Entropy and Free Energy in Structural Biology

From Thermodynamics to Statistical Mechanics to Computer Simulation
1. Auflage 2020
ISBN: 978-0-367-40692-9
Verlag: CRC Press

Buch, Englisch, 396 Seiten, Format (B × H): 221 mm x 286 mm, Gewicht: 1257 g

Reihe: Foundations of Biochemistry and Biophysics

Computer simulation has become the main engine of development in statistical mechanics. In structural biology, computer simulation constitutes the main theoretical tool for structure determination of proteins and for calculation of the free energy of binding, which are important in drug design. Entropy and Free Energy in Structural Biology leads the reader to the simulation technology in a systematic way. The book, which is structured as a course, consists of four parts:

Part I is a short course on probability theory emphasizing (1) the distinction between the notions of experimental probability, probability space, and the experimental probability on a computer, and (2) elaborating on the mathematical structure of product spaces. These concepts are essential for solving probability problems and devising simulation methods, in particular for calculating the entropy.

Part II starts with a short review of classical thermodynamics from which a non-traditional derivation of statistical mechanics is devised. Theoretical aspects of statistical mechanics are reviewed extensively.

Part III covers several topics in non-equilibrium thermodynamics and statistical mechanics close to equilibrium, such as Onsager relations, the two Fick's laws, and the Langevin and master equations. The Monte Carlo and molecular dynamics procedures are discussed as well.

Part IV presents advanced simulation methods for polymers and protein systems, including techniques for conformational search and for calculating the potential of mean force and the chemical potential. Thermodynamic integration, methods for calculating the absolute entropy, and methodologies for calculating the absolute free energy of binding are evaluated.

Enhanced by a number of solved problems and examples, this volume will be a valuable resource to advanced undergraduate and graduate students in chemistry, chemical engineering, biochemistry biophysics, pharmacology, and computational biology.

Autoren/Hrsg.

Meirovitch, Hagai

Fachgebiete

• Naturwissenschaften Biowissenschaften Angewandte Biologie Biophysik
• Mathematik | Informatik EDV | Informatik Professionelle Anwendung Computersimulation & Modelle, 3-D Graphik
• Naturwissenschaften Physik Physik Allgemein

Inhaltsverzeichnis

Contents

Preface. xv

Acknowledgments.xix

Author.xxi

Section I Probability Theory

1. Probability and Its Applications. 3

1.1 Introduction. 3

1.2 Experimental Probability. 3

1.3 The Sample Space Is Related to the Experiment. 4

1.4 Elementary Probability Space. 5

1.5 Basic Combinatorics. 6

1.5.1 Permutations. 6

1.5.2 Combinations. 7

1.6 Product Probability Spaces. 9

1.6.1 The Binomial Distribution.11

1.6.2 Poisson Theorem.11

1.7 Dependent and Independent Events. 12

1.7.1 Bayes Formula. 12

1.8 Discrete Probability—Summary. 13

1.9 One-Dimensional Discrete Random Variables. 13

1.9.1 The Cumulative Distribution Function.14

1.9.2 The Random Variable of the Poisson Distribution.14

1.10 Continuous Random Variables.14

1.10.1 The Normal Random Variable. 15

1.10.2 The Uniform Random Variable. 15

1.11 The Expectation Value.16

1.11.1 Examples.16

1.12 The Variance.17

1.12.1 The Variance of the Poisson Distribution.18

1.12.2 The Variance of the Normal Distribution.18

1.13 Independent and Uncorrelated Random Variables. 19

1.13.1 Correlation. 19

1.14 The Arithmetic Average. 20

1.15 The Central Limit Theorem. 21

1.16 Sampling. 23

1.17 Stochastic Processes—Markov Chains. 23

1.17.1 The Stationary Probabilities. 25

1.18 The Ergodic Theorem. 26

1.19 Autocorrelation Functions. 27

1.19.1 Stationary Stochastic Processes. 28

Homework for Students. 28

A Comment about Notations. 28

References. 29

Section II Equilibrium Thermodynamics and Statistical Mechanics

2. Classical Thermodynamics. 33

2.1 Introduction. 33

2.2 Macroscopic Mechanical Systems versus Thermodynamic Systems. 33

2.3 Equilibrium and Reversible Transformations. 34

2.4 Ideal Gas Mechanical Work and Reversibility. 34

2.5 The First Law of Thermodynamics. 36

2.6 Joule’s Experiment. 37

2.7 Entropy. 39

2.8 The Second Law of Thermodynamics. 40

2.8.1 Maximal Entropy in an Isolated System.41

2.8.2 Spontaneous Expansion of an Ideal Gas and Probability. 42

2.8.3 Reversible and Irreversible Processes Including Work. 42

2.9 The Third Law of Thermodynamics. 43

2.10 Thermodynamic Potentials. 43

2.10.1 The Gibbs Relation. 43

2.10.2 The Entropy as the Main Potential. 44

2.10.3 The Enthalpy. 45

2.10.4 The Helmholtz Free Energy. 45

2.10.5 The Gibbs Free Energy. 45

2.10.6 The Free Energy, H(T,µ). 46

2.11 Maximal Work in Isothermal and Isobaric Transformations. 47

2.12 Euler’s Theorem and Additional Relations for the Free Energies. 48

2.12.1 Gibbs-Duhem Equation. 49

2.13 Summary. 49

Homework for Students. 49

References. 49

Further Reading. 49

3. From Thermodynamics to Statistical Mechanics.51

3.1 Phase Space as a Probability Space.51

3.2 Derivation of the Boltzmann Probability. 52

3.3 Statistical Mechanics Averages. 54

3.3.1 The Average Energy. 54

3.3.2 The Average Entropy. 54

3.3.3 The Helmholtz Free Energy. 55

3.4 Various Approaches for Calculating Thermodynamic Parameters. 55

3.4.1 Thermodynamic Approach. 55

3.4.2 Probabilistic Approach. 56

3.5 The Helmholtz Free Energy of a Simple Fluid. 56

Reference. 58

Further Reading. 58

4. Ideal Gas and the Harmonic Oscillator. 59

4.1 From a Free Particle in a Box to an Ideal Gas. 59

4.2 Properties of an Ideal Gas by the Thermodynamic Approach. 60

4.3 The chemical potential of an Ideal Gas. 62

4.4 Treating an Ideal Gas by the Probability Approach. 63

4.5 The Macroscopic Harmonic Oscillator. 64

4.6 The Microscopic Oscillator. 65

4.6.1 Partition Function and Thermodynamic Properties. 66

4.7 The Quantum Mechanical Oscillator. 68

4.8 Entropy and Information in Statistical Mechanics. 71

4.9 The Configurational Partition Function. 71

Homework for Students. 72

References. 72

Further Reading. 72

5. Fluctuations and the Most Probable Energy. 73

5.1 The Variances of the Energy and the Free Energy. 73

5.2 The Most Contributing Energy E*. 74

5.3 Solving Problems in Statistical Mechanics. 76

5.3.1 The Thermodynamic Approach. 77

5.3.2 The Probabilistic Approach. 78

5.3.3 Calculating the Most Probable Energy Term. 79

5.3.4 The Change of Energy and Entropy with Temperature. 80

References. 81

6. Various Ensembles. 83

6.1 The Microcanonical (petit) Ensemble. 83

6.2 The Canonical (NVT) Ensemble. 84

6.3 The Gibbs (NpT) Ensemble. 85

6.4 The Grand Canonical (µVT) Ensemble. 88

6.5 Averages and Variances in Different Ensembles. 90

6.5.1 A Canonical Ensemble Solution (Maximal Term Method). 90

6.5.2 A Grand-Canonical Ensemble Solution. 91

6.5.3 Fluctuations in Different Ensembles. 91

References. 92

Further Reading. 92

7. Phase Transitions. 93

7.1 Finite Systems versus the Thermodynamic Limit. 93

7.2 First-Order Phase Transitions. 94

7.3 Second-Order Phase Transitions. 95

References.

Über Autor(innen)

Hagai Meirovitch is professor Emeritus in the Department of Computational and Systems Biology at the University of Pittsburgh School of Medicine. He earned an MSc degree in nuclear physics from the Hebrew University, a PhD degree in chemical physics from the Weizmann Institute, and conducted postdoctoral training in the laboratory of Professor Harold A. Scheraga at Cornell University. His research focused on developing computer simulation methodologies within the scope of statistical mechanics, as highlighted below. He devised novel methods for extracting the absolute entropy from Monte Carlo samples and techniques for generating polymer chains, which were used to study phase transitions in polymers, magnetic, and lattice gas systems. These methods, together with conformational search techniques for proteins, led to a free energy-based approach for treating molecular flexibility. This approach was used to analyze NMR relaxation data from cyclic peptides and to study structural preferences of surface loops in bound and free enzymes. He developed a new methodology for calculating the free energy of ligand/protein binding, which unlike standard techniques, provides the decrease in the ligand’s entropy upon binding. Dr Meirovitch conducted part of the research depicted above, and other studies, at the Supercomputer Computations Research Institute of the Florida State University, Tallahassee.