# Long-distance propagation of forces in a cell

*Entry by Angelo Mao, AP 225, Fall 2010*

**Title:** Long-distance propagation of forces in a cell

**Authors:** Ning Wang, Zhigang Suo

**Journal:** Biochemical and Biophysical Research Communications

**Volume:** 328 (2005)

**Pages:** 1133–1138

## Summary

The researchers propose two theoretical models for the propagation of "locally applied forces" throughout the body of a cell. Contrary to preexisting models, these models balance the effects of the actin bundles running throughout the inside of the cells against the cytoskeleton network. These models predict that the effects of force propagation via stiff actin bundles would far supersede the effects of force dampening via the cell's cytoskeleton. The results of theory are confirmed by experimentally applying a "local" force on the cell surface and observing where force deformations occur. Both theoretical and experimental results contribute insight to force propagation in the cell, which, in turn, has implications for cell signaling, behavior, and integrity.

**Soft Matter Keywords:** cell, in vitro, actin, force, modulus

## Theoretical Summary

**Background**

The researchers modeled the cell's interior as being governed by primarily two forces: actin fibers that could act as "force guides" and transmit force from the surface, where it occurred, to distances as far away as the cell diameter; and the internal cytoskeleton (CSK) network, which worked to homogenize the cell. If the force of the CSK network predominated, then the cell should act homogeneously, and the force should dissipate over a distance on the order of the size of the local application.

**Theoretical models**

The researchers proposed two theoretical models: one for longitudinal stresses, called the *stiff fiber model*, and one for transverse forces, called the *prestressed string model*, as applied to the actin bundle.

In the *prestressed string model*, the prestress keeping the actin bundles taut exerts a force of <math> h_{b}^{2} \sigma_{b} \partial^{2} v/ \partial x^{2} dx</math> and is countered by the restoring force of the CSK, which is <math> (G_{m}v/h_{m})h_{b}dx </math>. Equating the two yields a characteristic length <math>L_{1} = \sqrt{ \alpha_{b} h_{b} h_{m}/G_{m}}</math>.

The *stiff fiber model* treats an actin bundle as a spring following Hooke's law, with tensile stress <math> \alpha = E_{b} \partial u/\partial x</math>. A spring-like motion by the actin bundle is countered by the CSK shear stress of <math>\tau = G_{m} u/h_{m}</math>. Equating the two yields the characteristic length <math>L_{2} = \sqrt{ E_{b} h_{b} h_{m} /G_{m}}</math>.

**Theoretical results**

The researchers found equations for displacement of both transverse and longitudinal disturbances, and determined that the CSK has little effect on force and displacement in actin bundles. The characteristic lengths are suggestive of this result, because the moduli of the actin bundles (<math>\alpha_{b}</math> and <math>E_{b}</math>) are <math>10^{3} - 10^{5}</math> greater than the modulus for the CSK, <math>G_{m}</math>. Thus, the length scale for displacement is on the order of the size of the cell and indicates that the force does not dissipate over a small distance.

## Experimental Summary

**Experimental setup**

The researchers used human airway smooth muscles transfected with yellow fluorescent protein (YFP)-actin and other fluorescent tags. A magnetic bead coated with peptides was allowed to bind to focal adhesion complexes on the surface of the cell and subsequently caused to spin. Using the fluorescently tagged nature of the proteins, the researchers were able to track the motion of actin bundles.

**Actin absent**

The model predicted that, in the absence of actin, CSK forces would predominate. This is seen in Fig. 3, in which the regions of CSK displacement are at a distance away from the magnetic bead that are on the order of the size of the "locally applied force."

**Changing prestress**

The model predicted that the characteristic lengths were affected strongly by the presence of prestress on actin bundles. Therefore, disrupting actomyosin interactions by overexpressing a known disruptant, caldesmon, would lower the actin bundle prestress and lead to more localized displacement. This can be seen in the left-hand image of Fig. 4. A drug calcium ionophore A-23187 was introduced to counter the effect of caldesmon, and this was accompanied by distant displacements, as seen in the right-hand image of Fig. 4.

## Discussion

Although previous work had been done on applying forces to the cell and observing force propagation, this experimental setup allowed narrowed the focus on "locally applied forces," which were much smaller than the size of the cell. The two theoretical models proposed by the researchers fit the data well qualitatively, although the article did not test their models quantitatively, as could by done in the measurement of actin displacement. Nevertheless, this research potentially sheds insight in numerous cellular phenomena because forces in the environment, whether chronic or acute, can strongly affect cell behavior, and has been shown to play a role in stem cell differentiation and cancer, for example.