STATISTICAL MECHANICS BEH.410 Tutorial Maxine Jonas February 14, 2003 Why Statistical Mechanics? Understand & predict the physical properties of macroscopic systems from the properties of their constituents Deterministic approach -need of 6N coordinates at t 0 : r i and v i -but typically N ≡ moles (10 23 ) ! “Ensemble” rather than microscopic detail … and its surroundings ?microcanonical, canonical, grand canonical What With Statistical Mechanics? Averages, distributions, deviation estimates… … of microstates: specification of the complete set of positions and momenta at any given time (points on the constant energy hypersurface for Hamiltonian dynamics) Ensemble average & ergodic hypothesis: A system that is ergodic is one which, given an infinite amount of time, will visit all possible microscopic states available to it. The First Law – Work Work, heat & energy = basic concepts Energy of a system = capacity to do work ? At the molecular level, difference in the surroundings state function – independent of how state was reached Energy transfer that makes use of… Heat … chaotic molecular motion Work … organized molecular motion Second Law – Gibbs ? Spontaneous processes increase the overall “disorder” of the universe ? Reasoning through an example - microstates to achieve macrostate Gibbs postulate: for an isolated system, all microstates compatible with the given constraints of the macrostate (here E, V and N) are equally likely to occur - Here 2 N ways to distribute N molecules into 2 bulbs Second Law - Probability L ? Number of (indistinguishable) ways of placing L of the N molecules in the left bulb: ? Probability W L / 2 N maximum if L = N / 2 9 With N = 10 23 , p ( L = R ± 10 -10 ) = 10 -434 possible but extremely unlikely Second Law - Entropy Boltzmann’s constant k = 1.38 x 10 -23 J.K -1 Principle of Fair Apportionment Multiplicity of outcomes Second Law - Entropy The absolute entropy is never negative S ≥ 0 S max at equilibrium 0 0rder 1.391.33 0.69 Flat distribution ≡ high S p i 1 0 0 0 0 0 n e s w 1/2 1/2 1/3 1/3 1/6 1/6 1/4 1/4 1/4 1/4 The Boltzmann Distribution Law ? Maximum entropy principle + constraints E 3 E 2 E 1 ? exponential distribution Partition function The Boltzmann Distribution Law (2) ? Q ≡ number of states effectively accessible to system E low T E high T E medium T ? Q ≡ connection between microscopic models & macroscopic thermodynamic properties and ? More particles have low energy: more arrangements that way The Helmholtz Free Energy ? Systems held at constant T → minimum free energy (≠ S max ) Equilibrium if F (T, V, N) minimum (T fixed at boundaries) Internal energy Entropy F = U - TS ? Example of ‘dimerization’ dim mon 0 -ε F (T) Fundamental Functions U (S, V, N) min (S and V at boundaries) calorimetry S (U, V, N) max (U and V at boundaries) cal. H (S, p, N) min (S and p at boundaries) calorimetry F (T, V, N) min (T and V at boundaries) Internal energy vs.entropy G (T, p, N) min (T and p at boundaries) Enthalpy vs. entropy Macromolecular Mechanics ? Why study the mechanics of biological macromolecules? - provide structural integrity and shape - coupling of geometry & dynamics ? what is possible - importance of conformation for ion channels, pumps… - motility - mechanotransduction, signaling The Gaussian Chain Model (Kuhn) ? Long floppy chain made of N rigid links of length b (free to swivel about joints, overlapping & crossing allowed) ? Valid for small displacements from equilibrium, not large extensions b R Entropic reasoning ? mechanical spring (straightening out ≡ decrease of entropy) The Worm-like Chain Model ? Self-avoiding linear chains (Flory, 1953) ? Freely-jointed chain model (Grosberg & Khoklov, 1988) ? Worm-like chain model: Bending stiffness of polymer on short length scales (diverges for x → L) Persistence length bending thermal (Kratky-Porod) s = L F x x Experimental Validation of Models Extension (x/L) For c e (pN) FJC Hooke’s law dsDNA fit to WLC model 100 10 1 0.1 0.01 0 0.2 0.4 0.6 0.8 1.0 1.2 After Bustamante et al., Current Opinion in Structural Biology, 2001 ? Single-molecule studies of DNA mechanics: Bustamante et al. (2000) Curr. Op. Struct. Biol., 10: 279 ? WLC interpolated: Marko & Siggia (1995) Macromolecules, 28: 8759 Effect of Force on Equilibrium 12 x 1 x 2 FF Reaction coordinate G 1 2 force no force ? Force tilts energy profile ? favors configuration Sources ? Boal D. (2002) Mechanics of the cell. Cambridge University Press ? Bustamante C. et al. (2000) Curr. Op. Struct. Biol., 10: 279 ? Dill K.A. & Bromberg S. (2003) Molecular driving forces: Statistical thermodynamics in chemistry and biology. Garland Science. ? Leland T.W. Basic principles of classical and statistical thermodynamics http://www.uic.edu/labs/trl/1.OnlineMaterials ? Mahadevan L. Macromolecular mechanics, class material. ? Marini D. (2002) Some thoughts on statistical mechanics (notes) ? Marko J.F. & Siggia E.D. (1995) Macromolecules, 28: 8759 ? Stanford encyclopedia of philosophy: plato.stanford.edu/entries ? Tuckerman’s lecture notes: www.nyu.edu/classes/tuckerman/stat.mech ? www.biochem.vt.edu/courses/modeling/stat_mechanics.html