Mathematical Modelling of Axon Guidance in Chemical Gradients

Correct wiring is crucial to the proper functioning of the nervous system. Key signals guiding growth cones, the motile tips of developing axons, to their targets are molecular gradients. The receptors on the membrane of the growth cone bind stochastically with guidance molecules in the environment, giving an estimate of the direction of the gradient. The growth cone then executes biased random movements to navigate toward the chemoattractant source or away from the chemorepellent source. This thesis addresses two questions: 1) how the positioning and diffusion of receptors on the growth cone membrane affect the accuracy of gradient estimation, 2) how trajectories are influenced by chemical gradients. Membrane receptors can diffuse freely on the membrane, smearing out the directional information about the gradient. It was unknown how the positioning and diffusion of receptors affect the estimate of the gradient. To address question (1) above, we utilise an ideal-observer approach, assuming that the growth cone can perform maximum likelihood estimation of the gradient based on the stochastic binding patterns with ligand molecules. The performance of gradient sensing is measured by the Fisher Information between the binding pattern and the gradient direction. The quality of gradient sensing decreases with higher diffusion constant of the receptors and improves with higher concentration. We then extend to a two-dimensional model of an elliptical growth cone with a general prior distribution. With a random uniform distribution of receptors, the shape of the growth cone can introduce bias in the gradient estimate. This bias can be corrected by a non-uniform distribution of receptors with higher density near the minor axis of the growth cone. Besides stochastic gradient sensing, growth cones also trace highly stochastic trajectories, and it is unclear how molecular gradients bias their movement. We then introduce a mathematical model of a correlated random walk based on persistence, bias and noise to describe growth cone trajectories, constrained directly by measurements of the detailed statistics of growth cone movements in both attractive and repulsive gradients in a microfluidic device. This model explains the long-standing mystery that average axon turning angles in gradients in vitro plateau very rapidly with time at relatively small values of 10-20◦. This work introduces the most accurate predictive model of growth cone trajectories to date, and calls into question the ability of molecular gradients alone to provide reliable guidance cues for growing axons.