and guiding locomotion in motile

cells, and so has motivated signiﬁcant interest from the mathematical biology community [16].

The attention of applied mathematicians is increasingly turning to understanding how various forms

of spatial heterogeneity inﬂuence dynamics and steady state behavior. Research that has emphasized

the effect of spatially varying kinetic parameters are the most obvious examples of this trend [21,41].

Over the last few years, the question of what role cell geometry plays in guiding polarization has drawn

considerable interest. While initial polarization models were reduced to 1D systems, computational

advances have enabled numerical studies on 2D and fully 3D domains [6,15]. Early steps in addressing

this problem have produced simulations that are highly suggestive that localization is closely tied to

surface curvature via minimization of interfacial length, but this principle has yet to be demonstrated

by formal analysis [14,20]. Very recently, a novel numerical framework was introduced allowing for

efﬁcient simulation of cell polarization models on surfaces of revolution, which in particular motivates

this particular study [37].

In this paper, we perform a formal asymptotic analysis of the surface-bound version of the wave-

pinning model ﬁrst proposed in Ref. [39], which features both mass-conservation and bulk-surface

coupling. This model has attracted considerable interest as a minimal theory for polarization, and has

been successfully applied to model systems such as the Rho-GTP pathway [56] or Ezrin polarization in

embryonic mouse cells [58]. The emergence of spontaneous symmetry breaking in this system has been

intensely studied, and the dependence of the initial instability of the spatially uniform solution on various

parameters is well-characterized [33,43,48,56]. While previous studies considered the asymptotics of

wave-pinning as surface diffusion becomes small, they have generally been concerned with 1D domains

or lower orders of perturbation than we discover are needed to capture the full effect of 3D-embedded

domain geometry [14,40]. Other works which have considered fully 3D domains have limited analysis to

linear stability studies [20,47]. Our research here uses the timescale separation techniques to probe the

generation and propagation of interfaces in two-species reaction-diffusion systems to the speciﬁc case of

the wave-pinning model (for related rigorous studies, see [12,27,51]). We note that for single bistable

reaction-diffusion equations in the Euclidean setting the generation and propagation of interfaces is by

now well understood mathematically [1,5,11]. In the context closely related to our problem, results on

propagation of interfaces for bistable reaction-diffusion equations on Riemannian manifolds were recently

obtained in Refs. [44,45].

In this work, we demonstrate that understanding the patterning outcome of polar domains on a

surface of arbitrary shape requires a thorough analysis of the long-timescale behavior of the associated

mass-conserving reaction-diffusion equation. Working in the limit of slow surface diffusion, we establish a

separation of the dynamics into three distinct time scales: ﬁrst, the initial generation of the interface, then,

uniform growth or shrinking of the domains until their areas converge to the steady state values dependent

only on the global parameters, and ﬁnally, evolution of the interface by area-preserving geodesic curvature

ﬂow that results in a ﬁnite union of geodesic disks as time goes to inﬁnity. In doing so, we demonstrate

that stable steady states with multiple disjoint interfaces exist on biologically plausible domain shapes,

something impossible without geometric effects.

This paper is organized as follows. In Sec. 2, we formulate our model of polarization on a closed

surface and deﬁne three sub-problems whose limiting behavior should describe the effective dynamics on

GENERATION AND MOTION OF INTERFACES 3

different asymptotic timescales. In Sec. 3, we carry out a preliminary analysis of existence and stability

of the uniform states and, in particular, identify the parameter regimes in which only nonuniform steady

states can be stable. In Secs. 4 through 6, we asymptotically derive the limiting sub-problems as the

surface diffusion coefﬁcient tends to zero and examine their dynamical behavior. Finally, in Sec. 7 we

perform a number of numerical tests to corroborate the predictions of the asymptotic theory, and in Sec. 8

we make our concluding remarks.

2. Model Summary.

Let

Ω⊂R3

be an open, bounded, connected set with a sufﬁciently regular

boundary ∂Ω. A basic model of cell polarization can be written as (see [15] for more details):

∂tB=DB∇2

∂ΩB+kbβ+Bν

Gν+BνC−kdBin ∂Ω×(0,T),(2.1)

∂tC=DC∇2

ΩC,in Ω×(0,T),(2.2)

DC(∇C·ˆn)|∂Ω=−kbβ+Bν

Gν+BνC+kdB.in ∂Ω×(0,T),(2.3)

B(·,0) = B0in ∂Ω,(2.4)

C(·,0) = C0in Ω.(2.5)

Here,

(2.1)

describes a reaction-diffusion process on the surface

∂Ω

of the surface-bound protein concen-

tration

, while

(2.2)

gives bulk diffusion of the concentration

of the same protein in the bounded volume

Ω

, and

(2.3)

is the boundary condition coupling the two. The operator

∇2

∂Ω

is the Laplace–Beltrami

operator on the surface, while

∇2

Ω

refers to the standard 3D Laplacian. While we start with a model where

membrane-bound dynamics is already purely two-dimensional, the validity of this class of model as a

limiting case of a membrane of ﬁnite thickness was demonstrated in Ref. [31].

We are principally interested in the regime where

DCDB

, and in this regime it is reasonable to treat

the bulk concentration as spatially uniform [15]. As mass is conserved globally, we deﬁne the quantity

Ctot

as the total number of protein molecules divided by the bulk volume

Ω

, so that the bulk concentration

can be expressed as

(2.6) C=Ctot −1

VZ∂Ω

BdS.

Substituting this back into (2.1) yields a non-local equation for surface-bound species

∂tB=DB∇2

∂ΩB−kdB+kbβ+Bν

Gν+BνCtot −1

VZ∂Ω

BdSon ∂Ω×(0,T).(2.7)

This equation is rendered dimensionless by a rescaling

B(x,t)→u(˜x,˜

, with

u=kdB/(kbCtot)

t=kdt

and ˜x=x/√A∈∂˜

Ω, where A=|∂Ω|is the surface area of ∂Ω, and ˜

Ω=Ω/√Ais a rescaling of Ωthat

ensures that the rescaled domain

Ω

has boundary of unit area. Dropping the tildes and the subscript

∂Ω

for simplicity of notation from now on, we arrive at the dimensionless form

∂tu=δ2∇2u−u+f(u)(1−αU)on ∂Ω×(0,T),(2.8)

U(t) = Z∂Ω

u(x,t)dS for t∈(0,T),(2.9)

u(x,0) = g(x)for x∈∂Ω,(2.10)

4 P. W. MILLER ET AL.

with the dimensionless parameters1

δ=rDB

kdA,α=kbA

kdV,γ=kdG

kbCtot

,(2.11)

and where we deﬁned

f(u) = β+uν

γν+uν

for further notational convenience. Together with the already

dimensionless parameters

and

, the parameters

and

deﬁne the parameter space for our problem.

As was already noted, we have |∂Ω|=1 now.

The long timescale dynamics of this problem for a purely 1D model was previously studied via

asymptotic expansions in [40], and later on a disc in [14]. In Ref. [15], exact solutions were constructed

for the special case of a spherical domain and

ν=∞

. Here we conﬁne ourselves to the speciﬁc case

ν=2

for the sake of analytical tractability, but treat general spatial domains. A few useful remarks

on this system can be made based upon existing results. First,

(2.1)

–

(2.5)

represents a special case of

a more general class described in Ref. [52], and per the results therein, there exists a unique classical

solution

(B,C)

such that the functions

and

are smooth and uniformly bounded. Further, Theorem

2.6

of Ref. [26] demonstrates that our shadow system

(2.8)

–

(2.10)

likewise has a unique weak solution,

and that solution is the limit of the solutions to

(2.1)

–

(2.5)

DB→∞

, justifying the use of this reduced

approach.

In the asymptotic analysis described in the following sections, we ﬁnd it convenient to frame our

problem as three equivalent problems parametrized by δ1:

(Pδ

0)∂tuδ

0=δ2∇2uδ

0−uδ

0+f(uδ

0)1−αZ∂Ω

uδ

0dS,uδ

0(x,0) = gδ

0(x),

(Pδ

1)∂tuδ

1=δ∇2uδ

1+δ−1−uδ

1+f(uδ

1)1−αZ∂Ω

uδ

1dS,uδ

1(x,0) = gδ

1(x),

(Pδ

2)∂tuδ

2=∇2uδ

2+δ−2−uδ

2+f(uδ

2)1−αZ∂Ω

uδ

2dS,uδ

2(x,0) = gδ

2(x).

Each problem describes the dynamics at the timescale of a different order in

(Pδ

is the problem on the

original

O(1)

timescale,

(Pδ

is the problem on the

O(δ−1)

long timescale and

(Pδ

is the problem on the

O(δ−2)

longer timescale. Crucially, we will show that as

δ→0

the initial condition of the second and

third problem may be taken to be the inﬁnite time limit of the solution at the previous stage, thus allowing

to connect the solutions at different timescales. In our analysis below, we will formally obtain the limit

behavior of each of these problems when δ→0.

3. Preliminaries.

In this section we introduce some basic facts about our system, and establish

some notation. We begin by examining the behavior of the spatially uniform steady states of

(2.8)

. These

states satisfy

(3.1) −u+f(u)(1−αu) = 0,

and with our choice of the nonlinearity (recall that ν=2) this reduces to a cubic equation

−u(u2+γ2)+(1−αu)(u2(1+β) + β γ2) = 0.(3.2)

We use a slightly different convention from [15], in which the deﬁnitions of

and

differ from the present ones by constant

factors.

GENERATION AND MOTION OF INTERFACES 5

Figure 3.1.

Boundaries between the bistable and monostable regimes plotted for different values of

. Within the

triangular-shaped regions, three positive roots of (3.2)exist, while outside there is only one.

For

α,β,γ>0

(3.2)

will have at least one real positive root

, with two additional positive roots

u±

emerging when its discriminant

∆(α,β,γ) = γ4(1−4β(αβ +α+1)(2β(αβ +α+1)−5))

−4γ6(αβ +1)3(αβ +α+1)−4β(β+1)3γ2>0.

(3.3)

We adopt the convention

u−<u0<u+

, with

u±,0

denoting any one of these roots for shorthand. Setting

∆=0

allows one to derive the conditions for the existence of three roots. We ﬁnd the requirements for

three real roots to be

0<β<1

8,(3.4)

0<α<1−8β

8β(1+β),(3.5)

γ−(α,β)<γ<γ+(α,β),(3.6)

where

γ±(α,β) = s1−4β(αβ +α+1)(2β(αβ +α+1)−5)±p(1−8β(αβ +α+1))3

8(αβ +1)3(αβ +α+1).

(3.7)

Slices of the boundary between parameter space regions are plotted in the

γ−α

plane in Fig. 3.1 for

various values of β.

We next examine the linear stability of the uniformly stable states in the usual manner, examining the

behavior of perturbations of the form

u±,0(x,t) = u±,0+εη±,0(x,t)

for

ε1

. Denoting the orthonormal

eigenfunctions of the Laplace-Beltrami operator as

vk(x)

such that

−∇2vk(x) = λkvk(x)

for

k=0,1,...

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Generationandmotionofinterfacesinamass-conservingreaction-diffusionsystemPearsonW.Miller,1DanielFortunato,2,MatteoNovaga3,StanislavY.Shvartsman1;4;5,CyrillB.Muratov3;6;1CenterforComputationalBiology,FlatironInstitute,NewYork,NY10010,USA2CenterforComputationalMathematics,FlatironInstitute,NewYork,NY...

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Generation and motion of interfaces in a mass-conserving reaction-diffusion system Pearson W. Miller1Daniel Fortunato2 Matteo Novaga3

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