A computational model of cell polarization and motility coupling mechanics and biochemistry

The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from molecular to tissue) as well as disparate time scales, with reaction kinetics on the order of seconds, and the deformation and motion of the cell occurring on the order of minutes. The computational difficulty, even in 2D, resides in the fact that the problem is inherently one of deforming, non-stationary domains, bounded by an elastic perimeter, inside of which there is redistribution of biochemical signaling substances. Here we report the results of a computational scheme using the immersed boundary method to address this problem. We adopt a simple reaction-diffusion system that represents an internal regulatory mechanism controlling the polarization of a cell, and determining the strength of protrusion forces at the front of its elastic perimeter. Using this computational scheme we are able to study the effect of protrusive and elastic forces on cell shapes on their own, the distribution of the reaction-diffusion system in irregular domains on its own, and the coupled mechanical-chemical system. We find that this representation of cell crawling can recover important aspects of the spontaneous polarization and motion of certain types of crawling cells.

💡 Research Summary

The paper presents a two‑dimensional computational framework that simultaneously captures cell polarity signaling and the mechanics of cell shape change, using the immersed boundary (IB) method to handle a deformable, elastic cell perimeter. The authors model the cell cortex as a network of linear springs that define an elastic contour; forces generated by the springs are spread onto a fixed Eulerian fluid grid, allowing the contour to move and deform while the underlying reaction‑diffusion equations are solved on the same grid. Inside the contour, a minimal two‑component reaction‑diffusion system (an activator A and an inhibitor B) is employed. The activator exhibits self‑activation and is suppressed by the inhibitor, while the inhibitor diffuses faster (D_B > D_A). This FitzHugh‑Nagumo‑type kinetics produces a bistable pattern that spontaneously breaks symmetry, yielding a localized high‑A region that serves as a “polarization point.”

Coupling between chemistry and mechanics is achieved by converting the local concentration of A into a protrusive traction force that acts normal to the membrane. The magnitude of this force is proportional to A via a tunable coefficient α. The protrusive force competes with the elastic restoring force of the spring network (characterized by spring constant k). By integrating the chemical and mechanical sub‑systems with distinct time scales—seconds for reaction‑diffusion, minutes for membrane deformation—the model reproduces the emergence of a persistent front and rear, and the resulting directed migration of the cell.

Three simulation scenarios are explored. (1) Pure mechanics: with the reaction‑diffusion system disabled, an initially perturbed spring contour relaxes toward a circular shape, demonstrating the intrinsic tendency of the elastic perimeter to minimize curvature. (2) Pure chemistry: on a fixed circular domain, the activator‑inhibitor system self‑organizes into a stable polar spot, confirming that internal feedback loops can generate polarity without any external cue. (3) Fully coupled system: the polar spot drives a localized protrusive force, causing the elastic contour to extend at the front and retract at the rear. The cell translates in the direction of the high‑A region, and its speed and shape depend sensitively on α, k, and the diffusion ratio D_B/D_A. Excessive elasticity suppresses migration, while insufficient elasticity leads to unrealistic over‑extension, highlighting the need for a balanced mechanical–chemical feedback.

Key insights include: (i) spontaneous polarity can arise solely from reaction‑diffusion dynamics, supporting the hypothesis that stochastic intracellular fluctuations are sufficient to bias cell direction; (ii) the balance between protrusive traction and elastic resistance determines both the steady‑state shape and migration velocity; (iii) the diffusion disparity between activator and inhibitor controls the size and stability of the polar domain, suggesting that differential mobility of signaling molecules (e.g., membrane‑bound versus cytosolic proteins) is a critical regulator of persistent migration.

The work advances the state of the art by integrating a deformable domain solver with intracellular signaling, a combination that has been largely absent from previous models that treat mechanics and chemistry separately. The authors discuss extensions to three dimensions, incorporation of richer signaling networks such as Rho‑GTPase cascades, and quantitative validation against live‑cell imaging data. Such developments could make the framework a powerful tool for studying diverse processes—immune cell chemotaxis, cancer cell invasion, wound‑healing fibroblast migration—where the interplay of mechanical forces and biochemical polarity is essential.