1
|
Lundborg M, Lidmar J, Hess B. On the Path to Optimal Alchemistry. Protein J 2023; 42:477-489. [PMID: 37651042 PMCID: PMC10480267 DOI: 10.1007/s10930-023-10137-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 07/04/2023] [Indexed: 09/01/2023]
Abstract
Alchemical free energy calculations have become a standard and widely used tool, in particular for calculating and comparing binding affinities of drugs. Although methods to compute such free energies have improved significantly over the last decades, the choice of path between the end states of interest is usually still the same as two decades ago. We will show that there is a fundamentally arbitrary, implicit choice of parametrization of this path. To address this, the notion of the length of a path or a metric is required. A metric recently introduced in the context of the accelerated weight histogram method also proves to be very useful here. We demonstrate that this metric can not only improve the efficiency of sampling along a given path, but that it can also be used to improve the actual choice of path. For a set of relevant use cases, the combination of these improvements can increase the efficiency of alchemical free energy calculations by up to a factor 16.
Collapse
Affiliation(s)
| | - Jack Lidmar
- Department of Physics, KTH Royal Institute of Technology, 10691, Stockholm, Sweden
| | - Berk Hess
- Department of Applied Physics, KTH Royal Institute of Technology, 10691, Stockholm, Science for Life Laboratory, Solna, Sweden.
| |
Collapse
|
2
|
Blaber S, Sivak DA. Skewed thermodynamic geometry and optimal free energy estimation. J Chem Phys 2020; 153:244119. [PMID: 33380076 DOI: 10.1063/5.0033405] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023] Open
Abstract
Free energy differences are a central quantity of interest in physics, chemistry, and biology. We develop design principles that improve the precision and accuracy of free energy estimators, which have potential applications to screening for targeted drug discovery. Specifically, by exploiting the connection between the work statistics of time-reversed protocol pairs, we develop near-equilibrium approximations for moments of the excess work and analyze the dominant contributions to the precision and accuracy of standard nonequilibrium free-energy estimators. Within linear response, minimum-dissipation protocols follow the geodesics of the Riemannian metric induced by the Stokes friction tensor. We find that the next-order contribution arises from the rank-3 supra-Stokes tensor that skews the geometric structure such that minimum-dissipation protocols follow the geodesics of a generalized cubic Finsler metric. Thus, near equilibrium, the supra-Stokes tensor determines the leading-order contribution to the bias of bidirectional free-energy estimators.
Collapse
Affiliation(s)
- Steven Blaber
- Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
| | - David A Sivak
- Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
| |
Collapse
|
3
|
Schön JC. Optimal Control of Hydrogen Atom-Like Systems as Thermodynamic Engines in Finite Time. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1066. [PMID: 33286835 PMCID: PMC7597135 DOI: 10.3390/e22101066] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/27/2020] [Revised: 09/17/2020] [Accepted: 09/19/2020] [Indexed: 11/16/2022]
Abstract
Nano-size machines are moving from only being topics of basic research to becoming elements in the toolbox of engineers, and thus the issue of optimally controlling their work cycles becomes important. Here, we investigate hydrogen atom-like systems as working fluids in thermodynamic engines and their optimal control in minimizing entropy or excess heat production in finite-time processes. The electronic properties of the hydrogen atom-like system are controlled by a parameter κ reflecting changes in, e.g., the effective dielectric constant of the medium where the system is embedded. Several thermodynamic cycles consisting of combinations of iso-κ, isothermal, and adiabatic branches are studied, and a possible a-thermal cycle is discussed. Solving the optimal control problem, we show that the minimal thermodynamic length criterion of optimality for finite-time processes also applies to these cycles for general statistical mechanical systems that can be controlled by a parameter κ, and we derive an appropriate metric in probability distribution space. We show how the general formulas we have obtained for the thermodynamic length are simplified for the case of the hydrogen atom-like system, and compute the optimal distribution of process times for a two-state approximation of the hydrogen atom-like system.
Collapse
Affiliation(s)
- Johann Christian Schön
- Max Planck Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany
| |
Collapse
|
4
|
Kilburg D, Gallicchio E. Analytical Model of the Free Energy of Alchemical Molecular Binding. J Chem Theory Comput 2018; 14:6183-6196. [DOI: 10.1021/acs.jctc.8b00967] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023]
Affiliation(s)
- Denise Kilburg
- Department of Chemistry, Brooklyn College of the City University of New York, Brooklyn, New York 11210, United States
| | - Emilio Gallicchio
- Department of Chemistry, Brooklyn College of the City University of New York, Brooklyn, New York 11210, United States
| |
Collapse
|
5
|
Andresen B. Aktuelle Trends in der Thermodynamik in endlicher Zeit. Angew Chem Int Ed Engl 2011. [DOI: 10.1002/ange.201001411] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Bjarne Andresen
- Niels‐Bohr‐Institut, Universität Kopenhagen, Universitetsparken 5, DK‐2100 Kopenhagen (Dänemark) http://www.fys.ku.dk/∼andresen/BAhome/welcome.html
| |
Collapse
|
6
|
Andresen B. Current trends in finite-time thermodynamics. Angew Chem Int Ed Engl 2011; 50:2690-704. [PMID: 21374763 DOI: 10.1002/anie.201001411] [Citation(s) in RCA: 98] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2010] [Indexed: 11/08/2022]
Abstract
The cornerstone of finite-time thermodynamics is all about the price of haste and how to minimize it. Reversible processes may be ultimately efficient, but they are unrealistically slow. In all situations-chemical, mechanical, economical-we pay extra to get the job done quickly. Finite-time thermodynamics can be used to develop methods to limit that extra expenditure, be it in energy, entropy production, money, or something entirely different. Finite-time thermodynamics also includes methods to calculate the optimal path or mode of operation to achieve this minimal expenditure. The concept is to place the system of interest in contact with a time-varying environment which will coax the system along the desired path, much like guiding a horse along by waving a carrot in front of it.
Collapse
|
7
|
Okumura H. Optimization of partial multicanonical molecular dynamics simulations applied to an alanine dipeptide in explicit water solvent. Phys Chem Chem Phys 2010; 13:114-26. [PMID: 21038036 DOI: 10.1039/c0cp00371a] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The partial multicanonical algorithm for molecular dynamics and Monte Carlo simulations samples a wide range of an important part of the potential energy. Although it is a strong technique for structure prediction of biomolecules, the choice of the partial potential energy has not been optimized. In order to find the best choice, partial multicanonical molecular dynamics simulations of an alanine dipeptide in explicit water solvent were performed with 15 trial choices for the partial potential energy. The best choice was found to be the sum of the electrostatic, Lennard-Jones, and torsion-angle potential energies between solute atoms. In this case, the partial multicanonical simulation sampled all of the local-minimum free-energy states of the P(II), C(5), α(R), α(P), α(L), and C states and visited these states most frequently. Furthermore, backbone dihedral angles ϕ and ψ rotated very well. It is also found that the most important term among these three terms is the electrostatic potential energy and that the Lennard-Jones term also helps the simulation to overcome the steric restrictions. On the other hand, multicanonical simulation sampled all of the six states, but visited these states fewer times. Conventional canonical simulation sampled only four of the six states: The P(II), C(5), α(R), and α(P) states.
Collapse
Affiliation(s)
- Hisashi Okumura
- Research Center for Computational Science Institute for Molecular Science Okazaki, Aichi 444-8585, Japan.
| |
Collapse
|
8
|
Jansen M, Doll K, Schön JC. Addressing chemical diversity by employing the energy landscape concept. Acta Crystallogr A 2010; 66:518-34. [DOI: 10.1107/s0108767310026371] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2009] [Accepted: 07/04/2010] [Indexed: 11/11/2022] Open
Affiliation(s)
- Martin Jansen
- Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
| | | | | |
Collapse
|
9
|
Minh DDL, Chodera JD. Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages. J Chem Phys 2009; 131:134110. [PMID: 19814546 PMCID: PMC2771048 DOI: 10.1063/1.3242285] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2009] [Accepted: 09/13/2009] [Indexed: 11/14/2022] Open
Abstract
Existing optimal estimators of nonequilibrium path-ensemble averages are shown to fall within the framework of extended bridge sampling. Using this framework, we derive a general minimal-variance estimator that can combine nonequilibrium trajectory data sampled from multiple path-ensembles to estimate arbitrary functions of nonequilibrium expectations. The framework is also applied to obtain asymptotic variance estimates, which are a useful measure of statistical uncertainty. In particular, we develop asymptotic variance estimates pertaining to Jarzynski's equality for free energies and the Hummer-Szabo expressions for the potential of mean force, calculated from uni- or bidirectional path samples. These estimators are demonstrated on a model single-molecule pulling experiment. In these simulations, the asymptotic variance expression is found to accurately characterize the confidence intervals around estimators when the bias is small. Hence, the confidence intervals are inaccurately described for unidirectional estimates with large bias, but for this model it largely reflects the true error in a bidirectional estimator derived by Minh and Adib.
Collapse
Affiliation(s)
- David D L Minh
- Laboratory of Chemical Physics, NIDDK, National Institutes of Health, Bethesda, Maryland 20892, USA.
| | | |
Collapse
|
10
|
|
11
|
Schön JC, Jansen M. Determination, prediction, and understanding of structures, using the energy landscapes of chemical systems – Part II. ACTA ACUST UNITED AC 2009. [DOI: 10.1524/zkri.216.7.361.20362] [Citation(s) in RCA: 46] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Abstract
Abstract
In the past decade, new theoretical approaches have been developed to determine, predict and understand the struc-ture of chemical compounds. The central element of these methods has been the investigation of the energy landscape of chemical systems. Applications range from extended crystalline and amorphous compounds over clusters and molecular crystals to proteins. In this review, we are going to give an introduction to energy landscapes and methods for their investigation, together with a number of examples. These include structure prediction of extended and mo-lecular crystals, structure prediction and folding of proteins, structure analysis of zeolites, and structure determination of crystals from powder diffraction data.
Collapse
|
12
|
Minh DDL. Density-dependent analysis of nonequilibrium paths improves free energy estimates. J Chem Phys 2009; 130:204102. [PMID: 19485432 PMCID: PMC2832053 DOI: 10.1063/1.3139189] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2009] [Accepted: 04/29/2009] [Indexed: 11/15/2022] Open
Abstract
When a system is driven out of equilibrium by a time-dependent protocol that modifies the Hamiltonian, it follows a nonequilibrium path. Samples of these paths can be used in nonequilibrium work theorems to estimate equilibrium quantities such as free energy differences. Here, we consider analyzing paths generated with one protocol using another one. It is posited that analysis protocols which minimize the lag, the difference between the nonequilibrium and the instantaneous equilibrium densities, will reduce the dissipation of reprocessed trajectories and lead to better free energy estimates. Indeed, when minimal lag analysis protocols based on exactly soluble propagators or relative entropies are applied to several test cases, substantial gains in the accuracy and precision of estimated free energy differences are observed.
Collapse
Affiliation(s)
- David D L Minh
- Laboratory of Chemical Physics, NIDDK, National Institutes of Health, Bethesda, Maryland 20892-0520, USA.
| |
Collapse
|
13
|
Calculation of the Free Energy and the Entropy of Macromolecular Systems by Computer Simulation. REVIEWS IN COMPUTATIONAL CHEMISTRY 2007. [DOI: 10.1002/9780470125892.ch1] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register]
|
14
|
Atilgan E, Sun SX. Equilibrium free energy estimates based on nonequilibrium work relations and extended dynamics. J Chem Phys 2004; 121:10392-400. [PMID: 15549919 DOI: 10.1063/1.1813434] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Jarzynski's relation and the fluctuation theorem have established important connections between nonequilibrium statistical mechanics and equilibrium thermodynamics. In particular, an exact relationship between the equilibrium free energy and the nonequilibrium work is useful for computer simulations. In this paper, we exploit the fact that the free energy is a state function, independent of the pathway taken to change the equilibrium ensemble. We show that a generalized expression is advantageous for computer simulations of free energy differences. Several methods based on this idea are proposed. The accuracy and efficiency of the proposed methods are evaluated with a model problem.
Collapse
Affiliation(s)
- Erdinç Atilgan
- Department of Mechanical Engineering and Whitaker Insitute of Biomedical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
| | | |
Collapse
|
15
|
Ytreberg FM, Zuckerman DM. Efficient use of nonequilibrium measurement to estimate free energy differences for molecular systems. J Comput Chem 2004; 25:1749-59. [PMID: 15362132 DOI: 10.1002/jcc.20103] [Citation(s) in RCA: 63] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
A promising method for calculating free energy differences DeltaF is to generate nonequilibrium data via "fast-growth" simulations or by experiments--and then use Jarzynski's equality. However, a difficulty with using Jarzynski's equality is that DeltaF estimates converge very slowly and unreliably due to the nonlinear nature of the calculation--thus requiring large, costly data sets. The purpose of the work presented here is to determine the best estimate for DeltaF given a (finite) set of work values previously generated by simulation or experiment. Exploiting statistical properties of Jarzynski's equality, we present two fully automated analyses of nonequilibrium data from a toy model, and various simulated molecular systems. Both schemes remove at least several k(B)T of bias from DeltaF estimates, compared to direct application of Jarzynski's equality, for modest sized data sets (100 work values), in all tested systems. Results from one of the new methods suggest that good estimates of DeltaF can be obtained using 5-40-fold less data than was previously possible. Extending previous work, the new results exploit the systematic behavior of bias due to finite sample size. A key innovation is better use of the more statistically reliable information available from the raw data.
Collapse
Affiliation(s)
- F Marty Ytreberg
- Center for Computational Biology and Bioinformatics, University of Pittsburgh, 200 Lothrop St., Pittsburgh, Pennsylvania 15261, USA.
| | | |
Collapse
|
16
|
Zuckerman DM, Woolf TB. Overcoming finite-sampling errors in fast-switching free-energy estimates: extrapolative analysis of a molecular system. Chem Phys Lett 2002. [DOI: 10.1016/s0009-2614(01)01397-5] [Citation(s) in RCA: 56] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
|
17
|
Reinhardt WP, Miller MA, Amon LM. Why is it so difficult to simulate entropies, free energies, and their differences? Acc Chem Res 2001; 34:607-14. [PMID: 11456478 DOI: 10.1021/ar950181n] [Citation(s) in RCA: 46] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The classical 19th century thermodynamic inequalities of Clausius and Helmholtz are applied to the calculation of entropy and free energy changes by computer simulation. The irreversibility of finite-time thermodynamic paths is exploited to obtain upper and lower bounds on these quantities. Schrödinger's microscopic interpretation of heat and work provides the basis for a literal implementation of the key historical concepts on the computer using the Monte Carlo algorithm of Metropolis. Coupling schemes, paths, and reference states are variationally optimized to improve the convergence of the simulated properties, and a newly introduced variational flexibility, metric scaling, is overviewed. Reasons for expecting limiting power laws for the convergence are outlined.
Collapse
Affiliation(s)
- W P Reinhardt
- Department of Chemistry, Box 351700, University of Washington, Seattle, Washington 98195-1700, USA
| | | | | |
Collapse
|
18
|
General Discussion. Faraday Discuss 1997. [DOI: 10.1039/fd106389] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
|