Master defense by Bjarke Todbjerg Nielsen

Non-Relativistic Submanifolds and Fluid Dynamics - With Applications to Biological Membranes

In this thesis a theory of submanifolds in Newton-Cartan geometry is developed. Using this theory we build a framework for non-relativistic hydroelastic theories by formulating an action which is sensitive to both hydrodynamic degrees of freedom and to deformation, via the extrinsic curvature. Within this novel framework we formulate concrete membrane models with and without a dynamical fluid. These models are inspired by the Canham-Helfrich model[1] and are proposed as a description of lipid membranes formed by the spontaneous aggregation of amphiphilic molecules into a fluid bilayer.

We show that the static model reproduces known solutions of biophysical relevance [2], namely spherical, toroidal and biconcave discoid (i.e. erythrocyte-like) geometries. For the toroidal case the model gives the specific ratio R/r = sqrt(2) between major and minor radii which was also obtained by e.g. [2] and is known to be in excellent agreement with experiments [3].

We also propose certain models of lipid bilayers which include the motion of the fluid by allowing for fluid velocity-dependent scalars in the partition function and by assuming the flow to be stationary, meaning that it is directed along an isometry of the geometry. We find a solution to such a model given by a spherical vesicle with a stationary, azimuthal fluid flow.

[1] W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments,” Z. Naturforsch. C 28 (1973), no. 11, 693–703.

[2] Z. C. Tu and Z. C. Ou-Yang, “Recent theoretical advances in elasticity of membranes following Helfrich's spontaneous curvature model,” arXiv:1405.0651.

[3] M. Mutz and D. Bensimon, “Observation of toroidal vesicles,” Phys. Rev. A 43 (Apr, 1991) 4525–4527.