Geometrical Determinants of Neuronal Actin Waves

Hippocampal neurons produce in their early stages of growth propagative, actin-rich dynamical structures called actin waves. The directional motion of actin waves from the soma to the tip of neuronal extensions has been associated with net forward growth, and ultimately with the specification of neurites into axon and dendrites. Here, geometrical cues are used to control actin wave dynamics by constraining neurons on adhesive stripes of various widths. A key observable, the average time between the production of consecutive actin waves, or mean inter-wave interval (IWI), was identified. It scales with the neurite width, and more precisely with the width of the proximal segment close to the soma. In addition, the IWI is independent of the total number of neurites. These two results suggest a mechanistic model of actin wave production, by which the material conveyed by actin waves is assembled in the soma until it reaches the threshold leading to the initiation and propagation of a new actin wave. Based on these observations, we formulate a predictive theoretical description of actin wave-driven neuronal growth and polarization, which consistently accounts for different sets of experiments.

1. The growth velocity is a↵ected by the width of the adhesive stripe on which the neurite is growing.
We will consider the simplest scenario where the growth material synthetized at the soma level has to be distributed along the available surface at the neurite growth-cone, leading to a larger (smaller) velocity v tip when the width is smaller (wider). 2. The neurites have a certain probability to turn into axons, which signals for the whole cell to polarize.
This polarization event seems to depend on the absolute length of the neurite.
3. Following the polarization event the neurite that was "chosen" to become the axon may continue to grow at a di↵erent growth velocity v tip (with 1). The other neurite(s) continues to grow at a reduced growth velocity v tip (0   1), following the polarization event. We now turn these assumptions into a calculation of the mean neurite length on the patterns (denoted x :xy and x :yx, i.e. 2 :26, 6 :62, 2 :62 and 6 :26 as described in the main text) For this we need to assume some probability density function that describes the probability for the neurite to undergo polarization, as a function of its length. Based on the notion of a critical length for this transition, we consider the following simple step-like cumulative probability to polarize where L pol is the critical polarization length and pol gives the variance of the probability distribution function around this critical length.
We then divide the calculation into two cases, for the case of the right neurite polarizing, and for the left. We now demonstrate the calculations that result from our model for the x :xy patterns. A similar calculation follows along the same lines for the x :yx patterns. For polarization along the right neurite, the length of the right neurite is given by the following where l is the length of the x-segment on the right side, x pol is the location of the polarization event on the right neurite, T is the time at which the observation is performed and v x,y are the tip velocities on the respective width. The di↵erent times are defined as The maximal value of x pol for the right neurite is Over the possible range of x pol we need to integrate the lengths in Eq.3 multiplied by the probability that the right neurite polarizes at position x pol , which is given by p pol (x) (Eq.2). In addition we need to multiply by the probability that the competing neurite has not already polarized itself by this time. This probability is given by the cumulative probability up to the position of the tip on the left neurite x pol,L corresponding to x pol on the right neurite where t pol stands for t pol,1 , t pol,2 given in Eqs.4,5 respectively, and x pol,L = t pol v x . The final mean length that we get is where the overall probability for polarization of the right neurite is given by For completeness, the overall probability for polarization of the left neurite is given by where : In a similar manner we calculate the length of the left neurite when the right one is polarized (hL L,R i) and the lengths for the case of polarization on the left neurite (i.e. hL L,L i, hL R,L i). In order to get the total mean length along each side, we also need to consider the lengths of the neurites when no side has polarized. For the x : xy these lengths are given by and where the probability for no polarization is given by : where P R is given in Eq.11. Summing the lengths on each neurite from all the possible polarization outcomes gives the final mean lengths :   i.e. "left following left" and "right following right" events. We similarly defined the number of alternative actin waves N(Al) as "left following right" and "right following left" events. We then defined the "alternate repetition probability"as .
The repetition probability would be 50% in case of random shedding, and 100% in case of perfect alternative shedding. In bipolar neurons developing on 2μm (blue symbols) and 6µm (gold symbols) wide stripes, the experimental values of P are 74.0±5.7% and 63.2±15.2%, respectively, both significantly different from random. These values confirm a preferential alternative actin waves distribution, although this phenomenon is much accentuated for the thinner stripes. **, p=0.008, n=296, 5 cells and *, p=0.047, n=90, 6 cells. Two-tailed unpaired non parametric Mann-Whitney U test were used.