Elucidating the interactions between the adhesive and transcriptioanl functions of beta-catenin in normal and cancerous cells

Wnt signalling is involved in a wide range of physiological and pathological processes. The presence of an extracellular Wnt stimulus induces cytoplasmic stabilisation and nuclear translocation of beta-catenin, a protein that also plays an essential role in cadherin-mediated adhesion. Two main hypotheses have been proposed concerning the balance between beta-catenin's adhesive and transcriptional functions: either beta-catenin's fate is determined by competition between its binding partners, or Wnt induces folding of beta-catenin into a conformation allocated preferentially to transcription. The experimental data supporting each hypotheses remain inconclusive. In this paper we present a new mathematical model of the Wnt pathway that incorporates beta-catenin's dual function. We use this model to carry out a series of in silico experiments and compare the behaviour of systems governed by each hypothesis. Our analytical results and model simulations provide further insight into the current understanding of Wnt signalling and, in particular, reveal differences in the response of the two modes of interaction between adhesion and signalling in certain in silico settings. We also exploit our model to investigate the impact of the mutations most commonly observed in human colorectal cancer. Simulations show that the amount of functional APC required to maintain a normal phenotype increases with increasing strength of the Wnt signal, a result which illustrates that the environment can substantially influence both tumour initiation and phenotype.

Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

Gap junctions and emergent rhythms

Gap junction coupling is ubiquitous in the brain, particularly between the dendritic trees of inhibitory interneurons. Such direct non-synaptic interaction allows for direct electrical communication between cells. Unlike spike-time driven synaptic neural network models, which are event based, any model with gap junctions must necessarily involve a single neuron model that can represent the shape of an action potential. Indeed, not only do neurons communicating via gaps feel super-threshold spikes, but they also experience, and respond to, sub-threshold voltage signals. In this chapter we show that the so-called absolute integrate-and-fire model is ideally suited to such studies. At the single neuron level voltage traces for the model may be obtained in closed form, and are shown to mimic those of fast-spiking inhibitory neurons. Interestingly in the presence of a slow spike adaptation current the model is shown to support periodic bursting oscillations. For both tonic and bursting modes the phase response curve can be calculated in closed form. At the network level we focus on global gap junction coupling and show how to analyze the asynchronous firing state in large networks. Importantly, we are able to determine the emergence of non-trivial network rhythms due to strong coupling instabilities. To illustrate the use of our theoretical techniques (particularly the phase-density formalism used to determine stability) we focus on a spike adaptation induced transition from asynchronous tonic activity to synchronous bursting in a gap-junction coupled network.