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Lecture #3 Notes

Lecture #3 Notes - Physics of Cellular Materials...

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Physics of Cellular Materials: Biomembranes Tom Chou 1 1 Dept. of Biomathematics, UCLA, Los Angeles, CA 90095-1766 (Dated: December 6, 2002) Here I will review the mathematics and statistical physics associated with surfaces that form the boundaries of cells and internal organelles. Mechanical models such as Canham-Helfrich energy, bilayer couple (BC), and the area difference elasticity (ADE) models will be presented. The concept of entropic tension will also be discussed. I. MEMBRANES Membranes define the boundaries of cells and internal organelles such as mitochondria, ER, Golgi bodies, chloro- plasts, etc. They are comprised of amphiphillic molecules, those with parts that favor associations with water, and parts that “repel” water. Lipid bilayers are essentially two-dimensional incompressible fluids with a molecular thickness that imparts bending elasticity. Figure 1: A schematic of the membrane surfaces in a typical cell. A closer view of the plasma membrane shows a large number of proteins and structures associated with the lipid bilayer. A molecular dynamics view of a small patch of lipid bilayer reveals the molecular structure. A Thermodynamics Before presenting models of membrane bending, we consider the concept of surface tension of a membrane by looking at its thermodynamics. The origins of tension for material membranes like lipid bilayers is physically different from the familiar notion used to study e.g. the liquid-vapor interface. At an air-water interface, surface tension arises because of the difference between the energies of a water-water interaction compared to a water-air molecule interaction. At an air-water interface, the water molecules are coordinated with only about half the number of nearest neighbor water molecules. Bulk water has a lower energy than the waters at the interface. Thus, an interfacial energy arises due to the number of water-air interactions. Now consider lipid membranes. Lipid bilayers are inextensible so the number of lipid head-water interactions is relatively fixed unless the lipid molecules exchange with lipid molecules in the bulk. Therefore, we can define independent conjugate tensions for both material area A , and the projected area A p . Various free energies, or thermodynamic potentials, can be derived and used depending on the experimental measurement. Consider the patch of membrane shown in Fig. 2(b). It has a fairly fixed material area A since membranes are nearly inextensible. Therefore, to a very good approximation, A will be proportional to the total number of lipids in the patch
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2 A p A (b) A A’>A (a) Figure 2: (a) Molecular surface tension. (b) Fluctuations of a fixed material surface. of membrane. This patch is fixed on a frame of area A p , the projected area of the membrane patch. Both areas may be relevant. When one is considering mechanical pulling experiments on a thermally fluctuating membranes, the pulling force is the conjugate variable to A p , while if one is concerned with say adsorption of protein on a membrane surface, the relevant area is A . An inextensible material area
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