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 [22
]. PI4KIIIβ is one of four mammalian kinases, PI4KIIα, PI4KIIβ, PI4KIIIα and PI4KIIIβ, which generate PI4P (Phosphatidylinositol 4-phosphate) from PI (phosphatidylinositol) [25
]. Our work with this kinase and an oncogenic protein that activates it, eEF1A2, suggested that eEF1A2 and PI4KIIIβ stimulated filopodia production by activating the production of PI(4,5)P2
abundance regulates filopodia by recruiting actin-remodeling proteins to the migratory leading edge [27
]. While the effect that PI4KIIIβ expression had on filopodia was visually striking and qualitatively apparent, quantitative description proved difficult because of the highly variable appearance of filopodia in a given cell population. Filopodia numbers vary per cell; their lengths in a single cell frequently span more than an order of magnitude and very long filopodia sometimes appear even in unstimulated cells. In the end, we adopted a system that approximated our qualitative visual evaluation [22
]. We scored cells that had 10 or more filopodia > 3 μm in length as positive and the remainder as negative [22
]. Based on this criterion, PI4KIIIβ expression has a demonstrable and significant numerical effect on filopodia production [22
]. However, this descriptive system was unsatisfying because there is nothing intrinsically unique about filopodia longer than 3 μm nor is having 10 or more long filopodia of obvious biologic importance. In this study, we hoped to identify objective and quantitative parameters of the filopodial system to determine whether or not any given stimulus was altering filopodia production. Based on our current analysis, we propose that μ the peak of the density of the lognormal distribution represents a useful quantitation parameter of the filopodial network. The robust nature of the lognormal distribution (Figure ) among independent replicates of the same cell population indicates that it appears to be a tightly regulated feature of the cell type. Moreover, individual cells of the same population have a similar CDF distribution (Figure ). Based on our analysis of the bradykinin, poly-D-lysine perturbation, and PI4KIIIβ expression, we believe that counting ~300 filopodia in a population will allow quantitative determination of the effect that a given stimulus has on filopodial appearance based on alterations in μ.
The extensive filopodia length data that we have collected provide an empirical base on which to test existing theoretical models of filopodia growth [14
]. Our empirical data does not closely match many existing theoretical models. For example, Lan and Papoian predict that the frequency distribution of filopodia lengths will be tight and will peak at ~0.6 μm [16
] while our empirical peak is ~3 μm. Moreover, we frequently observe filopodia >5 μm in length, which is not readily accounted for in their work. The presence of these long filopodia is also not in agreement with Atligan et al., who propose that mechanical buckling forces provide strong limits on filopodia growth beyond a length of 1.7 μm [14
]. It is possible that adhesion between filopodia and the growth substrate may reduce the effect that buckling forces have in retarding filopodial growth, but this remains to be empirically tested. While Mogilner & Rubinstein predict that most of the filopodia will be of 2 μm in length [15
], in closer agreement with our studies, their modeling does not account for the lognormal distribution of the filopodial nor the large numbers of long filopodia that we observe. Mogilner & Rubinstein postulate that three different parameters limit filopodia growth dependent on filopodial length [15
]. According to their model, membrane resistance limits filopodia below 0.4 μm in length, between 0.4-1.5 μm filopodia length is limited by buckling, while longer filopodia growth is limited by the diffusion of G-actin. More recent modeling has suggested that the generation of long filopodia (~4-6 μm length) may be the result of active G-actin transport within filopodia or the loss of capping protein function [17
]. However, we consistently observe filopodia in the > 6 μm range, therefore we hypothesize that additional factors must be at work.
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 [14
]. Because of the directionality of the filopodial actin fibers, filopodial growth directly reflects actin polymerization at the tips. Many studies support this model [4
], but a recent cryo-electron tomographic analysis of filopodia in Dictyostelium
suggests otherwise [28
]. This ultrastructural analysis indicates that filopodia are composed of discontinuous actin filament bundles ~100 nm in length. The discontinuous nature of the actin filaments within these filopodia could therefore allow for longer filopodia because the buckling forces that affect individual filaments would be predicted to be smaller. However, the commonality of this structure in filopdia in other cell types remains to be determined.
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 [13
]. The cell-type specificity of filopodial quantitative parameters indicates that differing biochemical pathways are at play in individual cell lines. Another limitation of our study is that we have not measured filopodia in living cells. Individual filopodia undergo phases of growth, stasis and retraction during their lifespan [13
]. Our use of fixed (non-living) cells, treats filopodia as stationary objects and, in a sense, ignores their dynamism. To help circumvent this issue, we have collected data from a large population of cells. Because we have made no attempt to synchronize or otherwise manipulate the filopodial growth cycle, our collective dataset represents filopodia in all their dynamic phases. Large-scale analysis of filopodia in living cells will be necessary to betterunderstand and measure filopodial dynamics.
The inter-filopodial distance separation data that we collected also allow sus to test the predictions made by Mogilner et al. [15
]. Mogilner based their model on previous work by Svitkina et al. [11
], which provided evidence that filopodia are initiated from the fusion of cytosolic actin fibers. These lamellar actin fibers fuse into a λ-shaped precursor and subsequent actin polymerization creates a filopodium. Based on the distribution of λ-precursors and their lateral motion, Mogilner modeled interfilopodial spacing to a range of 1-3 μm, with a tight distribution. Another theoretical study, based on the idea that membrane protein adhesion complexes regulate the initiation of protrusive structures, also suggests that filopodia will have separation distances in this range [29
]. However, we observe that filopodia are often widely spaced, frequently having separation distances of 10 μm or more. It should be noted that lambda precursors are not the only proposed pathway of filopodia initiation, and filopodia may also form from de novo
nucleation by Formin proteins independent of lamellar actin strands [4
]. Moreover, filopodial fusion, an event predicted [14
] but not yet reported may also function to increase inter-filopodial distances. These may account for our large filopodial spacing.
We were initially surprised to find that PI4KIIIβ expression not only increases filopodial length, but also increases their separation (Figure ). Since no biochemical regulators of interfilopodial separation have been identified to date, it is not immediately apparent how PI4KIIIβ increases this parameter. However, it is possible that concomitant with an increase in filopodial length, PI4KIIIβ may be depleting a pool of G-actin or actin polymerizing factors that control filopodial initiation. On the other hand, our work indicates that filopodia length and separation are independent variables (Figure ), suggesting they are regulated by different mechanisms. This is also further buttressed by our observation that PI4KIIIβ affects both length and separation, while bradykinin and poly-D-lysine only affect filopodial length.
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 [20
]. In this work, a Gaussian distribution accounts for filopodia up to ~0.3 μm in length and then an exponential distribution of longer filopodia generates a heavy right tail. Qualitatively, this is consistent with our data. The presence of some experimental skew even in our log-transformed length dataset indicates that the lognormal does not wholly account for the data. Indeed, fitting a distribution of the type described by Gov [27
] results in a tighter fit to the data (results not shown). However, it is not clear that the biophysical model studied by Gov has relevance to our data, as the lengths that we observe are an order of magnitude larger than the ones modeled by Gov. Moreover, we found that much of the skew in the log-transformed data is due to cell-to-cell variation. Some individual cells showed positive skew, but others showed negative skew. As such, we did not feel there was sufficient support to adopt the four-parameter Gov model [27
] or even a three-parameter skew-lognormal model for filopodial 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 [30
]. Fruit and flower sizes also show lognormal distributions [32
]. With respect to filopodia, the cellular significance of lognormal length and separation distance distributions is unclear. However, this distribution is highly robust and resistant to perturbation. PI4KIIIβ expression and bradykinin treatment affect filopodia length but do not change the lognormal distribution. This suggests that the lognormal distribution is robust and likely reflects strong biophysical constraints on the pathways controlling filopodial dynamics. Filopodia length distribution is lognormal from the smallest length that we confidently detect (0.4 μm) up to almost 100 μm. Generally speaking, a lognormal distribution can arise as the product of a number of random variables. There are dozens, perhaps hundreds of proteins that affect actin polymerization and higher-ordered polymer assembly [1
]. The biochemical mechanism(s) through which their combinatorial action creates a lognormal distributed function is unclear to us. Nevertheless, fluctuations in the concentrations of these proteins may be among the factors influencing filopodia length and separation. Membrane forces are also likely to be important [19
]. Further theoretical modeling of the actin cytoskeleton is likely to be necessary to resolve this issue.