- •Foreword
- •Preface
- •Contents
- •Introduction
- •Oren M. Becker
- •Alexander D. MacKerell, Jr.
- •Masakatsu Watanabe*
- •III. SCOPE OF THE BOOK
- •IV. TOWARD A NEW ERA
- •REFERENCES
- •Atomistic Models and Force Fields
- •Alexander D. MacKerell, Jr.
- •II. POTENTIAL ENERGY FUNCTIONS
- •D. Alternatives to the Potential Energy Function
- •III. EMPIRICAL FORCE FIELDS
- •A. From Potential Energy Functions to Force Fields
- •B. Overview of Available Force Fields
- •C. Free Energy Force Fields
- •D. Applicability of Force Fields
- •IV. DEVELOPMENT OF EMPIRICAL FORCE FIELDS
- •B. Optimization Procedures Used in Empirical Force Fields
- •D. Use of Quantum Mechanical Results as Target Data
- •VI. CONCLUSION
- •REFERENCES
- •Dynamics Methods
- •Oren M. Becker
- •Masakatsu Watanabe*
- •II. TYPES OF MOTIONS
- •IV. NEWTONIAN MOLECULAR DYNAMICS
- •A. Newton’s Equation of Motion
- •C. Molecular Dynamics: Computational Algorithms
- •A. Assigning Initial Values
- •B. Selecting the Integration Time Step
- •C. Stability of Integration
- •VI. ANALYSIS OF DYNAMIC TRAJECTORIES
- •B. Averages and Fluctuations
- •C. Correlation Functions
- •D. Potential of Mean Force
- •VII. OTHER MD SIMULATION APPROACHES
- •A. Stochastic Dynamics
- •B. Brownian Dynamics
- •VIII. ADVANCED SIMULATION TECHNIQUES
- •A. Constrained Dynamics
- •C. Other Approaches and Future Direction
- •REFERENCES
- •Conformational Analysis
- •Oren M. Becker
- •II. CONFORMATION SAMPLING
- •A. High Temperature Molecular Dynamics
- •B. Monte Carlo Simulations
- •C. Genetic Algorithms
- •D. Other Search Methods
- •III. CONFORMATION OPTIMIZATION
- •A. Minimization
- •B. Simulated Annealing
- •IV. CONFORMATIONAL ANALYSIS
- •A. Similarity Measures
- •B. Cluster Analysis
- •C. Principal Component Analysis
- •REFERENCES
- •Thomas A. Darden
- •II. CONTINUUM BOUNDARY CONDITIONS
- •III. FINITE BOUNDARY CONDITIONS
- •IV. PERIODIC BOUNDARY CONDITIONS
- •REFERENCES
- •Internal Coordinate Simulation Method
- •Alexey K. Mazur
- •II. INTERNAL AND CARTESIAN COORDINATES
- •III. PRINCIPLES OF MODELING WITH INTERNAL COORDINATES
- •B. Energy Gradients
- •IV. INTERNAL COORDINATE MOLECULAR DYNAMICS
- •A. Main Problems and Historical Perspective
- •B. Dynamics of Molecular Trees
- •C. Simulation of Flexible Rings
- •A. Time Step Limitations
- •B. Standard Geometry Versus Unconstrained Simulations
- •VI. CONCLUDING REMARKS
- •REFERENCES
- •Implicit Solvent Models
- •II. BASIC FORMULATION OF IMPLICIT SOLVENT
- •A. The Potential of Mean Force
- •III. DECOMPOSITION OF THE FREE ENERGY
- •A. Nonpolar Free Energy Contribution
- •B. Electrostatic Free Energy Contribution
- •IV. CLASSICAL CONTINUUM ELECTROSTATICS
- •A. The Poisson Equation for Macroscopic Media
- •B. Electrostatic Forces and Analytic Gradients
- •C. Treatment of Ionic Strength
- •A. Statistical Mechanical Integral Equations
- •VI. SUMMARY
- •REFERENCES
- •Steven Hayward
- •II. NORMAL MODE ANALYSIS IN CARTESIAN COORDINATE SPACE
- •B. Normal Mode Analysis in Dihedral Angle Space
- •C. Approximate Methods
- •IV. NORMAL MODE REFINEMENT
- •C. Validity of the Concept of a Normal Mode Important Subspace
- •A. The Solvent Effect
- •B. Anharmonicity and Normal Mode Analysis
- •VI. CONCLUSIONS
- •ACKNOWLEDGMENT
- •REFERENCES
- •Free Energy Calculations
- •Thomas Simonson
- •II. GENERAL BACKGROUND
- •A. Thermodynamic Cycles for Solvation and Binding
- •B. Thermodynamic Perturbation Theory
- •D. Other Thermodynamic Functions
- •E. Free Energy Component Analysis
- •III. STANDARD BINDING FREE ENERGIES
- •IV. CONFORMATIONAL FREE ENERGIES
- •A. Conformational Restraints or Umbrella Sampling
- •B. Weighted Histogram Analysis Method
- •C. Conformational Constraints
- •A. Dielectric Reaction Field Approaches
- •B. Lattice Summation Methods
- •VI. IMPROVING SAMPLING
- •A. Multisubstate Approaches
- •B. Umbrella Sampling
- •C. Moving Along
- •VII. PERSPECTIVES
- •REFERENCES
- •John E. Straub
- •B. Phenomenological Rate Equations
- •II. TRANSITION STATE THEORY
- •A. Building the TST Rate Constant
- •B. Some Details
- •C. Computing the TST Rate Constant
- •III. CORRECTIONS TO TRANSITION STATE THEORY
- •A. Computing Using the Reactive Flux Method
- •B. How Dynamic Recrossings Lower the Rate Constant
- •IV. FINDING GOOD REACTION COORDINATES
- •A. Variational Methods for Computing Reaction Paths
- •B. Choice of a Differential Cost Function
- •C. Diffusional Paths
- •VI. HOW TO CONSTRUCT A REACTION PATH
- •A. The Use of Constraints and Restraints
- •B. Variationally Optimizing the Cost Function
- •VII. FOCAL METHODS FOR REFINING TRANSITION STATES
- •VIII. HEURISTIC METHODS
- •IX. SUMMARY
- •ACKNOWLEDGMENT
- •REFERENCES
- •Paul D. Lyne
- •Owen A. Walsh
- •II. BACKGROUND
- •III. APPLICATIONS
- •A. Triosephosphate Isomerase
- •B. Bovine Protein Tyrosine Phosphate
- •C. Citrate Synthase
- •IV. CONCLUSIONS
- •ACKNOWLEDGMENT
- •REFERENCES
- •Jeremy C. Smith
- •III. SCATTERING BY CRYSTALS
- •IV. NEUTRON SCATTERING
- •A. Coherent Inelastic Neutron Scattering
- •B. Incoherent Neutron Scattering
- •REFERENCES
- •Michael Nilges
- •II. EXPERIMENTAL DATA
- •A. Deriving Conformational Restraints from NMR Data
- •B. Distance Restraints
- •C. The Hybrid Energy Approach
- •III. MINIMIZATION PROCEDURES
- •A. Metric Matrix Distance Geometry
- •B. Molecular Dynamics Simulated Annealing
- •C. Folding Random Structures by Simulated Annealing
- •IV. AUTOMATED INTERPRETATION OF NOE SPECTRA
- •B. Automated Assignment of Ambiguities in the NOE Data
- •C. Iterative Explicit NOE Assignment
- •D. Symmetrical Oligomers
- •VI. INFLUENCE OF INTERNAL DYNAMICS ON THE
- •EXPERIMENTAL DATA
- •VII. STRUCTURE QUALITY AND ENERGY PARAMETERS
- •VIII. RECENT APPLICATIONS
- •REFERENCES
- •II. STEPS IN COMPARATIVE MODELING
- •C. Model Building
- •D. Loop Modeling
- •E. Side Chain Modeling
- •III. AB INITIO PROTEIN STRUCTURE MODELING METHODS
- •IV. ERRORS IN COMPARATIVE MODELS
- •VI. APPLICATIONS OF COMPARATIVE MODELING
- •VII. COMPARATIVE MODELING IN STRUCTURAL GENOMICS
- •VIII. CONCLUSION
- •ACKNOWLEDGMENTS
- •REFERENCES
- •Roland L. Dunbrack, Jr.
- •II. BAYESIAN STATISTICS
- •A. Bayesian Probability Theory
- •B. Bayesian Parameter Estimation
- •C. Frequentist Probability Theory
- •D. Bayesian Methods Are Superior to Frequentist Methods
- •F. Simulation via Markov Chain Monte Carlo Methods
- •III. APPLICATIONS IN MOLECULAR BIOLOGY
- •B. Bayesian Sequence Alignment
- •IV. APPLICATIONS IN STRUCTURAL BIOLOGY
- •A. Secondary Structure and Surface Accessibility
- •ACKNOWLEDGMENTS
- •REFERENCES
- •Computer Aided Drug Design
- •Alexander Tropsha and Weifan Zheng
- •IV. SUMMARY AND CONCLUSIONS
- •REFERENCES
- •Oren M. Becker
- •II. SIMPLE MODELS
- •III. LATTICE MODELS
- •B. Mapping Atomistic Energy Landscapes
- •C. Mapping Atomistic Free Energy Landscapes
- •VI. SUMMARY
- •REFERENCES
- •Toshiko Ichiye
- •II. ELECTRON TRANSFER PROPERTIES
- •B. Potential Energy Parameters
- •IV. REDOX POTENTIALS
- •A. Calculation of the Energy Change of the Redox Site
- •B. Calculation of the Energy Changes of the Protein
- •B. Calculation of Differences in the Energy Change of the Protein
- •VI. ELECTRON TRANSFER RATES
- •A. Theory
- •B. Application
- •REFERENCES
- •Fumio Hirata and Hirofumi Sato
- •Shigeki Kato
- •A. Continuum Model
- •B. Simulations
- •C. Reference Interaction Site Model
- •A. Molecular Polarization in Neat Water*
- •B. Autoionization of Water*
- •C. Solvatochromism*
- •F. Tautomerization in Formamide*
- •IV. SUMMARY AND PROSPECTS
- •ACKNOWLEDGMENTS
- •REFERENCES
- •Nucleic Acid Simulations
- •Alexander D. MacKerell, Jr.
- •Lennart Nilsson
- •D. DNA Phase Transitions
- •III. METHODOLOGICAL CONSIDERATIONS
- •A. Atomistic Models
- •B. Alternative Models
- •IV. PRACTICAL CONSIDERATIONS
- •A. Starting Structures
- •C. Production MD Simulation
- •D. Convergence of MD Simulations
- •WEB SITES OF INTEREST
- •REFERENCES
- •Membrane Simulations
- •Douglas J. Tobias
- •II. MOLECULAR DYNAMICS SIMULATIONS OF MEMBRANES
- •B. Force Fields
- •C. Ensembles
- •D. Time Scales
- •III. LIPID BILAYER STRUCTURE
- •A. Overall Bilayer Structure
- •C. Solvation of the Lipid Polar Groups
- •IV. MOLECULAR DYNAMICS IN MEMBRANES
- •A. Overview of Dynamic Processes in Membranes
- •B. Qualitative Picture on the 100 ps Time Scale
- •C. Incoherent Neutron Scattering Measurements of Lipid Dynamics
- •F. Hydrocarbon Chain Dynamics
- •ACKNOWLEDGMENTS
- •REFERENCES
- •Appendix: Useful Internet Resources
- •B. Molecular Modeling and Simulation Packages
- •Index
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Figure 12 Pair correlation functions of ClEH and NEH along the reaction path.
˚
are almost the same (2.258 A) in aqueous solution, whereas those in the gas phase are
˚
1.890 and 2.490 A, respectively, which is consistent with Hammond’s postulate [32]. The great stabilization of the product in aqueous solution is explained from changes
in PCFs along the reaction path, which are shown in Figure 12. The PCF of ClEH obviously illustrates the progress of hydrogen bonding along the reaction coordinate: For the reactant no peak corresponding to the ClEH interaction is observed, but a distinct peak
˚
around 2.2 A gradually appears with slight inward shifts as the reaction proceeds. In contrast, the hydrogen bond between the ammonia N and water H is observed to break.
˚
The first peak around 2.0 A corresponds to the hydrogen bonding of a water hydrogen to the lone pair electron in nitrogen. The lone pair electron participates in a new chemical bond with C of CH3Cl, and the peak disappears as the reaction proceeds. The formation of ClEH hydrogen bonds (solvation) and breaking of NEH bonds are key features in the understanding of solvent effects on the reaction mechanism.
IV. SUMMARY AND PROSPECTS
In this chapter, we have reviewed the RISM-SCF/MCSCF method, which combines electronic structure and liquid-state theories to deal with the chemistry of solutions. The ability
RISM-SCF/MCSCF for Processes in Solutions |
437 |
of the method to treat solvent effects on chemical processes has been demonstrated. Electronic structure plays a primary role in determining the structure of a molecule. However, changes in the electronic energy associated with a chemical process are comparable, in many cases, with those due to solvation in solution. This subtle balance between changes in electronic energy and in the solvation free energy sometimes causes drastic changes in the stability of chemical species, as we have seen in several examples. The solvent effect even reverses the equilibrium between a reactant and a product.
As has been repeatedly stated throughout the chapter, the theory provides good qualitative descriptions to solution chemistry in most cases, but quantitative agreement between theoretical and experimental results is only moderate. It is always possible to improve the results numberwise by tuning the molecular parameters and/or by introducing empirical parameters into either or both elements of the theory, MO and RISM. This direction, in fact, has been pursued by Gao and coworkers [33], who have shown almost perfect quantitative agreement between the theory and experiments by replacing the ab initio method by the semiempirical approach for the MO part and by adjusting the Lennard-Jones parameters of atoms. The effort should be greatly appreciated, because it could have demonstrated the capability of the combined MO and RISM method to account for experimental data at least at the same quantitative level as the continuum model descriptions. However, it will not be an ultimate goal of the combined quantum and statistical mechanics theory. The real strength of the theory lies in the fact that it does not require in principle any adjustable or empirical parameter to describe complicated solution chemistry. The theory could be or should be naturally improved by theoretical development of either or both elements of the method and by coupling them, not by adjusting or introducing empirical parameters. Since considerable efforts have been continuously devoted to improvement in both theoretical fields in chemical physics, there is no doubt that the RISM-SCF/MCSCF approach will find greater application in the future.
There are several directions conceivable to extend further the horizon of the theory. One such direction is to seek an experimental method to prove electron distributions in a molecule in solution: the partial atomic charges are an effective representation of the electron distribution. As has been described in the preceding sections, the RISM-SCF/ MCSCF method provides information on the electron distribution that is more detailed than information on the dipole moment. Therefore, if we could find a means to observe the electron distribution, it will provide more detailed information on molecular structure in solution. It will also provide a reliable tool for testing theory experimentally. One possible candidate among experimental methods to observe the electron distribution may be the NMR chemical shift, because the chemical shift is a manifestation of changes in the screening of nuclear magnetic fields due to electron clouds. It is highly desirable to establish a theory to bridge the NMR chemical shift and the RISM-SCF/MCSCF method.
In the first example of applications of the theory in this chapter, we made a point with respect to the polarizability of molecules and showed how the problem could have been handled by the RISM-SCF/MCSCF theory. However, the current level of our method has a serious limitation in this respect. The method can handle the polarizability of molecules in neat liquids or that of a single molecule in solution in a reasonable manner. But in order to be able to treat the polarizability of both solute and solvent molecules in solution, considerable generalization of the RISM side of the theory is required. When solvent molecules are situated within the influence of solute molecules, the solvent molecules are polarized differently depending on the distance from the solute molecules, and the solvent can no longer be ‘‘neat.’’ Therefore, the polarizable model developed for neat liquids is not valid. In such a case, solvent–solvent PCF should be treated under the solute
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field, which is typical of in-homogeneous liquids. The density functional theory in classical theory may be the best choice for extending the theory to such a problem [34].
Nonequilibrium processes including chemical reactions present a most challenging problem in theoretical chemistry. There are two aspects to chemical reactions: the reactivity or chemical equilibrium and the reaction dynamics. The chemical equilibrium of molecules is a synonym for the free energy difference between reactant and product. Two important factors determining the chemical equilibrium in solution are the changes in electronic structure and the solvation free energy. Those quantities can be evaluated by the coupled quantum and extended RISM equations, or RISM-SCF theory. Exploration of the reaction dynamics is much more demanding. Two elements of reaction dynamics in solution must be considered: the determination of reaction paths and the time evolution along the reaction path. The reaction path can be determined most naively by calculating the free energy map of reacting species. The RISM-SCF procedure can be employed for such calculations. If the rate-determining step of the reaction is an equilibrium between the reactant and the transition state, the reaction rate can be determined from the free energy difference of the two states based on transition state theory. On the other hand, for such a reaction in which the dynamics of solvent reorganization determines the reaction rate, the time evolution along the reaction path may be described by coupling RISM and the generalized Langevin equation (GLE) in the same spirit as the Kramers theory: The time evolution along a reaction path can be viewed as a stochastic barrier crossing driven by thermal fluctuations and damped by friction. Our treatment features the microscopic treatment of solvent structure on the level of the density pair correlation functions, which distinguishes it from earlier attempts that used phenomenological solvent models. One of the prerequisites for developing such a treatment is a theory to describe liquid dynamics on the molecular level. We recently proposed a new theory based on the interaction site model in which liquid dynamics is decoupled into the collective modes of density fluctuation: the acoustic and optical modes corresponding, respectively, to transnational and rotational motion of molecules [35]. From this point of view, transport coefficients such as the friction coefficient can be realized as a response of the collective modes of the solvent to perturbations due to solute. It is the first step in developing a theory of reaction dynamics to describe stochastic barrier crossing in terms of the collective fluctuations of solvent to reacting species along a properly defined reaction coordinate.
ACKNOWLEDGMENTS
We thank all our collaborators, especially Drs. Masaaki Kawata, Kazunari Naka, Tateki Ishida, and Akihiro Morita. We are also grateful to Profs. Masaru Nakahara, Yasuhiko Kondo, and Tadashi Okuyama, who have given invaluable advice and encouragement.
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