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Fig. 9-4. Schematic representation of an endothelial cell subjected to fluid shear stress. The force exerted on the surface must be transmitted through the cell by the cytoskeleton to the substrate to which the cell is attached. Some structural filaments attach to the cell nucleus, so that the force is transmitted through the nucleus as well as the extranuclear part of the cell. The lower portion of the figure suggests some of the biochemical cascades triggered by the force. From Davies, 1995.

Actin mechanics

The complicated combination of semiflexible filaments and binding proteins that form the structural matrix within a cell presents a formidable challenge to the scientific investigator. Many of the key interactions, such as the details of the FLNa binding to actin polymers, are still subjects of active research. The fraction of actin molecules polymerized into structural fibers varies from cell to cell, and the mean can range from 35 percent to 80 percent depending on the state of stress, the degree of cellcell attachment, and time. As will be described in the section on actin dynamics, the actin filaments polymerize and depolymerize continuously, so that the whole internal structure is replaced within a time scale that is tens of minutes to hours. To make the situation even more interesting, thermal fluctuations of the filaments could contribute substantially to the apparent cell stiffness. Yet just the simplicity of the randomness tempts one to find models that can at least scale the behavior of the mixture and arrive at working conclusions regarding the structural rigidity of the cell when exposed to external forces.

178 J.L. McGrath and C.F. Dewey, Jr.

It is attractive to look at two bodies of literature for examples from other fields. The first is polymer physics, where a body of literature is presented on various thermally driven models. These works are well summarized in the book by Boal (2002). Included in these models are floppy chains, semiflexible chains, and welded chains. A second field is the study of porous solids, which range from structured anisotropic systems such as honeycombs to foams with random isotropic bubble inclusions. This field is reviewed in the classic treatise by Gibson and Ashby in 1988, with a second edition in 1997 (Gibson and Ashby, 1997; Gibson and Ashby, 1988).

It is possible to derive a simple model for the elastic modulus of the actin cytoskeleton by picturing it as a collection of relatively stiff elements attached to one another with stiff joints. In quantitative terms, following nomenclature from the polymer literature (Gittes and MacKintosh, 1998),1 this means that the so-called persistence length over which the filaments will bend because of thermal agitation, lp, is long compared to the distance between attachment points (or mesh size), l, and l is in turn large compared to the characteristic thickness of the structural elements, t. Then the intracellular structure can be compressed, stretched, and sheared by applying forces to the filament ensemble through its attachments to the membrane, to the substrate, and to the adjacent cells.

This approach was taken by Satcher and Dewey (1996) who used the analogy of a porous solid to obtain numerical values for the stiffness of cellular actin networks. A simple representative model originally proposed by Gibson and Ashby (1988) considers a three-dimensional rectangular meshwork of short rods connected to the lateral sides of adjacent elements as shown in Fig. 9-5. The key feature of the model is that the structural elements are placed into bending by applied forces. Although the bonds between the filaments are shown in the model as being rigid and the prototypical geometry is taken to be cubic cells, similar scaling of the rigidity of the matrix would be expected if the joints were simply pinned and the prototypical geometry were triangular. In that case, individual elements would buckle with applied stress. In the matrix described by Fig. 9-5, the ratio of the apparent density of the matrix ρ to the density of the solid filament material, ρs , varies as (ρ /ρs ) (t/ l )2.

Comparing the idealization of Fig. 9-5 with the actual cytoskeletal configuration of Fig. 9-1B, one can see many simplifications. First, the angles with which the filaments come together to make the matrix vary considerably. One should recognize that Fig. 9-1B is a view looking down into a three-dimensional structure, and the actual lattice is much more open than is apparent from the micrograph. The vast majority of junctions have an angle between the intersecting filaments that is greater than 45 degrees, and many approach right angles. What is important to the representation is that it puts individual structural elements into bending, thereby causing a deflection δ that can be simply computed from beam theory. The scaling of the geometry then allows the overall elastic modulus to be computed as a function of the density of filaments and their bending stiffness. Because the characteristic length l of the cellular structures is on the order of 100 nm and the individual structural elements have a typical dimension

tof about 7 nm, one would expect the beams to be reasonably stiff in bending.

1Gittes and MacKintosh use the symbols ξ for the characteristic dimension of the lattice and a for the characteristic thickness of the structural element.

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Fig. 9-5. The porous solid model of Gibson and Ashby applied to represent the structure of actin filaments within the cell. (a) is the undistorted lattice and (b) shows the action of an applied force. The vertices of the filaments are assumed to be at right angles and tightly bound for purposes of the calculation. In reality, the angles between filaments vary and the lattice spacing is not uniform; further, the bonds between individual filaments are not necessarily rigid, but may simply pin the joint. From Satcher and Dewey, 1996.

The results of the calculation show that strain ε in the lattice, ε = δ/ l, is proportional to the stress per unit area, σ , of the matrix, and the proportionality constant is the effective Young’s modulus, E , for the material. Using bending theory for the strain ε, it is found (Satcher and Dewey, 1996) that

(E /Es ) ≈ C1(ρ /ρs )2

where C1 is a constant and E is the Young’s modulus of solid actin, which is taken to be 2.3 GPa (Gittes et al., 1993). In order to complete the quantitative calculation, C1 must be determined. That was done using empirical tabulations for polymer foams ranging over many orders of magnitude in density and structural properties as represented in Gibson and Ashby (1988). The result is that C1 ≈ 1. An identical scaling law is obtained if the lattice element is put into shear rather than compression, with the front-running constant of 3/8 instead of one.