Pattern Formation Beyond Turing: Physical Principles of Mass-Conserving Reaction--Diffusion Systems

TL;DR

The article develops mass-conserving reaction–diffusion (McRD) theory as a universal framework for intracellular protein pattern formation, emphasizing mass redistribution and interface dynamics. By introducing the mass-redistribution potential $\eta$ and a phase-space viewpoint, it explains how local reactive equilibria couple to global patterns and derive mesoscale laws, including stationary interfaces, curvature-driven motion, and interrupted coarsening. The E. coli Min system is used as a paradigmatic test bed, with a skeleton model, MinE switching, and persistent MinE membrane binding capturing the diversity and robustness of patterns both in vivo and in vitro. The work reveals how universal nonequilibrium interface principles underlie complex biological patterning and offers a foundation for synthetic design and broader nonequilibrium physics insights.

Abstract

Intracellular protein patterns govern essential cellular functions by dynamically redistributing proteins between membrane-bound and cytosolic states, conserving their total numbers. This review presents a theoretical framework for understanding such patterns based on mass-conserving reaction--diffusion systems. The emergence, selection, and evolution of patterns are analyzed in terms of mass redistribution and interface motion, resulting in mesoscale laws of coarsening and wavelength selection. A geometric phase-space perspective provides a conceptual tool to link local reactive equilibria with global pattern dynamics through conserved mass fluxes. The Min protein system of \emph{Escherichia coli} provides a paradigmatic example, enabling direct comparison between theory and experiment. Successive model refinements capture both the robustness of pattern formation and the diversity of dynamic regimes observed \emph{in vivo} and \emph{in vitro}. The Min system thus illustrates how to extract predictive, multiscale theory from biochemical detail, providing a foundation for understanding pattern formation in more complex and synthetic systems.

Figures (4)

• Figure 1: Protein pattern formation exemplified by the E. coli Min system. (a) Prokaryotic cells comprise the membrane, cytosol, and nucleoid. Membrane-bound proteins diffuse slowly ($D_m$) compared to cytosolic proteins ($D_c$); diffusion in the nucleoid ($D_n$) may differ from that in the cytosol. Biochemical reactions (red arrows) regulate membrane association of proteins (purple). (b) The Min system operates via ATP-driven cycling of MinC (yellow), MinD (magenta), and MinE (cyan) between cytosol and membrane. Pattern formation requires only MinD and MinE; MinC inhibits Z-ring formation. (c) These interactions generate pole-to-pole oscillations in vivo. Shown is a kymograph of MinD fluorescence (white: high intensity; courtesy of Sourjik lab). (d) In vitro, MinD (green) and MinE (red) form dynamic patterns on supported lipid bilayers, including traveling waves (sketch with scale bar: $\sim50µ m$) Loose.etal2008.
• Figure 2: Mass-redistribution instability. (a) In a single well-mixed compartment, the reaction term $f$ (red arrows) converts the cytosol ($c$) and membrane ($m$) densities into each other until a reaction equilibrium is reached. (b) During the reactive relaxation in the single compartment, the total density ${\rho=m+c}$ remains constant, restricting the reactive flow to diagonals in the local phase space (grey lines). The family of reactive equilibria (black dots) for different total densities $n$ form the nullcline (black). (c) The diffusive coupling of two compartments via cytosolic exchange (neglecting membrane diffusion) induces a lateral instability if the nullcline slope is negative. The underlying cause is a positive feedback resulting in self-amplifying mass transport between the two compartments. (d) Including membrane diffusion, the mass-redistribution instability occurs for total densities ${\rho\in [\rho_-^\mathrm{lat},\rho_+^\mathrm{lat}]}$ at which the slope of the nullcline is smaller than $-D_m/D_c$ (green-shaded region). (e) The local phase space can also be analyzed using the total density $n$ and the mass-redistribution potential $\eta$ as coordinates. Mass redistribution induces a lateral instability if the nullcline in the $(\rho,\eta)$-phase space has a negative slope, also for finite membrane diffusion.
• Figure 3: Modules of the Min interaction network. (a) The skeleton model captures the core ATPase cycle of MinD. MinD-ATP attaches to the membrane and recruits itself. MinE is recruited as well and stimulates the ATPase activity of MinD, leading to the detachment of both proteins. (b) The skeleton model results in pole-to-pole oscillations in in vivo geometry (top). Simulation in filamentous cells uncovers the intrinsic wavelength of the pattern, resulting in a standing-wave pattern (bottom). The skeleton model captures the temperature dependence of the oscillations. Adapted from Ref. Halatek.Frey2012. (c) In in vitro geometries, the skeleton model captures chaotic, standing-wave, traveling-wave pattern, sensing the geometry via variations in the bulk--surface ratio. Adapted from Ref. Wurthner.etal2022. (d) The MinE-switch model extends the skeleton model by a conformational switch between reactive and latent MinE states in the cytosol. Reactive MinE is recruited much more quickly by membrane-bound MinD than MinE in the latent conformation. (e) The MinE-switch model well describes the phase diagram of Min pattern formation in filamentous E. coli bacteria. It captures the robustness of pattern formation in protein concentrations, the pattern types, as well as the wavelengths and oscillation periods. Adapted from Ref. Ren.etal2025. (f) In vitro, mutation of the MinE protein showed that impairing its conformational switch (reactive MinE) strongly reduces the range of pattern formation, in accordance with the prediction by the MinE-switch model. Adapted from Ref. Denk.etal2018. (g) MinE may persistently bind to the membrane via its membrane-targeting sequence. (h) The relevance of MinE membrane binding for the Min oscillation in vivo remains to be clarified by combined experiments and theory. (i) MinE membrane-binding allows for the formation of stationary patterns observed with wild-type proteins in vitro. Scale bars in panels (c,f,i): $50µ m$. Adapted from Ref. Weyer.etal2024b.
• Figure 4: Interfaces in McRD systems. (a) Two-component McRD systems form nonlinear patterns featuring interfaces between high- and low-density plateaus (or peaks, see (e)) that undergo coarsening. (b) In three-species McRD systems, antagonistic reactions generate distinct membrane domains (blue, red, yellow). (c) The Min system forms stationary mesh patterns, resembling 2D liquid foams in vitroGlock.etal2019a and in simulations Weyer.etal2024b, where MinE-rich branches (cyan) separate MinD-rich domains (magenta). On a sphere, MinE forms polyhedral meshes (cyan); MinD not shown. Experimental data from Ref. Glock.etal2019a. (d) Interfaces arise from balanced attachment and detachment zones, forming non-equilibrium steady states. In local $(m,c)$ phase space (right), the pattern lies on the flux-balance subspace (dashed), with plateau densities $m_\pm$ at its outer intersections with the nullcline. Balance of attachment (blue) and detachment (red) areas determines $\eta_\mathrm{stat}$. (e) Peak patterns occur when high-density plateaus are not reached. In phase space, the pattern ends before the right-most intersection. Larger peaks (thin, purple) correspond to lower stationary mass-redistribution potential (area comparison). (f) This potential dependence induces a mass-competition instability: small mass differences between peaks self-amplify (blue arrows). (g) Weakly broken mass conservation introduces net production (smaller peak) and degradation (larger peak) that counteract this instability (blue arrows). (h) As a result, coarsening halts above a wavelength $\Lambda_\mathrm{stop}$, set by the source strength; at larger wavelengths, domain splitting occurs above $\Lambda_\mathrm{split}$. (i) In 2D, spatial separation of attachment and detachment zones leads to curvature-dependent turnover, driving interface straightening (orange arrow). (j) This curvature dependence destabilizes fourfold vertices: increased detachment in curved regions (red) vs. flatter ones (blue) causes vertex splitting into pairs of triple junctions. Panels b,c,i,j adapted from Ref. Weyer.etal2024b.