Arnold Sommerfeld Center for Theoretical Physics, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany.
Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany.
Proc Natl Acad Sci U S A. 2022 Aug 16;119(33):e2206888119. doi: 10.1073/pnas.2206888119. Epub 2022 Aug 12.
Self-organized pattern formation is vital for many biological processes. Reaction-diffusion models have advanced our understanding of how biological systems develop spatial structures, starting from homogeneity. However, biological processes inherently involve multiple spatial and temporal scales and transition from one pattern to another over time, rather than progressing from homogeneity to a pattern. To deal with such multiscale systems, coarse-graining methods are needed that allow the dynamics to be reduced to the relevant degrees of freedom at large scales, but without losing information about the patterns at small scales. Here, we present a semiphenomenological approach which exploits mass conservation in pattern formation, and enables reconstruction of information about patterns from the large-scale dynamics. The basic idea is to partition the domain into distinct regions (coarse grain) and determine instantaneous dispersion relations in each region, which ultimately inform about local pattern-forming instabilities. We illustrate our approach by studying the Min system, a paradigmatic model for protein pattern formation. By performing simulations, we first show that the Min system produces multiscale patterns in a spatially heterogeneous geometry. This prediction is confirmed experimentally by in vitro reconstitution of the Min system. Using a recently developed theoretical framework for mass-conserving reaction-diffusion systems, we show that the spatiotemporal evolution of the total protein densities on large scales reliably predicts the pattern-forming dynamics. Our approach provides an alternative and versatile theoretical framework for complex systems where analytical coarse-graining methods are not applicable, and can, in principle, be applied to a wide range of systems with an underlying conservation law.
自组织模式形成对于许多生物过程至关重要。反应-扩散模型已经推进了我们对于生物系统如何从均匀状态发展出空间结构的理解。然而，生物过程本质上涉及多个时空尺度，并随着时间的推移从一种模式过渡到另一种模式，而不是从均匀状态发展到一种模式。为了处理这种多尺度系统，需要使用粗粒化方法，这些方法可以将动力学简化为大尺度上的相关自由度，而不会丢失小尺度上的模式信息。在这里，我们提出了一种半唯象方法，该方法利用了模式形成中的质量守恒，并能够从大尺度动力学中重建有关模式的信息。基本思想是将域划分为不同的区域（粗粒化），并确定每个区域中的瞬时弥散关系，这些关系最终会提供关于局部模式形成不稳定性的信息。我们通过研究 Min 系统来演示我们的方法，该系统是蛋白质模式形成的典范模型。通过进行模拟，我们首先表明 Min 系统在空间异质几何形状中产生多尺度模式。这一预测通过体外重建 Min 系统得到了实验证实。我们使用最近开发的用于质量守恒反应-扩散系统的理论框架，表明大尺度上总蛋白密度的时空演化可靠地预测了模式形成动力学。我们的方法为复杂系统提供了一种替代的、通用的理论框架，在这些系统中，分析性粗粒化方法不适用，并且原则上可以应用于具有基础守恒定律的广泛系统。
