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During both development and regeneration of the nervous system, neurons display complex growth dynamics, and several neurites compete to become the neuron’s single axon. Numerous mathematical and biophysical models have been proposed to explain this competition, which remain experimentally unverified. Large-scale, precise, and repeatable measurements of neurite dynamics have been difficult to perform, since neurons have varying numbers of neurites, which themselves have complex morphologies. To overcome these challenges using a minimal number of primary neurons, we generated repeatable neuronal morphologies on a large scale using laser-patterned micron-wide stripes of adhesive proteins on an otherwise highly non-adherent substrate. By analyzing thousands of quantitative time-lapse measurements of highly reproducible neurite growth dynamics, we show that total neurite growth accelerates until neurons polarize, that immature neurites compete even at very short lengths, and that neuronal polarity exhibits a distinct transition as neurites grow. Proposed neurite growth models agree only partially with our experimental observations. We further show that simple yet specific modifications can significantly improve these models, but still do not fully predict the complex neurite growth behavior. Our high-content analysis puts significant and nontrivial constraints on possible mechanistic models of neurite growth and specification. The methodology presented here could also be employed in large-scale chemical and target-based screens on a variety of complex and subtle phenotypes for therapeutic discoveries using minimal numbers of primary neurons.
During development and regeneration, neurons undergo several complex morphological and functional changes, including neurite outgrowth, axon specification, branching, and synaptogenesis. Subtle abnormalities in these processes have been implicated in several neurological disorders such as autism,1,2 schizophrenia,3 and epilepsy.4 Developing assays for repeatable recapitulation of these complex neuronal behaviors on a large scale could allow not only mechanistic studies but also large-scale high-content screens and therapeutic discoveries. However, due to the significant complexity and heterogeneity of neuronal dynamics, the development of such assays has proven challenging.
The dynamics of immature neurite growth and subsequent axon specification during development have been most extensively studied in primary hippocampal neurons from embryonic rats. Developing hippocampal neurons extend several immature neurites, one of which ultimately becomes the longer axon while the remaining neurites become the shorter dendrites,5,6 i.e., the neurons polarize. Axotomies of hippocampal neurons in culture have revealed that once one neurite grows 10–15 µm longer than the others, it usually becomes the axon.7,8 The process of axon specification is believed to rely on feedback loops among the neurites, in which longer neurites promote their own growth and inhibit the growth of other neurites.6,9 Several molecular components of these feedback loops have recently been identified.10,11
Previous studies of neurite growth have typically involved neurons growing freely on two-dimensional substrates5,11–16 or along intersecting stripes.17,18 However, measurements of reproducible neurite growth dynamics on these surfaces are difficult to perform because neurons have varying numbers of immature neurites, and because these neurites can grow in a wide variety of morphologies, allowing quantification of only simple aspects of neurite growth dynamics over many neurons. In addition, primary neurons are prohibitively difficult to isolate in large numbers, and they must also be cultured at a high cell density, thereby increasing the likelihood of cell-cell interactions and reducing the number of neurites that grow without contacting other cells.
Here, we circumvented these difficulties by using a simple strategy to generate highly reproducible neurite growth behavior in a high-throughput format. In recent years, numerous methods have been demonstrated that use surface patterning to guide neuron development.13,17,19–28 However, surface patterning has never been previously used to perform high-content screens for elucidating neurite dynamics. Here, we used a femtosecond laser beam to pattern micron-wide lines of poly-d-lysine (PDL) onto a poly(ethylene glycol) (PEG) monolayer (Fig. 1A and B). Hippocampal neurons preferentially adhered to the PDL lines, and the narrowness of these lines prevented each neuron from growing more than one neurite in each direction (Fig. 1C). By imaging the neurite growth dynamics of only about a hundred cells, we acquired thousands of data points with very high signal-to-noise ratio. In particular, we found that neurite growth accelerates until the point at which neurons polarize. We also showed that immature neurites compete, even at very short lengths, and that neuronal polarity exhibits a distinct phase transition as neurites grow.
A number of mathematically and biophysically inspired models of neurite outgrowth have been proposed in recent years.9,11,29–36 In these models, small differences in neurite lengths and growth rates are amplified over time until one neurite becomes the axon, which continues to grow at a fairly constant rate, while all the other neurites become shorter dendrites or retract.5 However, in the absence of large-scale quantitative experimental data, these different models have remained similarly plausible.
We selected three representative models, which we compared to our experimental findings. The simplest of these models describes neurite outgrowth as a competition for resources produced by the soma at a fixed rate and distributed to the different neurites as a function of their lengths.29 More detailed biophysical models include the growth-dependent transport and diffusion of raw materials for neurite growth between the soma and the neurite tips.30,31 Recently, a more complex and biochemically supported model was constructed based on a positive feedback loop between the GTPase HRas and phosphatidylinositol 3-kinase (PI3K) at the neurite tip.11
By finding the best fits of these models to our neurite growth data (i.e., neurite length as a function of time), we first determined the unknown parameters used in these models. We then examined how well these models predicted neurite growth and competition dynamics. None of these models was fully consistent with our measurements. We found that although specific modifications of the complex molecular model of Fivaz et al.11 allowed this model to better recapitulate our observations, these modified versions also did not completely agree with our results. Thus, such high-content measurements can put significant and nontrivial constraints on possible mechanistic models of neurite growth.
To create narrow protein stripes preventing growth of more than single neurites, an optical patterning technique was used with sub-micron resolution. Poly-d-lysine patterns were created using an improved version of laser-assisted protein adsorption by photobleaching (LAPAP).37,38 The original LAPAP technique does not allow reliable growth of neurites along well-defined protein micropatterns, since cell bodies and neurites also grew on parts of the substrate that are not patterned. In order to overcome this challenge, rather than patterning onto a glass substrate coated with bovine serum albumin,38 we developed a new method to pattern proteins onto a glass coated with a PEG monolayer (Fig. 1B). This significantly reduced non-specific cell adhesion and neurite growth, thereby allowing single neurites to grow on well-defined tracks and form highly repeatable morphological patterns. The glass was first cleaned with Nano-Strip (Cyantek) for 15 min, then rinsed in deionized water, air-dried, and placed in a toluene bath (Chromasolv Plus, Sigma). Subsequently, the bath was moved into a nitrogen bag, and PEG silane (2-[methoxy(polyethyleneoxy)propyl]trichlorosilane, Gelest) was added at a volumetric ratio of 2%. After 2 h, the bath was removed and the glass was rinsed in deionized water, air-dried, and placed in a vacuum oven for 2 h. Following removal from the vacuum oven, the glass was firmly attached to a silicone gasket of a 16-well plate (ProPlate slide module, Grace Bio-Labs).
We used a Ti : sapphire laser (Spectra-Physics) operating at 780 nm, near the two-photon absorption peak for fluorescein,39 to pattern fluorescein dye (100 µg/ml, J. T. Baker) onto PEG-coated glass. Laser patterning was performed using a Nikon Eclipse Ti microscope equipped with a real-time focus correction system and a 40× objective lens (NA 0.9, Nikon). The laser power at the bottom surface of the glass slide was measured to be 7 mW, as estimated by multiplying the power level at the back focal-plane of the objective lens with its optical transmissivity. Patterns were created by scanning the laser beam at a speed of 200 µm/s using galvo mirrors (Cambridge Technology, UK) with a spatial resolution of 250 nm.
The PDL patterns consisted of 1 µm wide and 2 mm long lines with a line-to-line separation of 100 µm (Fig. 1A). After fluorescein patterning, each well of the 16-well plate was washed with phosphate buffered saline (PBS, Sigma) and then incubated for 1 min with PDL (10 µg/ml, 1–4 kDa, Sigma) fluorescently tagged with DyLight 549 (Pierce) to facilitate visualization. The width of the lines was confirmed by fluorescent imaging. Variation in PDL concentration along individual lines was approximately 3%. Variation in average fluorescence between lines in the same well was also about 3%. Wells were washed again with PBS before cell plating.
Rat hippocampal cells were isolated and plated as described in the methods. Neurons were imaged after plating using a CoolSnap HQ2 CCD camera. The motorized microscope stage (Prior ProScan) automatically scanned the sample to capture the entire patterned region. Immunohistochemical stains of tubulin confirmed that single neurites, rather than neurite bundles, grew out from the neurons along the PDL lines (Fig. S1).
We made a total of 3872 neurite length measurements, where individual neurons were imaged at 1-hour intervals for 18 h. The 18-hour time window was chosen because the competition between neurites vanished beyond this point as we show below. A representative measurement of a neuron is shown in Fig. 2A. On average, neurite growth in both directions along PDL lines was symmetric, with mean values of 33.2 ± 0.6 µm and 32.9 ± 0.5 µm (SEM, Fig. 2C), indicating that neurite outgrowth was not biased due to any anisotropy of the PDL patterns.
Growth velocity and acceleration were calculated for each of the neurites as a function of time. For a neurite with length L(t), where t is the number of hours that have elapsed since the beginning of the observation, that neurite’s velocity L˙(t) (in units of µm/h) was calculated using the relation L˙(t) = [L(t+1) − L(t−1)]/2, and acceleration L¨(t) (in units of µm/h2) was calculated using L¨(t) = L(t+1) − 2L(t) + L(t−1).
Each neurite growth model we studied consists of several equations as well as parameters such as rate constants and diffusion/transport constants (Fig. 3), which were assumed to be the same for all neurons. Neurite lengths for the Khanin et al.29 and Samuels et al.30 models were calculated as a function of time using a fourth-order Runge–Kutta-Fehlberg differential equation solver (MATLAB function ode45), while a multistep stiff differential equation solver (MATLAB function ode15s) was used to calculate neurite lengths for the Fivaz et al. model.
Best-fit values for the unknown parameters as well as initial conditions were determined using an optimization scheme based on gradient descent, with additional constraints placed on the parameters as necessary, such as restricting values of physical quantities to be positive. Uncertainties in the estimation of these parameters were determined by randomly varying the initial conditions and the initial parameter values used in the optimization.
Importantly, the best-fit values of all unknown parameters were varied to assure that the general neurite growth behaviors (i.e., neurite growth rate, competition, polarization) that we study in the following sections are not affected by the choice of these parameters.
The simplest model for neurite growth, proposed by Khanin et al. (Fig. 3A) predicts the specification of a single axon among several competing neurites.29 The model assumes that the summation of the growth rate of all the neurites remains constant over time, and that the growth rate of an individual neurite increases with the neurite’s length. We determined the two parameters of this model to be v0 = 7.2 ± 0.4 µm/h and α = 1.22 ± 0.08 (SEM), where v0 is the characteristic growth rate, and the dimensionless α represents the strength of the competition between the neurites (Fig. 3A). This result falls within the regime where axon specification occurs (α > 1).29
Samuels et al. proposed a model of intermediate complexity (Fig. 3B), consisting of five coupled differential equations in the case of two competing neurites.30 A sample fitting of this model to neurite data is shown in Fig. 2B. The model was based on the diffusive and active transport of a rate-limiting factor for neurite growth, although the identity of this factor was unknown. Similar models have proposed that this rate-limiting material is tubulin.31,35 The Samuels model consists of six parameters: three dimensionless constants (χ1, χ2, and χ3), and a characteristic length (Lsc), time (tsc), and concentration (Csc). We reduced the number of these parameters to five by normalizing concentrations C0, C1, and C2 with respect to Csc. The measured values for the parameters were tsc=6.4 ± 0.5 h, Lsc = 51.3 ± 4.6 µm, χ1 = 5.7 ± 0.9, χ2 = 36.8 ± 3.8, and χ3 = 5.2 ± 1.3. These values for χ1 and χ2 fall within the regime where axon specification occurs.30
The third model we compared to our data was proposed by Fivaz et al., who identified a positive feedback loop between HRas and PI3K at the neurite tip as the primary recruiter of additional HRas to the neurite tip and as a driving mechanism for neurite growth and competition.11 We used similar values for all parameters to those suggested by the authors, as these already produced good fits to the measured neurite lengths as a function of time.11
To quantify neurite growth behavior, we first analyzed total neurite growth rate to see whether total growth remained constant or was accelerative, since different neurite growth models make different predictions on this point. As the neurites grow, they also compete against each other. Thus, neurite competition was the second metric used in our analysis. Finally, we studied the overall neuronal polarity, which we expected to become more observable as one neurite out-competed the other.
We observed that the total neurite growth rate increased with total neurite length, i.e., total neurite growth was accelerative (Fig. 4A), up to a length of approximately 80 µm. This acceleration was fastest when both neurites were very short, although there was greater uncertainty associated with these measurements.
To determine whether the models predicted this accelerative growth, initial neurite lengths were randomized between 1 and 5 µm based on our experimental observations. The initial normalized concentrations (C0, C1, C2) in the soma and neurite tips in the Samuels et al. model were randomized between 0 and 0.5 i.e., within the range of values produced by parametric fitting. The initial chemical concentrations (i.e., S, P1, P2, T1, T2, D1, D2) in the Fivaz et al. model were randomized between 0 and twice their respective equilibrium concentrations.
The Khanin et al. model predicts that the total neurite growth rate is constant and independent of total neurite length, and that there is no acceleration of total neurite growth (Fig. 4B).29 Thus, the Khanin et al. model is not consistent with our total neurite growth measurements.
Meanwhile, the Samuels et al. model, predicts that the total growth rate of neurons with short neurites rapidly increases (Fig. 4C). This prediction is consistent with our experimental results.
For the Fivaz et al. model, the acceleration of total neurite growth does not agree with our experimental data (Fig. 4D). Indeed, a sustained increase in growth rate is not possible in the Fivaz et al. model because the total amount of the growth-inducing factor HRas within each neuron is fixed by the initial conditions.
We then made several simple modifications to the Fivaz et al. model, since this model has a well supported molecular basis.11 These modifications were made independently of each other and were each capable of recapitulating accelerative total neurite growth (although their effects on other characteristics of neurite growth dynamics, which we discuss next, were significantly different). The first modification introduced a nuclear source term for HRas that declines as neuron matures (Fig. 4E), since the total neurite growth rate depends on the total amount of available HRas in the neuron. A second modification introduced a nuclear source term for PIP2, which could then be phosphorylated into PIP3 by PI3K at the neurite tip (Fig. 4F), as the neurite growth rates are proportional to the concentrations of PIP3 in the neurite tips. The third modification increased the rate of vesicle trafficking of HRas as the neurons developed (“dynamic trafficking” in Fig. 4G), since increasing the rate of HRas trafficking should also increase the total neurite growth rate. These modifications to the Fivaz et al. model are further detailed in the methods section.
Neurite growth rates depend on the concentrations of growth-limiting factors in the neurite tips.11,30,31 Neurites compete for these factors to increase their length,6 and neurite growth rate, L˙, reflects the concentrations of these factors. To quantify the dynamics of this competition (i.e. changes in concentrations of growth-limiting factors), we therefore evaluated correlations in the changes in neurite growth rates (i.e., acceleration, L¨). We define the neurite competition factor (NCF) between two neurites as the normalized product of their accelerations:
A positive NCF corresponds to competitive behavior between the neurites, while a negative NCF corresponds to cooperative behavior, as shown in Fig. 5.
We experimentally observed that as one neurite’s growth rate increased, the growth rate of the other neurite on average always decreased (i.e., positive NCF in Fig. 4H for all neurite lengths). The relative growth rates of neurites often alternated, such that both neurites grew to substantial lengths. We observed this competitive behavior over a range of neurite lengths, even when the total neurite lengths were as short as 20 µm and as long as 100 µm. This competition was strongest when the total neurite length was between 50 and 60 µm.
We next evaluated how well the different models predicted the observed competition between growing neurites. The Khanin et al. model predicts that the growth rate of one of the two neurites will be always increasing while the other is always decreasing. As a result, the model incorrectly predicts that the competition between neurites starts highest and monotonically decreases as neurites grow (Fig. 4I). This prediction is inherent to the Khanin et al. model, and does not depend on the choice of parameters v0 and α.
The Samuels et al. model also diverges significantly from the experimentally observed competitive behavior for shorter total neurite lengths (<50 µm) (Fig. 4J). The model inherently and incorrectly predicts that the growth rates of both neurites increase (negative NCF in Fig. 4J) until
where Li is the length of the shorter neurite, and Ci is the corresponding resource concentration at that neurite’s tip. The factor of 2 in the above relation was determined empirically from typical concentrations produced by our fitting analysis. The numerator and denominator on the right-hand side of eqn (2) are respectively proportional to the diffusion and active transport rates of the resource. When Li/Lsc is larger than the ratio of the two rates (i.e. diffusion is slower than active transport), the growth rate of the shorter neurite will begin to decrease, i.e., NCF becomes positive. This transition in NCF occurs when Li is approximately 15 µm, which implies a total neurite length of at least 30 µm, i.e., twice the length of the shorter neurite. This was in agreement with our numerical simulations of the model (Fig. 6J). However, it was inconsistent with our experimental data, in which NCF was positive even at total neurite lengths much shorter than 30 µm.
We found that the Fivaz et al. model correctly predicted competition between the neurites, even at shorter neurite lengths (i.e., positive NCF in Fig. 4 K). Our modifications of the model also predicted competition correctly at shorter neurite lengths (Fig. 4L-N), although the addition of a PIP source term appeared to result in an NCF that increased monotonically with neurite length (Fig. 4M). The addition of dynamic trafficking resulted in the strongest competitive trend and a maximal NCF closest to what we observed experimentally. However, the original Fivaz et al. model and all modifications of the model predicted a significantly positive NCF even at longer neurite lengths, inconsistent with our experimental observations.
Neurons polarize as their neurites grow, as discussed in the introduction, and such polarization was evident in our experiments as total neurite length exceeded 80 µm (Fig. 4O). Here, we define neuronal polarity as |L1−L2|/(L1+L2). The increase in polarity occurred quite suddenly and is suggestive of a phase transition, which occurs in various biological phenomena.40,41
The polarization process predicted by the different neurite growth models demonstrated a variety of behaviors. For comparison, we set the initial neurite lengths in all the models to be the same to match the initial mean polarity (~0.2) we measured experimentally.
No phase transition was evident in the Khanin et al. model, which produced a polarity that steadily increased with total neurite length (Fig. 4P). Polarity in the Samuels et al. model more closely resembled our experimental results (Fig. 4Q). Like the Khanin et al. model, the Fivaz et al. model produced an increasing polarity (Fig. 4R) inconsistent with our experiments.
Our modifications of the Fivaz et al. model also resulted in distinct polarity behaviors. Introduction of an HRas source term caused the neuron to polarize at shorter lengths (Fig. 4S), while the PIP source term appeared to lower the steady-state neuronal polarity (Fig. 4T). These were inconsistent with the polarization behavior we measured. Our experimental observations were best predicted by the dynamic vesicle-trafficking modification, which resulted in a nearly constant polarity of about 0.2 for total lengths under 100 µm, and an increasing polarity for lengths exceeding 100 µm (Fig. 4U). This apparent phase transition occurs when the vesicular trafficking rate exceeds a critical value of 0.6 vesicles/min (Fig. S2).
Large-scale studies of neurite dynamics have remained difficult to perform because neurites branch and grow in various morphologies. By growing neurons on protein micropatterns, we dramatically increased the repeatability of neurite growth behavior. This enabled us to measure and analyze the dynamics of neurite growth acceleration, neurite competition, and neuronal polarity with unprecedented reproducibility using a minimal number of primary neurons.
We specifically found that neurites compete with each other, even when total neurite length is as short as 20 µm. This is a non-obvious behavior, as certain models predict incorrectly that both neurites can accelerate simultaneously. We also found that total neurite growth is also accelerating at these shorter lengths. Then, as total neurite length increases beyond a critical length (~80 µm in our assays), several events occur simultaneously: the neuron polarizes (Fig. 4O), total neurite acceleration approaches zero while the total neurite growth rate reaches a steady-state value (Fig. 4A), and neurite competition ceases (Fig. 4H).
We were further able to evaluate several leading models of neurite growth by analyzing their ability to predict these fundamental biophysical behaviors of neurite growth and axon specification (Table 1). The Khanin et al. model was empirically hypothesized and is incapable of reproducing behaviors similar to our experimental observations, since it presumes a constant total growth rate. The model therefore exhibits zero total neurite acceleration (Fig. 4B). In addition, the model incorrectly predicts that the strongest competition (i.e., the highest NCF) occurs while neurites are shortest and that neuronal polarity smoothly increases with total neurite length (Fig. 4P).
Unlike the Khanin et al. model, the Samuels et al. model proposes a chemical mechanism for neurite growth. In this model, growth depends on the concentration of a specific chemical that is actively transported from the soma to the neurite tips, that passively diffuses between these locations, and that is consumed during neurite growth. At shorter lengths, diffusion between neurites occurs rapidly, allowing the neurites to effectively share a common pool of resources when they are short. As a result, short neurites grow cooperatively until reaching lengths at which active transport begins to dominate (see Fig. 4Q and eqn (2)). Thus, while this model correctly predicts accelerative total neurite growth (Fig. 4C), it results in cooperative rather than competitive behavior for neurites shorter than the length at which the active transport rate exceeds diffusion.
The Fivaz et al. model includes the transport of several chemicals known to be involved in the establishment of neuronal polarity, allowing it to exhibit more complex behaviors than those of the Samuels et al. model. While the Fivaz et al. model correctly predicts that both shorter and intermediate neurites compete, the model does not predict accelerative growth at short neurite lengths (Fig. 4D), since the total amount of HRas (necessary for neurite growth) is fixed in the model. By including source terms for HRas and PIP, we were able to recapitulate accelerative growth in this model (Fig. 4E and F). However, neither these modifications nor the original model predicts the sharp phase transition that occurs during neuronal polarization (Fig. 4R-T), suggesting that some additional mechanism is missing in this model. We hypothesized that a stronger nonlinearity between neurite length and neurite growth is necessary to induce such a sharp transition in polarity. As an example, we introduced such a nonlinear mechanism in the form of a time-dependent vesicle trafficking because neuronal polarization strongly depends on vesicle trafficking rate (Fig. S2). This modification successfully recapitulated the observed phase transition in polarity. We note that other nonlinear mechanisms are also plausable that could recapitulate such a sharp phase transition in polarity, and further biochemical assays are necessary to identify the nature of this nonlinear mechanism. Although our modifications significantly improved agreement of the Fivaz et al. model with our measurements, there were still minor but measurable differences (e.g., none of the modifications recapitulated the weaker competition between long neurites). Thus, our high-content measurements put significant and nontrivial constraints on possible mechanistic models of neurite growth that cannot be reconciled by simple modifications. While biochemical assays in the future will be necessary to validate any model, quantification of neurite dynamics by our technique allows discrimination of different mechanistic models.
It is possible that restricting neuronal adhesion and neurite growth to one dimension affects neurite competition and axon specification. However, we still observe all of the neurite growth and competition dynamics previously measured with standard two-dimensional cell culture techniques. Further-more, the mechanistic models we evaluated allow for any number of neurites, including two, and do not depend on the relative orientations of neurites. Thus, it is likely that the highly repeatable neurite dynamics we observe here extend to more complex morphologies of developing neurons.
This work represents the most quantitative description of neurite growth and competition to date. The protein patterning and analytical strategies developed here can be extended to analyze more complex neurite dynamics. These high-content measurement techniques can allow construction and validation of models for complex neuronal processes. They could also be used for large-scale chemical and target-based screens on a variety of complex phenotypes for therapeutic discoveries using a minimal number of primary neurons.
Hippocampi were dissected from embryonic day 18 Sprague-Dawley rats and placed in ice-cold Hank’s balanced salt solution (HBSS), buffered with 10 mM HEPES, pH 7.3. The tissue was transferred to an enzyme solution containing 100 units of papain in 5 ml of HBSS, 1 mM CaCl2, and 1 mM L-cystine, and subsequently incubated at 37 °C for 45 min. The hippocampi were then washed in Dulbecco’s modified Eagle’s medium containing 10% fetal bovine serum to deactivate the papain, and the media was replaced with Neurobasal-B27 (Invitrogen) containing 2 mM glutamine and 100 units/ml penicillin/streptomycin. The cells were triturated using a flame-polished pipette and counted.
200 ml of Neurobasal-B27 were added to each well containing the PDL patterns. Cells were then pipetted into the wells at a density of 5000 cells per well. The cells were maintained in an incubator at 37 °C with 5% CO2 and 100% humidity, and were briefly removed every hour for imaging.
Cells plated on PDL patterns for 12 h were fixed for 15 min in 4% paraformaldehyde and then rinsed with a solution of phosphate buffered saline with 0.05% Tween 20 (PBST). Next, the cells were permeabilized with 0.1% Triton X-100 for 10 min, and washed again with PBST.
Prior to antibody staining, the surface was blocked with 3% bovine serum albumin for 30 min. Cells were incubated in a 1 : 1000 dilution of mouse monoclonal anti-tubulin (primary antibody, R&D Systems) for 1 h, and then in a 1 : 500 dilution of Alexa Fluor 488 goat anti-mouse (secondary antibody, Invitrogen) for another hour, with PBST washing between steps. Finally, the cells were incubated in a 1 : 10 000 dilution of DAPI for nuclear staining, washed again, and fluorescently imaged.
The original model proposed by Fivaz et al. consisted of the equations shown in Fig. 3C, which are rewritten here with the stochastic anterograde vesicular trafficking terms Vi(t) included:
Fivaz et al. defined Vi(t) as follows:
where vesicles containing HRas release their contents at the neurite tip at times tf and σ is the temporal width of the HRas release by the vesicles. Like Fivaz et al., here we assume an average vesicle trafficking rate of 1 vesicle per minute for each neurite.
Our first modification of the model was to include a term representing the nuclear synthesis of HRas that declines as the neuron matures. This was accomplished by adding a decaying source term to the equation for S:
where the characteristic length λ was set to 50 µm. The magnitude γ of the source term was varied in Fig. 4.
The second modification was the introduction of a PIP2 source term in neurites that also declines as the neuron matures. We define Qi as the concentration of PIP2 in neurite i. The equation for P is then
and we introduce a similar equation for Q:
Again, λ was set to 50 µm, and the magnitude κ of the source term was varied in Fig. 4.
The third modification (“dynamic trafficking”) added time dependence to the vesicle trafficking rate. This rate was initially set to zero, and exponentially approached a steady-state value of 1 vesicle per minute per neurite with a characteristic time τ, which was varied in Fig. 4.
A number of models have been proposed in recent years to explain how neurites of developing and regenerating neurons grow and compete with each other. However, none of these models has been validated due to significant challenges in measuring neurite growth dynamics with sufficient repeatability to extract the fine dynamics of neurite growth. Using a novel analysis and protein patterning technique that restricts the growth of individual neurites on a substrate that otherwise prevents adhesion of neurons and neurites, we elicited and quantified several important yet previously unknown aspects of neurite competition and neuronal polarity. Our high-content analysis puts significant and nontrivial constraints on possible mechanistic models of neurite growth.
ZWG thanks A. D. Wissner-Gross for helpful discussions regarding the analyses and manuscript. ZWG was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program and by a Fannie and John Hertz Foundation/Myhrvold Family fellowship. MAS was supported by the Kennedy Memorial Trust. MFY was supported by a Packard Fellowship in Science and Engineering and an Alfred Sloan Award. This project was funded by NIH Director’s New Innovator Award (1-DP2-OD002989-01), part of the NIH Roadmap for Medical Research, and by NIH R01 EUREKA Award (1-R01-NS066352-01).
†Electronic supplementary information available: Fig. S1. Immunohistochemical staining of hippocampal neurons and their neurites. Fig. S2. Effect of vesicle trafficking rate on neuronal polarization in the Fivaz et al. model. See DOI: 10.1039/c0ib00058b