Numerical simulation of the dynamics of molecular markers involved in cell polarisation

A cell is polarised when it has developed a main axis of organisation through the reorganisation of its cytosqueleton and its intracellular organelles. Polarisation can occur spontaneously or be triggered by external signals, like gradients of signaling molecules ... In this work, we study mathematical models for cell polarisation. These models are based on nonlinear convection-diffusion equations. The nonlinearity in the transport term expresses the positive loop between the level of protein concentration localised in a small area of the cell membrane and the number of new proteins that will be convected to the same area. We perform numerical simulations and we illustrate that these models are rich enough to describe the apparition of a polarisome.Comment: 15 page

Protrusion fluctuations direct cell motion

Many physiological phenomena involve directional cell migration. It is usually attributed to chemical gradients in vivo. Recently, other cues have been shown to guide cells in vitro, including stiffness/adhesion gradients or micro-patterned adhesive motifs. However, the cellular mechanism leading to these biased migrations remains unknown, and, often, even the direction of motion is unpredictable. In this study, we show the key role of fluctuating protrusions on ratchet-like structures in driving NIH3T3 cell migration. We identified the concept of efficient protrusion and an associated direction index. Our analysis of the protrusion statistics facilitated the quantitative prediction of cell trajectories in all investigated conditions. We varied the external cues by changing the adhesive patterns. We also modified the internal cues using drug treatments, which modified the protrusion activity. Stochasticity affects the short- and long-term steps. We developed a theoretical model showing that an asymmetry in the protrusion fluctuations is sufficient for predicting all measures associated with the long-term motion, which can be described as a biased persistent random walk.Comment: Supplementary movies available upon reques

Signatures of motor susceptibility in the dynamics of a tracer particle in an active gel

We study a model for the motion of a tracer particle inside an active gel, exposing the properties of the van Hove distribution of the particle displacements. Active events of a typical force magnitude give rise to non-Gaussian distributions, having exponential tails or side-peaks. The side-peaks appear when the local bulk elasticity of the gel is large enough and few active sources are dominant. We explain the regimes of the different distributions, and study the structure of the peaks for active sources that are susceptible to the elastic stress that they cause inside the gel. We show how the van Hove distribution is altered by both the duty cycle of the active sources and their susceptibility, and suggest it as a sensitive probe to analyze microrheology data in active systems with restoring elastic forces.Comment: 4 pages, 4 figures and supplemental information (5 pages, 4 figures

Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration

Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction-diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other quantitative results, such as the selection of the most motile individuals (when the motility is bounded). The key argument for the construction and analysis of traveling fronts is the derivation of the dispersion relation linking the speed of the wave and the spatial decay. When the motility is unbounded we show that the position of the front scales as $t^{3/2}$. When the mutation rate is low we show that the canonical equation for the dynamics of the fittest trait should be stated as a PDE in our context. It turns out to be a type of Burgers equation with source term.Comment: 7 page