- •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
422 Hirata et al.
∂ |
∂Enuc |
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1 |
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∂ωαγ(k) |
ˆ |
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∂Ra |
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|
∫dk cˆαs(k)cˆγs′(k) |
ˆ |
χss′(k) |
∂Ra |
2(2π)3β |
∂Ra |
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|
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|
α,γ,s,s′ |
|
(13) |
|
γijhαij |
|
1 |
Γijkl(φiφj|φkφl)a Vtqa εij Sija |
|||||
|
2 |
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|
i, j |
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|
i, j,k,l |
|
i j |
The second term of the right-hand side of Eq. (13) corresponds to the change of the solute– solvent distribution function due to the modification of the intramolecular correlation function ω. Other notations used here have the usual meanings. It has been well recognized that the energy gradient technique in the ab initio electronic structure theory is a powerful tool for investigating the mechanism of chemical reactions of polyatomic systems, and it opens up a variety of applications to the actual chemical processes in solution: carrying out the geometric optimization of reactant, transition state, and product in the solvated molecular system; constructing the free energy surfaces along the proper reaction coordinates; computing the vibrational frequencies and modes; and so on.
In analyzing the computational results, the following quantities are very important:
Ereorg Esolute Eisolate |
(14) |
where Eisolate is the total energy of the solute molecule in an isolated condition and Esolute is the energy of the solute molecule defined above. The quantity Ereorg represents the reorganization energy associated with the relaxation or distortion of the electronic cloud and molecular geometry in solution.
Now we have the tools in hand to tackle various problems in solvated molecules. In the following sections, we present our recent efforts to explore such phenomena by means of the RISM-SCF/MCSCF method.
III.SOLVATION EFFECT ON A VARIETY OF CHEMICAL PROCESSES IN SOLUTION
A. Molecular Polarization in Neat Water*
The molecular and liquid properties of water have been subjects of intensive research in the field of molecular science. Most theoretical approaches, including molecular simulation and integral equation methods, have relied on the effective potential, which was determined empirically or semiempirically with the aid of ab initio MO calculations for isolated molecules. The potential parameters so determined from the ab initio MO in vacuum should have been readjusted so as to reproduce experimental observables in solutions. An obvious problem in such a way of determining molecular parameters is that it requires the reevaluation of the parameters whenever the thermodynamic conditions such as temperature and pressure are changed, because the effective potentials are state properties.
One of the most efficient ways to treat this problem is to combine the ab initio MO method and the RISM theory, and this has been achieved by a slight modification of the original RISM-SCF method. Effective atomic charges in liquid water are determined such that the electronic structure and the liquid properties become self-consistent, and along the route of convergence the polarization effect can be naturally incorporated.
The temperature dependence of the effective charges and dipole moment of water
* This discussion is based on Ref. 15.
RISM-SCF/MCSCF for Processes in Solutions |
423 |
T / K
Figure 1 Temperature dependence of the reorganization energy (Ereorg) and effective charges on oxygen atom based on ( , ■) SPC and ( , ) TIP3P models.
are plotted in Figure 1. The parameters associated with the short-range part of the interaction and geometry are borrowed from two typical models of water, SPC and TIP3P. In both models, the magnitudes of the effective charges and dipole moment monotonically decrease with increasing temperature. The results can be explained in terms of the increase in molecular motion, especially rotational motion, with increasing temperature. As the motion of a molecule (molecule A) increases, the average electrostatic field produced by the surrounding water molecules becomes less anisotropic, which decreases the polarity of molecule A. Conversely, the reaction field from the water (molecule A) become more isotropic, which decreases the polarity of other molecules.
The pair correlation function of water has a marked feature that distinguishes water from other liquids (Fig. 2). One of the important features characterizing the liquid water
˚
structure is a peak around r 1.8 A observed in the oxygen–hydrogen (O–H) pair, which is a direct manifestation of the hydrogen bond between a pair of water molecules. Another feature is the position of the second peak in the oxygen–oxygen (O–O) PCF, which is caused by the tetrahedral icelike coordination. Since the icelike structure becomes less pronounced as temperature increases because of the thermal disruption of the hydrogenbonded network, those features in PCF become less prominent.
B. Autoionization of Water*
A water molecule has amphoteric character. This means it can act as both an acid and a base. The autoionization equilibrium process in water,
* This discussion is based on Ref. 16.
424 |
Hirata et al. |
Figure 2 Pair correlation functions of O–O and O–H at ( ) 273.15 and (——) 375.15K computed with the parameters of the SPC water model.
H2O H2O s H3O OH |
(15) |
is one of the most important and fundamental reactions in a variety of fields in chemistry, biology, and biochemistry. The ionic product (Kw) and its logarithm defined by
Kw [H3O ][OH ], pKw log Kw |
(16) |
are measures of the autoionization. The quantity can be related to the free energy change (∆Gaq) associated with the reaction of Eq. (15) by the standard thermodynamic relation
∆Gaq 2.303 RT pKw |
(17) |
It is experimentally known that the pKw value shows significant temperature dependence, i.e., it decreases with increasing temperature [17]. However, there is no easy explanation for this phenomenon even from the phenomenological point of view. The free energy change consists of various contributions, including changes in the electronic energy and solvation free energy of the molecular species taking part in the reaction, which are related to each other. Therefore, a theory that accounts for both the electronic and liquid structures of water with a microscopic description of the reaction is required.
RISM-SCF/MCSCF for Processes in Solutions |
425 |
The free energy change associated with the reaction in Eq. (15) can be written in terms of the energy change associated with the reaction in vacuo (∆Gvac) and the free energy change of the reacting species due to solvation as
∆Gaq ∆Gvac δG(H3O ) δG(OH ) 2δG(H2O) |
(18) |
where δG(H3O ), δG(OH ), and δG(H2O) are, respectively, the free energy changes of H3O , HO , and H2O upon solvation. It is also possible to decompose ∆Gaq into intraand intermolecular contributions as
∆Gaq ∆Eelecvac ∆δGkin ∆δGreorg ∆δµ |
(19) |
where ∆Eelecvac , ∆δGkin, and ∆δGreorg are electronic energy in vacuo, kinetic free energy, and electronic reorganization energy, respectively, which are intramolecular contributions. ∆δµ is the solvation free energy change. (We use ∆ for changes of quantities associated with the chemical reaction and δ for changes due to solvation.)
The value of pKw at temperature T relative to that at T 273.15 K, given by
∆T pKw(T) pKw(T) pKw (273.15) |
(20) |
is further decomposed into four contributions corresponding to the free energy components:
∆T pKw(T) ∆T pKw,elecvac ∆T pKw,kin(T) ∆T pKw,reorg(T) |
(21) |
|
∆T pKw,δµ(T) |
||
|
The resultant ∆T pKw(T) values and their components are plotted in Figure 3. As shown
Figure 3 Temperature dependence of calculated pKw. Dashed line indicates experimental values.