OCCAM Common Room (RI2.28)

In biological cells, molecules are transported actively or by diffusion and react with each other when they are close.

The reactions occur with certain probability and there are few molecules of some chemical species. Therefore, a stochastic model is more accurate compared to a deterministic, macroscopic model for the concentrations based on partial differential equations.

At the mesoscopic level, the domain is partitioned into voxels or compartments. The molecules may react with other molecules in the same voxel and move between voxels by diffusion or active transport. At a finer, microscopic level, each individual molecule is tracked, it moves by Brownian motion and reacts with other molecules according to the Smoluchowski equation. The accuracy and efficiency of the simulations are improved by coupling the two levels and only using the micro model when it is necessary for the accuracy or when a meso description is unknown.

Algorithms for simulations with the mesoscopic, microscopic and meso-micro models will be described and applied to systems in molecular biology in three space dimensions.

Cell locomotion is essential for early development, angiogenesis, tissue regeneration, the immune response, and wound healing in multicellular organisms, and plays a very deleterious role in cancer metastasis in humans. Locomotion involves the detection and transduction of extracellular chemical and mechanical signals, integration of the signals into an intracellular signal, and the spatio-temporal control of the intracellular biochemical and mechanical responses that lead to force generation, morphological changes and directed movement. While many single-celled organisms use flagella or cilia to swim, there are two basic modes of movement used by eukaryotic cells that lack such structures -- mesenchymal and amoeboid. The former, which can be characterized as `crawling' in fibroblasts or `gliding' in keratocytes, involves the extension of finger-like filopodia or pseudopodia and/or broad flat lamellipodia, whose protrusion is driven by actin polymerization at the leading edge. This mode dominates in cells such as fibroblasts when moving on a 2D substrate. In the amoeboid mode, which does not rely on strong adhesion, cells are more rounded and employ shape changes to move -- in effect 'jostling through the crowd' or `swimming'. Here force generation relies more heavily on actin bundles and on the control of myosin contractility. Leukocytes use this mode for movement through the extracellular matrix in the absence of adhesion sites, as does Dictyostelium discoideum when cells sort in the slug. However, recent experiments have shown that numerous cell types display enormous plasticity in locomotion in that they sense the mechanical properties of their environment and adjust the balance between the modes accordingly by altering the balance between parallel signal transduction pathways. Thus pure crawling and pure swimming are the extremes on a continuum of locomotion strategies, but many cells can sense their environment and use the most efficient strategy in a given context. We will discuss some of the mathematical and computational challenges that this diversity poses.

A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed.

This modelling approach incorporates recent experimental findings on behaviour of locusts. It exhibits nontrivial dynamics with a "phase change" behaviour and recovers the observed group directional switching. Estimates of the expected switching times, in terms of number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations with nonlocal and nonlinear right hand side is derived and analyzed. The existence of its solutions is proven and the systemʼs long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model when the number of individuals grows. This is a joint work with Radek Erban.

Would you like to solve a partial differential equation efficiently with a relative error of 10% or would you prefer to wait a bit longer and solve it with an error of only 1% ? Is it sufficient to know that the error is about 1% (having no idea what the `about' means) or would you prefer to have reliable information that the error is guaranteed to be below the required tolerance?

Answering these questions is necessary for the efficient and reliable numerical solution of practically any mathematical problem. In the context of numerical solution of partial differential equations, the crucial tool is the adaptive algorithm with suitable error indicators and estimators. I will overview the adaptive algorithm and its variants. I will concentrate on the a posteriori error estimators with the emphasis on the guaranteed ones.

Colonies of motile microorganisms, the cytoskeleton and its components, cells and tissues have much in common with soft condensed matter systems (i.e. liquid crystals, amphiphiles, colloids etc.), but also exhibit behaviors that do not appear in inanimate matter and that are crucial for biological functions.

These unique properties arise when the constituent particles are active: they consume energy from internal and external sources and dissipate it by moving through the medium they inhabit. In this talk I will give a brief introduction to the notion of "active matter" and present some recent results on the hydrodynamics of active nematics suspensions in two dimensions.

I describe recent work on the synchronization of IP3R calcium channels in the interior of cells. Hybrid  models of calcium release couple deterministic equations for diffusion and reactions of calcium ions to stochastic gating transitions of channels. I discuss the validity of such models as well as numerical methods.Hybrid models were used to simulate cooperative release events for clusters of channels. I show that for these so-called puffs the mixing assumption for reactants does not hold. Consequently, useful definitions of averaged calcium concentrations in the cluster are not obvious. Effective reaction kinetics can be derived, however, by separating concentrations for self-coupling of channels and coupling to different channels.

Based on the spatial approach, a Markovian model can be inferred, representing well calcium puffs in neuronal cells. I then describe further reduction of the stochastic model and the synchronization arising for small channel numbers. Finally, the effects of calcium binding proteins on duration of release is discussed.

• Benjamin Franz - "Hybrid modelling of individual movement and collective behaviour"
• Ingrid Von Glehn - "Image Inpainting on Surfaces"
• Rita Schlackow - "Genome-wide analysis of transcription termination regions in fission yeast"