Bridging scales in a multiscale pattern-forming system

Abstract

Self-organized pattern formation is vital for many biological processes. Mathematical modeling using reaction-diffusion models has 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. One possibility to deal with multiscale systems is to use coarse-graining methods that allow the dynamics to be reduced to the relevant degrees of freedom at large scales. Unfortunately, these approaches have the major disadvantage that the eliminated scales cannot be reconstructed from the large-scale dynamics and thus one loses the information about the patterns. Here, we present an approach for mass-conserving reaction-diffusion systems that overcomes this issue and allows one to reconstruct information about patterns from the large-scale dynamics. We illustrate our approach by studying the Min protein 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. On the basis of 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. Since conservation laws are inherent in many different physical systems, we believe that our approach can be generalized and contribute to uncover underlying physical principles in multiscale pattern-forming systems.

Geometry-induced patterns through mechanochemical coupling

“…All these processes show that intracellular reaction-diffusion systems are quite generally able to control cell shape, as was recently demonstrated in a minimal reconstituted setup where the E. coli MinDE protein system induced lipid vesicle deformations even in the absence of cytoskeletal proteins [19][20][21]. Since, conversely, cell geometry can guide protein pattern formation [22][23][24][25][26][27], this generically gives rise to mechanochemical feedback loops [28][29][30][31].…”

“…The membrane shape deformations cause inhomogeneous membrane compressions or dilations, leading to (local) accumulation or dilution of particle densities. Since protein densities are important control parameters in massconserving reaction-diffusion systems [26,42,45,46], deformations of the one-dimensional manifold can qualitatively change the dynamics of protein pattern formation. In turn, if the local density of proteins also drives the dynamics of the one-dimensional manifold, then this leads to a feedback loop between shape changes of the manifold and reaction-diffusion dynamics.…”

Coarsening and wavelength selection far from equilibrium: a unifying framework based on singular perturbation theory

“…Nevertheless, the total density of the different conformations for each protein is conserved and follows a continuity equation because pattern formation is fast compared to protein turnover by gene expression and protein degradation. Thus, the total den-sities are still control parameters of the (local) dynamics, and the redistribution of the total densities is crucial to the system dynamics on long scales [35,36,108,109]. Both in Cahn-Hilliard and reaction-diffusion systems, broken mass conservation can be accounted for by source terms in the continuity equation(s).…”

“…Another aspect of domain geometry is bulk-surface coupling. Owing to the cycling of proteins between membrane-bound and cytosolic states, this is a fundamental property of many protein-based, pattern-forming systems, and has a profound impact on the patterns that emerge [5,26,109,[127][128][129]. However, the impact of bulk-surface coupling on wavelength selection remains largely unexplored and is an important open problem for future research.…”

Quantitative analysis of surface wave patterns of Min proteins

“…In cases where it is required to explore Min patterns across very large areas, it may furthermore be of interest to stitch multiple fields-of-view of Min patterns into larger images. (28, 26) If the microscope’s software does not offer an automatized solution for this, stitching can be achieved by good bookkeeping and a few lines of code. When planning to stitch images, it can be helpful to choose individual field-of-views in such a way that there is a bit of an overlap (e.g.…”