Mathematical modelling of cell migration in confined domains

Host laboratory and collaborators

Florence Hubert / I2M /

Marie-Pierre Valignat / LAI /

Julien Olivier / I2M /


The project aims at modelling the propulsion mechanism of amoeboid cells in confined geometries. On a 2D substrate, cells migrate by combining protrusive forces to deploy a strongly adhering lamellipodium at their front and contractile forces to detach their rear. In this scenario, the lamellipodium is highly spread and therefore very thin (around 200-400 nm). In contrast, when the same cells migrate through the constrictions of a 3D medium, the lamellipodium thickens to occupy the whole section of the constrictions (up to 3x3 µm2). Despite these major structural changes, cells still migrate efficiently. The precise mechanisms underlying this transition between 2D/3D migration modes remains poorly understood, but an hypothesis that will be explored in this project is that the lamellipodium is responsible for the main propulsive force. We will construct a mathematical and numerical model of the actin gel dynamics in the lamellipodium to reproduce the structural transition from 2D to 3D and estimate the resulting forces between the lamellipodium and its micro-environment. This model will be compared with novel experimental data on the lamellipod structural transition using advanced light-sheet microscopy in microfluidics, the internal actin dynamics by FRAP, and the forces developped by lamellipodium using pressure control systems and traction force microscopy in 2D and 3D.


Mathematical and numerical modelling (analysis of PDEs. numerical analysis, Finite Volumes Schemes). Amoeboid migration (actin dynamics, microfluidics)


• To identify biological scenarios that could account for the behavior of the migrating cell in confined environment and to formulate mathematical models coherent with these scenarios.
• To study these models theoretically and to implement numerical approximations of the models so as to be able to reproduce experimental setting. To compare the numerical experiments to the real-life ones
• Light-sheet microscopy experiments of cells exhibiting 2D/3D transition in microfluidic devices (using SoSPIM technology with collaborators in Bordeaux, internal actin dynamics characterization by FRAP in the lamellipod of cells exhibiting 2D/3D transition in microfluidic devices and force measurement using traction force microscopy on flat surface and in 3D microfluidic devices (newly developed by collaborators in Lyon)

Proposed approach (experimental / theoretical / computational)

This project will combine a theoretical approach, and a computational approach and an experimental approach. From a theoretical viewpoint, we will use the tools of PDE and more generally, the tools of deterministic modelling to construct our model. Then we will use theoretical tools to analyze the model so as to determine its basic and fine properties. This helps defining the limits of the model. From a computational standpoint, we will use the technology of Finite Volume schemes developed at I2M to construct a homebrew code (so as to be able to control all of its aspects). This code will be validated on various academic test case. Once the code is validated it will be calibrated with biophysical experiment. New experiments may be suggested by this first step of calibration. The applicant will be involved in the design process and the interpretation of new data.


Most of the steps in this project will require a close interaction between the two laboratories. Once a common language has been established, interactions will occur at:

• development of mathematical models
• development of a novel set of experimental data with original tool (microfluidics, 3D TFM, SoSPIM)
• interpretation of the experimental data used for the calibration of the model
• selection of the most relevant biological scenario
• design of new experiments based on questions arising from in silico data

Expected profile

The expected student should have experience in mathematical modelling coming from either a physics curriculum or an applied mathematics one. He or she should also have an interest in biological systems and their particular nature. Consequently, he or she should be eager to interact strongly with people from various field, be it mathematics, physics or biology.