Growth cone gradient detection: a Bayesian analysis

Molecular gradients are one of the sources of information growing axons use to find their targets in the developing nervous system. Using a novel chemotaxis assay we have recently shown that axonal growth cones can detect differences as small as one molecule across their spatial extent (Rosoff et al, Nat. Neurosci, 7:678-682, 2004). How is this astonishing sensitivity achieved, given inevitable statistical fluctuations in receptor binding? We develop a Bayesian account of ideal gradient detection by small sensing devices. We first consider the case of a one-dimensional device which is trying to decide whether a locally uniform external gradient points left or right. The sensor does not know the steepness of the gradient or the baseline concentration. The optimal strategy in this case is to choose the direction which is most likely to have generated the observed binding statistics, by calculating the ratio R of these probabilities. Given simple prior distributions for gradient steepness and absolute concentration, we determine log(R), and also its average, , which determines the sensor's sensitivity. These quantities can be used to make specific predictions about both the behaviour of axons in gradients, and the computations that should optimally be performed by the intracellular signalling networks underlying growth cone chemotaxis. Conversely, they provide a way of determining how specific chemotaxing cells are limited by internal signalling constraints over and above the fundamental physical constraints of stochastic receptor binding. We are currently generalizing these calculations to two-dimensional sensing devices, and testing these predictions by direct comparison with experimental data from our chemotaxis assay