
- •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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Hayward |
Figure 2 Internal RMSF of residues (average over heavy atoms) determined for human lysozyme by the X-ray normal mode refinement method applied to real X-ray data (heavy curve), in comparison with results from a normal mode analysis on a single isolated lysozyme molecule (lightweight curve). (From Ref. 33.)
ties in terms of the normal mode eigenvectors is not so natural. However, they still can be approximately expressed in terms of the variance and covariances of internuclear vectors of spin pairs, which are in turn expressed in terms of the lowest frequency normal modes. The method then proceeds in much the same way as in the X-ray case. The method was demonstrated on a solvated 25-residue zinc finger peptide on which a molecular dynamics simulation was performed. Low frequency normal modes were used to fit the order parameters for backbone CEH, NEH spin pairs calculated directly from the simulation. The predictive power of the method was tested by calculating the order parameters of spin pairs not included in the refinement. The results showed general agreement with the directly calculated quantities, although some large discrepancies were seen in individual cases. Applying the method to NOESY intensities allows one also to refine the predicted averaged structure and gain information on correlated motion in analogy to the X-ray refinement method.
C. Validity of the Concept of a Normal Mode Important Subspace
As already mentioned in Section I, normal mode analysis is based on a physical model that is quite far from reality for a biological molecule under physiological conditions. Although some studies found partial overlap between some of the lowest frequency modes and the functional mode determined from two X-ray conformers, in general it would be fanciful to expect anything more than a moderate correspondence to individual normal modes. However, the expectation that there exists a larger subspace spanned by the first M lowest frequency normal modes (where M may be between 10% and 20% of 3N) that is also spanned by the modes with the largest fluctuation in the real molecule is more realistic. If this were the case, then the low frequency normal modes would contain information on the modes of largest fluctuation in the real molecule, which would also be largely determined by barrier-crossing motions. The success of the normal mode refinement method itself is the ultimate test of the validity of this concept.
Normal Mode Analysis of Biological Molecules |
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V.NORMAL MODE ANALYSIS AND REALITY
A number of studies have compared normal mode analysis predictions with results from more realistic simulation techniques or experiments. These studies shed light on the nature of the conformational energy surface and the effect of solvent.
A. The Solvent Effect
Normal mode analyses are usually performed in a single minimum using vacuum force fields. Molecular dynamics simulations of proteins in vacuum, however, reveal not one minimum but multiple minima in the energy surface. One effect of adding solvent is to increase the density of local minima over that found in vacuum [52,53]. Extended RISM calculations have shown for the protein melittin that adding the solvation free energy to its conformational potential energy results in two minima [54] along the direction of the first normal mode, rather than one. In addition to the change in the conformational energy surface, the solvent causes damping and other hydrodynamic effects. One can attempt to incorporate these into a normal mode analysis by using a variant of the normal mode analysis developed by Lamm and Szabo [55], called the Langevin mode analysis. The resulting generalized eigenvalue problem involves the simultaneous diagonalization of the Hessian and the friction matrix. The resulting modes display the extra feature of damping. Time correlation functions and spectral densities can be calculated directly from the Langevin modes. This approach was applied to the protein crambin and a DNA duplex to reveal that a number of modes had overdamped motions [56]. One obstacle to performing a Langevin mode analysis is the accurate determination of the off-diagonal hydrodynamic terms in the friction matrix. The spectral density determined from inelastic neutron scattering experiments for BPTI shows a shallower rise from zero frequency than normal mode calculations predict [57,58]. It has been shown that this is directly due to frictional damping of the low frequency modes and can be reproduced by the Langevin mode analysis [52,53].
B. Anharmonicity and Normal Mode Analysis
The emerging model for protein dynamics is one that incorporates the dual aspects of motion within minima, combined with transitions between minima. The normal mode analysis can be seen as addressing directly only one of these two features. Ironically, however, one variant of normal mode analysis can be used to help address the other feature, namely the transition of energy barriers. To determine barrier heights for transitions occurring in a molecular dynamics simulation, instantaneous normal modes can be determined by diagonalizing instantaneous force matrices at selected configurations along the trajectory. Negative eigenvalues indicate local negative curvature possibly arising from energy barriers in the multiple-minima surface. Simulations performed at different temperatures can give information on the distribution of barrier heights [59,60].
The physical model of protein dynamics indicated above would be greatly simplified if all minima were identical. Janezic et al. [10] performed 201 normal mode analyses starting minimizations from frames along a 1 ns vacuum molecular dynamics simulation of BPTI. Comparing normal modes by taking inner products revealed that in general the normal modes remained stable, indicating similar minima. Using this assumption, a normal
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Hayward |
mode analysis performed in one single minimum can be used to analyze anharmonic effects in the molecular dynamics simulation. By performing a quasi-harmonic analysis of a 200 ps vacuum molecular dynamics simulation on BPTI and taking inner products between the normal mode and the quasi-harmonic modes or principal modes, fluctuations along the quasi-harmonic modes could be analyzed for their harmonic and anharmonic contributions [11]. The ‘‘anharmonicity factor’’ for the ith quasi-harmonic mode was defined as
αi2 |
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qi2 |
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(15) |
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qi2 har |
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where |
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3N 6 |
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har |
kB T |
gik2 |
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qi |
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(16) |
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ωk2 |
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k 1 |
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and gik is the inner product value between the kth normal mode and the ith quasi-harmonic mode. A value of 1.0 would indicate that the fluctuation in the mode can be predicted by the normal mode analysis. A value greater than 1.0 would indicate fluctuation beyond what could be predicted by the normal mode analysis. Modes with anharmonicity factors greater than 1.0 are termed ‘‘anharmonic modes’’; those with values equal to 1.0, ‘‘harmonic modes.’’ Figure 3 shows a plot of the anharmonicity factor versus quasi-
Figure 3 Anharmonicity factor versus quasi-harmonic mode number from a 200 ps vacuum simulation of BPTI. It can be seen that beyond about the 200th mode the anharmonicity factors are about 1.0, indicating harmonicity. Those below mode number 200 show progressively greater anharmonicity factors, indicating that they span a space within which energy barriers are crossed. A similar picture was found for a 1 ns simulation of human lysozyme in water [61]. (Adapted from Ref. 11.)
Normal Mode Analysis of Biological Molecules |
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harmonic mode number for BPTI. Only 12% of the modes were found to be anharmonic; the remaining 88% were harmonic modes. This analysis was also applied to a 1 ns molecular dynamics simulation of human lysozyme in water, where it was found that only the first 5% of modes were anharmonic [61]. These results suggest that the harmonic component calculated from a single normal mode analysis may be separable from the anharmonic component. Under the assumption that all minima are identical, and using the ‘‘dual aspect model’’ of protein dynamics, contributions arising from transitions between minima can be separated from contributions from harmonic motion within minima in an expression for the variance–covariance matrix of atomic fluctuations [61]. This model, the jumping-among-minima, or JAM, model, can be used to determine many features of the multiple-minima energy surface, including the distribution of minima and barrier heights.
VI. CONCLUSIONS
One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in terms of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62–64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate.
Various techniques exist that make possible a normal mode analysis of all but the largest molecules. These techniques include methods that are based on perturbation methods, reduced basis representations, and the application of group theory for symmetrical oligomeric molecular assemblies. Approximate methods that can reduce the computational load by an order of magnitude also hold the promise of producing reasonable approximations to the methods using conventional force fields.
Evidence exists that some of the softest normal modes can be associated with experimentally determined functional motions, and most studies apply normal mode analysis to this purpose. Owing to the veracity of the concept of the normal mode important subspace, normal mode analysis can be used in structural refinement methods to gain dynamic information that is beyond the capability of conventional refinement techniques.
Ironically, the normal mode analysis method can be used to determine properties of the multiple-minima energy surface of proteins.
Although not discussed in detail here, the normal mode analysis method has been used to calculate the electron transfer reorganization spectrum in Ru-modified cytochrome c [65,66]. In this application the normal mode analysis fits comfortably into the theory of electron transfer.
Despite its obvious limitations, normal mode analysis has found varied and perhaps unexpected applications in the study of the dynamics of biological molecules. In many