Cell Size, Mechanical Tension, and GTPase Signaling in the Single Cell

Abstract

Cell polarization requires redistribution of specific proteins to the nascent front and back of a eukarytotic cell. Among these proteins are Rac and Rho, members of the small GTPase family that regulate the actin cytoskeleton. Rac promotes actin assembly and protrusion of the front edge, whereas Rho activates myosin-driven contraction at the back. Mathematical models of cell polarization at many levels of detail have appeared. One of the simplest based on "wave-pinning", consists of a pair of reaction-diffusion equations for a single GTPase. Mathematical analysis of wave-pinning so far is largely restricted to static domains in one spatial dimension. Here we extend the analysis to cells that change in size, showing that both shrinking and growing cells can lose polarity. We further consider the feedback between mechanical tension, GTPase activation, and cell deformation in both static, growing, shrinking, and moving cells. Special cases (spatially uniform cell chemistry, absence or presence of mechanical feedback) are analyzed, and the full model is explored by simulations in 1D. We find a variety of novel behaviors, including "dilution-induced" oscillations of Rac activity and cell size, as well as gain or loss of polarization and motility in the model cell.

Membrane Tension Can Enhance Adaptation to Maintain Polarity of Migrating Cells

“…By reversing the transformation we can obtain the full moving-boundary solution. This method is similar to that used in other investigation of pattern formation on time-dependent domains [62][63][64][65].…”

“…with no-flux boundary conditions. We can determine the velocity field a(x, t) throughout the cell body from the forces imposed at the cell ends since we assume that the cell changes length isotropically (as in [62][63][64][65]).…”

Spots, stripes, and spiral waves in models for static and motile cells

“…Here we adhere to the first approach. The case of 1D dynamic cell size is considered in Buttenschön et al (2020).…”

“…Several models have been examined mathematically to describe how a single GTPase coupled to other effectors or influences could result in spatio-temporal patterns. These include a GTPase with sources and sinks (Verschueren and Champneys 2017, see also Champneys et al 2021), with feedback from F-actin (Holmes et al 2012a;Mata et al 2013), with mechanical tension (Zmurchok et al 2018) and with effects of changing cell size (Buttenschön et al 2019). Many of these were explored in reaction-diffusion (RD) equations within a 1D static single cell domain or with spatially uniform distribution in each of many cells (Zmurchok et al 2018).…”

Membrane tension can enhance adaptation to maintain polarity of migrating cells

“…with no-flux boundary conditions. We can determine the velocity field a(x, t) throughout the cell body from the forces imposed at the cell ends since we assume that the cell changes length isotropically (as in [62][63][64][65]). This assumption means that as the cell length, L(t) = x + (t) − x − (t), changes, each Lagrangian volume element will change proportionally to the total length change (this is a simplifying approximation that will be relaxed in the 2D model to follow).…”