2. Consider an engineering mechanics analysis of cellular mechanosensing via focal adhesions shown in the schematic on the following page. Characteristic elasticity of the extracellular matrix (ECM) is depicted as a spring with constant k that represents the stiffness of the ECM resisting motion at X₁. Viscous forces acting on the ECM and FAK molecule that resist motion at x₁ and X₂ are represented as dashpots with damping constants YECM and YFAK, respectively. As discussed in class, the cell probes the stiffness of the ECM by 'tugging' with actomyosin contractility (constantly applied but variable force, f, exerted at point x₂). FAK sensor activity is represented mathematically as a potential function that depends on the length x2 - x1, U(x2 -x₁). If the ECM deforms, assume that there is equivalent motion at x₁ and x₂ (no transference of load to the force-bearing FAK molecule). If the ECM does not deform due to stiffening, there is no motion at x1, resulting in a change in the length X2 - X₁. Ignoring thermal and stochastic effects that influence protein movement in the cell membrane, the two equations shown below provide a mathematical framework for analysis of motion at points X₁ and x₂ and the impact on FAK sensor potential. Assume the viscosity of the extracellular fluid and cytoplasm remain constant unless stated otherwise. (Model based on Bell and Terentjev, Focal Adhesion Kinase: The Reversible Molecular Mechanosensor, Biophysical Journal, 112(11): 2439-2450, 2017. https://doi.org/10.1016/j.bpj.2017.04.048.) B. C. √x-0 D. mu Кесм YECM integrin adhesion to ECM U(x₂-x₁) -FAK YFAK 'sensor potential Fonction actomyosin Hugging Force cytoskeleton dx, Y FAK Derive a relation for tug force when the ECM deforms. -Kecu X₁ + ECM extracellular matrix focal adhesion complex A. Using the system of two equations shown above, derive a relation for actomyosin tug force when the system is in mechanical equilibrium. dxa -du d(x₂-x₁) dt= dU d(x₂-x₁) +f Derive a relation for the spring constant required to activate the FAK sensor potential for the tug force applied in part B. Use this mathematical framework to explain how a transient increase in cytoplasmic viscosity near the focal adhesion complex would impact FAK sensor potential.