Oscillations and Bifurcation Structure of Reaction-Diffusion Model for Cell Polarity Formation

TL;DR

This work analyzes a mass-conserving reaction-diffusion model with bistable nonlinearity to understand cell polarity formation. It combines analytical reductions and numerical bifurcation analysis to identify four spatiotemporal patterns, including two polarity-oscillation modes arising from a diffusion-driven instability, and shows how weak extracellular signals can steer pattern location and dynamics. A concrete localized unimodal stationary pattern is derived in closed form, and Hopf and period-doubling bifurcations are shown to govern the emergence and evolution of oscillations. The results offer a minimal, parameter-robust framework linking wave-pinning, pattern formation, and extracellular control in cell polarity, with implications for interpreting more realistic molecular networks.

Abstract

We investigate the oscillatory dynamics and bifurcation structure of a reaction-diffusion system with bistable nonlinearity and mass conservation, which was proposed by [Otsuji et al, PLoS Comp. Biol. 3 (2007), e108]. The system is a useful model for understanding cell polarity formation. We show that this model exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusion-driven (Turing-like) instability. Moreover, we investigate the effects of extracellular signals on the cell polarity oscillations.

Figures (9)

• Figure 1: The dynamics of the ODE system \ref{['aaa1']} with \ref{['aaa2']}.
• Figure 2: Formation of a localized unimodal stationary pattern for $d_1 = 0.1$ when $\xi = 2.0$. The initial value $(u(x, 0), v(x, 0))$ is given by $(\xi/2, \xi/2)$ with a small random perturbation. (a) The dynamics of $u$-component. The values of $u(x,t)$ on $0 \leq x \leq 10$ and $0 \leq t \leq 200$ are represented by a 3D graph. The profile of $v(x, t)$ is omitted here because the amplitude and spatial variation of $v(x, t)$ for each $t$ are relatively small as compared to those of $u(x, t)$. (b) The profiles of $u(x,t)$ and $v(x,t)$ for $t = 200$, which can be regarded as the localized unimodal stationary pattern given by $(\bar{u}(x-c), \bar{v}(x-c))$ for some $c \in [-K/2, K/2)$. The rigid and dashed lines represent $u$- and $v$-components, respectively. The numerical solutions on $0 \leq x \leq 10$ presented in this figure can be regarded as those on $-5 \leq x \leq 5$ under the periodic boundary condition. It should be noted that we cannot predict the value of $c$, the position of the peak of the localized unimodal pattern in $u$-component, because of a small random perturbation to the initial value of a solution.
• Figure 3: Spatiotemporal patterns appear in the second stage of the dynamics of numerical solutions for some values of $d_1$ when $\xi = 0.6$. The initial value is given by $(\xi/2, \xi/2)$ with a small random perturbation. The values of $u(x,t)$ on $(x, t) \in [0, 10] \times [0, 1400]$ or $(x, t) \in [0, 10] \times [0, 250]$ are represented by a 3D graph. The profile of $v(x, t)$ is omitted here because the amplitude and spatial variation of $v(x, t)$ for each $t$ are relatively small as compared to those of $u(x, t)$. The numerical solution on $0 \leq x \leq 10$ presented in this figure can be regarded as that on $-5 \leq x \leq 5$ under the periodic boundary condition. (a) $d_1 = 0.1$, A localized unimodal stationary pattern; (b) $d_1 = 0.25$, An oscillatory pattern that exhibits an alternating repetition of spatially homogeneous patterns and localized unimodal patterns with two different positions of their peak. The distance between two peak positions is (almost) $K/2 = 5.0$. (c) $d_1 = 0.4$, An oscillatory pattern that exhibits an alternating repetition of spatially homogeneous patterns and localized unimodal patterns with unique peak positions. It should be noted that we cannot predict the position of the peak of each localized unimodal pattern in (a)--(c) because of a small random perturbation to the initial value of a solution. (d) $d_1 = 0.8$, A spatially homogeneous stationary pattern.
• Figure 4: Bifurcation diagram with respect to $d_1$ for $\xi=1.0$, where $d_1$ decreases from $d_1 = 0.8$. The horizontal and vertical axes indicate $d_1$ and the size of the $u$-component of solutions represented by $\frac{1}{K}\int_I u(x)dx$, respectively. The solid line indicates stable solutions, whereas the dashed line indicates unstable ones. The white square indicates the pitchfork bifurcation point. Unstable branches bifurcating from unstable branches for $d_1 \leq 0.30$ are omitted. Notice that the two branches bifurcating from a pitchfork bifurcation point are piled up and are displayed in this bifurcation diagram. The primary branch starting from $d_1 = 0.8$ represents a family of spatially homogeneous equilibria \ref{['equx']}, which correspond to spatially homogeneous stationary patterns as shown in Figure \ref{['zu3']}(d). The secondary branch bifurcating from the primary branch via the supercritical pitchfork bifurcation at $d_1 \approx 0.74$ represents a family of (localized) unimodal stationary patterns as shown in Figure \ref{['zu3']}(a).
• Figure 5: Bifurcation diagram with respect to $d_1$ for $\xi=0.64$, where $d_1$ decreases from $d_1 = 0.8$. This bifurcation diagram is presented in the same way as that in Figure \ref{['100']}. The horizontal and vertical axes indicate $d_1$ and the size of the $u$-component of solutions represented by $\frac{1}{K}\max_{t>0} \int_I u(x,t)dx$, respectively. The white and black squares indicate the pitchfork and Hopf bifurcation points, respectively. The curve represented by a family of the black circles $\bullet$ indicates the stable branch of periodic solutions. The primary branch starting from $d_1 = 0.8$ represents a family of spatially homogeneous equilibria \ref{['equx']}, which correspond to spatially homogeneous stationary patterns as shown in Figure \ref{['zu3']}(d). The secondary branch bifurcating from the primary branch via the supercritical pitchfork bifurcation at $d_1 \approx 0.74$ represents a family of (localized) unimodal stationary patterns as shown in Figure \ref{['zu3']}(a). The third branch (indicated by $\bullet$) bifurcating from the second branch via the Hopf bifurcations at $d_1 \approx 0.46$ and $d_1 \approx 0.21$ represents a family of limit cycles as shown in Figure \ref{['zu3']}(c).

Theorems & Definitions (8)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 4.1
• Remark 5.1
• Remark 5.2