Physical and Statistical Approaches to Biological Shape

This is my next major research focus, which I will be starting in earnest when I start working with Prof Mahadevan in October. We are interested in several problems in this area, but we are likely to start by focussing on the folding of the human brain.

The diversity of shapes found in biology is truly awe inspiring. Evolution has produced complex patterns and shapes, ranging in size from the molecular double spiral of DNA to the colossal branched structure of giant redwood trees. Biology is, to borrow the title of Richard Dawkins' latest book, "the greatest show on earth". However, to satisfactorily answer the question "why is it that shape?", which is surely one of the most fundamental questions about any biological object, one must understand much more than the mechanism of evolution. One must also understand how the shape actually grows, and why the shape in question is well suited to its environment. The answers to these biological questions often lie in physics and maths. I plan to look at the shape of biological organs, focussing on the heart, the brain and the lung. The heart is a tubular structure with valves and walls, the lung is a tree-like branched organ, and much of the brain is a highly folded sheet; in each case, geometry and physics come together with biology to form a functional unit. I will attempt to explain the origin of these shapes using models that combine the physics of elasticity with biological models of tissue growth. For example, the familiar folded structure of the brain is somewhat reminiscent of the long vertical folds in an open pair of curtains, so describing how the brain forms may be much like describing how sheets wrinkle.

Successfully describing the formation of any of the above organs could provide a tangible medical benefit. Understanding how the organ forms will provide insight into how anatomical defects arise, and a better understanding of what a normal organ looks like might also allow defects to be diagnosed from medical images earlier and more easily, giving more scope for treatment. In the case of the brain, there are some tantalizing preliminary suggestions that certain patterns in the folds are found in patients with mental disorders, and a fuller understanding of how the folded pattern forms might even allow these disorders to be diagnosed in an MRI scanner.

Cell Division and Specialization

For a fertilized egg to turn into a fully-fledged organism requires many iterations of cell division and specialization, and this process continues later in life in stem-cell maintained tissues such as skin. Failure of these processes can result in cancer. Biologists have expended a great deal of effort trying to understand these fundamental processes from a ``bottom up" perspective, attempting to identify regulatory gene networks and other signaling mechanisms. This approach has enjoyed some success but is consistently frustrated by the extreme complexity of the low level mechanisms. Condensed matter physics is full of sets of systems that exhibit very similar macroscopic behavior despite different microscopic details, and such behaviors are often best described by a ``top-down" approach, not asking what behavior a given set of microscopic rules give, but what constraints on the microscopic rules different behaviors require. I am attempting to use this philosophy to identify the macroscopically important features of cell division and differentiation rules.

This work is in collaboration with Ben Simons and several experimental biologists. Our approach is rooted in ``clone-fate'' data sets. These are from experiments in which a mother cell is marked, and then the number of its surviving daughter cells --- which form the ``clone'' --- is recorded at later times. At them moment I am working with data from a brain cancer tissue and a mutated skin system in which cell differentiation is severely inhibited. I am also working on analogous data (provided by Shankar Srinivas) from very early term mouse embryos (8-50 cells), attempting to ascertain when and why the very first cell specialisations are made.

Liquid Crystal Elastomers

The exotic elasticity of liquid crystal elastomers was the focus of my PhD thesis, which was supervised by Mark Warner.

Liquid crystal elastomers (LCEs) are remarkable materials that combine the mobile orientational order of liquid crystals with the extreme stretchiness of rubber. LCEs are mixtures of rigid liquid-crystal rods and long flexible polymer chains that have been chemically cross-linked to form a rubber network. When an LCE is hot the rods point in random directions and the material behaves like a traditional isotropic rubber, but when the elastomer is cooled, the rods align, and the elastomer stretches by up to several times its own length in the alignment direction. Since the hot state is isotropic, on cooling any direction can be chosen for the rods to align in, and hence there are many equivalent cold states each with a different alignment and a different stretch with respect to the hot state. Stretches that turn one such aligned cold state into another different but equivalent one cannot cost energy --- they are ``soft'' deformations. Soft deformations are common in fluids --- we usually say the fluid is flowing --- but very unusual in a solid. Indeed we can fairly say the resultant material is somewhere between a solid and a liquid. Much more information about LCE's can be found at www.lcelastomer.org.uk.

My research into LCEs has focused on two main topics. Firstly I have studied the limitations of the above symmetry argument for completely soft deformations of monodomain LCEs. The essential issue is that, if the hot state is truly isotropic, then, on cooling, the liquid crystal rods will choose different directions to align at different points in the sample, so the cold aligned state will be have a polydomain structure and there will be no overall deformation. The rods can be made to all align in the same direction by imprinting a direction on the isotropic state, producing a monodomain elastomer which does change length considerably on cooling. However, in this case the hot state is not truly isotropic so the argument for the existence of soft modes is compromised. I have used phenomenological arguments to show that the resulting monodomain will still show qualitatively soft behavior, and, at some stretches, the modulus for additional extension will vanish altogether. I have also shown that chiral nematic monodomains will exhibit strain-induced electrical polarization, unlike nematic liquids or conventional elastomers.

My second focus has been the polydomain LCEs that form in the absence of imprinting. I have shown that, if these systems are prepared correctly, they will exhibit elasticity that is softer than the more fashionable monodomains. Since these samples are also much easier to produce, this could have technological implications. I collaborated closely with Kaushik Bhattacharya on this work.