Mathematical characterisation of the transduction chain in growth cone pathfinding

Axon guidance by graded diffusible ligands plays an important role in the developing nervous system. Concentration gradients induce an asymmetric localisation of molecules in the axon tip, the growth cone, and the consequent internal polarised signalling pathway leads to rearrangement of the growth cone cytoskeleton and, ultimately, to motility. Here the authors provide a mathematical description of the growth cone transduction chain as a series of functional boxes characterised by input/output relations. The model relies on the assumption that the characteristic time of independent concentration measures by growth cone receptors, the characteristic time of growth cone internal reorganisation preceding motion and the characteristic time needed for a discernible axon turning belong to separated scales. The results give insight into the deterministic against stochastic regime of internal growth cone functions that are not readily accessible from experimental observations, pointing out a substantial equilibrium of the two contributions. The mathematical model predicts the decrease of the coefficient of variation of the signal moving down the functional chain leading to motion. Moreover, possible mechanisms that allow for buffering against noise are highlighted. These results have an interest also for the more experimentally minded reader, since they can be used to predict sample sizes for detecting significant differences in benchmark gradient assays.