Stochastic dynamics of sINgle cells: Growth, Emergence and Resistance

Introduction

This project aims to introduce new stochastic and deterministic models for biological and medical applications. We will analyze these models mathematically, derive qualitative properties of solutions, quantify the emergence of asymptotic regimes, and determine the limiting equations.

Motivation

Motivated by recent biological experiments involving single cell observations, we emphasize the effects of very small populations in various biological and medical contexts related to evolution, such as:

• Emergence of leukemia
• Antibiotics resistance

Mathematical Challenges

Our main mathematical challenge is to quantify such effects, particularly on macroscopic approximations. It is our hope that this will possibly shed some light on new therapeutic strategies.

Modeling Approach

In order to track individuals and account for small populations, we are naturally led to stochastic multiscale models. The limiting macroscopic equations should involve nonlocal nonlinear partial differential equations (PDE) with constraints and singularities.

Investigation Focus

We shall investigate in particular the impact of various time scales on macroscopic approximations of a new class of birth and death processes. This will lead to a new class of Hamilton-Jacobi (HJ) equations with constraints and singularities.

Preliminary Findings

Preliminary numerical simulations indicate that these models should exhibit many surprising asymptotic behaviors, such as cyclic behaviors that we shall attempt to derive rigorously.

Future Studies

We also plan to study the lineages of sampled individuals at a given observation time by determining mathematically their time reversal paths. This issue is of particular relevance when considering the effect of time-dependent environments, in which case the survival of individuals may only be explained by a very small number of initial individuals.

Long-term Objectives

One long-term objective consists of imagining evolutionary scenarios of resistances and better strategies for antibiotics or chemotherapy. This will be closely developed in collaboration with biologists and medical biologists.