Mathematical Modeling of Circular Dorsal Ruffles and Lamellipodial Dynamics in Single and Collective Cell Migration

Abstract

Cell motility is a phenomenon that has intrigued scientists for many years. Increasingly, researchers realize the need for quantitative analysis of both the mechanical as well as the biochemical aspects at multiple scales. The objective of this thesis is therefore to use mathematical and computational modeling to quantitatively study several specific processes in cell motility.

The reorganization of actin, being the building block of the cell cytoskeleton, is crucial in driving cell movement. A good appreciation of the biochemical nature of actin dynamics is essential in the understanding of cell migration. This was achieved by studying the dynamics of circular dorsal ruffles (CDR), an actin-based structure often seen in growth-factor stimulated migrating cells. The presence of CDRs has been shown to be the precursor to lamellipodia generation and cell motility. Experimentalists have found that the appearance of CDRs is often accompanied by the disappearance of actin-rich stress fibers. While the generation of CDRs can been attributed to the activation of the Rac, stress fibers have been shown to be stabilized by the presence of active Rho. I therefore represented the formation of CDRs, starting from growth factor induced Rac activation interacting with pre-existing Rho and the associated stress fibers, using a system of partial differential equations. The numerical simulation results showed that increasing the substrate stiffness, which led to increased stress fiber formation prior to stimulation, increased the lifetime of the CDR without altering the size of these structures. A simplified model, which involved Rac and a Rac inactivator, showed that the dynamics of CDRs can be likened to wave propagation in an excitable medium.

The study of CDRs showed that the actin cytoskeleton is highly dynamic, with many proteins regulating its activity. Yet, cell migration cannot be reenacted without considering the interaction of forces that drive motion. An important part of a migrating cell is the lamellipodium, a thin protrusive portion at the front of the migrating cell. I developed a model of lamellipodial dynamics that incorporated actin polymerization and forces exerted on the actin cytoskeleton. Through the use of a stretch-sensitive protein that responded to substrate stiffness, the model showed that the lamellipodium can exhibit periodic protrusion-retraction cycles, continuous protrusion and unstable retraction, depending on the substrate stiffness and the relative amounts of integrin and myosin activation. In particular, periodic behavior similar to that seen in recent experiments can be achieved when the substrate is sufficiently stiff.

Studying cell migration is incomplete without looking at how cells move when interacting with one another, which is usually the case in vivo. Therefore, I investigated the collective migration of cells on constrained substrates. Using a lattice-based computational method known as the Cellular Potts Model, I studied the collective migration of cells as a function of the substrate channel width and found that the collective migration velocity decreased with increasing channel width. Analysis of the velocity field showed that the component of the cell velocities perpendicular to the channel's long axis demonstrated increasing correlation length with channel width whereas the parallel component was unaffected. The decrease in velocity as the adhesive substrate channel width was increased was found to be a consequence of the ability of the cell to polarize during motion. This study showed that the study of collective cell migration can reveal long range migratory behaviour within tissues which single cell migration would not elucidate.