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.

Fig. 1.

Bayesian model of spatial gradient detection. (A) Model growth cone (for more detail see Fig. S1). Receptors on the surface bind ligand molecules probabilistically according to standard Michaelis–Menten kinetics. Signals from the bound receptors are then combined in the growth cone to optimally decide the most consistent gradient direction for that pattern of ligand binding. Although there are several intuitively obvious decision rules the growth cone could employ, determining the provably optimal strategy is nontrivial. We show that the optimal rule is to weight the signal from each bound receptor by its distance from the center of the growth cone (see SI Text). The sizes of the growth cones in the bubble represent the degree of belief of the growth cone in the 2 hypotheses for gradient direction based on that particular pattern of receptor binding. (B) Chemotactic sensitivity curves calculated analytically for the optimal decision rule for the gradient parameters used experimentally. The percentages refer to the fractional change in concentration across 10 μm. We set K_D = 0.3 nM based on the best fit between the model and the data (see Fig. 3).

Fig. 2.

Response of DRG explants to precisely controlled gradients of NGF. (A) Representative explants illustrating different guidance ratios (gradient is increasing upwards). (Scale bar, 400 μm.) (B) Higher-powered image of a typical subfield of neurites growing across the gradient (increasing upwards). (Scale bar, 250 μm.) (C) Explant asymmetry (guidance ratio) as a function of absolute concentration and gradient steepness (see Methods; n and values given in Table S1). Note that each curve peaked at approximately the same concentration, response dropped off faster at higher concentrations than lower concentrations, curve width increased with gradient steepness, and peak height tended to increase with steepness. Error bars are SEMs.

Fig. 3.

Match between model and data. Measured guidance ratio plotted against the signal-to-noise formula predicted by the model. Error bars are SEMs. The red line is a linear fit (Pearson's r = 0.90). For the dependence of the fit on K_D see Fig. S5B.