Sensing the shape of a cell with reaction diffusion and energy minimization

Abstract

Some dividing cells sense their shape by becoming polarized along their long axis. Cell polarity is controlled in part by polarity proteins, like Rho GTPases, cycling between active membrane-bound forms and inactive cytosolic forms, modeled as a “wave-pinning” reaction-diffusion process. Does shape sensing emerge from wave pinning? We show that wave pinning senses the cell’s long axis. Simulating wave pinning on a curved surface, we find that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, when we introduce roughness to our surfaces, shape sensing can be disrupted, and high-activity domains can become localized to locations other than the global peaks and valleys of the surface. On rough surfaces, the domains of the wave-pinning model are more robust in finding the peaks and troughs than the minimization rule, although both can become trapped in steady states away from the peaks and valleys. We can control the robustness of shape sensing by altering the Rho GTPase diffusivity and the domain size. We also find that the shape-sensing properties of cell polarity models can explain how domains localize to curved regions of deformed cells. Our results help to understand the factors that allow cells to sense their shape—and the limits that membrane roughness can place on this process.

Numerical investigations of the bulk-surface wave pinning model

“…When D b is large enough, polarisation patterns can be generated and these are subject to a slow transition of the polarised area across the surface Γ. This is an intrinsic feature of equation (3) for a(x, t), as recently shown, for instance by Singh et al [17] and Miller et al [10] for the reduced surface reaction-diffusion model (5).…”

“…It was suggested that the motion of the polarised patch across the domain corresponds to a minimisation of the perimeter of the interface separating the states a high and a low , under the constrain of constant polarised mass [7]. Recently, Singh et al [17] compared the reduced model equation ( 5) with a problem of perimeter minimisation with constant polarisation patch area, obtaining, for certain initial conditions, a very good agreement between the two solutions.…”

“…Hence, as already discussed, the expansion of the initial peaks strictly depends also on the curvature of the domain. Indeed, following the work by Singh et al [17] we might expect the model to evolve towards a single patch solution that minimises its interface perimeter. Interestingly, the bulk diffusion coefficient D b reveals to be crucial in determining the outcome of the competition.…”

Preferential localization of a single spot in reaction–diffusion systems on non-spherical surfaces

“…The LSA may also find applications in the analysis of shape sensing of a cell. 46 Broadly, the methods outlined in the paper can be incorporated into any RD-like systems involved in the analysis of various symmetry-breaking mechanisms in cellular processes [7][8][9]31,37,47,48 where the effect of geometry can be significant. 31,49 In brief, the work can provide useful insights into controlling cell polarity by tuning the RD parameters and may shed some fundamental insights on axis selection by geometry.…”

Migration and division in cell monolayers on substrates with topological defects

“…However, it may be essential to understand longer-scale perfect alignment on unpatterned substrates [4]. We also neglect potential couplings between cell shape and polarity [68][69][70], which can drive complex behaviors like cell circling and oscillation in response to fields, as observed in keratocytes [69]. We have neglected these factors because we have no evidence that fibroblasts show circular migration behavior similar to keratocytes.…”