# Tuesday

09:00 |
Registration | |

9:15 |
Welcome | |

9:30-10:20 |
Dembo | Ideas about the driving force for cell motion: protrusion and intrusion of the membrane |

10:20-10:50 |
Coffee Break | |

10:50-11:40 |
Oliver | Free boundary value problems related to cell motility |

11:40-12:30 |
Primi | A simple 2D-model of cell sorting induced by propagation of chemical signals along spiral waves |

12:30-14:30 |
Lunch | |

14:30-15:20 |
Doumic | Mathematical Models for Prion proliferation |

15:20-16:10 |
Baake | Ancestral processes for mutation-selection models of population genetics |

16:10-16:40 |
Coffee Break | |

16:40-17:30 |
Calvez | The geometry of one-dimensional self-attracting particles |

# Wednesday

9:00-9:50 |
Ben Ama | |

9:50-10:40 |
Röger | Self-assembling of amphiphiles |

10:40-11:10 |
Coffee Break | |

11:10-12:00 |
Witterstein | Compressible phase change flow (An application to stem cells) |

12:00-14:00 |
Lunch | |

14:00-14:50 |
Peruani | Clustering, pattern formation, and fluctuations in myxobacterial colonies |

14:50-15:40 |
Horstmann | Models for aggregation in population dynamics and their relations to signal enhancement processes and field theory |

15:40-16:10 |
Coffee Break | |

16:10-17:30 |
Luckhaus | Measure valued solutions to the degenerate Keller Segel system |

# Thursday

9:00-9:50 |
Stevens | Alignment as an example of selection in stage structured population models |

9:50-10:40 |
Marciniak-Czochra | Measure-valued solutions of a structured population model |

10:40-11:10 |
Coffee Break | |

11:10-12:00 |
Huisinga | Stochastic modelling in HIV disease |

12:00-14:00 |
Lunch | |

14:00-14:50 |
Champagnat | Quasi-stationary distributions for nearly neutral birth and death processes in dimension 2 |

14:50-15:40 |
Velásquez | Pattern formation for kinetic equations |

15:40-16:10 |
Coffee Break | |

# Abstracts

**Ellen Baake: Ancestral processes for mutation-selection models of population genetics**

We consider both deterministic and stochastic models that describe the genetics of populations under the joint action of mutation and selection. To this end, mutation and reproduction are modelled in terms of a mutation-selection differential equation, a multi-type branching process, and in terms of a Moran model with mutation and selection; in each case, both the forward and the backward directions of time are analyzed. The ancestral processes turn out as keys for the understanding of the interplay of mutation and selection. For the differential equation and the branching model, it leads to a general variational principle that relates the present population, the ancestral population, and the mean growth rate. In some important cases, this maximum principle boils down to a low-dimensional problem that can be solved explicitly. In the Moran model, a simplified version of the ancestral selection graph is used to reveal the additional effects of genetic drift, which cannot be captured by the deterministic or the branching approach.

**Vincent Calvez: The geometry of one-dimensional self-attracting particles**

Self-attracting particles can be modelled using a nonlinear Fokker-Planck equation, including a mean field term for the attracting potential. We focus here on the Keller-Segel model for biological cells (or the Smoluchowski-Poisson equation in astrophysics). This model has raised a lot of interest in the field of mathematical biology since it captures the critical mass phenomenon in a very simple manner: the number of cells determines whether some spatial structure emerges or not at the population level. This model 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 convex and concave contributions. Despite this lack of convexity we shall exhibit some underlying structure which makes this model somehow close to the gradient flow of a convex functional. We present in this talk some analysis for a one-parameter family of Keller-Segel models in one dimension of space: We show how the system’s behaviour (finite time blow-up, long-time asymptotics) can be simply re-interpreted in the gradient flow framework. A suitable numerical scheme and its consequences are also discussed. This is joint work with José Antonio Carrillo (ICREA, Univ. Autonoma Barcelona).

**Nicolas Champagnat: Quasi-stationary distributions for nearly neutral birth and death processes in dimension 2**

This is joint work with P. Diaconis (Stanford) and L. Miclo (Toulouse).

Birth and death (B&D) processes in dimension two are very natural to model the stochastic competition between two finite sub-populations in interaction. Such a process is called "neutral" when the two sub-populations are indistinguishable (none of them has a selective advantage with respect to the other). We restrict ourselves to finite, Markov processes. First, we show that the eigenvectors of the transition matrix of a neutral B&D process can be decomposed into two components, one independent of the precise dynamics, and the other depending of the population dynamics only through the total population size. We then explain the link with Dirichlet eigenvalues in sub-domains, which allows us to order the eigenvalues. These results are then applied to the study of quasi-stationarity for two-dimensional B&D processes that are nearly neutral. A distribution is called quasi-stationary if it is invariant conditionally on non-extinction. In particular, the limit distribution conditionally on non-extinction (sometimes called Yaglom limit) is characterized, and computed in the neighborhood of neutrality. Other applications of our results on the probability of fixation in nearly neutral populations will also be discussed.

**Micah Dembo: Ideas about the driving force for cell motion: protrusion and intrusion of the membrane**

We will discuss recent improvements in our understanding of how the forces that govern amoeboid motility are generated and controled, how such cell motility can be modeled in terms of field equations and how such equations can be solved numerically. These ideas will be used to clarify the basis for structure-function correlations in cell motility (why cells that move in a certain ways tend to have certain structural properties). This is facilitated by application of a new computational technique that allows realistic 3D simulations of cells migrating on flat substrates. With this approach and very minimal assumptions, some "cyber" cells spontaneously display the classic irregular protrusion cycles and hand-mirror morphology of a crawling fibroblast and others the steady gilding motility and crescent morphology of a fish Keratocyte. The kertocyte motif is caused by optimal recycling of the cytoskeleton from the back to the front so that more of the periphery can be devoted to protrusion. The calculations presented represent an important step toward bridging the gap between the integrated mechanics and biophysics of whole cells and the microscopic molecular biology of cytoskeletal components.

**Marie Doumic: Mathematical Models for Prion proliferation**

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.

**Dirk Horstmann: Models for aggregation in population dynamics and their relations to signal enhancement processes and field theory**

Modeling and simulating aggregation of mobile species as well as analyzing the resulting mathematical models is a very popular topic in Mathematical Biology. On the other hand some of these models like the "Keller-Segel type" chemotaxis models are quite similar to certain semiconductor models. Furthermore, their steady state analysis leads to nonlocal elliptic boundary value problems that are also related to field theory. Other models for aggregation in population dynamics are related to well-posed filters in signal enhancement processes. Since one aim of this workshop is to find out possible interactions between mathematics, physics and biology, this talk will recall some results that point out the connection between some aggregation models and one dimensional models in signal enhancement processes. Furthermore, it will present some new results for a class of parameter-dependent, nonlocal elliptic boundary value problems in a disk that corresponds to the steady state problem for some chemotaxis systems. If the appearing parameter is less than an explicit critical value, several uniqueness results will be established for a class of solutions that are invariant under the action of a Dihedral group. Finally, the consequences for the time asymptotic behavior of the solutions to the time dependent chemotaxis systems will be discussed.

**Anna Marciniak-Czochra: Measure-valued solutions of a structured population model**

Mathematical modelling of spread, growth and functional changes of cell populations demands taking into account biological and mechanical properties of individuals, such as number of particular receptors on the cell membranes, level of expression of some proteins, stage of cell differentiation or length of polymers. Time evolution of a heterogeneous population parametrised by the dynamically regulated properties of individuals can be described by so called structured population models, which are first order hyperbolic equations defined on R+.

In this talk a framework for the analysis of measure-valued solutions of the nonlinear structured population model is presented. Existence and Lipschitz dependence of the solutions on the model parameters and initial data are shown by proving convergence of a variational approximation scheme, de ned in the terms of a suitable metric space. The estimates for a corresponding linear model are obtained based on the duality formula for transport equations. An extension of a Wasserstein metric to the measures with integrable first moment is proposed to cope with the nonconservative character of the model. This metric is compared with a bounded Lipschitz distance, called also at metric.

The results are discussed in the context of applications to biological data. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data.

**Fernando Peruani: Clustering, pattern formation, and fluctuations in myxobacterial colonies**

Myxobacteria, as many other bacteria, exhibit a transition from unicellularity to multicellularity when the level of nutrients is low. This fascinating process starts with the onset of clustering and collective motion. In contrast to Dictyostelium discoideum and other microorganisms, myxobacteria aggregate and coordinate their motion, in this early stage, without making use of diffusing chemical signals. We show through experiments with the mutant A+S-Frz- of M. xanthus, as well as through theoretical models, that is the active motion of the cells plus their rod-like shape what presumably allows cells to exhibit such collective effects. We argue that the combination of an anisotropic cell shape and the bacterial self-propulsion leads to a primitive effective alignment mechanism. Provided the cell density is above a given threshold, a transition to clustering occurs. The cluster size statistics from experimental data can be reproduced by a simple model for self-propelled rods. Moreover, we indicate that several features of myxobacterial collective behavior are generic properties of self-propelled particle systems, in particular, clustering and giant fluctuations. However, the emerging (asymptotic) spatial patterns are strongly dependent on the particular symmetry of the myxobacterial cell-cell interactions, namely, apolar interactions. We show that such interactions can cause the formation of high density regions along which cells migrate, exhibiting a high degree of (nematic) order.

**Matthias Röger: Self-assembling of amphiphiles**

We present a model for the self-assembling property of amphiphilic molecules. Starting from a micro-scale description we derive an energy on a continuum level and show by a variational analysis that this energy prefers localized structures of a definite thickness. In a certain rescaling this preferred thickness vanishes and we obtain a classical bending energy.

**Gabriele Witterstein: Compressible phase change flow (An application to stem cells)**

In this talk we present diffusive interface models describing flows which undergo a phase change, for example caused by biochemical reactions in the material flow. In mathematical terms, the equations consist of the compressible Navier-Stokes system coupled with an Allen-Cahn equation, and are based on an energetic variational formulation. We investigate density-dependent viscosity and a density-dependent transition layer of thickness ? (?). We establish the existence of a unique weak global solution in one dimension. Further we consider two diffusive interface models by choosing different relaxation parameters. This leads to two completely different interface models. In the first one, the characteristics cross the interface, and in the second one, they do not.