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Entropy and Free Energy in Structural Biology: From Thermodynamics to Statistical Mechanics to Computer Simulation [Kõva köide]

  • Formaat: Hardback, 374 pages, kõrgus x laius: 254x178 mm, kaal: 453 g, 24 Tables, black and white
  • Sari: Foundations of Biochemistry and Biophysics
  • Ilmumisaeg: 03-Sep-2020
  • Kirjastus: CRC Press
  • ISBN-10: 0367406926
  • ISBN-13: 9780367406929
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  • Formaat: Hardback, 374 pages, kõrgus x laius: 254x178 mm, kaal: 453 g, 24 Tables, black and white
  • Sari: Foundations of Biochemistry and Biophysics
  • Ilmumisaeg: 03-Sep-2020
  • Kirjastus: CRC Press
  • ISBN-10: 0367406926
  • ISBN-13: 9780367406929
Teised raamatud teemal:

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.

Preface xv
Acknowledgments xix
Author xxi
Section I Probability Theory
1 Probability and Its Applications
3(30)
1.1 Introduction
3(1)
1.2 Experimental Probability
3(1)
1.3 The Sample Space Is Related to the Experiment
4(1)
1.4 Elementary Probability Space
5(1)
1.5 Basic Combinatorics
6(3)
1.5.1 Permutations
6(1)
1.5.2 Combinations
7(2)
1.6 Product Probability Spaces
9(3)
1.6.1 The Binomial Distribution
11(1)
1.6.2 Poisson Theorem
11(1)
1.7 Dependent and Independent Events
12(1)
1.7.1 Bayes Formula
12(1)
1.8 Discrete Probability--Summary
13(1)
1.9 One-Dimensional Discrete Random Variables
13(1)
1.9.1 The Cumulative Distribution Function
14(1)
1.9.2 The Random Variable of the Poisson Distribution
14(1)
1.10 Continuous Random Variables
14(2)
1.10.1 The Normal Random Variable
15(1)
1.10.2 The Uniform Random Variable
15(1)
1.11 The Expectation Value
16(1)
1.11.1 Examples
16(1)
1.12 The Variance
17(2)
1.12.1 The Variance of the Poisson Distribution
18(1)
1.12.2 The Variance of the Normal Distribution
18(1)
1.13 Independent and Uncorrelated Random Variables
19(1)
1.13.1 Correlation
19(1)
1.14 The Arithmetic Average
20(1)
1.15 The Central Limit Theorem
21(2)
1.16 Sampling
23(1)
1.17 Stochastic Processes--Markov Chains
23(3)
1.17.1 The Stationary Probabilities
25(1)
1.18 The Ergodic Theorem
26(1)
1.19 Autocorrelation Functions
27(1)
1.19.1 Stationary Stochastic Processes
28(1)
Homework for Students
28(1)
A Comment about Notations
28(1)
References
29(4)
Section II Equilibrium Thermodynamics and Statistical Mechanics
2 Classical Thermodynamics
33(18)
2.1 Introduction
33(1)
2.2 Macroscopic Mechanical Systems versus Thermodynamic Systems
33(1)
2.3 Equilibrium and Reversible Transformations
34(1)
2.4 Ideal Gas Mechanical Work and Reversibility
34(2)
2.5 The First Law of Thermodynamics
36(1)
2.6 Joule's Experiment
37(2)
2.7 Entropy
39(1)
2.8 The Second Law of Thermodynamics
40(3)
2.8.1 Maximal Entropy in an Isolated System
41(1)
2.8.2 Spontaneous Expansion of an Ideal Gas and Probability
42(1)
2.8.3 Reversible and Irreversible Processes Including Work
42(1)
2.9 The Third Law of Thermodynamics
43(1)
2.10 Thermodynamic Potentials
43(4)
2.10.1 The Gibbs Relation
43(1)
2.10.2 The Entropy as the Main Potential
44(1)
2.10.3 The Enthalpy
45(1)
2.10.4 The Helmholtz Free Energy
45(1)
2.10.5 The Gibbs Free Energy
45(1)
2.10.6 The Free Energy, H (T, μ)
46(1)
2.11 Maximal Work in Isothermal and Isobaric Transformations
47(1)
2.12 Euler's Theorem and Additional Relations for the Free Energies
48(1)
2.12.1 Gibbs-Duhem Equation
49(1)
2.13 Summary
49(1)
Homework for Students
49(1)
References
49(1)
Further Reading
49(2)
3 From Thermodynamics to Statistical Mechanics
51(8)
3.1 Phase Space as a Probability Space
51(1)
3.2 Derivation of the Boltzmann Probability
52(2)
3.3 Statistical Mechanics Averages
54(2)
3.3.1 The Average Energy
54(1)
3.3.2 The Average Entropy
54(1)
3.3.3 The Helmholtz Free Energy
55(1)
3.4 Various Approaches for Calculating Thermodynamic Parameters
56(1)
3.4.1 Thermodynamic Approach
56(1)
3.4.2 Probabilistic Approach
56(1)
3.5 The Helmholtz Free Energy of a Simple Fluid
57(1)
Reference
58(1)
Further Reading
58(1)
4 Ideal Gas and the Harmonic Oscillator
59(14)
4.1 From a Free Particle in a Box to an Ideal Gas
59(1)
4.2 Properties of an Ideal Gas by the Thermodynamic Approach
60(2)
4.3 The chemical potential of an Ideal Gas
62(1)
4.4 Treating an Ideal Gas by the Probability Approach
63(1)
4.5 The Macroscopic Harmonic Oscillator
64(1)
4.6 The Microscopic Oscillator
65(3)
4.6.1 Partition Function and Thermodynamic Properties
66(2)
4.7 The Quantum Mechanical Oscillator
68(3)
4.8 Entropy and Information in Statistical Mechanics
71(1)
4.9 The Configurational Partition Function
71(1)
Homework for Students
72(1)
References
72(1)
Further Reading
72(1)
5 Fluctuations and the Most Probable Energy
73(10)
5.1 The Variances of the Energy and the Free Energy
73(1)
5.2 The Most Contributing Energy E*
74(2)
5.3 Solving Problems in Statistical Mechanics
76(5)
5.3.1 The Thermodynamic Approach
77(1)
5.3.2 The Probabilistic Approach
78(1)
5.3.3 Calculating the Most Probable Energy Term
79(1)
5.3.4 The Change of Energy and Entropy with Temperature
80(1)
References
81(2)
6 Various Ensembles
83(10)
6.1 The Microcanonical (petit) Ensemble
83(1)
6.2 The Canonical (NVT) Ensemble
84(1)
6.3 The Gibbs (NpT) Ensemble
85(3)
6.4 The Grand Canonical (uVT) Ensemble
88(2)
6.5 Averages and Variances in Different Ensembles
90(2)
6.5.1 A Canonical Ensemble Solution (Maximal Term Method)
90(1)
6.5.2 A Grand-Canonical Ensemble Solution
91(1)
6.5.3 Fluctuations in Different Ensembles
91(1)
References
92(1)
Further Reading
92(1)
7 Phase Transitions
93(6)
7.1 Finite Systems versus the Thermodynamic Limit
93(1)
7.2 First-Order Phase Transitions
94(1)
7.3 Second-Order Phase Transitions
95(3)
References
98(1)
8 Ideal Polymer Chains
99(12)
8.1 Models of Macromolecules
99(1)
8.2 Statistical Mechanics of an Ideal Chain
99(2)
8.2.1 Partition Function and Thermodynamic Averages
100(1)
8.3 Entropic Forces in an One-Dimensional Ideal Chain
101(3)
8.4 The Radius of Gyration
104(1)
8.5 The Critical Exponent v
105(1)
8.6 Distribution of the End-to-End Distance
106(2)
8.6.1 Entropic Forces Derived from the Gaussian Distribution
107(1)
8.7 The Distribution of the End-to-End Distance Obtained from the Central Limit Theorem
108(1)
8.8 Ideal Chains and the Random Walk
109(1)
8.9 Ideal Chain as a Model of Reality
110(1)
References
110(1)
9 Chains with Excluded Volume
111(12)
9.1 The Shape Exponent v for Self-avoiding Walks
111(1)
9.2 The Partition Function
112(1)
9.3 Polymer Chain as a Critical System
113(1)
9.4 Distribution of the End-to-End Distance
114(1)
9.5 The Effect of Solvent and Temperature on the Chain Size
115(4)
9.5.1 9 Chains ind = 3
116(1)
9.5.2 6 Chains in d = 2
116(1)
9.5.3 The Crossover Behavior Around G
117(1)
9.5.4 The Blob Picture
118(1)
9.6 Summary
119(1)
References
119(4)
Section III Topics in Non-Equilibrium Thermodynamics and Statistical Mechanics
10 Basic Simulation Techniques: Metropolis Monte Carlo and Molecular Dynamics
123(16)
10.1 Introduction
123(1)
10.2 Sampling the Energy and Entropy and New Notations
124(1)
10.3 More About Importance Sampling
125(1)
10.4 The Metropolis Monte Carlo Method
126(3)
10.4.1 Symmetric and Asymmetric MC Procedures
127(1)
10.4.2 A Grand-Canonical MC Procedure
128(1)
10.5 Efficiency of Metropolis MC
129(2)
10.6 Molecular Dynamics in the Microcanonical Ensemble
131(3)
10.7 MD Simulations in the Canonical Ensemble
134(1)
10.8 Dynamic MD Calculations
135(1)
10.9 Efficiency of MD
135(2)
10.9.1 Periodic Boundary Conditions and Ewald Sums
136(1)
10.9.2 A Comment About MD Simulations and Entropy
136(1)
References
137(2)
11 Non-Equilibrium Thermodynamics--Onsager Theory
139(14)
11.1 Introduction
139(1)
11.2 The Local-Equilibrium Hypothesis
139(1)
11.3 Entropy Production Due to Heat Flow in a Closed System
140(1)
11.4 Entropy Production in an Isolated System
141(1)
11.5 Extra Hypothesis: A Linear Relation Between Rates and Affinities
142(2)
11.5.1 Entropy of an Ideal Linear Chain Close to Equilibrium
143(1)
11.6 Fourier's Law--A Continuum Example of Linearity
144(1)
11.7 Statistical Mechanics Picture of Irreversibility
145(2)
11.8 Time Reversal, Microscopic Reversibility, and the Principle of Detailed Balance
147(2)
11.9 Onsager's Reciprocal Relations
149(1)
11.10 Applications
150(1)
11.11 Steady States and the Principle of Minimum Entropy Production
151(1)
11.12 Summary
152(1)
References
152(1)
12 Non-equilibrium Statistical Mechanics
153(24)
12.1 Fick's Laws for Diffusion
153(5)
12.1.1 First Fick's Law
153(1)
12.1.2 Calculation of the Flux from Thermodynamic Considerations
154(1)
12.1.3 The Continuity Equation
155(1)
12.1.4 Second Fick's Law--The Diffusion Equation
156(1)
12.1.5 Diffusion of Particles Through a Membrane
156(1)
12.1.6 Self-Diffusion
156(2)
12.2 Brownian Motion: Einstein's Derivation of the Diffusion Equation
158(2)
12.3 Langevin Equation
160(9)
12.3.1 The Average Velocity and the Fluctuation-Dissipation Theorem
162(1)
12.3.2 Correlation Functions
163(1)
12.3.3 The Displacement of a Langevin Particle
164(2)
12.3.4 The Probability Distributions of the Velocity and the Displacement
166(2)
12.3.5 Langevin Equation with a Charge in an Electric Field
168(1)
12.3.6 Langevin Equation with an External Force--The Strong Damping Velocity
168(1)
12.4 Stochastic Dynamics Simulations
169(2)
12.4.1 Generating Numbers from a Gaussian Distribution by CLT
170(1)
12.4.2 Stochastic Dynamics versus Molecular Dynamics
171(1)
12.5 The Fokker-Planck Equation
171(3)
12.6 Smoluchowski Equation
174(1)
12.7 The Fokker-Planck Equation for a Full Langevin Equation with a Force
175(1)
12.8 Summary of Pairs of Equations
175(1)
References
176(1)
13 The Master Equation
177(12)
13.1 Master Equation in a Microcanonical System
177(1)
13.2 Master Equation in the Canonical Ensemble
178(2)
13.3 An Example from Magnetic Resonance
180(5)
13.3.1 Relaxation Processes Under Various Conditions
181(3)
13.3.2 Steady State and the Rate of Entropy Production
184(1)
13.4 The Principle of Minimum Entropy Production--Statistical Mechanics Example
185(1)
References
186(3)
Section IV Advanced Simulation Methods: Polymers and Biological Macromolecules
14 Growth Simulation Methods for Polymers
189(22)
14.1 Simple Sampling of Ideal Chains
189(1)
14.2 Simple Sampling of SAWs
190(2)
14.3 The Enrichment Method
192(1)
14.4 The Rosenbluth and Rosenbluth Method
193(2)
14.5 The Scanning Method
195(11)
14.5.1 The Complete Scanning Method
195(1)
14.5.2 The Partial Scanning Method
196(1)
14.5.3 Treating SAWs with Finite Interactions
197(1)
14.5.4 A Lower Bound for the Entropy
197(1)
14.5.5 A Mean-Field Parameter
198(1)
14.5.6 Eliminating the Bias by Schmidt's Procedure
199(1)
14.5.7 Correlations in the Accepted Sample
200(1)
14.5.8 Criteria for Efficiency
201(1)
14.5.9 Locating Transition Temperatures
202(1)
14.5.10 The Scanning Method versus Other Techniques
203(1)
14.5.11 The Stochastic Double Scanning Method
204(1)
14.5.12 Future Scanning by Monte Carlo
204(1)
14.5.13 The Scanning Method for the Ising Model and Bulk Systems
205(1)
14.6 The Dimerization Method
206(2)
References
208(3)
15 The Pivot Algorithm and Hybrid Techniques
211(6)
15.1 The Pivot Algorithm--Historical Notes
211(1)
15.2 Ergodicity and Efficiency
211(1)
15.3 Applicability
212(1)
15.4 Hybrid and Grand-Canonical Simulation Methods
213(1)
15.5 Concluding Remarks
214(1)
References
214(3)
16 Models of Proteins
217(14)
16.1 Biological Macromolecules versus Polymers
217(1)
16.2 Definition of a Protein Chain
217(1)
16.3 The Force Field of a Protein
218(1)
16.4 Implicit Solvation Models
219(1)
16.5 A Protein in an Explicit Solvent
220(1)
16.6 Potential Energy Surface of a Protein
221(1)
16.7 The Problem of Protein Folding
222(1)
16.8 Methods for a Conformational Search
222(3)
16.8.1 Local Minimization--The Steepest Descents Method
223(1)
16.8.2 Monte Carlo Minimization
224(1)
16.8.3 Simulated Annealing
225(1)
16.9 Monte Carlo and Molecular Dynamics Applied to Proteins
225(1)
16.10 Microstates and Intermediate Flexibility
226(1)
16.10.1 On the Practical Definition of a Microstate
227(1)
References
227(4)
17 Calculation of the Entropy and the Free Energy by Thermodynamic Integration
231(12)
17.1 "Calorimetric" Thermodynamic Integration
232(1)
17.2 The Free Energy Perturbation Formula
232(2)
17.3 The Thermodynamic Integration Formula of Kirkwood
234(1)
17.4 Applications
235(2)
17.4.1 Absolute Entropy of a SAW Integrated from an Ideal Chain Reference State
235(2)
17.4.2 Harmonic Reference State of a Peptide
237(1)
17.5 Thermodynamic Cycles
237(4)
17.5.1 Other Cycles
240(1)
17.5.2 Problems of TI and FEP Applied to Proteins
240(1)
References
241(2)
18 Direct Calculation of the Absolute Entropy and Free Energy
243(8)
18.1 Absolute Free Energy from <exp[ +ElkT]>
243(1)
18.2 The Harmonic Approximation
244(1)
18.3 The M2 Method
245(1)
18.4 The Quasi-Harmonic Approximation
246(1)
18.5 The Mutual Information Expansion
247(1)
18.6 The Nearest Neighbor Technique
248(1)
18.7 The MIE-NN Method
249(1)
18.8 Hybrid Approaches
249(1)
References
249(2)
19 Calculation of the Absolute Entropy from a Single Monte Carlo Sample
251(26)
19.1 The Hypothetical Scanning (HS) Method for SAWs
251(2)
19.1.1 An Exact HS Method
251(1)
19.1.2 Approximate HS Method
252(1)
19.2 The HS Monte Carlo (HSMC) Method
253(2)
19.3 Upper Bounds and Exact Functionals for the Free Energy
255(5)
19.3.1 The Upper Bound FB
255(1)
19.3.2 FB Calculated by the Reversed Schmidt Procedure
256(1)
19.3.3 A Gaussian Estimation of FB
257(1)
19.3.4 Exact Expression for the Free Energy
258(1)
19.3.5 The Correlation Between oA and FA
258(1)
19.3.6 Entropy Results for SAWs on a Square Lattice
259(1)
19.4 HS and HSMC Applied to the Ising Model
260(1)
19.5 The HS and HSMC Methods for a Continuum Fluid
261(5)
19.5.1 The HS Method
261(1)
19.5.2 The HSMC Method
262(2)
19.5.3 Results for Argon and Water
264(1)
19.5.3.1 Results for Argon
264(2)
19.5.3.2 Results for Water
266(1)
19.6 HSMD Applied to a Peptide
266(3)
19.6.1 Applications
269(1)
19.7 The HSMD-TI Method
269(1)
19.8 The LS Method
270(4)
19.8.1 The LS Method Applied to the Ising Model
270(2)
19.8.2 The LS Method Applied to a Peptide
272(2)
References
274(3)
20 The Potential of Mean Force, Umbrella Sampling, and Related Techniques
277(24)
20.1 Umbrella Sampling
277(1)
20.2 Bennett's Acceptance Ratio
278(3)
20.3 The Potential of Mean Force
281(4)
20.3.1 Applications
284(1)
20.4 The Self-Consistent Histogram Method
285(4)
20.4.1 Free Energy from a Single Simulation
286(1)
20.4.2 Multiple Simulations and The Self-Consistent Procedure
286(3)
20.5 The Weighted Histogram Analysis Method
289(8)
20.5.1 The Single Histogram Equations
290(1)
20.5.2 The WHAM Equations
291(2)
20.5.3 Enhancements of WHAM
293(2)
20.5.4 The Basic MBAR Equation
295(1)
20.5.5 ST-WHAM and UIM
296(1)
20.5.6 Summary
296(1)
References
297(4)
21 Advanced Simulation Methods and Free Energy Techniques
301(30)
21.1 Replica-Exchange
301(7)
21.1.1 Temperature-Based REM
301(4)
21.1.2 Hamiltonian-Dependent Replica Exchange
305(3)
21.2 The Multicanonical Method
308(4)
21.2.1 Applications
311(1)
21.2.2 MUCA-Summary
312(1)
21.3 The Method of Wang and Landau
312(3)
21.3.1 The Wang and Landau Method-Applications
314(1)
21.4 The Method of Expanded Ensembles
315(2)
21.4.1 The Method of Expanded Ensembles-Appl ications
317(1)
21.5 The Adaptive Integration Method
317(2)
21.6 Methods Based on Jarzynski's Identity
319(5)
21.6.1 Jarzynski's Identity versus Other Methods for Calculating AF
323(1)
21.7 Summary
324(1)
References
324(7)
22 Simulation of the Chemical Potential
331(12)
22.1 The Widom Insertion Method
331(1)
22.2 The Deletion Procedure
332(2)
22.3 Personage's Method for Treating Deletion
334(2)
22.4 Introduction of a Hard Sphere
336(1)
22.5 The Ideal Gas Gauge Method
337(1)
22.6 Calculation of the Chemical Potential of a Polymer by the Scanning Method
338(2)
22.7 The Incremental Chemical Potential Method for Polymers
340(1)
22.8 Calculation of u by Thermodynamic Integration
341(1)
References
341(2)
23 The Absolute Free Energy of Binding
343(24)
23.1 The Law of Mass Action
343(1)
23.2 Chemical Potential, Fugacity, and Activity of an Ideal Gas
344(1)
23.2.1 Thermodynamics
344(1)
23.2.2 Canonical Ensemble
344(1)
23.2.3 NpT Ensemble
345(1)
23.3 Chemical Potential in Ideal Solutions: Raoult's and Henry's Laws
345(1)
23.3.1 Raoult's Law
346(1)
23.3.2 Henry's Law
346(1)
23.4 Chemical Potential in Non-ideal Solutions
346(1)
23.4.1 Solvent
346(1)
23.4.2 Solute
347(1)
23.5 Thermodynamic Treatment of Chemical Equilibrium
347(1)
23.6 Chemical Equilibrium in Ideal Gas Mixtures: Statistical Mechanics
348(1)
23.7 Pressure-Dependent Equilibrium Constant of Ideal Gas Mixtures
349(1)
23.8 Protein-Ligand Binding
350(12)
23.8.1 Standard Methods for Calculating δA°
352(2)
23.8.2 Calculating δA° by HSMD-TI
354(2)
23.8.3 HSMD-TI Applied to the FKBP12-FK506 Complex: Equilibration
356(1)
23.8.4 The Internal and External Entropies
357(2)
23.8.5 TI Results for FKBP12-FK506
359(1)
23.8.6 δA° Results for FKBP12-FK506
359(3)
23.9 Summary
362(1)
References
362(5)
Appendix 367(2)
Index 369
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 ligands 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.