Biomolecular Thermodynamics: From Theory to Application (Foundations of Biochemistry and Biophysics)

This book introduces the concepts and practical tools necessary to understand the behavior of biological macromolecules at a quantitative level. It begins by describing biochemical phenomena using principles of classical and statistical thermodynamics. Unlike other books, this text goes beyond theory to explain in detail how the equations are applied to the analysis of experimental measurements. This emphasis on real-world applications is continued throughout and is a major feature of the book. Cover Half Title Title Page Copyright Page Dedication Contents Detailed Contents Series Preface Preface Acknowledgments Note to Instructors Author Chapter 1: Probabilities and Statistics in Chemical and Biothermodynamics Elementary Events Relationship between probabilities, random samples, and populations Set theory diagrams depict elementary and composite outcomes How Probabilities Combine Combining probabilities for mutually exclusive outcomes within a single events Combining probabilities for outcomes from multiple independent events “And” combinations of independent outcomes of separate events involve multiplication “Or” combinations of independent outcomes for separate events involve addition and multiplication Permutation Versus Composition Calculating the number of permutations Discrete probability distributions The binomial distribution The Poisson distribution The FFT (“for the first time”) and geometric distributions The multinomial distribution Average values for discrete outcomes Continuous Distributions The exponential decay distribution The Gaussian distribution Average values for continuous distributions Problems Chapter 2: Mathematical Tools in Thermodynamics Calculus in Thermodynamics Derivatives of single-variable functions Differentials involving single-variable functions Integrals of single-variable functions Derivatives of multivariable functions Maximizing (and minimizing) multivariable functions The gradient of multivariable functions Fitting Continuous Curves to Discrete Data The least-squares approach to compare discrete data and with a continuous curve Solving the least-squares optimization problem A visual picture of least-squares An analytical approach: Linear least squares A search approximation: Nonlinear least squares Pitfalls in NLLS Putting quantitative error bars on fitted parameters Analysis of parameter distributions using the “bootstrap” method Analysis of parameter uncertainties and comparing models using the f test Problems Appendix 2.1: Determining the Covariance Matrix in Least-Squares Fitting Appendix 2.2: Testing parameters and models with the .~2 and f-ratio probability distributions Chapter 3: The Framework of Thermodynamics and the First Law What Is Thermodynamics and What Does It Treat? Classical and statistical thermodynamics Dividing up the Universe: System and Surroundings Equilibrium, Changes of State, and Reversibility The meaning of equilibrium in classical thermodynamics Irreversible and reversible changes of state The surroundings as a “reservoir” Thermodynamic Variables and Equations of State Variables in thermodynamics Ways to derive the ideal gas equation of state A graphical representation of the ideal gas law The First law of Thermodynamics The first law in words The first law in equation form The importance of pathways in first law calculations Work The work associated with expansion of a gas The Reversible Work Associated with Four Fundamental Changes of State Work done in reversible constant-pressure (isobaric) expansion Work done in constant volume pressure change Work done in reversible isothermal expansion of an ideal gas Work done in reversible adiabatic expansion of an ideal gas Heat Types of heat capacities The Heat Flow Associated with the Four Fundamental Changes in an Ideal Gas Heat flow for reversible constant-pressure (isobaric) expansion of an ideal gas Heat flow for reversible constant-volume (isochoric) heating of an ideal gas Heat flow for reversible isothermal expansion of an ideal gas Heat flow for reversible adiabatic expansion of an ideal gas The Work Associated with the Irreversible Expansion of an Ideal Gas Adiabatic irreversible expansion of a small system against a mechanical reservoir Expansion to mechanical equilibrium with the surroundings Expansion to a stopping volume where psys > psurr Expansion against vacuum The Connection between Heat Capacities and State Functions The relationship between CV and Cp A Nonideal Model: The Van der Walls Equation of State Problems Chapter 4: The Second Law and Entropy Some Familiar Examples of Spontaneous Change Spontaneous Change and Statistics Spontaneous change and the mixing of simple fluids Spontaneous change and the dissipation of temperature differences The Directionality of Heat Flow at the Macroscopic (Classical) Level The Clausius and Kelvin statements of the second law Heat engines Carnot heat engines and efficiencies From the Carnot Cycle to More General Reversible Processes Entropy Calculations for Some Simple Reversible Processes Comparison of Reversible and Irreversible Cycles A General form for Irreversible Entropy Changes Entropy As a Thermodynamic Potential Entropy Calculations for Some Irreversible Processes Heat flow between two identical bodies at different temperatures Heat flow between a small body and a thermal reservoir Entropy Changes for Irreversible Expansions Entropy change for irreversible expansion to mechanical equilibrium with a constant pressure surroundings Entropy change for an irreversible expansion against vacuum Entropy and Molecular Statistics Statistical entropy and multiplicity Counting the microstates The statistical entropy of a simple chemical reaction The statistical entropy of an expanding two-dimensional lattice gas The statistical entropy of heat flow Entropy and Fractional Populations Problems Further Reading Chapter 5: Free Energy as a Potential for the Laboratory and for Biology Internal Energy As a Potential: Combining the First and Second Laws An example of the internal energy potential U = U(S,V) for an ideal gas Other Energy Potentials for Different Types of Systems A more formal approach to generate H, A, and G: The Legendre transforms Legendre transforms of multivariable functions Relationships among Derivatives from the Differential Forms of U, H, G, and A Derivatives of the energy potentials Maxwell relations Manipulation of thermodynamic derivative relations The behavior of energy derivatives in changes of state Contributions of Different Chemical Species to Thermodynamic State Functions—Molar Quantities Molar quantities Molar volumes for systems made of only one species Molar volumes in mixtures Mixing volume ideality Mixing volume nonideality Molar free energies: The chemical potential A Constraint on the Chemical Potentials: the Gibbs–Duhem Relationship The relationship between the chemical potential and other thermodynamic potential functions Maxwell relations involving variations in composition Partial Pressures of Mixtures of Gases Problems Appendix 5.1: Legendre Transforms of a Single Variable Chapter 6: Using Chemical Potentials to Describe Phase Transitions Phases and Their Transformations The Condition for Equilibrium Between two Phases How Chemical Potentials of Different Phases Depend on Temperature and Pressure: Deriving a T–p Phase Diagram for Water Temperature dependence of heat capacities for ice, water, and steam Temperature dependence of enthalpies and entropies of ice, liquid water, and steam Temperature dependences of chemical potentials for ice, water, and steam Temperature-driven phase transitions Adding the effects of pressure to chemical potential relationships Combining pressure and temperature into a single phase diagram Additional Restrictions from the Phase Diagram: The Clausius–Clapeyron Equation and Gibbs’ Phase Rule A three-dimensional representation of the Gibbs–Duhem equation Intersection of Gibbs–Duhem planes and the Clausius–Clapeyron equation The number of coexisting phases and the phase rule Problems Further Reading Chapter 7: The Concentration Dependence of Chemical Potential, Mixing, and Reactions The Dependence of Chemical Potential on Concentration Concentration scales The difference between concentrations and mole amounts Concentration dependence of chemical potential for an ideal gas Choosing standard states Standard states are like reference points on maps The concentration dependence of chemical potential for “ideal” liquid mixtures The Gibbs free energy of mixing of ideal solutions Concentration dependence of chemical potentials for nonideal solutions A Simple Lattice Model for Nonideal Solution Behavior Chemical potential on the molar scale Chemical Reactions A formalism for chemical reactions Reaction free energy from a finite difference approach Reaction free energy from differentials Similarities (and Differences) between Free Energies of Reaction and Mixing How Chemical Equilibrium Depends on Temperature How Chemical Equilibrium Depends on Pressure Problems Further Reading Historical Development Curvature, Convexity, and Phase Stability Reaction Thermodynamics Chapter 8: Conformational Equilibrium Macromolecular Structure A simple Two-State model for conformational transitions Simultaneous visualization of N and D The Thermal Unfolding Transition As a Way to Determine Kfold and .G° A simple geometric way to connect Yobs to Kfold An equation to fit unfolding transitions A Simple Model for Thermal Transitions: Constant .H° AND .S° Fitting Conformational Transitions to Analyze the Thermodynamics of Unfolding Extrapolation of conformational transitions A More Realistic Model FOR Thermal Unfolding of Proteins: The Constant Heat Capacity Model A form of the constant heat capacity model that is good for fitting data A form of the constant heat capacity model that is suited for entropy and enthalpy analysis Cold denaturation of proteins Measurement of Thermal Denaturation by Differential Scanning Calorimetry Chemical Denaturation of Proteins Problems Appendix 8.1: Differential Scanning Calorimetry References Chapter 9: Statistical Thermodynamics and the Ensemble Method The Relationship between the Microstates of Molecular Models and Bulk Thermodynamic Properties The Ensemble Method and the Ergodic Hypothesis The Ensemble Method of Building Partition Functions Isolated thermodynamic systems and the microcanonical ensemble The Microcanonical partition function and entropy A Microcanonical Ensemble from the Heat Exchange Model A four-particle isolated system An isolated system with the two subsystems combined Another way to look at the energy distribution Problems Chapter 10: Ensembles That Interact with Their Surroundings Heat Exchange and the Canonical Ensemble The Canonical Partition Function A canonical ensemble with just two microstates A canonical ensemble with three microstates, and Lagrange maximization A canonical ensemble with an arbitrary number m of microstates Thermodynamic variables and the canonical partition function The internal energy The entropy The thermodynamic value of ß Thermodynamic relations from the canonical partition function A Canonical Ensemble Representing a Three Particle Isothermal System Population distributions as a function of temperature Bulk thermodynamic properties for the three-particle ladder The Isothermal–Isobaric Ensemble and Gibbs Free Energy The isothermal–isobaric partition function The Lagrange multipliers of the isothermal–isobaric partition function, and relationship to thermodynamic quantities Problems Chapter 11: Partition Functions for Single Molecules and Chemical Reactions A Canonical Partition Function for a System with One Molecule Temperature dependence of the molecular partition function Thermodynamic quantities from q The Relationship between the Molecular and Canonical Partition Functions Indistinguishable particles An Isothermal–Isobaric Molecular Partition Function A Statistical Thermodynamic Approach to Chemical Reaction Single Molecule Ensembles for Chemical Reactions Building a Single Molecule Reaction Partition Function Building a Multimolecular Reaction Partition Function Using reaction partition functions Problems Chapter 12: The Helix–Coil Transition General reaction partition functions for the helix–coil transition The Noncooperative Homopolymer Model The Noncooperative Heteropolymer Model Coupling between the Sites and the Basis for Cooperativity Coupling between Residues through “Nearest-Neighbor” Models The zipper model—a “one-helix” approximation for cooperative homopolymers An exact nearest-neighbor description using matrices Fraction helix from the matrix partition function Extending the matrix approach to accommodate sequence variation Problems Appendix 12.1: Other Representations of the Helix–Coil Transition Chapter 13: Ligand Binding Equilibria from a Macroscopic Perspective Ligand Binding to a Single Site Fractional saturation and average ligation number for the single-site model Graphical representation of the fractional saturation The binding capacity Practical Issues in Measuring and Analyzing Binding Curves Discrete data and data selection Indirect measurement of Total (rather than free) ligand as an independent variable Measurements of binding are subject to experimental error Binding of Multiple Ligands A Macroscopic Representation of Multiple Ligand Binding Average ligation number and fractional saturation The Binding Polynomial P: A Partition Function for Ligand Binding A concise relationship between the fractional saturation and P Populations from the binding polynomial The binding capacity from the binding polynomial An Example—The Macroscopic Binding of Two Ligands K2>>K1: Positive cooperativity K1>>K2: Negative cooperativity (or heterogeneous sites) Binding capacity representation of two-step binding Hill plots as a graphical representation of cooperativity An analytical formula for the Hill coefficient A simple limiting model for the Hill coefficient Strengths and limitations of the macroscopic approach Binding to Multiple Different Ligands: “Heterotropic” Binding Effects of thermodynamic cycles on the stepwise constants A General Framework to Represent Thermodynamic Linkage between Multiple Independent Ligands Linkage coefficients for the simple two-site heterotropic model Problems Chapter 14: Ligand Binding Equilibria from a Microscopic Perspective An Example of General Microscopic Binding: Three Ligand Binding Sites A stepwise microscopic description An overall microscopic description A geometric picture of the relationship between stepwise and overall microscopic constants Binding polynomials in terms of microscopic constants Generating P using stepwise microscopic stepwise constants Generating P using overall microscopic constants Simplifications to Microscopic Binding Models Binding to three identical, independent sites Saturation curves for three independent identical sites Binding to s Identical, Independent Sites Binding to Two Classes of Independent Sites Binding to Identical Coupled Sites A simple model for identical coupling: One interaction per ligand A more involved model for identical coupling: Interactions among all sites Explicit Structural Models for Coupling Among Binding Sites An example of rotational symmetry: Binding to a hexagon Allostery in Ligand Binding A general (macrostate) allosteric model A two-conformation, two-site general allosteric model Microscopic allosteric models and approximations Allosteric models involving subunits The KNF model Problems References Symmetry in Macromolecular Structure Allosteric Models Appendix: How to Use Mathematica Bibliography Index "An impressive text that addresses a glaring gap in the teaching of physical chemistry, being specifically focused on biologically-relevant systems along with a practical focus ... the ample problems and tutorials throughout are much appreciated."--Tobin R. Sosnick, Professor and Chair of Biochemistry and Molecular Biology, University of Chicago"Presents both the concepts and equations associated with statistical thermodynamics in a unique way that is at visual, intuitive, and rigorous. This approach will greatly benefit students at all levels." -Vijay S. Pande, Henry Dreyfus Professor of Chemistry, Stanford University"a masterful?tour de force ... Barrick's rigor and scholarship come through in every chapter."-Rohit V. Pappu, Edwin H. Murty Professor of Engineering, Washington University in St. LouisThis book provides a comprehensive, contemporary introduction to developing a quantitative understanding of how biological macromolecules behave using classical and statistical thermodynamics. The author focuses on practical skills needed to apply the underlying equations in real life examples. The text develops mechanistic models, showing how they connect to thermodynamic observables, presenting simulations of thermodynamic behavior, and analyzing experimental data. The reader is presented with plenty of exercises and problems to facilitate hands-on learning through mathematical simulation. Douglas E. Barrick is a professor in the Department of Biophysics at Johns Hopkins University. He earned his Ph. D. in biochemistry from Stanford University, and a Ph. D. in biophysics and structural biology from the University of Oregon " an impressive text that addresses a glaring gap in the teaching of physical chemistry, being specifically focused on biologically-relevant systems along with a practical focus. the ample problems and tutorials throughout are much appreciated." Tobin R. Sosnick, Professor and Chair of Biochemistry and Molecular Biology, University of Chicago "Presents both the concepts and equations associated with statistical thermodynamics in a unique way that is at visual, intuitive, and rigorous . This approach will greatly benefit students at all levels." Vijay S. Pande, Henry Dreyfus Professor of Chemistry, Stanford University " a masterful tour de force . Barrick's rigor and scholarship come through in every chapter." Rohit V. Pappu, Edwin H. Murty Professor of Engineering, Washington University in St. Louis This book provides a comprehensive, contemporary introduction to developing a quantitative understanding of how biological macromolecules behave using classical and statistical thermodynamics. The author focuses on practical skills needed to apply the underlying equations in real life examples. The text develops mechanistic models, showing how they connect to thermodynamic observables, presenting simulations of thermodynamic behavior, and analyzing experimental data. The reader is presented with plenty of exercises and problems to facilitate hands-on learning through mathematical simulation. Douglas E. Barrick is a professor in the Department of Biophysics at Johns Hopkins University. He earned his Ph.D. in biochemistry from Stanford University, and a Ph.D. in biophysics and structural biology from the University of Oregon.