Session A: Mathematical Biology

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Davide Ambrosi

(Politecnico di Milano, Italia)

The electromechanical coupling in cardiac mechanics

The coupling between cardiac mechanics and electric signalling can be addressed in a two possible alternative framework: the electrical potential dictates the active stress or the active strain. In this talk the explore the second (less popular) strategy, obtained thanks a factorization of the deformation tensor. After rewriting the equations in a deforming domain, we prove that neglecting the deformation of the substrate induces a non-negligible error in predicting celerity and width of the pulse, while the amplitude and the stability conditions are not affected by the electro--mechanical decoupling. Exact solutions of travelling front type can be found with explicit form of the celerity and steepness. The role of visco-elasticity and the characterization of unstable initial conditions are discussed with the support of numerical simulations, both in one and two spatial dimension.

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Vincent Calvez

(CNRS & ENS-Lyon, France)

The geometry of one-dimensional self-attracting particles

(joint work with José Antonio Carrillo, ICREA, Univiversidad Autònoma Barcelona)

We wish to present in this talk a brief overview of recent developments in the analysis of the generalized Keller-Segel system:

\[

\partial_t \rho(t,x) = \Delta\rho^m(t,x) + \chi \nabla\cdot \left(\rho(t,x) \nabla\left(\dfrac{|x|^{k}}{k}*\rho(t,x)\right)\right)\, , \quad t>0\, , \quad x\in \mathbb{R}^N\, .

\]

Here $\rho(t,x)$ denotes the density of diffusive cells which are self-attracted through a mean field potential. This model has raised a lot of interest in the field of mathematical biology since it generally exhibits a dichotomy between global existence (dispersion of the cells) and finite time blow-up (aggregation of the cells). We will explain how this system possesses the structure of a gradient flow for the Wasserstein metric on the space of probability densities. Interestingly enough the energy functional is the sum of two opposite contributions, being respectively convex and concave, and homogeneous. We will show some consequences of this gradient flow interpretation, concerning finite time blow-up and long time asymptotics.

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Maria Rita D'Orsogna

(California State University at Northridge, Los Angeles, USA)

Modelling viral entry dynamics

Successful viral infection of a healthy cell requires complex host-pathogen interactions. In this talk we focus on the dynamics specific to the HIV virus entering a eucaryotic cell. We model viral entry as a stochastic engagement of receptors and coreceptors on the cell surface. We also consider the transport of virus material to the cell nucleus by coupling microtubular motion to the concurrent biochemical transformations that render the viral material competent for nuclear entry. We discuss both mathematical and biological consequences of our model, such as the formulation of an effective integrodifferential boundary condition embodying a memory kernel and optimal timing in maximizing viral probabilities.

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Marie Doumic-Jauffret

(INRIA, France)

Aggregation/fragmentation models for protein polymerization

Abnormal protein polymerization is at the origin of a number of neuro-degenerative diseases, e.g. Prion (Madcow), Huntington or Alzheimer diseases. This phenomenon can be described mathematically either in a discrete or in a continuous setting, by equations of fragmentation-coagulation type. We review here some results and some open problems on the study of prion proliferation equations.

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Laurent Dumas

(Université Paris VI, France)

A numerical simulation of the human arterial network based on non invasive measurements

The numerical simulation of the human arterial network of a given patient in order to find the main characteristics of its arteries and thus to prevent from cardiovascular diseases is a major subject of interest. A simple fluid-structure interaction model for the simulation of blood flow in arteries is studied here. Derived from the general 3D equations, it computes the section area of the artery and the volumetric flux of the flow for each longitudinal position and time. The inverse problem corresponding to the identification of the main parameters of this model is successfully solved by using a reduced number of non invasive measurements. A discussion on the amount of information needed is in particular studied. Some first results based on experimental measurements with a Doppler technique called echotracking are also presented.

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Klemens Fellner

(University of Cambridge, UK)

Stability of stationary states of non-local evolution equations

(joint work with Gael Raoul, CMLA, ENS Cachan)

We study non-local evolution equations for a density of individuals, which interact through a given symmetric potential. Such models appear in many applications such as swarming and flocking, opinion formation, inelastic materials, .... In particular, we are interested in interaction potentials, which behave locally repulsive, but aggregating over large scales. A particular example for such potentials was recently given in models of the alignment of the directions of filaments in the cytoskeleton.

We present results on the structure and stability of steady states. We shall show that stable stationary states of regular interaction potentials generically consist of sums of Dirac masses. However the amount of Diracs depends delicately on the interplay between local repulsion and aggregation. In particular we shall see that singular repulsive interaction potentials introduce diffusive effects in the sense that stationary state are rendered continuously.

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Thomas Hillen

(University of Alberta, Canada)

Pointwise weak steady states of transport equations for cell movement in network tissue

(joint work with K. Painter (Heriot-Watt), P. Hinow (Milwaukee) and Z.A. Wang (Vanderbilt))

Some metastatic cancer cells use an interesting mechanism to move through network tissue, such as collagen networks. I will present a transport equation model, which describes the movement of these cells along network fibers and the network degradation due to proteolytic enzymes which are produced by the cells. In numerical implementation we find that the resulting models shows generation of networks which resemble vasculature formation. We believe that pointwise weak steady states are the organizing centres behind these network patterns. I will introduce the notion of pointwise weak steady states and discuss their use to understand network formation.

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Pierre-Emmanuel Jabin

(Université de Nice Sophia-Antipolis, France)

Selection-mutation dynamics for the evolution of traits in a population

This talk aims at introducing some models describing the evolution of a large population of individuals with different biological "traits". The reproduction rate of each individual depends on its trait and on the competition for ressources within the population. In addition mutations may be included, enabling new traits to develop. These models leads to difficult asymptotic problems as the time scale of the mutations is usually much larger than the time scale of births and death. One such asymptotic will in particular be investigated, corresponding to frequent small mutations.

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Doron Levy

(University of Maryland, USA)

Group Dynamics in Phototaxis

Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis." Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. Following these observations, in this talk we will present a hierarchy of mathematical models for phototaxis: a stochastic model, an interacting particle system, and a system of PDEs. We will discuss the models, their simulations, and our theorems that show how the system of PDEs can be considered as the limit dynamics of the particle system. Time-permitting, we will overview our recent results on particle, kinetic, and fluid models for phototaxis. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute), Tiago Requeijo (Math, Stanford), and Seung-Yeal Ha (Seoul, Korea).

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Sébastien Martin

(Université Paris-Sud 11, France)

Modelling of air flows and gas exchange in the human lung

This talk aims at describing some models of the respiratory system. The mechanical response of the lung can be described by either sophisticated models based on (Navier-)Stokes flows or simple mass-spring models, that are accurate enough to predict the behaviour of the system during a complex dynamical situation (the forced inspiratory-expiratory maneuver that is performed during spirometric evaluations). Oxygen absorption models are also described in order to observe the effects of changes in the parameters of the mechanical model in the efficiency of the lung. In particular, the models suggest that the airway smooth muscle (which is directly related to the bronchial rigidity) could have a structural function, to confer transport optimized mechanical properties to the bronchial tree.

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Nicolas Meunier

(Université Paris V, France)

Analysis of self-organization systems for cell polarization

(Joint work with V. Calvez and R. Voituriez)

In this work, we investigate the dynamics of a modified Keller-Segel type model. On the contrary to the classical configuration, the chemical production term is located on the boundary. In the one- dimensional case and in a particular case in dimension two, we prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass and they blow-up in finite time above the critical mass. Furthermore, in the one-dimensional case, using entropy techniques, we provide quantitative convergence results for the subcritical case. This work is completed with a more realistic model (still one-dimensional) for modeling purpose. In this new setting, the chemical is supplied by a quantity which evolves by exchanging particles at the boundary. Finally some results are given for the two-dimensional case and we also provide some links with cell polarisation that is an essential step for many biological processes and that is involved for instance in cell migration, division, or morphogenesis.

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Julie Mitchell

(University of Wisconsin - Madison, USA)

Using clustering and optimization in flexible protein docking

Modeling molecular flexibility continues to be a frontier in predicting how two proteins bind. Allowing full dynamic flexibility can be excessive, yet neither traditional rigid body docking nor small-scale rearrangements of amino acid side chains can capture many biologically relevant motions. We propose an intermediate approach, which is to combine rigid optimization with discrete deformations that represent a protein's ``natural'' motions. Through exhaustive low-resolution search, hierarchical clustering, and global optimization within each energy basin, we are able to achieve accurate docking predictions. Our optimization approach employs a Convex Global Underestimation method that has been adapted for energy functions defined on $R^3 \times SO(3)$.

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Gael Raoul

(ENS-Cachan, France)

Kimura's model: An integro-differential model to study evolution

Kimura's model is a simple intero-differential model used by theoretical biologists to study evolution. Simulations for this model can be run, and they show that that the population usually converges to a finite sum of Dirac Masses. This phenomenon is interpreted by biologists as a speciation process. We study the stability of such populations, and link our results to other existing theories used by biologists to study evolution.

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Christian Schmeiser

(University of Vienna, Austria)

Nonlinear friction in the cytoskeleton as a macroscopic limit of

the activity of cytoskeletal proteins

The mechanical stability of polymer networks in the cytoskeleton relies on the action of cross-linking proteins and transmembrane proteins creating adhesion to the substrate. It will be demonstrated that their averaged effect can be described as friction, when cross-linking and adhesion molecules are assumed as elastic and when their building and breaking are described as stochastic processes.

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Alexandre Vidal

(Université d'Evry Val d'Essonne, France)

A model of the Gonadotropin Releasing Hormone secretion by hypothalamic neuron clusters

The secretion of Gonadotropin Releasing Hormone (GnRH) by specific hypothalamic neurons plays a major role in the neuroendocrine control of the reproduction function in mammals. While the qualitative properties of the physiological secretion pattern is common to all females, the quantitative properties characterize the specificities of the ovulatory cycle in each species. The difficulty to obtain in vivo time series of GnRH release by the hypothalamus spurs us on to develop a controllable model to study this complex biological system.

I will describe the qualitative properties of the physiological GnRH secretion pattern and introduce the types of specifications that the biological knowledge can specify in various species. I will present a model of the GnRH neuron secreting activities based on reaction-diffusion coupling of fast-slow oscillators. Applying bifurcation theory and a fast-slow analysis to a reduced model, I will show the model ability to reproduce the qualitative and species-dependent quantitative properties of the GnRH secretion pattern. Finally, I will display and discuss the synchronization/desynchronization alternation of the cluster activities reproducing subtle observable features but biologically misunderstood.

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