Cytoskeletal bundle mechanics

Abstract

The mechanical properties of cytoskeletal actin bundles play an essential role in numerous physiological processes, including hearing, fertilization, cell migration, and growth. Cells employ a multitude of actin-binding proteins to actively regulate bundle dimensions and cross-linking properties to suit biological function. The mechanical properties of actin bundles vary by orders of magnitude depending on diameter and length, cross-linking protein type and concentration, and constituent filament properties. Despite their importance to cell function, the molecular design principles responsible for this mechanical behavior remain unknown. Here, we examine the mechanics of cytoskeletal bundles using a molecular-based model that accounts for the discrete nature of constituent actin filaments and their distinct cross-linking proteins. A generic competition between filament stretching and cross-link shearing determines three markedly different regimes of mechanical response that are delineated by the relative values of two simple design parameters, revealing the universal nature of bundle-bending mechanics. In each regime, bundle-bending stiffness displays distinct scaling behavior with respect to bundle dimensions and molecular composition, as observed in reconstituted actin bundles in vitro. This mechanical behavior has direct implications on the physiological bending, buckling, and entropic stretching behavior of cytoskeletal processes, as well as reconstituted actin systems. Results are used to predict the bending regimes of various in vivo cytoskeletal bundles that are not easily accessible to experiment and to generate hypotheses regarding implications of the isolated behavior on in vivo bundle function.

FIGURE 1

Fiber bundles consisting of F-actin. (A) Ciliary bundle from the sensory epithelium of a bullfrog saccule consisting of ∼60 stereocilia (courtesy of David P. Corey and John A. Assad). (B) Filopodium protruding from the lamellipodium of a mouse melanoma cell (reproduced from Svitkina et al. (81) by copyright permission of The Rockefeller University Press). (C) Epithelial microvilli. (D) Drosophila neurosensory micro- and macrochaete bristles (reproduced from Tilney et al. (82) with the permission of The American Society for Cell Biology).

FIGURE 2

Theoretical bundle model. (A) Cross-linked fiber bundle with N = 16 fibers. Discrete cross-links couple nearest-neighbor fibers mechanically in stretching and bending. (B) (left) Deformed backbone of a fiber bundle subject to in-plane bending; (middle) close-up view of three typical fibers showing fiber and cross-link deformations in (faded gray lines) decoupled and (solid black lines) fully coupled bending; (right) transverse distributions of fiber axial displacement, and (arrows) mean axial displacement, in (faded gray lines) decoupled and (solid black lines) fully coupled bending.

FIGURE 3

Theoretical bundle-bending stiffness. (A) Dependence of normalized bending stiffness, on filament number, N, for various constant values of the fiber coupling parameter, α = (bottom to top). Thick lines denote (bottom) decoupled and (top) fully coupled bending regimes. (B) Dependence of on α at constant (bottom to top). Dotted lines correspond to Timoshenko theory predictions. Inset: Dependence of the crossover values, α, of the fiber coupling parameter on bundle filament number, N, at the (bottom curves) decoupled-to-intermediate and (top curves) fully coupled-to-intermediate regime crossovers for (squares) pinned and (circles) clamped boundary conditions. Solid lines indicate N-independent and linear-in-N scaling. Crossover values of α are defined by the value of α at which is within a factor of two of its limiting decoupled and fully coupled values.

FIGURE 4

Experimental and theoretical bending stiffness of fascin cross-linked actin bundles for N = 27 ± 6. Experimental bundle stiffness (symbols) is measured using a microemulsion droplet system for a range of fascin concentrations with corresponding mean spacings, δ: (squares) 40 nm, (circles) 56 nm, (diamonds) 68 nm, (pointed-up-triangles) 225 nm, (pointed-down-triangles) 412 nm, as described in Claessens et al. (29). Bundle length is varied in an uncorrelated fashion by a factor of over two. Cross-linker axial spacing is calculated using a simple Langmuir isotherm approximation, (83,84), where is the minimum in-plane spacing between ABPs in hexagonally ordered actin bundles (31) and is the fascin-actin dissociation constant (83,84). Theoretical bundle stiffness (solid line) is calculated using Eq. 2 with c = 5 (Appendix) assuming N = 27, and bounding curves (dashed lines) that account for experimental uncertainty are calculated using N = 21 and N = 33.

FIGURE 5

Bundle-bending stiffness state diagram for various cytoskeletal bundles. Dashed lines denote crossovers between (I) decoupled, (II) shear-dominated, and (III) fully coupled bending regimes. (a) Acrosomal process of the horseshoe crab sperm cell (64); (b) vertebrate hair cell stereocilia (2,3,66); (c) brush-border microvilli (2,3,85); (d) stress fibers; (e) filopodia (16); (f) Drosophila neurosensory bristles (59); and (g) outer pillar hair cell MT bundles (25). Spacing between ABPs is taken to be the minimal in-plane value for hexagonally packed bundles, (31). Extensional stiffnesses are for F-actin (41) and MTs (40), respectively.