Elasticity and differential geometry of biological shapes

Ulrich Schwarz, Heidelberg University

Abstract:

Nature provides us with many examples of interesting and non-trivial shapes on all scales. Below the micrometer scale, we deal with biopolymers and biomembranes, which are in everlasting motion due to the thermal forces from their natural environment. On the scale of tens of micrometers, we encounter the manifold shapes taken by biological cells, for example the biconcave shape of a healthy red blood cells and the many strange shapes they can acquire under disease conditions like sickle cell anemia. On the scale of organisms, we encounter such surprising phenomena as the curling of hair or the rippling of plant leaves. In this lecture series, we will discuss some of these examples from the viewpoint of theoretical physics. In most cases, we will deal with one- or two-dimensional objects in three-dimensional space. We will introduce the appropriate concepts from differential geometry and elasticity theory, and then derive the corresponding shape equations using variational calculus. For polymers and membranes, we will discuss how to calculate the partition sum using path integrals. We might also discuss the effect of biological growth and numerical approaches based on the finite element method (FEM).

We will touch on the following general subjects from mathematics and physics:

differential geometry
classical field theory
continuum mechanics
elasticity theory
hydrodynamics
finite element method (FEM)
statistical physics
path integrals
calculus of variations
differential equations

Required background:

A basic knowledge in statistical physics is helpful

Literature:

J Jost, Differentialgeometrie und Minimalflächen, 1st edition, Springer 1994
Gerald Lim HW, Michael Wortis and Ranjan Mukhopadhyay, Red blood cell shapes and shape transformations, pages 83-254 in: G Gompper and M Schick, Soft matter volume 4: lipid bilayers and red blood cells, Wiley-VCH 2008
Leo van Hemmen, Theoretische Membranphysik: vom Formenreichtum der Vesikel, Script TU Munich 2001, , http://www.t35.physik.tu-muenchen.de/addons/publications/Hemmen-2001.pdf
U Seifert, Configurations of fluid membranes and vesicles, Advances in physics 46: 13-137, 1997, http://www.tandfonline.com/doi/abs/10.1080/00018739700101488#.VDK8TOI2408
D Nelson, S Weinberg and T Piran, Statistical mechanics of membranes and surfaces, World Scientific 2004
B Audoly and Y Pomeau, Elasticity and Geometry, Oxford University Press 2010
D Boal, Mechanics of the cell, 2nd edition, Cambridge University Press 2010
R Phillips and coworkers, Physical biology of the cell, 2nd edition, Taylor and Francis 2012