Micromechanical model for elasticity of the cell cytoskeleton

Semiflexible polymer networks, such as cell cytoskeleton, differ significantly from their flexible counterparts in their deformation energy storage mechanism. As a result, the network elasticity is governed by both enthalpic and entropic variations. In addition, the enthalpic effect shows two distinct regimes of energy storage mechanism, the affine and nonaffine regimes. In the past, computation-based modeling on random networks, such as the Mikado model, was used to demonstrate the physical mechanism of mechanical deformation of semiflexible networks. These models are computationally intensive and hence are difficult to apply to studying whole cells. In this paper, we develop a micromechanical model to predict the average macroscopic elastic properties of a random, semiflexible, biopolymer network. The model employs a unit cell consisting of four semiflexible chains and four equivalent axial-bending springs. The proposed unit-cell-based micromechanical model represents a statistically average realization of the actual network and gives the average mechanical properties, such as the shear modulus. Comparisons between the model predictions and Mikado model results confirm that this micromechanical model captures the essential deformation physics revealed from previous studies on the actual network and is capable of predicting the transition between nonaffine and affine deformations. This model can be used to develop efficient continuum constitutive models of the cytoskeleton in the future.

Figure 1

Electron microscope image of semidilute solution of cytoskeleton [32].

Figure 2

2D random network of fibers, $L/l_c=29.09$, Mikado model [13].

Figure 3

Unit cell idealization of a cytoskeleton.

Figure 4

Actual and effective lengths of the main chains.

Figure 5

(a) Cross-link arrangements in the actual network. (b) Average attachment of the cross-linked elements on the main chain. The last two attachments are attached to the main chain in the unit cell; all other attachments are replaced by axial-bending springs. (c) Equivalent spring [the equivalent of all the springs in (b)] and its attachment to the main chain.

Figure 6

(Color online) (a) Normalized shear modulus $G/G_{\text{affine}}$ vs $l_b/L$ and $L/l_c=29.09$; (b) axial energy fraction for the same case. The actual network prediction is from the Mikado model [13].

Figure 7

(Color online) (a) Nondimensional shear modulus vs $L/l_c$ and for $l_b/L=0.006$; (b) relative energy fraction for the same case.

Figure 8

(Color online) Normalized shear modulus vs $L/l_c$, for three different values of $l_b/L$.

Figure 9

(Color online) Collapse of the curves. With $z=0.5882$ in Eq. (18), all the curves collapse onto one master curve.