Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a one-dimensional (1-D) model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a two-dimensional (2-D) model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov-Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns. Our work clarifies the effect of geometry and dimensionality on behaviors observed in mass-conserved cell polarity models.

中文翻译：

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细胞内极化和振荡的耦合膜-本体反应-扩散模型中的模式形成。

反应扩散系统已被广泛用于研究细胞生物学中的时空现象，例如细胞极化。耦合体表面模型自然包括细胞溶质和膜结合极性分子的区室化。在这里，我们研究了质量守恒的膜体模型中极性蛋白 Cdc42 的分布，并探讨了扩散和空间维度对时空模式形成的影响。我们首先分析了裂殖酵母中 Cdc42 振荡的一维 (1-D) 模型，该模型由体域中的两个扩散方程与非线性常微分方程组成，用于细胞两端的结合动力学。在 1-D 中，我们的分析揭示了对称和非对称稳态的存在，以及慢-快系统典型的反相弛豫振荡。然后，我们将分析扩展到具有圆形体几何形状的二维 (2-D) 模型，对于该模型，物种可以在细胞内扩散或与膜结合并经历非线性反应扩散过程。我们还考虑了一个非局部偏微分方程系统，它在快速体扩散的极限下近似于二维膜体模型的动力学。在所有三个模型变体中，我们发现质量守恒选择了简单地重新分配质量的空间模式的扰动。在 1-D 中，只能发生单元两端之间的反相振荡，同相振荡被排除在外。在更高的维度上，没有观察到径向对称的振荡。相反，唯一的不稳定性是对称性破坏，要么对应于静止的图灵不稳定性，导致静止模式的形成，或振荡图灵不稳定性，导致行波和驻波。Codimension-two Bogdanov-Takens 分岔发生在两个不同的不稳定性重合时，导致行波减慢并最终成为静止模式。我们的工作阐明了几何形状和维度对质量守恒细胞极性模型中观察到的行为的影响。