Department of Biochemistry, Microbiology & Immunology, University of Ottawa, 451 Smyth Road, Ottawa, Ontario, K1H 8M5, Canada

Ottawa Hospital Research Institute, 501 Smyth Road, Ottawa, Ontario, K1H 8L6, Canada

Abstract

Background

Filopodia are actin-based cellular projections that have a critical role in initiating and sustaining directional migration in vertebrate cells. Filopodia are highly dynamic structures that show a rich diversity in appearance and behavior. While there are several mathematical models of filopodia initiation and growth, testing the capacity of these theoretical models in predicting empirical behavior has been hampered by a surprising shortage of quantitative data related to filopodia. Neither is it clear how quantitatively robust the cellular filopodial network is and how perturbations alter it.

Results

We have measured the length and interfilopodial separation distances of several thousand filopodia in the rodent cell line Rat2 and measured these parameters in response to genetic, chemical and physical perturbation. Our work shows that length and separation distance have a lognormal pattern distribution over their entire detection range (0.4 μm to 50 μm).

Conclusions

We find that the lognormal distribution of length and separation is robust and highly resistant to perturbation. We also find that length and separation are independent variables. Most importantly, our empirical data is not entirely in agreement with predictions made based on existing theoretical models and that filopodial size and separation are an order of magnitude larger than what existing models suggest.

Background

When mammalian cells migrate, they do so by generating protrusive actin structures in the form of advancing lammellipodia or filopodia

The simple composition of filopodia belies the complex biochemical events that shape their initiation and growth. The pathways controlling the assembly of mature filopodia are controversial, and two different models, convergent elongation and

In mammalian cells, filopodia have a strikingly varied appearance and behavior. Their lengths span greater than two orders of magnitude and they can grow to 50 μm or more in size

Methods

Cell lines and treatments

Rat2 fibroblasts were purchased from American Type Culture Collection (Manassa, VA) and cultured in Dubecco's Modified Eagle Medium High Glucose 1X from Gibco, Invitrogen (Grand Island, NY) containing 10% Fetal Bovine Serum (FBS) (Gibco) and 1% antibiotic/antimycotic (Gibco). The cultured cells were incubated in 10 cm plates at 37°C in 5% CO→_{2}. Cells were treated with bradykinin at 100 ng/ml for 30 minutes using DMSO as a vehicle. For poly-D-lysine experiments, cover slips were coated with 50 μg/ml poly-D-lysine for 2 hours prior to cell plating. Rat2 fibroblast cells ectopically expressing PI4KIIIβ and empty vector controls have been previously described

Immunofluorescence

Rat2 cells were grown to 70-80% confluency, trypsinized with 0.05% 1X Trypsin-EDTA (Gibco), diluted 1:100 and plated in 6 well plates containing glass coverslips (Fisher; Pittsburg, PA). 24 hrs later, cells were fixed in 3.7% paraformaldehyde for 20 minutes, permeabilized with 0.5% Triton-X for 15 minutes and left overnight in IF Buffer (130 mM NaCl, 7 mM Na→_{2}HPO_{4}, 3.5 mM NaH_{s}PO_{4, }7 mM NaN_{3}, 0.2% Triton X-100, 0.1% BSA, 0.05% Tween-20, ph 7.4). The following day, cells were stained for 1 hr with Phalloidin-488 (Invitrogen) diluted 1:200 in 1X PBS (pH7.4) and subsequently stained with Hoescht-405 (Invitrogen) diluted 1:40 in 1X PBS. Coverslips were mounted on glass microscope slides (Fisher) with Fluorescent Mounting Medium (Dako; Carpinteria, CA). Images of single Rat2 cells were obtained from an Olympus Fluoview FV1000 laser scanning confocal microscope. Openlab Software (Improvision, MA) was used to measure filopodia lengths and separation.

Data Analysis

For each length or distance data set, histograms were plotted on a logarithmic axis, with bins of equal width in log-space. For both visualization and statistical fitting purposes, as described below, the empirical cumulative distribution function, F(x), is defined as the fraction of the data having a value strictly less than x. The empirical probability density function, which was used only for visualization purposes, was taken to be the Parzen windows estimator with a radius parameter of h = 0.25 applied in the log-transformed space. That is, if x_{1 }... x_{n }are the original data and y_{1 }... y_{n }are the transformed data (y_{i }= log_{10}x_{i}) then the probability density function is f(y) = c(y)/n, where c(y) is the number of points y_{1 }... y_{n }for which the absolute difference to point y is less than or equal to h.

We fit different distributions to the data by comparison of idealized and empirical cumulative distribution functions. Let G(x, θ) denote the cumulative distribution function of a statistical distribution with parameter or parameters θ. We judged that filopodia lengths or interfilopodial distances less than 0.4 μm could not be reliably quantified from the images. So, no such measurements were included in our data set. To fit θ based on the data we first defined the "cut-off cumulative distribution function" as G_{C}(x, θ) = 0 if x ≤ 0.4 and G_{C}(x, θ) = (G(x, θ)-G(0.4, θ))/(1-G(0.4, θ)) if x > 0.4. The cut-off function recognizes that our data collection procedure does not record any values smaller than 0.4 μm; in essence, any part of the statistical distribution falling below that threshold is zeroed out and the remainder of the distribution is rescaled so that it integrates to one. We define the error of parameters θ as the sum of squared residuals: E(θ) = Σ_{X}(F(x)- G_{C}(x, θ)), where the sum is over x = 10^{-0.40}, 10^{-0.39}, 10^{-0.38}, ..., 10^{1.60 }for length data and x = 10^{-0.40}, 10^{-0.39}, 10^{-0.38}, ..., 10^{2.30 }for distance data. Parameters θ are fit by minimizing the error E(θ). We fit four different families of distributions in this way: the exponential, which has probability density function g(x, λ) = λ exp(-λx) and cumulative distribution function G(x, λ) = 1- exp(-λx); the powerlaw, which has probability density function g(x, x_{min}, α) = ((α-1)/x_{min})(x/x_{min})^{-α }and cumulative distribution function G(x, x_{min}, α) = 1 - (x/x_{min})^{1-α }for x ≥ x_{min}; the Gaussian, which has probability density function g(x, μ, σ) = (2πσ^{2})^{-1/2 }exp(-(x-μ)^{2}/2σ^{2}); and the lognormal, which has probability density function g(x, μ, σ) = (2πσ^{2}x^{2})^{-1/2 }exp(-(ln(x)-μ)^{2}/2σ^{2}) for x > 0.

Results

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

The actin cytoskeleton in Rat2 cells

**The actin cytoskeleton in Rat2 cells**. Rat2 fibroblast cell stained for actin (white) and DNA (blue). The leftmost panel shows transverse actin stress fibers (S) and filopodia (F) as hair-like projections from the cell perimeter. The central panel shows counting of individual filopodia lengths (L1, L2, L3, L4) and distance separation (D1, D2, D3). The right panel shows a mitotic cell with retraction fibers (R) indicated. The red scale bar is 10 μm.

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}
_{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

Filopodia length distribution is unimodal and best fits a lognormal model

**Filopodia length distribution is unimodal and best fits a lognormal model**. **(A) **Histogram of all the filopodia lengths (n = 1,682), and independent experiments 1, 2 and 3 with n = 745, n = 573 and n = 364 respectively. **(B) **Empirical cumulative distribution functions (CDFs) of lengths with the lognormal and other statistical models for all the experiments combined and each independent experiment 1, 2 and 3. **(C) **Empirical PDFs and CDFs of filopodia lengths for 53 cells.

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 ^{2})^{-1/2}exp(-(x-μ)^{2}/2σ^{2})) or power law (p(x) = ((α-1)/x_{min})(x/x_{min})^{-α}) distribution (Figure

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

Filopodia distance separation distribution is unimodal and best fits a lognormal model

**Filopodia distance separation distribution is unimodal and best fits a lognormal model**. **(A) **Histogram of all the filopodia distances (n = 1,670), and independent experiments 1, 2 and 3 with n = 741, n = 569 and n = 360 respectively. **(B) **Empirical cumulative distribution functions (CDFs) of distances with the lognormal and other statistical models for all the experiments combined and each independent experiment 1, 2 and 3. **(C) **Empirical PDFs and CDFs of filopodia distances for 53 cells.

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

Filopodia length and distance separation are independent

**Filopodia length and distance separation are independent**. **(A) **Scatter plot of filopodia lengths against the neighboring distances shows no correlation. Identically coloured data points represent measurements from the same cell. **(B) **Scatter plot of the distances between filopodia versus the distances between the next filopodia in a clockwise direction around the edge of the cell. **(C) **Scatter plot of the lengths of filopodia versus the lengths of the next filopodia in a clockwise direction around the edge of the cell.

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

As shown in Figure

Robust nature of the filopodial length and distance separation lognormal distribution

**Robust nature of the filopodial length and distance separation lognormal distribution**. Perturbation analysis of Rat2 cells after genetic, chemical and physical induction of filopodia. **(A) **PПI4KIIIβ expression induces filopodia and increases both length and separation distance relative to the empty vector control. Rat2 cells stably expressing PI4KIIIβ and controls have been previously described **(B) **Bradykinin, a chemical inducer of filopodia, increased the length but had no effect on the interfilopodial distances compared to the DMSO control. Rat2 cells were treated with 100 ng/mL bradykinin. **(C) **Poly-D-lysine, a physical inducer of filopodia, increased the length of filopodia modestly but had no effect on distances.

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

Lastly, we chose to analyze the relationship between length and separation distance in the perturbed cells. Figure

Filopodia length and distance separation are independent after perturbation

**Filopodia length and distance separation are independent after perturbation**. Graph of filopodial length and separation distance for PI4KIIIβ, Bradykinin and Poly-D-lysine perturbations. Identically coloured data points represent measurements from the same cell.

Discussion and Conclusions

Among the goals for this study was to identify a method to quantitatively describe the filopodial system in a given cell population. Our interest in this idea first arose when we began to quantitate the effect that PI4KIIIβ expression had on filopodia in the mammalian breast cancer cell line BT549 _{2 }
_{2 }abundance regulates filopodia by recruiting actin-remodeling proteins to the migratory leading edge

The extensive filopodia length data that we have collected provide an empirical base on which to test existing theoretical models of filopodia growth

It is worth mentioning that existing models of filopodia formation are based on the assumption that the actin filaments within an individual filopodium are as long as the filopodium itself

It is important to note some limitations in our current study. Firstly, we have relied exclusively on the Rat2 cell line and other cells may behave differently. The B16 melanoma line is commonly used to study filopodia and these cells show much smaller filopodia and more uniform length distribution relative to Rat2

The inter-filopodial distance separation data that we collected also allow sus to test the predictions made by Mogilner et al.

We were initially surprised to find that PI4KIIIβ expression not only increases filopodial length, but also increases their separation (Figure

We find that both filopodial length and separation distance have a lognormal distribution. Earlier biophysical modeling of filopodia-like structures in lymphocytes shorter than 1.1 μm has suggested the restraining force of the membrane might account for a heavy right-tailed length distribution

The lognormal distribution is not uncommon in biology. For example, species abundance distributions and long-term survival in breast cancer are lognormal functions

Authors' contributions

ANH carried out the cell biology studies, performed the microscopy and measurements, participated in the data analysis and drafted the manuscript; AAM generated cell lines, assisted in the cell biology, participated in the microscopy and measurements and assisted in manuscript revision; TJP participated in the design of the study, performed the mathematical analysis of the data, participated in manuscript writing and revision and generated the figures; JML conceived of the study, participated in its design and coordination, participated in the data analysis and participated in manuscript writing and figure generation. All authors read and approved the final manuscript.

Acknowledgements

Supported by grants from the Natural Sciences and Engineering Research Council of Canada (JML, TJP). AAM is supported by a fellowship from the Canadian Institute of Health Research. We thank D. Bickel, J. Copeland, H. McBride, S. Michnick, D. Pinke, and S. Shaikh for helpful discussion and reading of this manuscript. We also thank H. McBride for help with and use of the confocal microscope.