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

Abstract

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the wellunderstood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for wavelength-selection dynamics in (nearly) mass-conserving twocomponent reaction-diffusion systems independent of the specific mathematical model chosen. This encompasses both the dynamics of the mesa-and peak-shaped patterns found in these systems. Our analysis uncovers a diffusion-and a reaction-limited regime of the dynamics, which provides a systematic link between the dynamics of mass-conserving reaction-diffusion systems and the Cahn-Hilliard as well as conserved Allen-Cahn equations, respectively. A stability threshold in the family of stationary patterns with different wavelengths predicts the wavelength selected for the final stationary pattern. At short wavelengths, self-amplifying mass transport between single pattern domains drives coarsening while at large wavelengths weak source terms that break strict mass conservation lead to an arrest of the coarsening process. The rate of mass competition between pattern domains is calculated analytically using singular perturbation theory, and rationalized in terms of the underlying physical processes. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. Moreover, we compare these expressions with numerical results and find excellent agreement throughout the diffusion-and reaction-limited regimes of the dynamics, including the crossover region. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multi-component systems with potentially several conservation laws.

Geometry-induced patterns through mechanochemical coupling

“…One might therefore wonder why we do not observe multiscale patterns here. The reason is that the two-component system has only one stable attractor (mesa or peak pattern) [42,67,69], which significantly limits the phenomenology. One could, however, readily apply our approach to the Min dynamics by replacing the reaction-diffusion component in our model with the biochemical reaction network of the Min system.…”

“…Hence, for our choice of parameters, the reaction-diffusion model always produces mesa patterns (lower and upper plateau in the density profile which are connected by an interface, see Fig. 1a), because the flux-balance subspace intersects the reactive nullcline at three points [42,67,69] (see Fig. 1a,b).…”

“…2 The wavelength of this initial pattern is well approximated by λ c . However, this initial pattern is not stable and slowly coarsens to a single peak or interface [64][65][66][67][68][69].…”

Particle-based simulations reveal two positive feedback loops allow relocation and stabilization of the polarity site during yeast mating

“…Several more abstract minimalistic models, containing as few as two chemical species with non-linear positive feedback, were also studied to gain a deeper understanding of how cells might use Rho-family GTPases to generate spatial patterns. Because of their mathematical tractability, these models produced important insights into the mechanisms underlying establishment, competition and coexistence of GTPase clusters [14,15,30,[57][58][59][60][61]. While theoretical work has begun to establish conditions that support polarity establishment and coarsening, less attention has been given to identifying mechanisms that produce movement or relocation of the polarity site.…”

Particle-based simulations reveal two positive feedback loops allow relocation and stabilization of the polarity site during yeast mating

“…Several more abstract minimalistic models, containing as few as two chemical species with non-linear positive feedback, were also studied to gain a deeper understanding of how cells might use Rho-family GTPases to generate spatial patterns. Because of their mathematical tractability, these models produced important insights into the mechanisms underlying establishment, competition and coexistence of GTPase clusters [14,15,30,[57][58][59][60][61]. While theoretical work has begun to establish conditions that support polarity establishment and coarsening, less attention has been given to identifying mechanisms that produce movement or relocation of the polarity site.…”