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Cytoskeletal bundle bending, buckling, and stretching behavior

F-actin bundles constitute principal components of a multitude of cytoskeletal processes including stereocilia, filopodia, microvilli, neurosensory bristles, cytoskeletal stress fibers, and the sperm acrosome. The bending, buckling, and stretching behaviors of these processes play key roles in cellular functions ranging from locomotion to mechanotransduction and fertilization. Despite their central importance to cellular function, F-actin bundle mechanics remain poorly understood. Here, we demonstrate that bundle bending stiffness is a state-dependent quantity with three distinct regimes that are mediated by bundle dimensions in addition to crosslink properties. We calculate the complete state-dependence of the bending stiffness and elucidate the mechanical origin of each. A generic set of design parameters delineating the regimes in state-space is derived and used to predict the bending stiffness of a variety of F-actin bundles found in cells. Finally, the broad and direct implications that the isolated state-dependence of F-actin bundle stiffness has on the interpretation of the bending, buckling, and stretching behavior of cytoskeletal bundles is addressed.


Filamentous actin (F-actin) is a stiff biopolymer that is tightly crosslinked in vivo
by actin-binding proteins (ABPs) to form stiff bundles that form major constituents of a multitude of slender cytoskeletal processes including stereocilia, filopodia, microvilli, neurosensory bristles, cytoskeletal stress fibers, and the acrosomal process of sperm cells ( Fig. 1) (1, 2). The mechanical properties of these cytoskeletal processes play key roles in a broad range of cellular functions-the bending stiffness of stereocilia mediates the mechanochemical transduction of mechanical stimuli such as acoustic waves to detect sound and motion (3,4), the critical buckling load of filopodia and acrosomal processes determines their ability to withstand compressive mechanical forces generated during cellular locomotion and fertilization (5)(6)(7), and the entropic stretching stiffness of cytoskeletal bundles mediates cytoskeletal mechanical resistance to cellular deformation (8). Thus, a detailed understanding of F-actin bundle mechanics is fundamental to gaining a mechanistic understanding of cellular function.
Cells utilize a myriad of ABPs to assemble and crosslink F-actin filaments into bundles of precisely regulated dimensions that range dramatically from several (microvilli, stereocilia, stress fibers) to tens (acrosome, filopodia) and even hundreds (macrochaete neurosensory bristles in Drosophila) of microns and from tens (filopodia, microvilli) to hundreds (stereocilia, neurosensory bristles) of constituent filaments (1,2).
Of the multitude of ABPs expressed by the cell, only a small subset is used to crosslink neighboring F-actin filaments in cytoskeletal bundles. Fascin is the predominant ABP in filopodia and neurosensory bristles, plastin is prevalent in microvilli and stereocilia, scruin in the limulus sperm acrosome, and α-actinin predominates in cytoskeletal stress fibers. Each ABP has a distinct mechanical shear stiffness that has been demonstrated to strongly affect F-actin bundle bending stiffness in vitro (9)  beam. It is not obvious a priori, however, whether these drastically different regimes of bending stiffness are determined only by ABP type, or whether bundle dimensions play a commensurate leading role. Moreover, an additional bending regime that is intermediate to decoupled and fully coupled bending has been observed in F-actin (9), microtubule (12,13), and carbon nanotube (14) bundles, however its nature and mechanical origin remain obscure.
In this article, we demonstrate that F-actin bundles have three distinct bending regimes that are mediated by both ABP type and equally importantly by bundle dimensions-namely diameter and length. We isolate the origin of the third, intermediate regime to decoupled and fully coupled bending and demonstrate that it interestingly exhibits a bending stiffness that, unlike the other regimes, is proportional to crosslink shear stiffness and bundle length. We derive a generic set of design parameters that delineates the three bending regimes and use it to make novel predictions for the bending behavior of cytoskeletal bundles that are not easily amenable to experimental measurement. Finally, the direct and broad implications that these results have on the interpretation of the bending, buckling, and stretching behavior of the multitude of cytoskeletal bundles found in cells are addressed.

Model
We consider generic fiber bundles of length L that consist of n cubically-or hexagonally-packed fibers, as is typical of highly crosslinked F-actin, MTs, and SWNTs (10,15,16) (Fig. 2A) δ . Crosslinks therefore constrain transverse fiber deflections to be equal but allow interfiber relative slip. The consideration of ordered fiber bundles simplifies our analyses to in-plane bending of N n = fiber layers that are crosslinked to their nearest neighbors in-and out-of-plane ( Fig. 2A), where the corresponding 3D bundle bending stiffness is related simply to its 2D counterpart by, While the stretching and bending stiffness of F-actin (19)(20)(21), MTs (13,20), and SWNTs (22,23) are experimentally known, the shear stiffness of fiber crosslinks is often unknown. One exception is provided by the recent measurements of Claessens et al., (9), in which an effective k was measured for the ABPs plastin, fascin, and α-actinin. In other cases, k may in principle be calculated directly using atomistic-based simulation methods or measured using micromanipulation techniques. The spacing between fibers, t, can be measured from crystal structures (15,24,25) and δ can be determined from chemical equilibrium and fiber packing considerations (9,16).
Some biological crosslinks such as the ABPs fascin and plastin have finite offrates, In addition to finite off-rates, real crosslinks also have a finite extensibility k ⊥  (19,20). The minimum axial distance between coplanar crosslinks in hexagonally-packed F-actin bundles is 37.5 nm (16). The associated transverse fluctuation of F-actin is 1 nm r ⊥ ∼ , which is much less than the inter-axial spacing between fibers, ( ) 10 nm electron microscopy (16), and the assumption of inextensible crosslinks is justified.
The three-dimensional bundle bending stiffness can in general be expressed as a function of all the independent parameters in the model, ( , , , , , , ) dimensionless form may be written, * and the limit of short crosslinks ( ) f t d has been applied. We will shortly demonstrate, however, that * B κ depends only on the two independent parameters, n and the fiber-coupling parameter, The fiber coupling parameter is evidently a measure of the competition between crosslink shearing and fiber stretching, where / L δ is the number of crosslinks per fiber.

Numerical analysis
To elucidate the detailed mechanics of fiber-bundle bending, we begin by examining the bending response of model fiber bundles subject to simple three-point bending computationally using the finite element (FE) method (Materials and Methods). § § In analogy with experiment, B κ is evaluated as a function of increasing fiber number n, for bundles of fixed α, which is akin to fixing the fiber and crosslink properties (Fig. 3A). Decoupled bending characterized by linear scaling is observed for small α and fully coupled bending for large α. Interestingly, between these two limits we also observe an intermediate range of α that displays a smooth crossover from quadratic- § § Three-point beam bending refers to pinning or clamping a beam at its ends and applying a transverse point load at its center. The resultant load-deflection yields a measure of its apparent bending stiffness (Materials and Methods).
to linear-scaling in n. This is in contrast to a bending stiffness that is characterized by an a α < < (4,5). This crossover was recently observed experimentally in F-actin bundles crosslinked by fascin using a controlled variation of n at fixed fascin concentration (9). Re-plotting * δε is a characteristic deformation that is constant along the bundle axis but may differ between fiber layers, k.
The crossover between the fully coupled and intermediate regimes is then determined by the point at which crosslink shearing becomes favorable to fiber stretching, is the variation in stretching energy and δν be evaluated, which we turn to next.
The mean axial force in the k th fiber is related via Hooke's law to its mean axial strain, ( ) , which in the fully coupled regime increases linearly with distance, y, from the bundle neutral axis, ( ) , which is given by, ( ) , , where a constant characteristic radius of curvature has been assumed in evaluating υ , consistent with the present scaling picture. Note the differences between the expressions for the fiber axial force and the crosslink shear force: The former increases through the bundle cross-section whereas the latter increases along the bundle axis.
Variations in fiber stretching and crosslink shearing energy associated with the imposed relaxation ( ) k δε may now be calculated using the above results to yield, which may be re-written, which has been derived previously for the special case of two filaments in the context of DNA mechanics (17). It depends on j q through the non-dimensional factor 2 ( ) ( ) /12 c q qL = and on the design parameters n and α isolated previously using scaling analysis. In three-point bending at zero temperature the bending stiffness is well- The q-dependence of B κ demonstrates that it is an apparent material property that depends on the nature in which the bundle is probed. This is in contrast to a standard worm-like polymer, which is defined as having an intrinsic bending stiffness that is mode-number-independent (31). Thus, inference of B κ from "macroscopic" bundle observables such as the mean-square end-to-end distance, the zero-temperature forcedeflection relation, or the fluctuation spectrum by associating the bundle with an equivalent worm-like polymer will yield different apparent values for B κ (30).

Connection to Timoshenko theory
Fiber bundles consisting of SWNTs (14,32) and MT protofilaments (13, 33) have recently been analyzed using Timoshenko beam theory *** . In this approach, the heterogeneous microstructure of the bundle is ignored so that the bundle can instead be treated as a single homogeneous medium with effective macroscopic geometric and mechanical properties. The bundle stiffness computed from Timoshenko theory for threepoint bending with pinned boundary conditions may be written (34), Thus, Timoshenko theory converges to the same fully coupled bundle bending stiffness as the microscopic-based theory when, ( ) n α , and crosses-over to the sheardominated regime when, ( ) n α ∼ (Fig. 3B). Unlike the microscopic theory, however, Timoshenko theory does not asymptote to the decoupled bending regime when, ( 1) α , and it is only asymptotically correct for large bundles, ( 1) n , because it does not explicitly account for the heterogeneous underlying fiber structure of the bundle (Fig.   3B).

Application to F-actin bundles
The bending stiffness of F-actin bundled by fascin, plastin, α-actinin, or nonspecific PEG-induced depletion forces was recently measured experimentally using an in vitro droplet assay in which F-actin bundles form compact stable rings (9). In that work, the dependence of bundle stiffness on bundle diameter n was systematically explored for several ABP concentrations. Here, we focus on fascin and instead explore the effects of Macroscopic measurements of its bending stiffness have been made using hydrodynamic flow (11), where it was determined that the bundle exhibits fully coupled bending. This independent macroscopic observation is consistent with the a priori prediction of the fiber-based model, in which the ranges in α and n are determined from the parameters probed experimentally (Fig. 4). The shear stiffness of fascin is used as an estimate for scruin, although the molecular structure and interfilament packing of the latter suggest that it is considerably stiffer (15).
Vertebrate hair cell stereocilia are finger-like projections in the inner ear that serve as mechanochemical transducers for sound and motion (Fig. 1A). Ranging from 1-10 μm in length, each stereocilium consists of up to 900 hexagonally-packed F-actin filaments crosslinked predominantly by plastin (1,2,35). Macroscopic measurements of the bending stiffness of hair cell stereocilia bundles and of the root of individual stereocilia made using microneedle manipulation (4) yielded decoupled bending behavior. Together with their short length, the low stiffness of plastin, 6 10 N/m k − ≤ (9), places their theoretical stiffness deep in the decoupled regime, consistent with these experimental observations (Fig. 4).
Knowledge of the microstructure and the filament and crosslink mechanical properties may be used to also make novel predictions of B κ for cytoskeletal processes that have not been measured experimentally. Of course, in vivo F-actin bundles are typically crosslinked by more than one ABP type (36), however one ABP is prevalent in each process and is therefore expected to dominate the bundle response (1, 2). (9) and function mechanically to enhance the tensile stiffness of cells. Each of these processes is predicted to exhibit decoupled bending due to its relatively short length. Filopodia are active Factin bundles present at the leading edge of motile cells and neuronal growth cones that increase in length during locomotion and growth (2) (Fig. 1B). Consisting of 10-30 filaments, they are predominantly crosslinked by fascin and typically range from 1-10 μm, but may reach lengths of up to 30-40 μm in certain cases such as in the sea urchin embryo (5,37). As a final F-actin bundle example, we consider the 11 fascin-crosslinked bundles constituting the Drosophila neurosensory bristle. Each bundle is ≈400 microns long and contains 500-700 filaments in macrochaetes (38, 39) (Fig. 1C).

Implications for in situ mechanical function
Filopodia and the sperm acrosome are amongst two F-actin-based cytoskeletal processes that are subject to potentially high compressive forces in vivo during locomotion, growth, and fertilization (5,6,40). Bundles subject to axial compression in situ will lead to structural failure at a critical load that is determined by the Euler buckling limit,  ABPs are complex, hierarchically structured macromolecules that may dissociate and rebind as well as exhibit highly nonlinear force-extension response depending on the time-and length-scales probed (41). Accordingly, the coupling parameter α is in fact a nonlinear function that depends on the degree and time-scale of crosslink deformation.
Thus, bundles in one bending regime may potentially switch to other regimes depending on the deformations imposed in situ, and this may be an important modeling consideration. Investigation of the nonlinear-and time-dependent-response of fiber bundles provides a rich avenue of investigation that will require careful and controlled experimentation together with atomistic modeling to unravel in the future.

Conclusion
Crosslinked F-actin bundles are key structural components of a broad range of cytoskeletal processes. To date, a common conception has been that these bundles display two limiting bending behaviors that depend solely on the stiffness of intervening where k is normalized properly to account for discretization. Three-point bending is   (12). A carbon nanotube fiber bundle: (g) SWNTs bundled noncovalently by van der Waals interactions. Reprinted with permission from (14). Copyright (1999) by the American Physical Society.    (14); and (i) irradiationcrosslinked SWNT bundles (32). Spacing between ABPs is taken to be the minimal in-plane value for hexagonally-packed bundles, 37.5 nm δ = (16). Extensional stiffnesses are,  (14), respectively.