### Quantitation of Filopodia

Filopodia span a wide range of observable lengths and individual cells show high variability in the size and number of filopodia they possess. To understand filopodia in their cellular context, we observed filopodia production in rodent fibroblast Rat2 cells. We chose this cell line because it is non-cancerous and individual cells have filopodia that span nearly two orders of magnitude in length. The appearance of the actin cytoskeleton in typical Rat2 cells is shown in Figure 1. In interphase, two types of linear actin polymers are commonly seen, stress fibers (S) and filopodia (F). Stress fibers traverse the cell in a lengthwise manner. Filopodia, on the other hand, are visible as linear projections from the cell body that emanate from multiple places and proceed in multiple directions. Filopodia are distinguishable from the less frequently observed and visibly similar retraction fibers. Retraction fibers are seen primarily in mitotic cells but also appear in cells in interpahse, at the trailing edge during migration. Based on our previously published work with living Rat2 cells [23], filopodia can be visually distinguished from retraction fibers (R) based on their relative thickness and extended presence behind the plasma membrane. We have purposely excluded mitotic cells from our analysis to avoid potential confusion between filopodia and retraction fibers. Moreover, Rat2 cells are relatively non-migratory so they have very few retraction fibers relative to filopodia in non-mitotic cells.

To quantitate filopodial properties in Rat2 cells, we used image analysis software to manually trace the lengths of individual filopodia in fixed Rat2 cells. The length of a filopodium was extrapolated from the pixel length of the trace line. Based on the resolution of our fluorescence microscopy system, we estimate that we can accurately determine the length of filopodia > 0.4 μm in length. Filopodia shorter than this cannot accurately be distinguished from lamellar actin structures and therefore were not counted. We also measured the distance that separates a given filopodium from its nearest neighbor. Cells visualized were non-mitotic and not visibly attached to other cells but were otherwise randomly chosen. The cell population as a whole was in a logarithmic phase of growth and no attempt was made to synchronize filopodia growth cycles. As such, the filopodia that we measure represent structures in undetermined phases of growth, shrinkage and stasis. We collected this data for all filopodia in the individual cells that we imaged. Thus, each filopodium is defined by a length (L_{
x
}) and a separation distance (D_{x}) measurement.

### Filopodia lengths are distributed lognormally

We compiled filopodia length measurements from three independent experiments. We counted filopodia from a total of 52 Rat2 cells (experiment 1 = 25; experiment 2 = 18; experiment 3 = 10). The total number of filopodia was 1,682 (experiment 1 = 745; experiment 2 = 573; experiment 3 = 364). As shown in Figure 2A, filopodia distribution in the total data set is unimodal with a mean of 2.70 μm. The length distribution of the individual experiments was also unimodal with a respective mean of 2.79 μm, 2.49 μm, and 2.84 μm for Experiments 1, 2 and 3. Approximately 82% of the filopodia fall within the range of 1 μm to 10 μm in length.

We next determined the statistical model that would best fit the empirical cumulative probability distribution (CDF) of filopodia lengths and distances. We found that the length distribution of the collective dataset was best modeled as a lognormal distribution (p(x)= (2πσ^{2}x^{2})^{-1/2} exp(-(ln(x)-μ)^{2}/2σ^{2})) (Figure 2B). That is, the logarithm of the length is approximately normally distributed. The dataset is poorly modeled as an exponential (p(x) = λ exp(-λx)), Gaussian (p(x) = (2πσ^{2})^{-1/2}exp(-(x-μ)^{2}/2σ^{2})) or power law (p(x) = ((α-1)/x_{min})(x/x_{min})^{-α}) distribution (Figure 2B). The power law and exponential distributions fit least well, as they are incapable of capturing the unimodality of the observed data. The exponential, however does provide a reasonable fit for the distribution of filopodia larger than ~1.5 μm. The Gaussian is the next most accurate, capturing the unimodal data, but it overestimates the left tail while underestimating the right tail. The lognormal captures both unimodality and the heavy right tail. The datasets of individual experiments are also fit well by lognormal distributions (Figure 2A, B), as are the length distributions from each individual cell (Figure 2C). The similarity in CDF distribution between individual cells in a population indicates that the system regulating filopodia length shows robust behavior in the Rat2 population.

### Filopodia distance separations are distributed lognormally

As we did for filopodia lengths, we compiled the data for the separation distances between adjacent filopodia. As with filopodia length, the separation distance is unimodal in both the total data set and in the three separate experiments (Figure 3A). The mean distance for the collective dataset was 6.18 μm and experiments 1, 2 and 3 had respective means of 5.52 μm, 5.11 μm, and 9.23 μm. When we calculated the CDF for the distance distribution, lognormal was the best fit of the distribution data (Figure 3B). As is the case of filopodial length, the separation dataset is poorly modeled as an exponential, Gaussian, or power law distribution (Figure 3B). The power law distribution is the poorest fit, while an exponential distribution may fit the distribution of filopodia that are separated by 10 μm or more. Nearly all of the cells in a Rat2 population show a good lognormal fit of separation distance data. The similarity in CDF distribution between individual cells in a population indicates that the system regulating filopodia distance separation shows robust behavior between cells. 74% of the interfilopodial distance separation falls within the range of 1 μm to 10 μm.

### Length and separation distance are independent variables

The polymerization of actin polymers within a filopodium depends on an intracellular pool of G-actin. It is possible that as an individual filopodium grows, it might locally deplete the G-actin pool around it and thereby interfere with *de novo* filopodia creation or actin polymerization in pre-existing filopodia. If this were the case, then there may be some empirical relationship between filopodia length and separation. To test this idea, we determined whether or not the length of an individual filopodium is detectably correlated with separation distance between its neighbours (Figure 4A). The figure shows the length of individual filopodia versus the average separation distance between its two nearest filopodia plotted on a log-log scale. On the whole, however, there is no substantial correlation between filopodial length and interfilopodial separation distance (r~0.02). On a per cell basis, there are some cells that show a weak negative correlation between length and separation (r~-0.6) and some with a weak positive correlation (r~0.25). We next determined whether or not there was any substantial correlation between the distance separating adjacent filopodia and whether there might be correlation between the lengths of filopodial neighbours. Such a correlation would be predicted should the concentration of a G-actin pool be a limiting factor in either the initiation of an individual filopodium or in its total length. Figure 4B shows that there is a mild correlation between the separation distance of adjacent filopodia. A similar weak correlation exists between the length of filopodial neighbours (Figure 4C). This suggests that any spatial constraints linking filopodia length and separation are likely to be quite small and, together with Figure 4A, suggests that filopodial length and separation distance are likely to be independent variables.

### Perturbation Analysis of Filopodia

We next wanted to investigate how the filopodia system quantitatively responds to perturbation. There are many agents that have been described to be inducers of filopodia formation, but high-quality empirical measurement of what these agents do to filopodia are not common. Since we have been able to mathematically describe the filopodia system with some degree of confidence, we are now able to define how known filopodial perturbations affect the system as a whole. We chose to alter filopodia production in three distinct manners: genetically, chemically and physically. For the genetic perturbation, we engineered Rat2 cells to ectopically express the lipid kinase PI4KIIIβ, which we have reported stimulates filopodia production [22]. To chemically induce filopodia, we used the peptide hormone bradykinin, which induces filopodia through activation of G-protein coupled receptors [24]. To physically induce filopodia, we coated the growth substrate with poly-D-lysine, which could increase filopodia size by increasing the positive charge of the substrate and enhancing adhesion.

As shown in Figure 5A, expression of PI4KIIIβ causes a large increase in the length of filopodia. The mean length in PI4KIIIβ-expressing cells was 5.13 μm, significantly longer than the 2.03 μm mean length in the vector-only controls (t-test, p < 0.0001). The length distribution remains unimodal, and an increase in the number of long filopodia (10 μm - 100 μm) is visible. The longest filopodium in PI4KIIIβ expressing cells was 65.71 μm. Interestingly, the separation distance between the filopodia also increases following PI4KIIIβ expression and the mean separation in PI4KIIIβ-expressing cells was 12.00 μm, significantly higher than the 4.16 μm distance in vector-only controls (t-test, p < 0.00005). Importantly, even though the length and separation of filopodia have increased substantially, the distribution of both parameters remains lognormal. This indicates that the lognormal distribution is a robust aspect of filopodia length and separation distance control.

Bradykinin treatment causes an increase in filopodia length, albeit to a much lesser extent than PI4KIIIβ expression. The mean length of bradykinin treated filopodia was 5.19 μm, significantly longer than the 3.95 μm mean length in DMSO treated controls (t-test, p < 0.04773). The change that bradykinin makes to filopodia length distribution is primarily in the longer filopodia as 27% of filopodia in bradykinin treated cells were > 6 μm in length, compared to only 17% (48/282) of filopodia in the DMSO controls. The mean separation did not change appreciably, with a mean distance of 4.97 μm in the bradykinin-treated cells compared to 5.16 μm in the DMSO-treated cells. The lack of significant change in distance separation (t-test, p < 0.4386) further strengthens our assertion that filopodia length and distance separation are independent variables. As in the case with PI4KIIIβ expression, the distributions of length and separation following bradykinin remain unimodal and are best fit by lognormal distributions.

The effects of poly-D-lysine on filopodia were very modest (Figure 5C). The mean length of filopodia in cells grown on poly-D-lysine was 3.13 μm which is not much longer than the 2.82 μm mean filopodia length of Rat2 cells grown on plain glass slides. This difference is statistically significant (t-test, p < 0.02968) due to the large number of samples, but it is not readily apparent in the PDF and CDF distribution. Like bradykinin, no change in filopodia separation distances is apparent (5.90 μm compared to 5.49 μm). The distribution of filopodia is unimodal and fits a lognormal distribution. Collectively, the effects of genetic, chemical and physical filopodia inducers show that increases in filopodia length do not apparently alter the lognormal distribution pattern of filopodia length nor the lognormal distribution of the distances that separate them.

Lastly, we chose to analyze the relationship between length and separation distance in the perturbed cells. Figure 6 shows this relationship, plotted on a log-log scale, for all three perturbations. In the case of bradykinin and poly-D lysine, there was no obvious relationship between length and separation. In this respect, these two perturbations do not cause changes from the wild-type situation. In the case of PI4KIIIβ expression, there is a weak, albeit statistically significant, positive correlation (r~0.39). This appears to result from two individual cells (coloured black and purple) with very long and highly separated filopodia. Filopodia length and separation are not highly correlated in these two cells, but the magnitude of the length and separation measurements leads to an apparent correlation in the overall population. As such, we conclude that length and filopodial separation remain independent variables even following perturbation.