# Actin Filament Length Tunes Elasticity of Flexibly Cross-Linked Actin Networks

Wiki entry by Emily Gehrels, Fall 2012

Based on the article: Kasza, K.E., et.al. (2010). Actin Filament Length Tunes Elasticity of Flexibly Cross-Linked Actin Networks. Biophysical Journal, Vol 99 (4), pg 1091-1100.

## Background

All cells contain a cytoskeleton. This cytoskeleton is made up of actin monomers that have been polymerized into filamentous actin (F-actin), which are bound together with cross-linking proteins. A lot of research has gone into trying to understand how the components and structure of the cytoskeleton lead to the mechanical behaviors that are observed in nature.

There have been several studies performed on F-actin networks that have been reconstituted in vitro using small, rigid cross-links between the actin filaments. These systems show the response of stiffening with increasing applied stress. In these systems the stiffening comes from the F-actin itself. F-actin is a semiflexible polymer that is not straight, but contains bends at room temperature. The stiffening with applied strain occurs when the bends in the filaments are first straightened and then pulled on when fully straight. These systems, however, implicitly assume that the properties of the F-actin dominate the properties of the network as a whole, ignoring the cross-linking proteins. In reality cross-linking proteins can be large and flexible themselves. The goal of this paper is to study the contribution of flexible cross-linking proteins to the overall mechanical properties of actin networks. This paper focuses on one particular large, flexible cross-link that is commonly found in cells, filamin. Actin networks containing filamin behave differently than those containing short, rigid cross-linking proteins. These networks containing filamin are soft, but can can support large shear stresses, producing behavior that is more similar to that of cells than the behavior previously studied in rigidly cross-linked systems.

## Results

Figure 1 Dependence on L of the linear elastic modulus measured at a frequency of 0.1 Hz, $G_{o}$. (A) For filamin networks shown for varying values of $R_{F}$. (B) Rigidly cross-linked networks show qualitatively different behavior. (C) $G_{o}$ for filamin networks at different $R_{F}$ collapse onto a single curve when plotted versus $R_{F}L^2$.
Figure 2 (A) Schematic of network of stiff polymers of mean length L connected by flexible cross-links. (B and C) Schematic of stiff polymer and attached cross-links in a network before (B) and after (C) shear.

By adding gelsolin to an F-actin network, the average length of actin strands is shortened, allowing for dynamic control of the lengths of the actin filaments in the network. This allows for experiments determining the dependence of the network behavior on the length of the actin filaments.

Using stress-controlled rheometry the elastic modulus (G) of the samples can be measured. The value of G for different lengths of actin filaments (L) in A) filamin networks with different molar ratios of filamin to actin ($R_{f}$ ) B) rigidly cross_linked networks can be seen in Figure 1. Note the large difference between the dependence of the elastic modulus on the length of the actin fibers between the two types of samples. The model that is consistent with the length dependence of G shown in Figure 1A described the filamin strands as soft, flexible, linear polymers attached to relatively stiff actin strands (Figure 2A). In this model the behavior of the network under strain is dominated by the stretching of the cross-links. As seen in Figure 2 B and C the cross-links are stretched by different amounts depending on their positions along the actin filament. This dependence on the distance from the center of the actin is what causes the explicit L dependence in the elastic modulus of the form:

$G_{o}=\frac{1}{8} \rho nkL\sim R_{F}L^2$

as seen in Figure 1 C where measurements taken with different values of $R_{F}$ collapse onto a single curve when plotted against $R_{F}L^2$.

The nature of the elasticity of the network is further tested by measuring the nonlinear elastic properties using strain ramps. Strain ramps are a measurement of the stress ($\sigma$) of the system as a result of increasing the strain ($\gamma$) at a fixed rate. The derivative of the stress with respect to the strain (K) is used to determine the nonlinear behavior of the system. The results of the measurements for both filamin cross-linked networks and rigidly cross-linked networks (not shown here) show linear increase in stress with strain (K=1) up to a certain value of strain ($\gamma_{c}$). At $\gamma_{c}$ both networks begin to stiffen and continue to do so until they break. The differences between the two systems is that rigidly cross-linked networks show the same behaviors independent of the value of the length of the actin stands (L), whereas filamin cross-linked networks display increasing values of $\gamma_{c}$ and smaller amounts of stiffening for increasing L values. This behavior for the filamin network is not consistent with the model assuming a stress-strain relation arising from the straightening and then stretching of actin filaments as the rigidly cross-linked results imply. Instead, the stiffening of the network arises from the stiffening of the cross-links themselves as given by:

$\gamma_{c}=4\frac{l_{o}}{L}$

where $l_{o}$ is the contour length of the filamin, which denotes the length at which the polymer reached its maximum extension.

## Conclusion

Both the linear and nonlinear behavior of filamin-gelsolin-F-actin networks are in agreement with a model that assumes that the elasticity of the network is dominated by the elasticity of the cross-links. There is an explicit dependence in both of these behaviors on the length of the actin fibers, which in this model are treated as rigid rods compared to the flexible filamin cross-links. This addition of filamin into the actin network in place of rigid cross-links creates a system that more readily mimics behaviors of real biological cells.