Würthner Laeschkir, Goychuk Andriy, Frey Erwin
Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich, Germany.
Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 Munich, Germany.
Phys Rev E. 2023 Jul;108(1-1):014404. doi: 10.1103/PhysRevE.108.014404.
Intracellular protein patterns regulate a variety of vital cellular processes such as cell division and motility, which often involve dynamic cell-shape changes. These changes in cell shape may in turn affect the dynamics of pattern-forming proteins, hence leading to an intricate feedback loop between cell shape and chemical dynamics. While several computational studies have examined the rich resulting dynamics, the underlying mechanisms are not yet fully understood. To elucidate some of these mechanisms, we explore a conceptual model for cell polarity on a dynamic one-dimensional manifold. Using concepts from differential geometry, we derive the equations governing mass-conserving reaction-diffusion systems on time-evolving manifolds. Analyzing these equations mathematically, we show that dynamic shape changes of the membrane can induce pattern-forming instabilities in parts of the membrane, which we refer to as regional instabilities. Deformations of the local membrane geometry can also (regionally) suppress pattern formation and spatially shift already existing patterns. We explain our findings by applying and generalizing the local equilibria theory of mass-conserving reaction-diffusion systems. This allows us to determine a simple onset criterion for geometry-induced pattern-forming instabilities, which is linked to the phase-space structure of the reaction-diffusion system. The feedback loop between membrane shape deformations and reaction-diffusion dynamics then leads to a surprisingly rich phenomenology of patterns, including oscillations, traveling waves, and standing waves, even if these patterns do not occur in systems with a fixed membrane shape. Our paper reveals that the local conformation of the membrane geometry acts as an important dynamical control parameter for pattern formation in mass-conserving reaction-diffusion systems.
细胞内蛋白质模式调节着各种重要的细胞过程，如细胞分裂和运动，这些过程通常涉及细胞形状的动态变化。细胞形状的这些变化反过来可能会影响模式形成蛋白的动态变化，从而导致细胞形状与化学动力学之间形成一个复杂的反馈回路。虽然有几项计算研究探讨了由此产生的丰富动力学，但潜在机制尚未完全理解。为了阐明其中一些机制，我们在动态一维流形上探索了一个细胞极性的概念模型。利用微分几何的概念，我们推导出了在随时间演化的流形上控制质量守恒反应扩散系统的方程。通过对这些方程进行数学分析，我们表明膜的动态形状变化可以在膜的部分区域诱导模式形成不稳定性，我们将其称为区域不稳定性。局部膜几何形状的变形也可以（在局部区域）抑制模式形成，并在空间上移动已有的模式。我们通过应用和推广质量守恒反应扩散系统的局部平衡理论来解释我们的发现。这使我们能够确定一个简单的几何诱导模式形成不稳定性的起始标准，该标准与反应扩散系统的相空间结构相关。膜形状变形与反应扩散动力学之间的反馈回路进而导致了令人惊讶的丰富模式现象学，包括振荡、行波和驻波，即使这些模式在具有固定膜形状的系统中不会出现。我们的论文表明，膜几何形状的局部构象作为质量守恒反应扩散系统中模式形成的一个重要动态控制参数。
