A Bayesian model predicts the response of axons to molecular gradients

Abstract

Axon guidance by molecular gradients plays a crucial role in wiring up the nervous system. However, the mechanisms axons use to detect gradients are largely unknown. We first develop a Bayesian ''ideal observer'' analysis of gradient detection by axons, based on the hypothesis that a principal constraint on gradient detection is intrinsic receptor binding noise. Second, from this model, we derive an equation predicting how the degree of response of an axon to a gradient should vary with gradient steepness and absolute concentration. Third, we confirm this prediction quantitatively by performing the first systematic experimental analysis of how axonal response varies with both these quantities. These experiments demonstrate a degree of sensitivity much higher than previously reported for any chemotacting system. Together, these results reveal both the quantitative constraints that must be satisfied for effective axonal guidance and the computational principles that may be used by the underlying signal transduction pathways, and allow predictions for the degree of response of axons to gradients in a wide variety of in vivo and in vitro settings.axon guidance ͉ chemotaxis ͉ growth cone ͉ nerve growth factor ͉ nerve regeneration E ndogenous chemical gradients are a key source of information used by developing axons when wiring up the nervous system. Furthermore, artificially generated gradients are a potential therapy for restoring connectivity after neural injury. Many of the molecular gradients that direct axons in the developing nervous system have recently been identified, together with some of the signaling pathways through which they operate (1-8). However, our understanding of the mechanisms by which axons actually detect gradients remains qualitative. This limits our ability to predict both the response of axons when gradients are perturbed and the optimal parameters for promoting regrowth after injury.To be guided by a gradient, axons must be able to detect small spatial variations in receptor binding. This requires integrating signals from spatially distributed receptors to make a decision as to the direction of the gradient. Resources within the growth cone can then be marshaled appropriately by this information, for instance, via the production of steep gradients of intracellular signaling molecules (8). Although there is evidence for a role for gradients of molecules such as Ca 2ϩ in this latter step (9, 10), very little is known about the computations required to accurately make the initial decision.What constrains the ability of an axon to make a decision regarding gradient direction? Both experimental and computational work addressing chemotaxis in related systems such as bacteria, leukocytes, and Dictyostelium has identified the fundamental role of noise in limiting gradient perception. Noise can arise from low numbers of ligand molecules, from the stochastic nature of receptor binding, and from intracellular signaling pathways (11-16). Such constraints must also apply to axonal gradient sensing (17...

Analyzing neurite outgrowth from explants by fitting ellipses

“…Examples of the resulting explant images are shown in Fig. 1; for detailed experimental methods see Mortimer et al (2009). As confirmed by confocal imaging (data not shown) most of the neurite outgrowth was concentrated in roughly a two-dimensional plane, and thus the two-dimensional epifluorescence imaging was sufficient to capture most of the information about neurite outgrowth.…”

“…As described in Mortimer et al (2009), images were first thresholded in intensity to produce a binary image. The pixels representing the explant body were then removed, leaving just pixels representing neurites.…”

“…To compare with the results of the ellipse-fitting we used the "total outgrowth" and "guidance ratio" values previously calculated for this dataset in Mortimer et al (2009). First, each thresholded image was divided into an up-gradient half and a down-gradient half by a line perpendicular to the gradient direction passing through the centroid of the explant body.…”

“…That is, are larger explants simply expanded versions of smaller explants, or is there a systematic change in shape with size? In previous work, explant size was characterized by summing all pixels representing neurites to give the neurite outgrowth Mortimer et al (2009). We therefore plotted each ellipse parameter against this outgrowth.…”

“…Up to now this analysis has usually involved the extraction of only one or two parameters, such as total neurite pixels and the ratio of pixels representing neurite growth from one side or region of the explant compared to the other (e.g. Bilsland et al, 1999;Caton et al, 2000;Rosoff et al, 2004;Mortimer et al, 2009). Although such measures can be robust and informative for addressing certain questions, they are clearly a rather crude characterization of the overall pattern of neurite outgrowth from explants.…”