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RESEARCH PAPERS

# Mechanics of Biomacromolecular Networks Containing Folded Domains

[+] Author and Article Information
H. Jerry Qi1

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 and Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309qih@colorado.edu

Christine Ortiz

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Mary C. Boyce

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Note that refolding events could easily be included by the addition of the frequency of “backwards” events term, i.e., the frequency of refolding. In this paper, we focus on monotonic loading and thus the probability of refolding events is negligible and is thus neglected.

Alternatively, following random walk statistics, $ro$ can be taken to be $r0=Nl$ when the network formation is a random process; this is the common assumption in rubber elasticity. The non-zero initial length of $r0$ prescribes a pretension in the network chains; the pretension is balanced by equal and opposite intermolecular interactions as in rubbery solids; or by interactions with the lipid bilayer, other molecules and/or the cytosol.

The Kuhn length for the networked spectrin is approximately the length of a folded domain, whereas that used in fits of the single molecule tests is approximately the residue length; we speculate this difference to be due to the heterodimer structure of the networked spectrin as well as due to the preparation of single molecules for testing.

1

To whom correspondence should be addressed.

J. Eng. Mater. Technol 128(4), 509-518 (Apr 27, 2006) (10 pages) doi:10.1115/1.2345442 History: Received November 04, 2005; Revised April 27, 2006

## Abstract

The force-extension behavior of single modular biomacromolecules is known to exhibit a characteristic repeating pattern of a nonlinear rise in force with imposed displacement to a peak, followed by a significant force drop upon reaching the peak. This “saw-tooth” pattern is a result of stretch-induced unfolding of modules along the molecular chain and is speculated to play a governing role in the function of biological materials and structures. In this paper, constitutive models for the large strain deformation of networks of modular macromolecules are developed building directly from statistical mechanics based models of the single molecule force-extension behavior. The proposed two-dimensional network model has applicability to biological membrane skeletons and the three-dimensional network model emulates cytoskeletal networks, natural fibers, and soft biological tissues. Simulations of the uniaxial and multiaxial stress-strain behavior of these networks illustrate the macroscopic membrane and solid stretching conditions which activate unfolding in these microstructures. The models simultaneously track the evolution in underlying microstructural features with different macroscopic stretching conditions, including the evolution in molecular orientation and the forces acting on the constituent molecular chains and junctions. The effect of network pretension on the stress-strain behavior and the macroscopic stress and strain conditions which trigger unfolding are presented. The implications of the predicted stress-strain behaviors on a variety of biological materials are discussed.

###### FIGURES IN THIS ARTICLE
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Copyright © 2006 by American Society of Mechanical Engineers
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## Figures

Figure 1

(a) Schematic of a single modular polymer chain where the black rectangles represent folded modules which are connected by unfolded regions. (b) A representative folded module; here, structure of an individual Drosophiliaα-spectrin segment 14 repeat domain which exhibits a three-helix bundle (1).

Figure 2

Schematic of a two-dimensional network: (a) The four-chain network, (b) the four-chain network after deformation

Figure 3

Schematic of the three-dimensional eight-chain network: (a) The eight-chain network; (b) The eight-chain network after deformation

Figure 4

FJC model compared to experiment: Spectrin force versus extension behavior; simulation uses a single valued energy barrier; experimental data from Rief (2), the inset shows the dependence of unfolding force on extension rate

Figure 5

Force versus extension curves from simulations of stretching multimolecule. The force is reported as force per unit molecule.

Figure 6

(a) Uniaxial tensile nominal stress versus stretch behavior of a uniform and distributed network membrane, where the distributed network behavior is also shown at different strain rates; (b) Evolution in chain stretch ratio and chain angle with macroscopic stretch λ; (c) Membrane shear stress-tanγ behavior; (d) Evolution of chain stretch ratio and chain angles with tanγ

Figure 7

Eight-chain network model results for: (a) uniaxial tension nominal stress versus stretch behavior; (b) evolution in chain stretch ratio and chain angle with macroscopic stretch during uniaxial tension; (c) equibiaxial tension nominal stress versus stretch behavior; (d) evolution in chain stretch ratio and chain angle with macroscopic stretch during equibiaxial tension

Figure 8

Effect of pretension on the uniaxial stress-stretch behavior of a three-dimensional network (at the strain rate of 0.1∕s for the distributed network)

Figure 9

Comparison of FJC model and WLC model predictions for uniaxial membrane nominal stress-strain behavior

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