Mathematical Modeling of Biological Interfacial Phenomena

We review recent results showing that phase separation phenomena and the formation of sharp interfaces is a natural feature of classical aggregation-diffusion models (such as the Patlak-Keller-Segel model) at appropriate scales. Furthermore, we will see that the motion of such interfaces is typically driven by surface tension mechanisms and can be described by classical free boundary problems such as mean-curvature flows and Hele-Shaw free boundary problems. 

Most biochemical reactions in living cells are not closed systems; they interact with their surroundings by exchanging energy and materials. At a mesoscopic scale, the quantity of each chemical can be modeled by random time-changed Poisson processes. Understanding macroscopic behaviors is facilitated by a nonlinear reaction rate equation that describes species concentrations. In the thermodynamic limit, the large deviation rate function from the chemical master equation is governed by a Hamilton–Jacobi equation.  We decompose the general macroscopic reaction rate equation into an Onsager-type strong gradient flow, supplemented by conservative dynamics. We will also present findings on the large deviation principle and the importance sampling of transition paths that connect metastable states in chemical reactions.

The interplay between the complex microscopic structures and microscopic (cellular) processes enables many biological tissues to combine the ability to resist high pressure and mechanical forces with the flexibility required for large expansions and growth. To analyse the interactions between the mechanics, microstructure, and the chemistry we derive microscopic models for plant biomechanics, assuming that the elastic/viscoelastic properties of plant cell walls depend on the chemical processes and chemical reactions depend on the mechanical stresses. Multiplicative decomposition of the deformation gradient into elastic and growth parts is used to model the stress or strain based growth. To analyse and simulate the complex multiscale models, the macroscopic effective equations are derived using periodic or stochastic homogenization techniques. Numerical solutions for the macroscopic models demonstrate the impact of the microscopic structure and tissue heterogeneity on deformation and growth.

When studying invasion fronts in systems of reaction-diffusion equations one often considers asymptotic limits where a system parameter is taken to be asymptotically large or small.  In this limit, it is sometimes possible to obtain a reduced equation which is more amenable to analysis.  This talk will focus on mathematical techniques to justify this reduction.  Using a combination of geometric singular perturbation theory and persistence results under regular perturbations, we show that the pushed and pulled fronts obtained in the reduced system persist for the full system. The primary example to be considered is a logistic Keller-Segel equation with chemorepulsion.  This is joint work with Montie Avery and Arnd Scheel.  

"The amoeba Dictyostelium discoideum exhibits aerotaxis: in experiments, when a colony is covered, cells quickly consume the available oxygen and then move outwards at a constant speed to find new oxygen.  In this talk, we discuss a reaction-diffusion equation, arising from a large population limit of an stochastic differential equation, for this behavior that was proposed by Demircigil and Tomašević.  This model is quite degenerate, making even the well-posedness difficult to show. The main goal of the talk is to discuss the existence, uniqueness, and long-time behavior of solutions.  We show a transition in the character of the behavior as the strength of the aerotactic response varies.  Our results use crucially the shape defect function, an object that measures the distance from the solution to the traveling wave in a novel way related to Stein's method in probability.  This is a joint work with Mete Demircigil."

I will present a joint work with Jean-Baptiste Saulnier
(Marseille), Michèle Romanos (Lyon) and Tâm Mignot (Marseille). We
revisited the modeling of the so-called rippling phase of the soil
bacteria Myxococcus xanthus lifecycle. The emergence of
counter-propagating waves has attracted great attention from the
mathematical and biophysical communities in the past two decades.
Combining recent biological insights, high-resolution microscopy,
trajectory analysis, kinetic modeling and numerical simulations, we
proposed an integrated scenario for the collective behavior of this
fascinating micro-organism, in which traffic congestion seems to play a
key role.

The motile micro-organisms such as Escherichia coli, sperm, or some seaweed are usually modeled by self-propelled particles that move with the run-and-tumble process. Individual-based stochastic models are usually employed to model the aggregation phenomenon at the boundary, which is an active research field that has attracted a lot of biologists and biophysicists. Self-propelled particles at the microscale have complex behaviors, while characteristics at the population level are more important for practical applications but rely on individual behaviors. Kinetic PDE models that describe the time evolution of the probability density distribution of the motile micro-organisms are widely used. However, how to impose the appropriate boundary conditions that take into account the boundary aggregation phenomena is rarely studied. In this talk, we propose the boundary conditions for a confined run-and-tumble model (CRTM) for self-propelled particle populations, study its long time behavior. We establish the relation between CRTM and the confined Fokker–Planck model and confined diffusion model. Then discuss their possible clinical applications.