Closing the loop: lamellipodia dynamics from the perspective of front propagation

A continuum mechanical model of cell motion driven by a biphasic traction stress

The aim of this paper is to place the cell locomotion problem within the general framework of classical continuum mechanics, and while doing so, to account for the deformation of the actin network in the cytoskeleton; the myosin activity on the lamellum including its effect on depolymerization at the trailing edge; model the stress-dependent driving forces and kinetic laws controlling polymerization at the leading edge, depolymerization at the trailing edge and ATP hydrolysis consistently with the dissipation inequality; and, based on the observations in Gardel et al. (Gardel et al. 2008 J. Cell Biol. 183, 999-1005 (doi:10.1083/jcb.200810060)), include a biphasic velocity-dependent traction stress acting on the actin network. While we chose certain specific models for each of these, in part to allow for an analytical solution, the generality of the framework allows one to readily introduce different constitutive laws to describe these phenomena as might be needed, for example, to study some different type of cells. As described in §5, the predictions of the model compare well with observations such as the magnitude of the very different actin retrograde speeds in the lamellum and lamellipodium including their jump at the interface, the magnitude of the cell speed, and the relative lengths of the lamellipodium and lamellum.

Thermodynamically consistent treatment of the growth of a biopolymer in the presence of a smooth obstacle interaction potential

We investigate the effect of filament-obstacle interactions on the force-velocity relation of growing biopolymers, via calculations explicitly treating obstacle diffusion and stochastic addition and subtraction of subunits. We first show that the instantaneous subunit on- and off-rates satisfy a rigorous thermodynamic relationship determined by the filament-obstacle interaction potential, which has been violated by several calculations in the literature. The instantaneous rates depend not only on the average force on the obstacle but also on the shape of the potential on the nanometer length scale. Basing obstacle-induced reduction of the on-rate entirely on the force, as previous work has often done, is thermodynamically inconsistent and can overestimate the stall force, sometimes by more than a factor of two. We perform simulations and analytic calculations of the force-velocity relation satisfying the thermodynamic relationship. The force-velocity relation can deviate strongly from the Brownian-Ratchet predictions. For shallow potential wells of depth ∼5k_{B}T, which might correspond to transient filament-membrane attachments, the velocity drops more rapidly than predicted by the Brownian-Ratchet model, in some cases by as much as a factor of 50 at an opposing force of only 1 pN. On the other hand, the zero-force velocity is much less affected than would be expected from naive use of the Boltzmann factor. Furthermore, the growth velocity has a surprisingly strong dependence on the obstacle diffusion coefficient even when the dimensionless diffusion coefficient is large. For deep potential wells, as might result from strong filament-membrane links, both the on- and off-rates are reduced significantly, slowing polymerization. Such potentials can sustain pulling forces while polymerizing but only if the attractive well is relatively flat over a region comparable to or greater than the monomer size. For double-well potentials, which have such a flat region, the slowing of polymerization by external pushing force is almost linear up to the stall force in some parameter ranges.

Correlated random walks inside a cell: actin branching and microtubule dynamics

Correlated random walks (CRW) have been explored in many settings, most notably in the motion of individuals in a swarm or flock. But some subcellular systems such as growth or disassembly of bio-polymers can also be described with similar models and understood using related mathematical methods. Here we consider two examples of growing cytoskeletal elements, actin and microtubules. We use CRW or generalized CRW-like PDEs to model their spatial distributions. In each case, the linear models can be reduced to a Telegrapher's equation. A combination of explicit solutions (in one case) and numerical solutions (in the other) demonstrates that the approach to steady state can be accompanied by (decaying) waves.

Cell protrusion and retraction driven by fluctuations in actin polymerization: A two-dimensional model

Animal cells that spread onto a surface often rely on actin-rich lamellipodial extensions to execute protrusion. Many cell types recently adhered on a two-dimensional substrate exhibit protrusion and retraction of their lamellipodia, even though the cell is not translating. Travelling waves of protrusion have also been observed, similar to those observed in crawling cells. These regular patterns of protrusion and retraction allow quantitative analysis for comparison to mathematical models. The periodic fluctuations in leading edge position of XTC cells have been linked to excitable actin dynamics using a one-dimensional model of actin dynamics, as a function of arc-length along the cell. In this work we extend this earlier model of actin dynamics into two dimensions (along the arc-length and radial directions of the cell) and include a model membrane that protrudes and retracts in response to the changing number of free barbed ends of actin filaments near the membrane. We show that if the polymerization rate at the barbed ends changes in response to changes in their local concentration at the leading edge and/or the opposing force from the cell membrane, the model can reproduce the patterns of membrane protrusion and retraction seen in experiment. We investigate both Brownian ratchet and switch-like force-velocity relationships between the membrane load forces and actin polymerization rate. The switch-like polymerization dynamics recover the observed patterns of protrusion and retraction as well as the fluctuations in F-actin concentration profiles. The model generates predictions for the behavior of cells after local membrane tension perturbations.

Curvature-Induced Spatial Ordering of Composition in Lipid Membranes

Phase segregation of membranal components, such as proteins, lipids, and cholesterols, leads to the formation of aggregates or domains that are rich in specific constituents. This process is important in the interaction of the cell with its surroundings and in determining the cell's behavior and fate. Motivated by published experiments on curvature-modulated phase separation in lipid membranes, we formulate a mathematical model aiming at studying the spatial ordering of composition in a two-component biomembrane that is subjected to a prescribed (imposed) geometry. Based on this model, we identified key nondimensional quantities that govern the biomembrane response and performed numerical simulations to quantitatively explore their influence. We reproduce published experimental observations and extend them to surfaces with geometric features (imposed geometry) and lipid phases beyond those used in the experiments. In addition, we demonstrate the possibility for curvature-modulated phase separation above the critical temperature and propose a systematic procedure to determine which mechanism, the difference in bending stiffness or difference in spontaneous curvatures of the two phases, dominates the coupling between shape and composition.

Crawling and turning in a minimal reaction-diffusion cell motility model: Coupling cell shape and biochemistry

We study a minimal model of a crawling eukaryotic cell with a chemical polarity controlled by a reaction-diffusion mechanism describing Rho GTPase dynamics. The size, shape, and speed of the cell emerge from the combination of the chemical polarity, which controls the locations where actin polymerization occurs, and the physical properties of the cell, including its membrane tension. We find in our model both highly persistent trajectories, in which the cell crawls in a straight line, and turning trajectories, where the cell transitions from crawling in a line to crawling in a circle. We discuss the controlling variables for this turning instability and argue that turning arises from a coupling between the reaction-diffusion mechanism and the shape of the cell. This emphasizes the surprising features that can arise from simple links between cell mechanics and biochemistry. Our results suggest that similar instabilities may be present in a broad class of biochemical descriptions of cell polarity.

The mechanisms of spatial and temporal patterning of cell-edge dynamics

Adherent cells migrate and change their shape by means of protrusion and retraction at their edges. When and where these activities occur defines the shape of the cell and the way it moves. Despite a great deal of knowledge about the structural organization, components, and biochemical reactions involved in protrusion and retraction, the origins of their spatial and temporal patterns are still poorly understood. Chemical signaling circuitry is believed to be an important source of patterning, but recent studies highlighted mechanisms based on physical forces, motion, and mechanical feedback.