Discrete cilia modelling with singularity distributions

We discuss in detail techniques for modelling flows due to finite and infinite arrays of beating cilia. An efficient technique, based on concepts from previous ‘singularity models’ is described, that is accurate in both near and far-fields. Cilia are modelled as curved slender ellipsoidal bodies by distributing Stokeslet and potential source dipole singularities along their centrelines, leading to an integral equation that can be solved using a simple and efficient discretisation. The computed velocity on the cilium surface is found to compare favourably with the boundary condition. We then present results for two topics of current interest in biology. 1) We present the first theoretical results showing the mechanism by which rotating embryonic nodal cilia produce a leftward flow by a ‘posterior tilt,’ and track particle motion in an array of three simulated nodal cilia. We find that, contrary to recent suggestions, there is no continuous layer of negative fluid transport close to the ciliated boundary. The mean leftward particle transport is found to be just over 1 μm/s, within experimentally measured ranges. We also discuss the accuracy of models that represent the action of cilia by steady rotlet arrays, in particular, confirming the importance of image systems in the boundary in establishing the far-field fluid transport. Future modelling may lead to understanding of the mechanisms by which morphogen gradients or mechanosensing cilia convert a directional flow to asymmetric gene expression. 2) We develop a more complex and detailed model of flow patterns in the periciliary layer of the airway surface liquid. Our results confirm that shear flow of the mucous layer drives a significant volume of periciliary liquid in the direction of mucus transport even during the recovery stroke of the cilia. Finally, we discuss the advantages and disadvantages of the singularity technique and outline future theoretical and experimental developments required to apply this technique to various other biological problems, particularly in the reproductive system

Recapitulation of signals regulating embryonic bone formation during postnatal growth and in fracture repair

A number of proteins have recently been identified which play roles in regulating bone development. One important example is Indian hedgehog (Ihh) which is secreted by the prehyprtrophic chondrocytes. Ihh acts as an activator of a second secreted factor, parathyroid hormone-related protein (PTHrP), which, in turn, negatively regulates the rate of chondrocyte differentiation. Here we examine the expression of these genes and their molecular targets during different stages of bone development. In addition to regulating PTHrP expression in the perichondrium, we find evidence that Ihh may also act on the chondrocytes themselves at particular stages. As bone growth continues postnatally in mammals and the developmental process is reactivated during fracture repair, understanding the molecular basis regulating bone development is of medical relevance. We find that the same molecules that regulate embryonic endochondral ossification are also expressed during postnatal bone growth and fracture healing, suggesting that these processes are controlled by similar mechanisms