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A Coarse-Grained Model for Force-Induced Protein Deformation and Kinetics

Helene Karcher ^*, Seung E. Lee ^*, Mohammad R. Kaazempur-Mofrad  and Roger D. Kamm ^*

^* Department of Mechanical Engineering and Division of Biological Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; and Department of Bioengineering, University of California, Berkeley, California 94720

Correspondence: Address reprint requests to Prof. Roger D. Kamm, 500 Technology Square, Rm. NE47-315, Massachusetts Institute of Technology, Cambridge, MA 02139. Tel.: 617-253-5330; Fax: 617-258-5239; E-mail: rdkamm{at}mit.edu.

Force-induced changes in protein conformation are thought to be responsible for certain cellular responses to mechanical force. Changes in conformation subsequently initiate a biochemical response by alterations in, for example, binding affinity to another protein or enzymatic activity. Here, a model of protein extension under external forcing is created inspired by Kramers' theory for reaction rate kinetics in liquids. The protein is assumed to have two distinct conformational states: a relaxed state, C₁, preferred in the absence of external force, and an extended state, C₂, favored under force application. In the context of mechanotransduction, the extended state is a conformation from which the protein can initiate signaling. Appearance and persistence of C₂ are assumed to lead to transduction of the mechanical signal into a chemical one. The protein energy landscape is represented by two harmonic wells of stiffness ₁ and ₂, whose minima correspond to conformations C₁ and C₂. First passage time t_f from C₁ to C₂ is determined from the Fokker-Plank equation employing several different approaches found in the literature. These various approaches exhibit significant differences in behavior as force increases. Although the level of applied force and the energy difference between states largely determine equilibrium, the dominant influence on t_f is the height of the transition state. Distortions in the energy landscape due to force can also have a significant influence, however, exhibiting a weaker force dependence than exponential as previously reported, approaching a nearly constant value at a level of force that depends on the ratio ₁/₂. Two model systems are used to demonstrate the utility of this approach: a short -helix undergoing a transition between two well-defined states and a simple molecular motor.

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