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, 89 (2), 782-95

The Physics of Filopodial Protrusion

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The Physics of Filopodial Protrusion

A Mogilner et al. Biophys J.

Abstract

Filopodium, a spike-like actin protrusion at the leading edge of migrating cells, functions as a sensor of the local environment and has a mechanical role in protrusion. We use modeling to examine mechanics and spatial-temporal dynamics of filopodia. We find that >10 actin filaments have to be bundled to overcome the membrane resistance and that the filopodial length is limited by buckling for 10-30 filaments and by G-actin diffusion for >30 filaments. There is an optimal number of bundled filaments, approximately 30, at which the filopodial length can reach a few microns. The model explains characteristic interfilopodial distance of a few microns as a balance of initiation, lateral drift, and merging of the filopodia. The theory suggests that F-actin barbed ends have to be focused and protected from capping (the capping rate has to decrease one order of magnitude) once every hundred seconds per micron of the leading edge to initiate the observed number of filopodia. The model generates testable predictions about how filopodial length, rate of growth, and interfilopodial distance should depend on the number of bundled filaments, membrane resistance, lamellipodial protrusion rate, and G-actin diffusion coefficient.

Figures

FIGURE 1
FIGURE 1
Organization and characteristic scales of filopodia and lamellipodia.
FIGURE 2
FIGURE 2
Dependence of the critical length, at which the filopodium would buckle, on the number of cross-linked and not cross-linked filaments (solid curves), as predicted by Eq. 2. The dotted lines show the predicted length range for the characteristic numbers of the filaments.
FIGURE 3
FIGURE 3
(A) Computed shape of the actin filaments bundled by short elastic links. (B) The computed buckling force (scaled by the buckling force for one 2-μm-long filament) is plotted as the function of the number of bundled filaments N for the average number of cross-links per 1 μm of each filament's length equal to 0.5, 1, 1.5 (circles) and to 10, 11, 12 (stars). At small cross-linking density, function I(N) is approximated well by I(N) = N (dashed line). At large cross-linking density, function I(N) is approximated well by I(N) = 0.5 × N2 (solid curve). (C) The buckling force was computed as the function of the average number of cross-links per 1 μm of each filament's length for the bundles consisting of N = 3, 6, 9, 12 filaments. Then, for each value of the cross-linking density, the computed force dependence on N was fitted with function A × Nx. The exponent x is plotted as the function of the average number of cross-links per 1 μm of each filament's length (squares). The plot confirms that at small cross-linking density I(N) ∼ N, whereas at large cross-linking density I(N) ∼ N2.
FIGURE 4
FIGURE 4
(A) Results of computer simulations of the 2-D G-actin distribution (a(x, t)) in the filopodium and the small adjacent part of the lamellipodium (distance is in microns; G-actin concentration illustrated with shading is in nondimensional units, see Appendix II for details). Approximate analytical one-dimensional solution described in the text coincides with the computer-simulated G-actin concentration in the filopodium. Both in the lamellipodium, and the filopodium, the linear gradient of G-actin develops in the direction of protrusion. One sample lamellipodial filament at the leading edge is shown. (B) Filopodial length as a function of time predicted by the solution of Eqs. 3 and 4. (C) Stationary filopodial length as a function of the number of bundled filaments for three different values of the lamellipodial filaments' critical angle, as predicted by Eq. 8. Higher value of the critical angle corresponds to slower lamellipodial protrusion.
FIGURE 5
FIGURE 5
(A) Predicted filopodial length limited by the membrane resistance, buckling, and G-actin diffusion as a function of the number of bundled filaments (θc = 80°). (B) Length distribution for 26 filopodial protrusions gleaned from Fig. 2 of Oldenbourg et al. (24).
FIGURE 6
FIGURE 6
(A) Illustration of the lateral drift. Dashed lines represent the lamellipodial leading edge at four consecutive moments of time. Barbed ends of the individual filaments and the Λ-precursor change their position along the leading edge as the edge protrudes. The rest of the figure illustrates the dynamics of the actin bundles: bundling of the individual filaments (rate b), maturation of the precursors into the filopodium (rate m), merging of two precursors into the filopodium (rate r1), merging of the precursor with the filopodium (rate r2), and merging of two filopodia into one (rate r3). (B) Distance distribution between neighboring filopodia for 26 protrusions gleaned from Fig. 2 of Oldenbourg et al. (24). (C) Results of Monte Carlo simulations of the initiation, lateral drift, maturation, and merging of Λ-precursors (light gray) and filopodia (dark gray). (Horizontal axis) Distance along the lamellipodial leading edge in microns; the vertical axis is time in seconds. (D) Distance distribution between neighboring filopodia based on the stochastic model simulations. (E) Results of Monte Carlo simulations (dots) confirm the analytical prediction that the density of filopodia along the leading edge is proportional to the square root of the rate of initiation of Λ-precursors. We used the value Vd = 0.06 μm/s; b is plotted in units of 0.25 μm−1s−1. (F) Results of Monte Carlo simulations (dots) confirm the analytical prediction that the density of filopodia along the leading edge is inversely proportional to the square root of the drift rate. We used the value b = 0.02 μm−1s−1; Vd is plotted in units of 0.2 μm/s.
FIGURE 7
FIGURE 7
Nondimensionalized linear densities of fascin along the filopodial actin bundle. The horizontal axis shows distance in microns; the origin corresponds to the filopodial tip. (Solid line) Fascin associated with F-actin; (dashed line) “activated” (decreasing) and “inactivated” (increasing) fascin; (dotted line) fascin associated with F-actin along the Λ-precursor's bundle.

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