Following previous results showing that neurite elongation was slowed down when neurites were allowed to spread on 6 μm, as compared to 2 μm wide stripes (Tomba et al., 2014), we studied the temporal rate of actin wave production in similar geometries of adhesion. Linear adhesive patterns with widths of 2 or 6 μm, respectively, were produced, leading to 2-branched neurons displaying diametrically opposed neurites spreading on 2 or 6 μm wide stripes (these designs are referenced later on by the nomenclature 2-2 or 6-6, respectively). We selected young stage 2 neurons (Dotti et al., 1988), i.e., neurons of rather symmetric neurite length taken at 1–2 days in vitro (1-2 DIV) displaying a ratio between the longest and the shortest neurite lower than 1.5, except for transient and reversible neurite asymmetry due to soma displacement (see supplementary movie). Such an early stage of development is considered as the most relevant time window to assess the role of actin waves (Winans et al., 2016). We also designed a pattern combining diametrically opposed 2 and 6 μm wide stripes. A central disk of adhesion of 15 μm diameter was introduced in this asymmetric pattern to set the location of the soma and therefore the width of both neurites. In our nomenclature, this pattern is labeled “2:6,” with the mark “:” standing for the 15 μm disk. Figure 1A displays images of neurons grown on the three types of patterns, 2-2, 6-6, and 2:6, all exhibiting a single actin wave. It should be noted that, in the 2:6 patterns, we could not obtain well-formed 6 μm wide stumps before the 2 μm wide branch gets rather long, as illustrated by the cell shown in Figure 1A. This point is highlighted in Fig.S2A displaying the average neurite lengths for all conditions at 2 DIV and showing that 2:6 patterns inherently produce the most asymmetric neurons in length.

To evaluate the rate of actin wave initiation in these three geometrical configurations (i.e., 2-2, 6-6, and 2:6 patterns), we computed a quantitative characteristic, the Inter Wave Interval (IWI), corresponding to the time interval between the initiation of two consecutive actin waves within a given neurite. In the case of stripes with uniform width, data from both neurites of each of the analyzed cells were merged. For an overview of the way actin waves distribute temporally between the two branches, see Fig. S3 displaying the directional correlations between two subsequent actin waves.

The graph of Figure 1B summarizes our results. Strikingly, the mean interval between consecutive actin wave production events seems to scale with the pattern width. Actin waves are indeed about three times less frequent on wide neurites that elongate on 6 μm wide stripes, as compared to neurite growing on 2 μm wide stripes. This finding is further strengthened by the results obtained on the 2:6 patterns. We observed no significant differences between the IWI's distributions from the 2 μm wide branch of the 2:6 pattern and the distribution of IWIs obtained on 2 μm wide stripes (2-2 pattern). The same observation was made for the 6 μm wide branches. In other words, each branch of the 2:6 pattern behaved independently of the other: this suggests that actin waves are initiated according to a temporal pattern defined locally at the neurite level.

Actin waves travel distally along neurites but usually emerge near the soma. We therefore wondered if the width of the proximal segment alone was responsible for the observed distributions of IWIs. To answer this question, we designed variants of the previous patterns combining 2 and 6 μm wide stripes along a single branch (see Tomba et al., 2014). For instance, a 2 μm wide stump was inserted at the basis of a 6 μm wide stripe, yielding a geometry labeled as “−26” (see Figure 2A for an example of a neuron grown on a 2:26 pattern, according to the terminology defined above). The reverse situation, i.e., a 6 μm wide stump inserted at the basis of a 2 μm wide stripe was also used (and referenced as “−62”). Finally, to ensure that actin waves were initiated on stripes of uniform width near the soma, the lengths of the 2 or 6 μm wide proximal stumps were taken to be at least two times longer than the typical longitudinal actin wave dimension, i.e., about 10 μm (Figure 1A).

The graphs of Figure 2B show that there is no significant difference between the rate of actin wave production along neurites elongating on 2-2 and −26 stripes. The same observation was made between 6-6 and −62 geometries. This indicates that the geometrical control of the periodicity of actin waves is localized within the region of emergence of these structures near the soma.

It was reported in the seminal papers of Ruthel and Banker (Ruthel and Banker, 1999) that actin waves were triggering a retraction of the neurite tip while moving forward, then its extension. The combination of both phenomena resulted in a net forward growth of the tip. A similar observation was recently reported by Winans et al. (2016) who stated that, at 1 DIV, 90% of actin waves yield neurite outgrowth when reaching the tip, suggesting the central role of actin waves in the early stages of neuronal growth. Here, we computed the distributions of the net displacement of the neurite tip associated to actin waves produced on 2 and 6 μm wide stripes, respectively (only actin waves that could be followed up to the tip where taken into account). In all the experiments selected, the onset of actin waves and their position during their propagation were measured (Figures 3A,B). The progressive retrograde motion of the tip seemed well correlated with the time span of these propagative structures. Defining the duration of the extension phase was less straightforward. We noted however that similar time constants seemed to be involved in the retraction and extension processes, as illustrated by the rather symmetric shapes of the corresponding peaks in the curves representing the tip position versus time (Figure 3B). We thus decided to compute the tip extension during a time span equal to the duration of its retraction, as expressed by the identical widths of the red and blue rectangles in the inset of Figure 3B. Other criteria could have been chosen, e.g., to wait for a stabilization of the position of the tip consecutive to the burst of growth triggered by the actin wave, but then our observation would have depended strongly on the time elapsed between two successive actin waves on the same neurite. The relatively short duration chosen to compute the neurite extension phase might consistently lead to an underestimation of the total neurite elongation. However, our procedure should provide a precise proxy for neurite elongation that allows a comparison between different adhesive patterns.

Using the above procedure, we found that the increment of growth per actin wave λ_w was larger for 6 μm wide than for 2 μm wide stripes, i.e., λ₆ = 1.46 μm (n = 57, 5 cells) and λ₂ = 0.61 μm (n = 201, 5 cells), respectively, as illustrated in Figure 3C. Note that previous work reported larger values of net growth in the range of 2–4.5 μm per wave, using a longer time window at the expense of a less clearly defined criterion (Ruthel and Banker, 1999). But even in the presence of a proportional measurement bias, as described just above, the ratio between these numbers should correctly quantify the ratio of growth induced by the waves on each of these stripes.

The above experiments suggest a local control of actin wave production, at the neurite basis. In order to test this observation in a different context, we explored a configuration in which the neurite width is set at a fixed value (2 μm), and the number of neurites varies. We therefore designed adhesive geometries with a central disk (diameter 15 μm) and 2 μm thin lines symmetrically distributed around this disk in order to produce neurons with 2, 3, or 4 branches (see Figure 4A for the sketches of the adhesive patterns and images of typical cells).

The distribution of IWI computed in two different ways, i.e., at the neurite and cell levels, similarly to Figure 1, 2, respectively), is reported in Figure 4B for each geometry. Interestingly, the mean values of IWI at the neurite level increases nearly proportionally with the number of branches, whereas the global rate of production of actin waves computed at the cell level does not vary significantly with the number of branches. Note that this invariance of the mean IWI computed at the scale of the entire neuron versus the number of branches is associated with a conservation of the total neurite length at 2 DIV (Fig. S2B).

In the present work, we used our findings about the geometrical determinants of actin waves obtained from simple patterns to build a new mechanistic model of neuronal growth and polarization, assuming that these growth and polarization and mainly driven by actin waves. This model was then compared with the experimental data previously obtained at 3 DIV on the more complex sets of patterns described above.

More specifically, we modeled the elongation of the neurites along the adhesive stripes from the following assertions, motivated by the accumulated experimental observations:

These assertions are mathematically expressed as described below.

The average growth velocity v of the neurite tip in condition of controlled neuronal geometries is ruled by the following set of equations:

where ω_w is the production rate of actin waves on an initial stripe segment of width w, λ_w the growth increment per actin wave and α a free parameter expressing the proportional measurement bias discussed in section 2.2.

In addition, this velocity v_w will change by a concentration/dilution factor θ, if the growing tip has moved past a stripe width change, to account for the conservation of the material conveyed by actin waves, i.e.,

θ>1 (resp. θ < 1) when the actin wave passes from a wide (resp. thin) region to a thinner (resp. wider) region, so that the material delivered to the leading edge at the tip is concentrated (resp. diluted) and therefore gives a larger (resp. smaller) net growth.

Following polarization, the equations displayed below will rule the growth of axons and of future dendrites:

Experimentally, β and γ express the change in actin wave initiation mean rate following the event of axonal determination in our present model, as addressed in the following section.

First, we model L_pol by a Gaussian distribution with mean L_pol = 50 μm and standard deviation σ_pol = 20 μm, as already considered in the work of Tomba et al. (2014). This choice matches with the observed length of the initial axonal segment, a structure implied in action potential generation.

The set of parameters associated with actin waves are the following: α, ω_w, λ_w, θ, β, γ with w = 2 or 6 μm. We have already experimentally determined ω_w and λ_w for both values of w (see Figures 1, 3, respectively).

To determine β and γ, we seek the rates of production of actin waves in the nascent axon and in the future dendrite, as compared to the rate obtained in undifferentiated, symmetric neurons (cf. the data shown in Figure 1). We selected 1 DIV neurons with already well-developed symmetric neurites and followed them up to 2 DIV (see Fig. S4A). Interestingly, we observed no obvious change in the periodicity of actin waves at the soma level when neurons evolve from a symmetric to an asymmetric morphology evoking a transition to a polarized state (see Fig. S4B). This observation suggests a constraint on β and γ, such that γ+β2=1.

To go further, we then considered another set of patterned neurons on stripes characterized by asymmetric lengths (with one neurite at least 50% longer that the other) and already long neurites (> 50 μm), so that one neurite should have differentiated into an axon. We measured the IWI of each branch and found a mean value of βγ=r≈2.2 ± 0.57, meaning that axons convey roughly twice as many actin waves as dendrites (Fig. S5).

Combined with the relation γ+β = 2, this yields: γ=21+r≈0.6, β = rγ ≈ 1.4.

Finally, we expect the dilution factor θ to be at most equal to the ratio wiwf of the widths before/after the transition region, as the material is delivered and spread over the leading edge that spans the stripe width. In our model, θ will remain a free fitted parameter. Similarly, the value of the parameter α that tunes the absolute value of neurite growth is also an adjustable parameter of the model.

The data of Tomba et al. (2014) were used to challenge our new model. We used the above numbers deduced from the experiments of the present work. In addition, we let θ (i.e., the dilution/concentration factor of the wave material when the neurite width changes) and α (i.e., the width-independent coefficient regarding the amplitude of the tip net growth induced by an actin wave) as the only two free parameters. The comparison between the model and the experimental data is shown in Figures 5, 6. Note that axons were identified through Tau-1 immunolabelling (see Fig. S6 for a few examples of stained neurons).