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PLoS One
2013 Jan 01;83:e59010. doi: 10.1371/journal.pone.0059010.

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Islands of conformational stability for filopodia.

Daniels DR
,
Turner MS
.

Abstract
Filopodia are long, thin protrusions formed when bundles of fibers grow outwardly from a cell surface while remaining closed in a membrane tube. We study the subtle issue of the mechanical stability of such filopodia and how this depends on the deformation of the membrane that arises when the fiber bundle adopts a helical configuration. We calculate the ground state conformation of such filopodia, taking into account the steric interaction between the membrane and the enclosed semiflexible fiber bundle. For typical filopodia we find that a minimum number of fibers is required for filopodium stability. Our calculation elucidates how experimentally observed filopodia can obviate the classical Euler buckling condition and remain stable up to several tens of μm. We briefly discuss how experimental observation of the results obtained in this work for the helical-like deformations of enclosing membrane tubes in filopodia could possibly be observed in the acrosomal reactions of the sea cucumber Thyone, and the horseshoe crab Limulus. Any realistic future theories for filopodium stability are likely to rely on an accurate treatment of such steric effects, as analysed in this work.

Figure 1. Sketch of a helically deformed membrane enclosing a helical fiber bundle.The membrane radius is given by , the helical polymer radius is given by , and is the radial size of the enclosed polymer filament bundle.

Figure 2. Contour plot of the total energy per unit contour length from Eq.(12).The energy is plotted as a function of the enclosed filament helical radius , and the extension factor . The membrane bending modulus is and the surface tension is . The same values of and are used in all subsequent figures. The number of filaments in this case is given by . These parameters do not give rise to a local energy minimum. The contours near the top of the plot have values around , those contours near the middle , and the nearest to bottom contours on the plot , (at room temperature).

Figure 3. Contour plot of the total energy per unit contour length from Eq.(12).The energy is plotted as a function of the enclosed filament helical radius , and the extension factor . The number of filaments in this case is given by . These parameters give rise to a local energy minimum at: and , corresponding to one helical winding per of contour length. Both the contours near the top and bottom of the plot have values around , while the closed contour near the middle has a value of .

Figure 4. Contour plot of the total energy per unit contour length from Eq.(12).The energy is plotted as a function of the enclosed filament helical radius , and the extension factor . The number of filaments in this case is given by . These parameters give rise to a local energy minimum at: and , corresponding to one helical winding per of contour length. The closed contour near the top of the plot has a value of , while the contours close to the bottom of the plot have values .

Figure 5. Plot of the extension factor along the axis versus the number of filaments .The values plotted correspond to the energetic minima of the total energy per unit contour length from Eq.(12), for a given number of filaments .

Figure 6. Plot of the polymer helical radius versus the number of filaments .The values plotted correspond to the energetic minima of the total energy per unit contour length from Eq.(12), for a given number of filaments . For comparison, note that .

Figure 7. Plot of the polymer helical winding length versus the number of filaments .The winding length values plotted correspond to the polymer contour length required for one complete helical turn in order to maintain stability of the filopodium.

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