Statistical Mechanics: Linking Microscopic to Macroscopic Thermodynamics - Prof. Gabriele , Study notes of Biochemistry

Download Statistical Mechanics: Linking Microscopic to Macroscopic Thermodynamics - Prof. Gabriele and more Study notes Biochemistry in PDF only on Docsity! 1 University of Washington - Department of Chemistry Chemistry 453 - Spring Quarter 2008 Lecture 1 3/31/08 Text Reading: Ch. 11 p. 614-616; 646-649 A. Introduction to Statistical Mechanics Biological systems are complex macroscopic systems composed of many components or particles (atoms, molecules, etc.). Only recently have we been able, though single- molecule imaging techniques, to observe individual biological molecules at work • Thermodynamics deals with the properties of macroscopic systems at equilibrium (Chem 452) • Properties of macroscopic systems at equilibrium are described by the usual state variables and state functions of equilibrium thermodynamics (covered in Ch. 1-5) including pressure P, volume V, temperature T, mass M, entropy S, Gibbs’ free energy G, internal energy E, etc. • The properties of systems composed of a small number of components …or the individual particles themselves, are described in terms of mechanical properties including: physical coordinates (x,y,z), momenta (px, py, pz), mass, potential and kinetic energies (classical mechanics) or, for subatomic properties, through the quantum mechanical wave equation and its evolution (e.g. dipole moments) • When we study a biological system, we measure macroscopic properties (temperature, pressure, sedimentation in a gradient, separation in a gel or through a chromatographic material); what are these observables telling us about the properties of the molecules that compose the system? Example 1: RNA melting and its three-dimensional structure – RNA adopts complex three-dimensional structures that are key to its ability to carry out biological functions such as protein synthesis or RNA splicing. We study these structures by a variety of methods, most of which monitor macroscopic properties of the system we study. For example, we heat an RNA in solution and observe changes in the UV absorption spectrum as the temperature increases. We are monitoring the unfolding of the RNA from a highly structured conformation containing high degrees of tertiary structure, then the unfolding of the secondary structure, finally the complete loss of order within the single- stranded RNA chain. We control the temperature (macroscopic property), we measure the UV spectrum of an RNA solution containing very large numbers of RNA molecules of identical sequence. What we would really like to do is be able to relate these macroscopic properties we measure with the microscopic properties of the RNA we study because these are potentially functionally important (why does the UV spectrum change upon changes in temperature and what does that tell us about RNA structure; how stable is its structure; how does it depend on sequence; does it change under different physiological conditions; etc.) Example 2: DNA size, shape and separation – DNA from a bacteriophage (e.g. T4) is a double helical molecule with a molecular weight of 108. In solution, this molecule can 2 form a very wide range of conformations, from a tight coil to an extended form. When we measure how fast it diffuses or how it sediments in a gradient (e.g. if we want to separate and purify this DNA), these properties depend on the average dimension and shape of the molecule, which in turn depend on its size and on the range of possible conformations of this molecule. We want to characterize these properties as well as we can, for example because we want to exploit our ability to separate different DNA molecules based on their size and shape for the purpose of purifying (or sequencing) it. How to relate microscopic (i.e. mechanical) properties to macroscopic (thermodynamic) properties is a central problem in the physical and chemical sciences. Statistical methods (statistical thermodynamics) relate the average action of individual molecules to measurable macroscopic properties • In principle, macroscopic properties like pressure could be calculated from a description of the mechanical properties for all the particles in a system (e.g. a molecular dynamics calculation). In general, this is not practical or even possible because of the large number (e.g. 6.02x1023) of particles that typically compose chemical and biochemical systems. • Thus, macroscopic properties of a system are related to microscopic properties of individual molecules using statistical methods. • Thermodynamic properties such as energy, entropy or free energy are directly related to the average energy of each molecule in a system and how molecules are distributed among all possible energies • Heat and work done when there is a change in the system an be calculated from the changes in the energy levels and subsequent changes in the distribution of molecules among the energy levels Example 3 – Helix/coil transition in polypeptides – The disruption of hydrogen bonds that leads to the helix/coil transition of a polypeptide chain (e.g. at increasing temperature) can be understood in terms of the statistical distribution of bonds formed and broken at a given temperature, which depends on the energy difference between the hydrogen bonded and non-hydrogen bonded states of the polypeptide The field that uses statistical methods to calculate macroscopic properties of large collections of particles, molecules, atoms, etc. based on the microscopic mechanical properties of the individual components of the system is called statistical mechanics (or statistical thermodynamics). By doing so, we can relate the structure and interactions of biological molecules (which ultimately is what we wish to know) with the macroscopic properties we often measure or exploit during separation Mechanical Properties Thermodynamic Properties of Molecules of Systems Coordinates xi, yi, zi Temperature T Momenta pxi, pyi, pzi → Statistical → Pressure P Masses mI Mechanics Mass M Kinetic Energies EKi Entropy S, Free Energy G Potential Energies Uij Internal Energy E