Transcript Slide 1
Summer 2008 Sylabus Biophysics II Cell Biophysics English: RM224, 15:15-18:30 Lecture notes with the according references will be published in the www. 1. 2. 3. 4. 5. 6. Basic Cell Biology Membrane Biophysics Active and Passive Physics of the Cytoskeleton Intracellular Transport Neurophysics Photosynthesis 3. Active and Passive Physics of the Cytoskeleton 3.1 Fundamental Polymer Physics Literatur: • M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Press • M. Doi, Introduction to Polymer Physics, Oxford Press Flory-Huggins solution theory Flory-Huggins solution theory is a mathematical model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change ΔGm for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments. The thermodynamic equation for the Gibbs free energy change accompanying mixing at constant temperature and (external) pressure is We first calculate the entropy of mixing, the increase in the uncertainty about the locations of the molecules when they are interspersed. In the pure condensed phases — solvent and polymer — everywhere we look we find a molecule.[3] Of course, any notion of "finding" a molecule in a given location is a thought experiment since we can't actually examine spatial locations the size of molecules. The expression for the entropy of mixing of small molecules in terms of mole fractions is no longer reasonable when the solute is a macromolecular chain. We take account of this dissymmetry in molecular sizes by assuming that individual polymer segments and individual solvent molecules occupy sites on a lattice. Each site is occupied by exactly one molecule of the solvent or by one monomer of the polymer chain, so the total number of sites is N1 is the number of solvent molecules and N2 is the number of polymer molecules, each of which has x segments. In addition to the entropic effect, we can expect an enthalpy change. There are three molecular interactions to consider: solvent-solvent w11, monomer-monomer w22 (not the covalent bonding, but between different chain sections), and monomer-solvent w12. Each of the last occurs at the expense of the average of the other two, so the energy increment per monomer-solvent contact is The total number of such contacts is where z is the coordination number, the number of nearest neighbors for a lattice site, each one occupied either by one chain segment or a solvent molecule. That is, xN2 is the total number of polymer segments (monomers) in the solution, so xN2z is the number of nearest-neighbor sites to all the polymer segments. Multiplying by the probability φ1 that any such site is occupied by a solvent molecule, we obtain the total number of polymer-solvent molecular interactions. Theta solvent The conformation assumed by a polymer chain in dilute solution can be modeled as a random walk of monomer subunits using a . However, this model does not account for steric effects. Real polymer coils are more closely represented by a self-avoiding walk because conformations in which different chain segments occupy the same space are not physically possible. This “excluded volume” effect causes the polymer to expand. Chain conformation is also affected by solvent quality. The intermolecular interactions between polymer chain segments and coordinated solvent molecules have an associated energy of interaction which can be positive or negative. If a solvent is “good,” interactions between polymer segments and solvent molecules are energetically favorable, and will cause polymer coils to expand. If a solvent is “poor,” polymer-polymer self-interactions are preferred, and the polymer coils will contract. The quality of the solvent depends on both the chemical compositions of the polymer and solvent molecules and the solution temperature. If a solvent is precisely poor enough to cancel the effects of excluded volume expansion, the “theta (θ) condition” is satisfied. For a given polymer-solvent pair, the theta condition is satisfied at a certain temperature, called the “theta (θ) temperature.” A solvent at this temperature is called a theta solvent. In general, measurements of the properties of polymer solutions depend on the solvent. However, when a theta solvent is used, the measured characteristics are independent of the solvent. They depend only on short-range properties of the polymer such as the bond length, bond angles, and sterically favorable rotations. The polymer chain will behave exactly as predicted by the random walk or ideal chain model. This makes experimental determination of important quantities such as the root mean square end-to-end distance or the radius of gyration much simpler. Polymer blends Free energy of mixing two polymers A and B: G = RT [1/xA ФA ln ФA + 1/xB ФB ln ФB + Χ ФA ФB ] xA, xB >>1, X<0 Long synthetic polymers do not mix! Only the biopolymers in the cell mix! Worm-like chain The worm-like chain (WLC) model in polymer physics is used to describe the behavior of semi-flexible polymers; it is sometimes referred to as the Kratky-Porod worm-like chain model. Several biologically important polymers can be effectively modeled as worm-like chains, including actin filaments, DNA, and intermediate filaments. The WLC model envisions an isotropic rod that is continuously flexible; this is in contrast to the freely-jointed chain model that is flexible only between discrete segments. The worm-like chain model is particularly suited for describing stiffer polymers, with successive segments displaying a sort of cooperativity: all pointing in roughly the same direction. At room temperature, the polymer adopts a conformational ensemble that is smoothly curved; at T = 0 K, the polymer adopts a rigid rod conformation. For a polymer of length l, parametrize the path of the polymer as , allow t(s) to be the unit tangent vector to the chain at s, and r(s) to be the position vector along the chain. end-to-end distance: The persistence length P is a basic mechanical property quantifying the stiffness of a long polymer. Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persistence length, the properties can only be described statistically, like a three-dimensional random walk. Formally, the persistence length is defined as the length over which correlations in the direction of the tangent are lost. Let us define the angle Θ between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance s away from position 0. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance. A piece of cooked spaghetti has a persistence length on the order of 10 cm. Double-helical DNA has a persistence length of about 50 nanometers. This can be used to show that a Kuhn segment is equal to twice the persistence length of a worm-like chain. Stretching Worm-like Chain Polymers Laboratory tools such as atomic force microscopy (AFM) and optical tweezers have been used to characterize the force-dependent stretching behavior of the polymers listed above. An interpolation formula that describes the extension x of a WLC with contour length L0 and persistence length P in response to a stretching force F is where kB is the Boltzmann constant and T is the absolute temperature (Bustamante, et al., 1994; Marko et al., 1995). In the particular case of stretching DNA in physiological buffer (near neutral pH, ionic strength approximately 100 mM) at room temperature, the compliance of the DNA along the contour must be accounted for. This enthalpic compliance is accounted for by adding a stretch modulus K0 to the above relation: where a typical value for the stretch modulus of double-stranded DNA is around 1000 pN and 45 nm for the persistence length (Wang, et al., 1997). Persistence length, actin End-to-end distance: J. Wilhelm and E. Frey, Radial distribution function of semiflexible polymers, Phys. Rev. Lett. 77, 2581 (1996). L. LeGoff, O. Hallatschek, E. Frey, and F. Amblard, Tracer Studies on F-Actin Fluctuations, Phys. Rev. Lett. 89, 258101 (2002). a b 5 µm 5 µm c d 5 µm 10 µm 5 µm Semiflexible polymers: Persistence length, actin: 7-9 µm Persistence length, actin + phalloidin: 15-16 µm Analysis of thermal fluctuations: Isambert, H., P. Venier, A. C. Maggs, A. Fattoum, R. Kassab, D. Pantaloni, and M.-F. Carlier. 1995. Flexibility of actin filaments derived from thermal fluctuations. Effect of bound nucleotide, phalloidin, and muscle regulatory proteins. J. Biol. Chem. 270: 11437-11444. 10 µm Homework 8 • Why are block co-polymers a strategy to overcome the fact that polymers do not mix? • Why do cytoskeletal polymers mix?