Protein Forced Unfolding and Its Effects on the Finite Deformation Stress-Strain Behavior of Biomacromolecular Solids
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Protein Forced Unfolding and Its Effects on the Finite Deformation Stress-Strain Behavior of Biomacromolecular Solids H. Jerry Qi1,3, Christine Ortiz,2 Mary C. Boyce1 Departments of Mechanical Engineering1, and Materials Science and Engineering2, Massachusetts Institute of Technology, Cambridge, MA 02139 Department of Mechanical Engineering, University of Colorado, Boulder, CO 803093 ABSTRACT Many proteins have been experimentally observed to exhibit a force-extension behavior with 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 (a "saw-tooth" profile) due to successive unfolding of modules during extension. This behavior is speculated to play a governing role in biological and mechanical functions of natural materials and biological networks composed of assemblies of such protein molecules. In this paper, a constitutive model for the finite deformation stress-strain behavior of crosslinked networks of modular macromolecules is developed. The force-extension behavior of the individual modular macromolecule is represented using the Freely Jointed Chain (FJC) statistical mechanics model together with a two-state theory to capture unfolding. The single molecule behavior is then incorporated into a formal continuum mechanics framework to construct a constitutive model. Simulations illustrate a relatively smooth “yield”-like stress-strain behavior of these materials due to activate unfolding in these microstructures. INTRODUCTION Many types of proteins are known to have a modular multi-domain architecture composed of folded modules covalently linked in series (Fig. 1). Single molecule extension experiments using atomic force microscopy and related techniques have identified successive force-induced unfolding of individual folded domains[1-3] resulting in a force-extension behavior with a “sawtooth” profile; a characteristic repeating pattern of a nonlinear rise in force to a peak, followed by a force drop after each peak [1-3]. Upon reaching a critical force level (which is dependent upon the extension rate), a module will unfold, increasing the overall chain contour length and thus, increasing the configurational space available to accommodate the extended chain length. Due to the entropic nature of the single molecule extension behavior, the increase in configurational space results in a decrease in the force level for a given chain end-to-end distance and thus, the observed drop in force upon unfolding. Understanding how the “saw-tooth” single molecule force-extension behavior is translated into an overall material stress-strain behavior is important in order to understand and predict the behavior of natural materials and biological structures composed of these molecules. This paper focuses on the development of a constitutive model for the finite deformation stress-strain behaviors of biological systems composed by modular macromolecules. The FJC[4] model together with two-state transition theory [
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