- •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
Internal Coordinate Simulation
q˙n 1/2 Tn 1/2Mn 11/2 p˜n 1/2 pn 1/2 Mn 1/2 q˙n 1/2
fn 1/2 |
h |
pn 1/2 p˜n 1/2 |
|
2 |
|||
|
|
qn 1 qn q˙n 1/2 h
127
(9d)
(9e)
(9f)
(9g)
where T is the matrix of the corresponding projection operator. It is computed for a halfstep conformation with correctly closed rings. These additional computations only slightly reduce the net performance because it is still largely determined by the cost of evaluation of atom–atom forces [54].
V.PRACTICAL EXAMPLES
A. Time Step Limitations
Testing time step limitations plays an important role in ICMD because, in many cases, several alternative models of the same system can be constructed, with different spectra of fast motions. In general, in MD the step-size maximum depends on the system being studied, and for a given algorithm it is determined by its ability to conserve the total energy in microcanonical ensemble conditions [26,30]. For the leapfrog discretization the most appropriate method consists in checking the variation of the average total energy computed with different time steps [55]. The same test trajectory is computed starting from the same constant-energy hypersurface, and the average total energy is compared with the ‘‘ideal’’ value, i.e., its low time step limit. With growing time steps, the average total energy normally deviates upward, and a deviation of 0.2D[U], where U is the potential energy and D[ ] denotes the operator of time variance, is taken as an upper acceptable level. The step-size maximum thus determined is denoted as hc and is said to be ‘‘characteristic.’’
Figure 4 shows the results of two such time step tests for a hydrated B-DNA dodecamer duplex [54]. All bases were rigid except for rotation of thymine methyls. Bond lengths were fixed as well as all bond angles except those centered at sugar ring atoms. It is always interesting to check whether the time step is limited by harmonic or anharmonic motions. To distinguish them, virtually harmonic conditions are simulated by reducing the temperature to very low values so that the time step becomes limited by the highest frequency normal mode. In Figure 4a, for instance, the two traces corresponding to low and normal temperatures leave the band of acceptable deviation with a similar time step of around 4 fs, which indicates the harmonic nature of the limitation. The expected fastest harmonic mode in this case is the scissors HECEH vibration with a frequency around 1600 cm 1, which according to theory [55] should limit hc to approximately 3.6 fs. In order to raise hc to the level of 10 fs, inertias of hydrogen-only rigid bodies, as well as rigid bodies in flexible rings, are increased by different empirically adjusted increments. In the case of a scissors hydrogen, for example, an inertia Iij µδij is added at the position
δ µ ˚ 2
of the carbon atom, where ij is the Kronecker delta and 9 amu A . This means that the hydrogen is no longer considered as a point mass but as a rigid body of the same, but redistributed, mass, which helps to scale down the scissors frequency by a factor of 3.
128 |
Mazur |
Figure 4 Time step dependence of the average total energy for two models of a partially hydrated dodecamer DNA duplex. Thinner traces show results for virtually harmonic conditions when temperature was lowered to 1 K. The DNA molecule has fixed bond lengths, rigid bases, and fixed valence angles except for the intraand extracyclic bond angles in sugars. (a) No modifications of inertia;
(b) inertia modified as explained in the text. (From Ref. 54.)
Testing of the resulting model system is shown in Figure 4b. We see that both the low and room temperature hc values have increased to the desired 10 fs level.
Modification of inertia of hydrogen-only rigid bodies is a simple and safe way to balance different frequencies in the system, and it usually allows one to raise hc to 10 fs. Unfortunately, the further increase appears problematic because of various anharmonic effects produced by collisions between non-hydrogen atoms [48].
B. Standard Geometry Versus Unconstrained Simulations
In our last example we return to the issue of the possible damaging effects of the standard geometry constraints. Two long trajectories have been computed for a partially hydrated dodecamer DNA duplex of the previous example, first by using ICMD and second with Cartesian coordinate molecular dynamics without constraints [54]. Both trajectories started
˚
from the same initial conformation with RMSD of 2.6 A from the canonical B-DNA form. Figure 5 shows the time evolution of RMSD from the canonical A and B conformations. Each point in the figure corresponds to a 15 ps interval and shows an average RMSD value. We see that both trajectories approach the canonical B-DNA, while the RMSD
Internal Coordinate Simulation |
129 |
Figure 5 Time dependence of RMSD of atomic coordinates from canonical A- and B-DNA forms in two trajectories of a partially hydrated dodecamer duplex. The A and B (A and B correspond to A and B forms) trajectories started from the same state and were computed with internal and Cartesian coordinates as independent variables, respectively. (From Ref. 54.)
from A-DNA increases and reaches the level corresponding to the difference between the
˚
canonical A and B forms. The RMSD from B-DNA falls below the 2 A level, and in both
˚
cases the final RMSD from the crystallographic conformation is around 1.3 A. The RMSD
˚
between the two final computed states is around 1.1 A, which is within the range of short time scale fluctuations in dynamics, while the overall drift from the initial state goes
˚
beyond 2.5 A.
These two duplex models have 646 and 2264 internal degrees of freedom, respectively. In spite of this large difference they show very similar behavior in terms of atomic position fluctuations as well as in terms of fluctuations of torsions, sugar pseudorotation, and DNA helical parameters [54]. Apparently, the standard geometry model, which is allowed to move only along narrow paths in the full unconstrained configurational space, still keeps enough low energy tracks to sample from the main areas defined by a given temperature of 300 K. This example shows that the differences between the trajectories computed by ICMD and Cartesian MD at least are not readily seen, and, probably, for many applications they are not essential. It should be noted at the same time that the Cartesian coordinate trajectory was computed with a lower time step of 2 fs and took nearly five times as much computer time.
VI. CONCLUDING REMARKS
Internal coordinate molecular modeling is an efficient instrument with specific advantages that make it an indispensable complement to other existing approaches. It is best suited for simulation and analysis of large-scale structural transformations in biomacro-mole- cules, and at present ICMD is generally considered the most powerful tool in conforma-
130 |
Mazur |
tional searches, notably in NMR-based structural refinement [56]. Its application to physical problems involves certain controversial and unclear aspects that hold significant theoretical interest for future studies. The slow but steady progress in the development of these methods in recent years suggests that their performance and scope of application will continue to grow.
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