Resumen de Flocking and pattern formation in active particles and epithelial tissues
Carolina Trenado Yuste
Active matter is a term used to describe a novel class of living and non-living systems characterized by having a huge number of active units capable of propel themselves by consuming energy from their environment and converting it into motion. The observations of the dynamics in coordinated motions motivate the study of active matter. Cooperative migrations of living entities are, for example, the synchronous motion of a school of fish swimming in the ocean, a group of starlings moving coherently, or the collective migration of epithelial cells trying to close a wound. These so-called flocking phenomena have in common that individual units move in group or flock and generate large-scale complex dynamical motions forming patterns of different shapes. When describing the observations of collective motion, it is important to obtain the best definition of the individual trajectories of the particles. This is an intricate task due to the large number of entities in each system. Thus, to understand flocking, which is an emergent property of the collectivity, it is convenient to study mathematical models built with minimal ingredients but capturing the essence of the phenomena.
The first studies that gave a minimal framework to explain the flocking behavior of living systems were agent-based models. In particular, the Vicsek model (VM) describes N particles that move with equal speed in a box with periodic boundary conditions. The velocity of each entity adopts the average direction of its close neighbors plus a random perturbation inherent to each individual. The VM and its variants lie within the so-called self-propelled particle models (SPP).
These systems display phase transitions from disordered to coherent behavior depending on the parameters of the model. In particular, for the VM, when the alignment noise is sufficiently small or the particle density is high enough, particles move coherently as a swarm. Indeed, below a critical size of the box, flocking occurs as a continuous bifurcation from a disordered state, with uniform particle density, to an ordered state. For a box size larger than critical, the bifurcation is discontinuous and a variety of patterns are possible. In the literature, it is possible to find a wide range of variations of the VM along three specific pathways, namely, (i) the variation in the polarity of the particles, (ii) the interaction between them (repulsion or attraction forces) and (iii) the external medium where they move.
I. Bifurcation study of the standard 2D VM. To study flocking in the 2D standard VM and to interpret numerical simulations, it is convenient to derive kinetic or hydrodynamic equations that can be studied with analytical tools. Among kinetic equations, Ihle has derived one that is closest to the discrete space-time dynamics of the VM. This kinetic equation for the one-particle distribution function is discrete in time and space, it preserves the number of particles, and it always has a simple uniform solution corresponding to the disordered state of the system. Thus, we have carried out a novel linear stability analysis of the uniform distribution for small wave numbers and we have studied the possible bifurcations issuing from it. The uniform distribution is linearly stable provided all solution modes of the linearized kinetic equation have multipliers on the unit circumference or inside it. There is always a neutrally stable mode corresponding to particle conservation. When the control parameter (e.g., the alignment noise) crosses a critical value, a multiplier exits the unit circle thereby destabilizing the uniform distribution. The corresponding complex valued mode, together with the real valued particle conservation mode, yield the bifurcation (amplitude) equations. They consist of a continuity equation for a density disturbance and a 2D vector equation for a current density. As the control parameter departs from its critical value, these equations contain two different time scales. In the short time scale, the equations are hyperbolic and parabolic in the longer scale. The hyperbolic equations have oscillatory solutions with many incommensurate frequencies: linearizing about a constant solution, we obtain a Klein-Gordon equation with periodic boundary conditions. Space independent solutions of the amplitude equations exist on the parabolic time scale and satisfy the usual normal form of a pitchfork bifurcation. Simulations of the original VM exhibit resonance phenomena linked to the oscillations on the hyperbolic time scale predicted by our analysis when the alignment rule is modified by adding a time periodic function.
Although collective motion is considered to be a profitable state for the entities of the system, recently studies have found that there exist more intricate individual interests between the entities inside the organization remaining them together but not with a collective purpose. For example, instead of moving linearly, E. Coli bacteria swim clockwise in circular trajectories near walls or in some insect species, the cannibalism is a nutritional behavior that makes them to move in a coordinated migration. When insects as desert locusts are in a swarm, they try to bite each other but with the risk to be bitten and although this behavior should makes them to be separated, it produces a collective migration.
II. Bifurcation study of the VM with modified alignment rule. Following the same methodology used to studying flocking in the two-dimensional VM, we have explored a different mechanism to attain synchronous rotation in small clusters. We consider a two-dimensional (2D) modified VM with forward update. 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. This choice makes the VM similar to the Kuramoto model of phase synchronization with conformist and contrarian oscillators. To interpret and understand the results of our numerical simulations, we study the discrete space-time kinetic equation with the modified alignment rule. For the uniform distribution, the multipliers of destabilizing modes cross the unit circle through minus one (contrarian rule, which gives rise to a period doubling bifurcation) or through two complex conjugate values (almost contrarian rule, which gives rise to a Hopf bifurcation). When contrarian compulsions are prevalent, increasing the alignment noise may transform incoherent particle motion to a phase displaying time periodic polarization with period doubling. If we relax the contrarian rule measured counterclockwise from the average direction, the flocking order parameter may oscillate periodically in time (with period different from 2). Active particles perform rotations or oscillations besides the collective translation characterizing the ordered phase of the standard VM. The resulting bifurcation equations describing these patterns are related to the time dependent Landau-Ginzburg equation.
The discipline of active matter also studies systems across scales, from sub-cellular processes to the dynamics of tissues and organs. The dynamics and the collective strategies of cells are crucial in processes as morphogenesis, tissue repair and embryogenesis. Cells are always competing for space to ensure tissue cohesion. In a normal situation, cells in the tissue are in a stable homeostatic configuration. However, there exist some circumstances (as in cancer metastasis or wound healing) when tissues become unstable and cells acquire a confluent motion. Collective cell migration poses challenging fundamental questions within the fields of soft and active matter, namely the characterization of fluid, solid or glass-like behavior associated with flocking and jamming-unjamming transitions.
III. Numerical simulations of confluent cellular motion and TDA of moving interfaces. The active vertex model (AVM) is a convenient computational tool in studies of confluent motion of epithelial cells. In this field, there are many experiments and different mathematical and computational models. Based on the physics of foams, the AVM considers epithelial cells to be non-overlapping two-dimensional convex polygons with a preferred area and perimeter. Preserving this Voronoi tessellation produces forces and line tensions that are incorporated to the particle dynamics of the cell centers. The latter are in a Delaunay triangulation and are subject to forces that constrain cells to have target areas and perimeter lengths. In our version of the AVM, cell centers are also subject to other forces that try to align their velocities to those of neighboring cells (as in the VM), and to friction with the substrate, inertia, and stochastic Ornstein-Uhlenbeck noisy forces. In our dynamics, cells are not self-propelled. The AVM is based on the earlier vertex model, which is a quasi-static and tries to maintain the cellular sheet in mechanical equilibrium: the dynamics is characterized by minimization of an energy that depends on the area, perimeter and junction of each polygon or cell.
We have simulated the vertex model jointly with the dynamics explained above in two different cases related to wound healing and to invasion of one cell population 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 (AMA).
For the case (i), the experimental setup consists of a micro-fabricated stencil that keeps the growing cell culture equally distributed. Once the cell culture is ready, the stencil is removed. The advantages of this experimental protocol unlike other experiments are that cells are not damaged and the edges of the monolayers are defect-free. Numerical simulations of our AVM show that the cells on a monolayer boundary are larger than those in its interior if the dynamics contains inertia, whereas the opposite is true in the overdamped case. In the experiments carried out by the Silberzan group, boundary cells have larger area than those inside the tissue, thereby confirming the need for inertia. In addition, numerical simulations show that a freely expanding cellular monolayer develops fingers, which agrees with experiments of cells invading empty spaces. Fingering phenomena are thus explained without needing leader cells with a different phenotype.
Case (ii) is motivated by recent experiments by Moitrier et al. in the Silberzan group. They have reported confrontation assays between antagonistically migrating cell sheets. In their experiment, each cell population grows into one of the compartments separated by a cell-free gap. The normal cell type was always seeded in the left compartment of the culture insert, while the transformed cells were seeded in the right compartment. Cells were left to incubate overnight until fully attached - then, the culture insert was removed, leaving a free space between the two monolayers, which could then migrate towards each other to close this gap. The two confluent cellular monolayers advance toward the intermediate empty space, collide and Ras or transformed cells population displaces the wild type one.
For our simulations of the AVM, we consider wild type or normal cell in Human Embryonic Kidney cells (HEC) to be solid-like whereas invading Ras cells are fluid-like and push the former backward. As time elapses, there are cell exchanges and islands of one cell type form inside the tissue of the other cells, which characterizes a flocking liquid state. However, in AMA with Madin-Darby Canine Kidney (MDCK) cells, the roles are inverted: Ras cells are solid-like and wt cells are fluid-like. The precise form of the separating interface among monolayers of different cell type depends on cell parameters governing segregation vs aggregation of these cells. We characterize it by topological data analysis (TDA). A measure of cellular diversity in the junction tensions produces islands of one type of cells inside the monolayer of the other cells, which is reflected in TDA of both numerical simulations and experiments. We focus on specific parts of selected snapshots of images from experiments and then on time series of images from numerical simulations. While we have few images of interfaces from experiments, we can generate arbitrarily many from numerical simulations. Having many images, the automatic TDA tool enables us to describe in detail the topological changes of the interfaces and to implement hierarchical clustering strategies, thereby classifying the evolving interface structures.
