Chapter 4 Alchemical Free Energy Calculations: Ready for Prime Time?

doi:10.1016/S1574-1400(07)03004-6

Annual Reports in Computational Chemistry

Volume 3, 2007, Pages 41-59

https://doi.org/10.1016/S1574-1400(07)03004-6 Get rights and content

Publisher Summary

In an alchemical transformation, a chemical species is transformed into another via a pathway of nonphysical (alchemical) states. Many physical processes, such as ligand binding or transfer of a molecule from gas to solvent, can be equivalently expressed as a composition of such alchemical transformations. Often, these alchemical processes are much more amenable to computational simulation than the physical process itself, especially in complex biochemical systems. A relative ligand-binding affinity, for example, may be computed via a thermodynamic cycle by alchemically transforming one ligand to another both bound to a receptor and in solution. There are other successful nonalchemical approaches for the computation of free energy differences, such as phase equilibrium Monte Carlo methods for modeling multicomponent fluids and potential of mean force methods. Alchemical approaches, however, generally allow for the larger range of conformational complexity typical of biochemical systems. This chapter discusses the efficiency and convergence of free energy methods rather than on accuracy, which is a function of the force field.

Introduction

Two major goals of computational chemistry are to provide physical insight by modeling details not easily accessible to experiment and to make predictions in order to aid and guide experiment. Both of these goals frequently involve the calculation of free energy differences, since the free energy difference of a chemical process governs the balance of the different chemical species present and the amount of chemical work available.

The ability to rapidly and accurately calculate free energy changes in complex biochemical systems would make possible the computational design of new chemical entities, which has the potential to revolutionize a number of fields. Pharmaceutical chemistry would benefit through virtual high throughput screening, computational lead optimization, and virtual specificity screens, saving money and time on early product development [1]. Chemical biology would benefit from the creation of molecules to modulate the function of specific proteins in desired ways, or by the design of enzymes to catalyze particular reactions. Reliable methods for efficient free energy computation would be widely useful in many other fields, such as bioremediation and materials design.

Section snippets

Background

When free energy methods were first applied to problems in drug discovery in the early 1990's, there was a great deal of excitement. This excitement cooled considerably when it became clear that free energies could not reliably be obtained for important applications such as ligand binding to receptors for drug design 2., 3., 4.. Inefficient early methods and limited computer power meant that converged free energies in complex heterogeneous systems were simply not achievable. Additionally, many

Equilibrium Methods

In practice, an equilibrium alchemical free energy calculation is separated into multiple stages: (1) selection of alchemical intermediates (see Section 5); (2) generation of uncorrelated samples at each intermediate (see Section 6); and (3) estimation of the free energy difference between the states of interest using one of the analysis methods listed below. Each of these three stages present a number of different choices, and comparing among all possible options becomes exceedingly difficult.

Nonequilibrium Methods

Free energy differences can also be computed from nonequilibrium simulations switching between two Hamiltonians, using measurements of the work $W_{0 \to 1}$ performed on the system during the switching process 41., 42., 43., 44., 45., 46.. The Jarzynski relation [41] $Δ F = - β^{−1} \ln {〈 e^{- β W_{0 \to 1}} 〉}_{0}$ and its subsequent generalization by Crooks [42] $Δ F = - β^{−1} \ln \frac{{〈 f (W_{0 \to 1}) 〉}_{0}}{{〈 f (- W_{1 \to 0}) 〉}_{1}}$ hold for arbitrary $f (W)$ if the system is initially prepared in equilibrium and switching between the two Hamiltonians proceeds with the same

Intermediate States

To compute free energy differences between states with little overlap, it will usually be more efficient to compute the free energy along a pathway of intermediate states. Free energy calculations can be made significantly more efficient by optimizing the choice of intermediate states for increased phase space overlap 36., 59., 60., 61.,

The simplest path to construct is linear in the two end point Hamiltonians: $H (λ) = λ H_{1} + (1 - λ) H_{0} .$ However, there is strong consensus that when annihilating or

Sampling

For both equilibrium (Section 3) and nonequilibrium (Section 4) alchemical free energy methods, it is necessary to generate a number of uncorrelated configurations at one or more Hamiltonians. In many cases, a large number of such configurations must be generated to obtain sufficiently precise results. This is the time-consuming step in most free energy calculations, as slow or infrequent conformational changes, like protein or ligand conformational rearrangement in binding free energy

Applications

In this review, we have attempted to demonstrate that the potential for precise and efficient alchemical free energy calculations has greatly increased in recent years. However, to what extent is it possible to say that free energy calculations have begun to fulfill this potential? What is the evidence in the last several years that free energy methods actually are “ready for prime time?”

Conclusion

There is still much work to do in determining optimal free energy methods, and even more in making them easy for the average practitioners to apply. With the proliferating combinations of methods, more systematic comparisons are mandatory. Most free energy methods work efficiently on low-dimensional toy problems that have been common for testing these methods, such as coupled harmonic oscillators or one-dimensional potentials, or small molecular changes like enlarging particle radii. But since

Acknowledgements

The authors would like to thank Chris Oostenbrink, Jed Pitera, and M. Scott Shell for many useful comments on earlier drafts of this review.

References (147)

T. Rodinger et al.
Enhancing the accuracy, the efficiency and the scope of free energy simulations
Curr. Opin. Struct. Biol.
(2005)
D.A. Kofke
Free energy methods in molecular simulation
Fluid Phase Equilibria
(2005)
K. Raha et al.
Calculating binding free energy in protein-ligand interaction
Annu. Rep. Comp. Chem.
(2005)
B.O. Brandsdal et al.
Free energy calculations and ligand binding
Adv. Prot. Chem.
(2003)
C.H. Bennett
Efficient estimation of free energy differences from Monte Carlo data
J. Comp. Phys.
(1976)
M. Souaille et al.
Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations
Comp. Phys. Comm.
(2001)
A.E. Roitberg
Non-equilibrium approaches to free energy calculations
Annu. Rep. Comp. Chem.
(2005)
C. Oostenbrink et al.
Calculating zeros: Non-equilibrium free energy calculations
Chem. Phys.
(2006)
T.C. Beutler et al.
Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations
Chem. Phys. Lett.
(1994)
R.C. van Schaik et al.
Ligand–receptor interactions
J. Mol. Biol.
(1993)

Y. Okamoto

Generalized-ensemble algorithms: Enhanced sampling techniques for Monte Carlo and molecular dynamics simulations

J. Mol. Graph. Model.

(2004)

A. Mitsutake et al.

Replica-exchange simulated tempering method for simulations of frustrated systems

Chem. Phys. Lett.

(2000)

U.H.E. Hansmann

Parallel tempering algorithm for conformational studies of biological molecules

Chem. Phys. Lett.

(1997)

Y. Sugita et al.

Replica-exchange molecular dynamics method for protein folding

Chem. Phys. Lett.

(1999)

K. Murata et al.

Free energy calculations for DNA base stacking by replica-exchange umbrella sampling

Chem. Phys. Lett.

(2004)

R. Bitetti-Putzer et al.

Generalized ensembles serve to improve the convergence of free energy simulations

Chem. Phys. Lett.

(2003)

M.K. Gilson et al.

A statistical-thermodynamic basis for computation of binding affinities: A critical review

Biophys. J.

(1997)

B. Roux et al.

Thermodynamic stability of water molecules in the bacteriorhodopsin proton channel: a molecular dynamics free energy perturbation study

Biophys. J.

(1996)

J. Wang et al.

Absolute binding free energy calculations using molecular dynamics simulations with restraining potentials

Biophys. J.

(2006)

W.L. Jorgensen

The many roles of computation in drug discovery

Science

(2004)

C. Chipot et al.

Free energy calculations: the long and winding gilded road

Mol. Simulation

(2002)

C. Chipot et al.

Introduction

A.E. Mark

Calculating free energy differences using perturbation theory

M. Leitgeb et al.

Alchemical free energy calculations and multiple conformational substates

J. Chem. Phys.

(2005)

N. Lu et al.

Variational formula for the free energy based on incomplete sampling in a molecular simulation

Phys. Rev. E

(2003)

H.J.C. Berendsen

Incomplete equilibration: A source of error in free energy computations

P.J. Steinbach et al.

New spherical-cutoff methods for long-range forces in macromolecular simulation

J. Comp. Chem.

(1994)

M. Lisal et al.

An examination of the five-site potential (TIP5P) for water

J. Chem. Phys.

(2002)

P. Mark et al.

Structure and dynamics of liquid water with different long-range interaction truncation and temperature control methods in molecular dynamics simulations

J. Comp. Chem.

(2002)

P.A. Kollman

Free energy calculations: Applications to chemical and biochemical phenomena

Chem. Rev.

(1993)

D.L. Beveridge et al.

Free energy via molecular simulation: Applications to chemical and biomolecular systems

Annu. Rev. Biophys. Biophys. Chem.

(1989)

W.L. Jorgensen

Free energy calculations: A breakthrough for modeling organic chemistry in solution

Acc. Chem. Res.

(1989)

D. Frenkel et al.

Understanding Molecular Simulation: From Algorithms to Applications

(2002)

M.S. Shell et al.

Calculating free energy differences using perturbation theory

E. Darve

Thermodynamic integration using constrained and unconstrained dynamics

R.W. Zwanzig

High-temperature equation of state by a perturbation method. I. Nonpolar gases

J. Chem. Phys.

(1954)

N.D. Lu et al.

Appropriate methods to combine forward and reverse free-energy perturbation averages

J. Chem. Phys.

(2003)

N.D. Lu et al.

Accuracy of free-energy perturbation calculations in molecular simulation. I. Modeling

J. Chem. Phys.

(2001)

N.D. Lu et al.

Accuracy of free-energy perturbation calculations in molecular simulation. II. Heuristics

J. Chem. Phys.

(2001)

M.R. Shirts et al.

Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods

Phys. Rev. Lett.

(2003)

M.R. Shirts et al.

Comparison of efficiency and bias of free energies computed by exponential averaging, the Bennett acceptance ratio, and thermodynamic integration

J. Chem. Phys.

(2005)

A.M. Ferrenberg et al.

Optimized Monte Carlo data analysis

Phys. Rev. Lett.

(1989)

S. Kumar et al.

The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method

J. Comp. Chem.

(1992)

C. Bartels et al.

Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations

J. Comp. Chem.

(1997)

E. Gallicchio et al.

Temperature weighted histogram analysis method, replica exchange, and transition paths

J. Phys. Chem. B

(2005)

J.G. Kirkwood

Statistical mechanics of fluid mixtures

J. Chem. Phys.

(1935)

H. Resat et al.

Studies on free energy calculations. I. Thermodynamic integration using a polynomial path

J. Chem. Phys.

(1993)

C.R.W. Guimaraes et al.

Thermodynamic analysis of thrombin inhibition by benzamidine and p-methylbenzamidine via free-energy perturbations: Inspection of intraperturbed-group contributions using the finite difference thermodynamic integration (FDTI) algorithm

J. Phys. Chem. B

(2002)

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Chapter 4 Alchemical Free Energy Calculations: Ready for Prime Time?

Publisher Summary

Introduction

Section snippets

Background

Equilibrium Methods

Nonequilibrium Methods

Intermediate States

Sampling

Applications

Conclusion

Acknowledgements

Curr. Opin. Struct. Biol.

Fluid Phase Equilibria

Annu. Rep. Comp. Chem.

Adv. Prot. Chem.

J. Comp. Phys.

Comp. Phys. Comm.

Annu. Rep. Comp. Chem.

Chem. Phys.

Chem. Phys. Lett.

J. Mol. Biol.

J. Mol. Graph. Model.

Chem. Phys. Lett.

Chem. Phys. Lett.

Chem. Phys. Lett.

Chem. Phys. Lett.

Chem. Phys. Lett.

Biophys. J.

Biophys. J.

Biophys. J.

The many roles of computation in drug discovery

Science

Free energy calculations: the long and winding gilded road

Mol. Simulation

Introduction

Calculating free energy differences using perturbation theory

Alchemical free energy calculations and multiple conformational substates

J. Chem. Phys.

Variational formula for the free energy based on incomplete sampling in a molecular simulation

Phys. Rev. E

Incomplete equilibration: A source of error in free energy computations

New spherical-cutoff methods for long-range forces in macromolecular simulation

J. Comp. Chem.

An examination of the five-site potential (TIP5P) for water

J. Chem. Phys.

Structure and dynamics of liquid water with different long-range interaction truncation and temperature control methods in molecular dynamics simulations

J. Comp. Chem.

Free energy calculations: Applications to chemical and biochemical phenomena

Chem. Rev.

Free energy via molecular simulation: Applications to chemical and biomolecular systems

Annu. Rev. Biophys. Biophys. Chem.

Free energy calculations: A breakthrough for modeling organic chemistry in solution

Acc. Chem. Res.

Understanding Molecular Simulation: From Algorithms to Applications

Calculating free energy differences using perturbation theory

Thermodynamic integration using constrained and unconstrained dynamics

High-temperature equation of state by a perturbation method. I. Nonpolar gases

J. Chem. Phys.

Appropriate methods to combine forward and reverse free-energy perturbation averages

J. Chem. Phys.

Accuracy of free-energy perturbation calculations in molecular simulation. I. Modeling

J. Chem. Phys.

Accuracy of free-energy perturbation calculations in molecular simulation. II. Heuristics

J. Chem. Phys.

Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods

Phys. Rev. Lett.

Comparison of efficiency and bias of free energies computed by exponential averaging, the Bennett acceptance ratio, and thermodynamic integration

J. Chem. Phys.

Optimized Monte Carlo data analysis

Phys. Rev. Lett.

The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method

J. Comp. Chem.

Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations

J. Comp. Chem.

Temperature weighted histogram analysis method, replica exchange, and transition paths

J. Phys. Chem. B

Statistical mechanics of fluid mixtures

J. Chem. Phys.

Studies on free energy calculations. I. Thermodynamic integration using a polynomial path