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# ps9 - 2.797/20.310/3.0536.024 Fall 2006 MOLECULAR CELLULAR...

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2.797/20.310/3.0536.024 Fall 2006 Problem Set # 9 Solutions to these problems will either be distributed or discussed in recitation. You do NOT need to hand in a solution for grading. Problem 1: A test for membrane bending stiffness Some cell membranes experience bending as a routine matter. For example, red blood cells have to bend in order to fit through a capillary that’s slightly smaller than the cell. White blood cells sometimes have to squeeze through small cracks between endothelial cells (lining the blood vessels) to reach a site of infection. Axons of some nerves that run through joints get folded whenever the joint is bent. In order to test the bending response of a cell membrane, one could isolate a small section of the cell membrane, clamp one end, and then apply a load at the other end. This is not readily done in real life, but for this question, ignore those technical details. f L x 1 x 2 block block t The blocks are immobile and do not deform so that both the displacement and slope of the membrane at that point is fixed and equal to zero. The thickness of the membrane is t, and the force applied to the membrane per unit length perpendicular to the page is f . The membrane can be assumed to be two-dimensional, with infinite length perpendicular to the page. Thermal fluctuations are also to be ignored in parts (a) and (b). 1

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a) Derive the equation of deformation of the membrane u 2 = u 2 (x 1 ), assuming the membrane is a uniform solid with elastic modulus E and total length L . b) Compare this solution to an order-of-magnitude solution obtained by approximating the governing equation used in (a). Are they similar in form? c) Now consider the effects of thermal fluctuations on the behavior of the membrane. Assuming that the dimension of the membrane perpendicular to the page in the
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ps9 - 2.797/20.310/3.0536.024 Fall 2006 MOLECULAR CELLULAR...

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