Flocking and pattern formation in active particles and epithelial tissues

Abstract

Collective behavior and, more specifically, flocking are phenomena observed in living systems, from bacterial colonies and spermatozoa, to larger systems such as insects and birds. These organizations exhibit changes from disordered to coherent behavior, which are examples of spontaneous symmetry-breaking out of equilibrium. Collective migrations in these systems can be predicted by simple models such as the Vicsek model (VM) or its variants, in which particles tend to align their velocities to an average of their neighbours’. The change from a disordered state to an ordered state can occur continuously or discontinuously and a variety of resulting patterns are possible. The study of mathematical models of these systems may reveal these changes to be bifurcations in their governing equations. We consider a system of particles moving within a two dimensional box with periodic boundary conditions. In Chapter 2, and following Ihle’s approach, we derive a kinetic equation for a one-particle distribution function in the limit of infinitely many particles by assuming molecular chaos. The kinetic equation is discrete in time and space and it always has a simple uniform solution that corresponds to the disordered state of the system. We have carried out a linear stability analysis of this state and studied the possible bifurcations issuing from it. In the usual case, particles align their velocities to their average velocity with any other particles within a circle of influence plus some angular noise, which has a uniform probability density. The spectrum of the linearized equation has always one multiplier on the unit circumference and there is another one that moves from inside to outside the unit circle as a control parameter crosses a critical value. We use bifurcation methods to derive amplitude equations that describe solutions issuing from the disordered state. The amplitude equations comprise a conservation law for a density disturbance coupled to a two dimensional vector equation for a current density. Analysis and numerical simulations of these equations show that their solutions exhibit an interplay between parabolic and hyperbolic behavior in two different time scales when the distance to the critical value of the bifurcation control parameter goes to zero. In this limit, there appear oscillation frequencies that give rise to resonance phenomena if the alignment rule contains a periodic function of time. Direct simulation of the VM confirms the existence of these resonances. In Chapter 3, we use the same methodology to study the effect of modifying the probability density of the noise in the alignment rule by which VM particles change their velocities. The mechanism of velocity synchronization consists of: active particles may be conformist and align their velocities to the average velocity of their neighbors, or be contrarian and move opposite to the average angle. Depending on the weights of conformist and contrarian or almost contrarian rules, we study the ordered state solutions of the amplitude equations corresponding to period-doubling, Hopf, or pitchfork bifurcations of the disordered state. In Chapter 4, we consider the collective migration of epithelial cell monolayers moving on a surface. This phenomenon is crucial for many relevant processes including wound healing, morphogenesis, and cancer-cell invasion during metastasis. There are many experiments on confluent cellular motion and different mathematical and computational models in the literature. A convenient model based on the physics of foams considers the cells as non-overlapping two dimensional convex polygons. In the active vertex model we study, the cell centers are in a Delaunay triangulation and are subject to forces that constrain them to have target areas and perimeter length, other forces that try to align their velocities to neighboring cells (as in the VM), friction with the substrate, inertia, and stochastic forces. We have simulated numerically this model in two different cases related to wound healing and to invasion of one cell collective by another one: (i) a cellular monolayer spreading on empty space, and (ii) the collision of two different cell populations in an antagonistic migration assay. For (i), we discuss how inertia is necessary to explain the larger size of cells in the boundary with respect to those in the interior of the layer. For (ii), we discuss which parameters of the model produce results that agree with experiments by P. Silberzan’s group. In both cases, the interfaces that separate cells from empty space or cells belonging to different populations are quite rough and may shed and absorb islands as time elapses. To analyze both images from experiments and results of numerical simulations, we use topological data analyses of the interfaces.