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Research Papers

Structural Mechanics Based Model for the Force-Bearing Elements Within the Cytoskeleton of a Cell Adhered on a Bed of Posts

[+] Author and Article Information
Amit Pathak

Department of Mechanical Engineering,  University of California, Santa Barbara, CA 93106

Christopher S. Chen

Department of Bioengineering,  University of Pennsylvania, Philadelphia, PA 19104

Anthony G. Evans

Department of Mechanical Engineering, Materials Department,  University of California, Santa Barbara, CA 93106

Robert M. McMeeking

Department of Mechanical Engineering, Materials Department, University of California, Santa Barbara, CA 93106; School of Engineering, University of Aberdeen, King’s College, Aberdeen, AB24 3UE, Scotland;  INM—Leibniz Institute for New Materials, Campus D2 2, 66123 Saarbrücken, Germany

J. Appl. Mech 79(6), 061020 (Sep 28, 2012) (8 pages) doi:10.1115/1.4006452 History: Received November 18, 2011; Revised January 18, 2012; Posted March 26, 2012; Published September 26, 2012; Online September 28, 2012

Mechanical forces play a vital role in the activities of cells and their interaction with biological and nonbiological material. Various experiments have successfully measured forces exerted by the cells when in contact with a substrate, but the intracellular contractile machinery leading to these actions is not entirely understood. Tan , (2003, “Cells Lying on a Bed of Microneedles: An Approach to Isolate Mechanical Force,” Proc. Natl. Acad. Sci. USA, 100 (4), pp. 1484–1489) use a bed of PDMS posts as the substrate for cells and measure the localized mechanical forces exerted by the cell cytoskeleton on the posts. In live cell experiments for this setup, post deflections are measured, and from these results the forces applied by the cell are calculated. From such results, it is desirable to quantify the contractile tensions generated in the force-bearing elements corresponding to the stress fibers within the cell cytoskeleton that generate the loads applied to the posts. The purpose of the present article is to consider the cytoskeleton as a discrete network of force-bearing elements, and present a structural mechanics based methodology to estimate the configuration of the network, and the contractile tension in the corresponding stress fibers. The network of stress fibers is modeled as a structure of truss elements connected among the posts adhered to a single cell. In-plane force equilibrium among the network of stress fibers and the system of posts is utilized to calculate the tension forces in the network elements. A Moore-Penrose pseudo-inverse is used to solve the linear equations obtained from the mechanical equilibrium of the cell-posts system, thereby obtaining a least squares fit of the stress fiber tensions to the post deflections. The predicted network of force-bearing elements provides an approximated distribution of the prominent stress fibers connected among deflected posts, and the tensions in each fibril.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

A schematic depiction of a two-dimensional cell lying on top of a bed of posts, where the posts located near the periphery of the cell get deflected inward due to cell contractility. The cytoskeletal mesh responsible for cell contractility is proposed to be equivalent to a truss network composed tensile structural elements connected among the deflected posts.

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Figure 2

(a) Top-view of the fibroblast cell adhered on a bed of PDMS posts, where the cytoskeleton is stained for F-actin. (b) Discrete truss elements are sketched arbitrarily by straight lines that represent the stress fibers equivalent to the overall actin mesh seen dispersed over the entire cell area. This network of truss elements is a pictorial representation of how an actual actin network can be replaced by finite number of discrete stress fibers that generates the equivalent effects of an actual cytoskeletal actin mesh. Here, 10 deflected posts at the periphery of the cell are marked.

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Figure 3

Vector representation of a fibril originating from post i and terminating at post k, in a 2D Cartesian coordinate system. Here, xi and xk are the coordinates of the top of the posts, Tj is the tension in the fibril, oriented at an angle θj, connected between posts i and k.

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Figure 4

A sample dataset for the top-view of the posts adhered to a single cell. Here, 6 posts deflected due to cell contractility are marked.

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Figure 5

Predicted structure of stress fibers on a 6 posts system corresponding to the experimental dataset (Fig. 4). Here, 14 equivalent fibrils represent the underlying actin mesh. Relative tensions in the stress fibers are illustrated according to the associated color map.

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Figure 6

Predicted network of stress fibers on a 10 posts system corresponding to the experimental dataset (Fig. 2). Here, 33 equivalent fibrils represent the underlying actin mesh. Relative tensions in the stress fibers are illustrated according to the associated color map.

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Figure 7

Visualization is shown of a fibroblast cell on a bed of posts. The actin fibers are stained in green. The arrows represent force vectors, showing the deflection of the posts with the lengths of the arrows proportional to the force exerted by the cell on the posts [24].

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