Bayesian alignment of continuous molecular shapes using random fields

Statistical methodology is proposed for comparing molecular shapes. In order to account for the continuous nature of molecules, classical shape analysis methods are combined with techniques used for predicting random fields in spatial statistics. Applying a modification of Procrustes analysis, Bayesian inference is carried out using Markov chain Monte Carlo methods for the pairwise alignment of the resulting molecular fields. Superimposing entire fields rather than the configuration matrices of nuclear positions thereby solves the problem that there is usually no clear one--to--one correspondence between the atoms of the two molecules under consideration. Using a similar concept, we also propose an adaptation of the generalised Procrustes analysis algorithm for the simultaneous alignment of multiple molecular fields. The methodology is applied to a dataset of 31 steroid molecules.

Modelling and analysis of planar cell polarity

Planar cell polarity (PCP) occurs in the epithelia of many animals and can lead to the alignment of hairs, bristles and feathers; physiologically, it can organise ciliary beating. Here we present two approaches to modelling this phenomenon. The aim is to discover the basic mechanisms that drive PCP, while keeping the models mathematically tractable. We present a feedback and diffusion model, in which adjacent cell sides of neighbouring cells are coupled by a negative feedback loop and diffusion acts within the cell. This approach can give rise to polarity, but also to period two patterns. Polarisation arises via an instability provided a sufficiently strong feedback and sufficiently weak diffusion. Moreover, we discuss a conservative model in which proteins within a cell are redistributed depending on the amount of proteins in the neighbouring cells, coupled with intracellular diffusion. In this case polarity can arise from weakly polarised initial conditions or via a wave provided the diffusion is weak enough. Both models can overcome small anomalies in the initial conditions. Furthermore, the range of the effects of groups of cells with different properties than the surrounding cells depends on the strength of the initial global cue and the intracellular diffusion.