# MIRaW - Mathematical Interdisciplinary Research at Warwick

### Monday 11 May 2009

Biomembranes: Modelling and Computation

Organisers: Charlie Elliott (Warwick), Barbara Niethammer (Oxford), Matthew Turner (Warwick)

Biomembranes: Modelling and Computation

*(Joint with EPSRC Symposium on Challenges in Scientific Computing and Oxford Centre for Nonlinear PDE)*

## Abstracts and Talks

**Chaouqi Misbah** (Grenoble)* **Modelling of vesicles under flow*

We present an overview on vesicle modeling under flow. The theory is based either on sharp interafce models (making use of the boundary integral formulation) or on phase-field or level set (diffuse interface) approaches. We describe various dynamics and rheology captured by these models and compare their advantages and drawbacks. We briefly discuss another alternative for solving Navier-Stokes equations, namely "Lattice Boltzmann model", and briefly present its verstaility as another promizing tool for studying, for example, blood rheology.

**Paul O'Shea** (Nottingham)*Towards modeling complexity in biomembranes; distribution functions and inverse problem solutions*

This talk will be introduced with presentation using some video animations etc of membrane structure and dynamics.

A short outline of the properties of membranes will be used to define the main challenges posed to attaining a full understanding of the function of biological membranes. Inconsistencies and a lack of information in some key areas will be shown to compromise some types of model-building. With these problems in mind, our approches to building thermodynamic and molecular dynamic models will be described together with their attendant shortcomings. In particular, it will be shown that computational approaches based just simpley on the propertis of membrane lipids may not accommodate the observed properties of biological membranes. Finally, solution of an inverse problem based on density expansions of distribution functions and the Ornstein-Zernike equation may offer some insight into how both proteins and lipids interact as an ensemble leading to a self-consistent description of several observed membrane properties. Emphasis throughout this presentation will be on the corellation of modeling approaches with experimental measurements

**Ellen Reister-Gottfried** (Stuttgart)* Thermal membrane fluctuations: the impact on lateral protein diffusion and specific adhesion
*Various processes in a biological membrane depend on the membrane shape. However, due to thermal activation the membrane shape is not constant but fluctuates in the course of time. In my talk I will present two examples for which we have performed analytical calculations and continuum simulations to identify the influence of thermal membrane fluctuations: As the first example I will consider lateral diffusion of a protein in a fluctuating membrane. In our model the protein couples to the membrane curvature. We analytically determine the lateral diffusion coefficient in the limit of small temperature to bending rigidity ratios. The comparison with simulations with realistic parameters shows good agreement. The second example I will present is specific membrane adhesion via receptor-ligand pairs. Comparing continuum simulations for a flexible, fluctuating membrane with results for a stiff membrane, reveals that the binding energy for the flexible membrane needs to be larger to achieve the same degree of adhesion. However, the dynamics of the adhesion process is considerably faster for a fluctuating membrane. Both in equilibrium and during adhesion we find indications of a membrane mediated attractive force between neighbouring bonds.

**Björn Stinner** (Warwick)*Surface finite elements for membranes with lateral phase separation *

The lipid molecules which biomembranes consist of may separate into coexisting phases, and an energy contribution additional to the bending energy arises from the phase interfaces. To compute equilibrium membrane shapes we relax suitable initial shapes by a kind of gradient flow dynamics. Using the phase field methodology, the intermembrane domains are described by thin layers, and the dynamics consists of a parabolic differential equation on the membrane surface for the phase separation coupled to a geometric evolution law for the membrane itself. The discretisation is based on representing the membrane by a triangulated surface on which parametric finite elements are defined. The convergence as the thickness of the interfacial layers tends to zero has been numerically investigated as well as the influence of various physical parameters.

**Matthew Turner** (Warwick)*Non equilibrium fluid membranes*

Fluid membranes, particularly those found in living cells, are rarely at equilibrium. They are subject to applied forces and flows and can change their shape as a result. We review some recent experimental results on fluid membranes deformed by growing protein fibers and go on to discuss analytical approaches to this problem, as well as the effect of membrane geometry on the diffusion of embedded molecules. Finally we discuss the process of demixing of multi-component membranes and comment on some controversies surrounding the "lipid raft" hypothesis.