Is Axon Guidance by Attraction and Repulsion, or by a Roll of the Dice?

Attractants and repellants guide axons to their targets.  On its journey, a migrating axon may be confronted with multiple attractive and repulsive guidance cues.  This presents a conundrum. How does the axon avoid a tug-of-war between attractants and repellants?  Does the strongest cue win?  Can one cue negate the effects of another?  Can an axon switch its responsiveness to cues until they all match?  

Our study suggests that the key to understanding this problem may lie within the realm of probability theory.

The theory behind combinatorial regulation of axon guidance

In previous posts I presented the hypothesis that the direction of axon guidance is not determined by directional responses to the guidance cues, but rather by the succession of randomly directed movement.  (See my post  "Random Walks, the Brain Initiative, and the Genius of Einstein's Brain".) 

The response to guidance cues is a random walk.  (See my post "Axon Guidance Meets Statistical Physics".) At each discrete time, the guidance cues regulate the probability of axon outgrowth in each direction.  In our study, we scored axon outgrowth from a neuron in four directions, thereby creating the probability distribution:

direction, X	probability
ventral	P(X = ventral)
anterior	P(X = ventral)
posterior	P(X = ventral)
dorsal	P(X = ventral)

  

In general, the probability distribution of variable X, the direction of outgrowth, is

It satisfies the following condition:

This just says that if all the probabilities for all the possible directions are added together the sum must equal 1.  

This little bit of probability theory is simple, but has profound implications.  It means that a guidance cue must affect the probability of outgrowth in more than one direction.  Since the sum of all the probabilities must equal 1, if a cue increases or decreases the probability of outgrowth in one direction it must alter the probability of outgrowth in another direction(s) as well.

The direction of guidance is determined by combinatorial regulation.  The directional effect of a cue depends on the effects that all the other cues combined have on the probabilities. 

The combinatorial regulation of axon guidance in practice

We can derive a probability distribution for the outgrowth of the axon in different mutants. From these probability distributions we can simulate a simple random walk.  The results give us a picture of how all the guidance cues collectively set the probabilities for outgrowth in each direction when the axon extended from the neurons's cell body. (See my post "A Random Walk into the Genetics of Axon Guidance".)  The simulations show the directional bias for the movement of the axon as it first extended.  By comparing simulations based on the probability distribution from different mutants we can discover how different guidance molecules act together to determine a directional bias.  

For each mutant 10 random walk simulations were plotted.  The results indicate the relative directional bias that is created by single and double mutations. 

In this illustration the directional bias created by the unc-6 mutant (blue), by the egl-20 mutant (green), and by the egl-20;unc-6 double mutant (aqua marine) are each different.  Therefore, the directional bias caused by the loss of UNC-6 requires EGL-20 and the directional bias caused by the loss of EGL-20 requires UNC-6. 

The illustration also shows that a cue can be required for guidance in different directions, depending on the presence of another cue or receptor.   UNC-6 is required for the ventral (down) directional bias observed in wildtype, since when UNC-6 function is missing in the unc-6 mutant (blue) the direction is anterior (left).  But UNC-6 is also required for a posterior (right) directional bias when SAX-3 is missing since there is a posterior directional bias in the sax-3 mutant (red), but not in the sax-3;unc-6 double mutant (magenta). Similarly, we can deduce from the single and double mutants that EGL-20 promotes the ventral bias in wild-type animals, the posterior bias in sax-3 mutants, and the anterior bias in unc-6 mutants.  

Why probability theory sweeps aside attraction and repulsion

Credit: Michael Heiss

Guidance cues induce or inhibit axon outgrowth.  A tug-of-war between attractive and repulsive outgrowth signals may occur when there are attractive and repellent cues.  

In our probabilistic approach, we describe all of these molecular activities in aggregate.  Attraction and repulsion are occurring at the molecular level,  but these events are rapidly fluctuating and their effects are unpredictable.  The directional outcome of these underlying events is observed at the macroscopic level of axon guidance.  Because the movement of the axon can be considered as a stochastic process,  probability theory can be used to describe axon guidance.    

It can be argued that describing axon guidance probabilistically without considering the underlying causal deterministic explanation is not satisfactory.  But systems that obey precise laws can and do behave in a random way.  The probabilistic approach may change our understanding about the practical effects of guidance cues in vivo.  For example, although guidance cues might act as attractants and repellants for axon outgrowth they do not necessarily act as attractants and repellants for the macroscopic movements of axon guidance. This has implications for understanding nervous system development and the possible therapeutic use of guidance cues to treat diseases and injury.

Credit: Curtis Perry

Although Albert Einstein thought about the nature of space-time in a statistical manner, he had a desire to understand the underlying causes that lead to the statistical approach.  He was dismayed when his friend Max Born proposed that quantum mechanics were to be understood as a probability without any causal explanation.   He wrote to Born, "The theory says a lot, but does not really bring us any closer to the secret of the Old One. I, at any rate, am convinced that He does not throw dice".

Reference:

Tang, X., & Wadsworth, W. (2014). SAX-3 (Robo) and UNC-40 (DCC) Regulate a Directional Bias for Axon Guidance in Response to Multiple Extracellular Cues PLoS ONE, 9 (10) DOI: 10.1371/journal.pone.0110031