Axon Guidance Meets Statistical Physics

The proposition that the response of an axon to guidance cues is a random walk provides a different perspective of axon guidance.

For the most part, Biologists like deterministic models, i.e. cause and effect.  From the deterministic point-of-view, axon guidance is caused when axon outgrowth activity occurs at the site where the neuron detects an external attractive guidance cue. 

But what if the direction of axon outgrowth activity were to rapidly fluctuate in different directions?  

In this case, it could be very difficult to measure how each outgrowth contributes to the direction of guidance.  This would be like trying to understand the direction of Brownian particle movement by measuring the effect of each impact of each liquid molecule with the particle. 

Random walks are a tool of statistical physics. A random walk can describe the macroscopic movement of an object that is influenced by numerous unpredictable events.  It is a probabilistic examination of the underlying events.  For example, temperature, heat, and entropy are a way of looking at the practical properties of the molecules in a system.  These are concepts that describe the behavior of the molecules in a large system. They clarify what we observe in practice.  The main purpose of statistical physics is to explain the properties of matter in aggregate, as dictated by the physical laws governing atomic motion.

Why is the perspective important?

From the deterministic point-of-view, we can better understand axon guidance by knowing how guidance molecules cause axon outgrowth.  Currently in the field of axon guidance research there is a strong emphasis on finding the molecular mechanisms that cause an attractive or repulsive response in the neuron.  The logic is that the type of outgrowth response determines the direction of guidance. 

The stochastic point-of-view shifts the emphasis.  The direction of axon guidance is not determined by the direction of axon outgrowth but rather by the succession of randomly directed movement.  Its mathematical, not biological! The question is,  how do the biological processes regulate the probability of outgrowth in different directions.    

Why we can propose that the response of an axon to guidance cues is a random walk. 

First, a few definitions: 

1. Stochastic Process - a collection of random variables
2. Random Variable - a variable that can take on a set of different possible values
3. Probability Distribution - possible values of a random variable and their associated probabilities
4. Random Walk - a mathematical formalization of a path that consists of a succession of random steps

Here is a picture of the same neuron in different animals.  In response to the extracellular guidance cues in the surrounding environment, the axon can grow out from the neuron's cell body in different directions; ventral (down), anterior (left), posterior (right) or dorsal (up).  

The direction of outgrowth is a variable, X.  If we pick an animal from the population, there is a probability that we will observe the axon growing out in either the ventral, anterior, posterior, or dorsal direction.  

We now have a probability distribution: 

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

The axon grows out over a period of time and you can always chop a period of time into shorter time intervals.  Now it can be deduced that the axon outgrowth takes place through a series of steps where the direction for each step is chosen at random according to the probability distribution.

One step at one discrete time where the axon has a probability of going in one direction (left).  A series of steps over time (right). 

This is now a stochastic sequence {Sn} defined by:

This is a simple random walk!  

Note that we don't "see" any random walks; we only suspect that it must occur.   We need evidence that the guidance response behaves as a random walk. 

References:

Kulkarni, G., Xu, Z., Mohamed, A., Li, H., Tang, X., Limerick, G., & Wadsworth, W. (2013). Experimental evidence for UNC-6 (netrin) axon guidance by stochastic fluctuations of intracellular UNC-40 (DCC) outgrowth activity Biology Open, 2 (12), 1300-1312 DOI: 10.1242/bio.20136346

Yang, Y., Lee, W., Tang, X., & Wadsworth, W. (2014). Extracellular Matrix Regulates UNC-6 (Netrin) Axon Guidance by Controlling the Direction of Intracellular UNC-40 (DCC) Outgrowth Activity PLoS ONE, 9 (5) DOI: 10.1371/journal.pone.0097258

Xu, Z., Li, H., & Wadsworth, W. (2009). The Roles of Multiple UNC-40 (DCC) Receptor-Mediated Signals in Determining Neuronal Asymmetry Induced by the UNC-6 (Netrin) Ligand Genetics, 183 (3), 941-949 DOI: 10.1534/genetics.109.108654