Metadynamics

Metadynamics (MTD; also abbreviated as METAD or MetaD) is a computer simulation method in computational physics, chemistry and biology. It is used to compute free energy and other state functions of a system, where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Alessandro Laio and Michele Parrinello in 2002^[1] and is usually applied within molecular dynamics simulations. MTD closely resembles a number of recent methods such as adaptively biased molecular dynamics,^[2] adaptive reaction coordinate forces^[3] and local elevation umbrella sampling.^[4] More recently, both the original and well-tempered metadynamics^[5] were derived in the context of importance sampling and shown to be a special case of the adaptive biasing potential setting.^[6] MTD is related to the Wang-Landau sampling.^[7]

Introduction

The technique builds on a large number of related methods including (in a chronological order) the deflation,^[8] tunneling,^[9] tabu search,^[10] local elevation,^[11] conformational flooding,^[12] Engkvist-Karlström^[13] and adaptive biasing force methods.^[14]

Metadynamics has been informally described as "filling the free energy wells with computational sand".^[15] The algorithm assumes that the system can be described by a few collective variables. During the simulation, the location of the system in the space determined by the collective variables is calculated and a positive Gaussian potential is added to the real energy landscape of the system. In this way the system is discouraged to come back to the previous point. During the evolution of the simulation, more and more Gaussians sum up, thus discouraging more and more the system to go back to its previous steps, until the system explores the full energy landscape -at this point the modified free energy becomes a constant as a function of the collective variables which is the reason for the collective variables to start fluctuating heavily. At this point the energy landscape can be recovered as the opposite of the sum of all Gaussians.

The time interval between the addition of two Gaussian functions, as well as the Gaussian height and Gaussian width, are tuned to optimize the ratio between accuracy and computational cost. By simply changing the size of the Gaussian, metadynamics can be fitted to yield very quickly a rough map of the energy landscape by using large Gaussians, or can be used for a finer grained description by using smaller Gaussians.^[1] Usually, the well-temperated metadynamics^[5] is used to change the Gaussian size adaptively. Also, the Gaussian width can be adapted with the adaptive Gaussian metadynamics.^[16]

Metadynamics has the advantage, upon methods like adaptive umbrella sampling, of not requiring an initial estimate of the energy landscape to explore.^[1] However, it is not trivial to choose proper collective variables for a complex simulation. Typically, it requires several trials to find a good set of collective variables, but there are several automatic procedure proposed: essential coordinates,^[17] Sketch-Map,^[18] and non-linear data-driven collective variables.^[19]

Multi-replica approach

Independent metadynamics simulations (replicas) can be coupled together to improve usability and parallel performance. There are several such methods proposed: the multiple walker MTD,^[20] the parallel tempering MTD,^[21] the bias-exchange MTD,^[22] and the collective-variable tempering MTD.^[23] The last three are similar to the parallel tempering method and use replica exchanges to improve sampling. Typically, the Metropolis–Hastings algorithm is used for replica exchanges, but the infinite swapping^[24] and Suwa-Todo^[25] algorithms give better replica exchange rates.^[26]

Algorithm

Assume, we have a classical ${\textstyle N}$ -particle system with positions at ${\textstyle \{{\vec {r}}_{i}\}}$ ${\textstyle (i\in 1...N)}$ in the Cartesian coordinates ${\textstyle ({\vec {r}}_{i}\in \mathbb {R} ^{3})}$ . The particle interaction are described with a potential function ${\textstyle V\equiv V(\{{\vec {r}}_{i}\})}$ . The potential function form (e.g. two local minima separated by a high-energy barrier) prevents an ergodic sampling with molecular dynamics or Monte Carlo methods.

Original metadynamics

A general idea of MTD is to enhance the system sampling by discouraging revisiting of sampled states. It is achieved by augmenting the system Hamiltonian ${\textstyle H}$ with a bias potential $V_{\text{bias}}$ :

H=T+V+V_{\text{bias}}

The bias potential is a function of collective variables ${\textstyle (V_{\text{bias}}\equiv V_{\text{bias}}({\vec {s}}\,))}$ . A collective variable is a function of the particle positions $({\vec {s}}\equiv {\vec {s}}(\{{\vec {r}}_{i}\}))$ . The bias potential is continuously updated by adding bias at rate $\omega$ , where ${\vec {s}}_{t}$ is an instantaneous collective variable value at time $t$ :

{\frac {\partial V_{\text{bias}}({\vec {s}}\,)}{\partial t}}=\omega \,\delta (|{\vec {s}}-{\vec {s}}_{t}|)

At infinitely long simulation time $t_{\text{sim}}$ , the accumulated bias potential converges to free energy with opposite sign (and irrelevant constant $C$ ):

V_{\text{bias}}({\vec {s}}\,)=\!\!\int _{0}^{t_{\text{sim}}}\!\!\!\omega \,\delta (|{\vec {s}}-{\vec {s}}_{t}|)\;dt\quad \Rightarrow \quad F({\vec {s}}\,)=-\!\!\!\!\lim _{t_{\text{sim}}\to \infty }\!\!V_{\text{bias}}({\vec {s}}\,)+C

For a computationally efficient implementation, the update process is discretised into $\tau$ time intervals ( $\lfloor \;\rfloor$ denotes the floor function) and $\delta$ -function is replaced with a localized positive kernel function $K$ . The bias potential becomes a sum of the kernel functions centred at the instantaneous collective variable values ${\vec {s}}_{j}$ at time $\tau j$ :

V_{\text{bias}}({\vec {s}}\,)\approx \tau \!\!\!\sum _{j=0}^{\left\lfloor {\frac {t_{\text{sim}}}{\tau }}\right\rfloor }\!\!\omega \,K(|{\vec {s}}-{\vec {s}}_{j}|)

Typically, the kernel is a multi-dimensional Gaussian function, which covariance matrix has diagonal non-zero elements only:

V_{\text{bias}}({\vec {s}}\,)\approx \tau \!\!\!\sum _{j=0}^{\left\lfloor {\frac {t_{\text{sim}}}{\tau }}\right\rfloor }\!\!\omega \exp \!\!\left(\!-{\frac {1}{2}}\left|{\frac {{\vec {s}}-{\vec {s}}_{j}}{\vec {\sigma }}}\right|^{2}\right)

The parameter $\tau$ , $\omega$ , and $\vec \sigma$ are determined a priori and kept constant during the simulation.

Implementation

Below there is a pseudocode of MTD base on molecular dynamics (MD), where $\{{\vec {r}}\}$ and $\{{\vec {v}}\}$ are the $N$ -particle system positions and velocities, respectively. The bias $V_{\text{bias}}$ is updated every $n=\tau /\Delta t$ MD steps, and its contribution to the system forces $\{{\vec {F}}\,\}$ is $\{{\vec {F}}_{\text{bias}}\}$ .

set initial  $\{{\vec {r}}\}$  and  $\{{\vec {v}}\}$  
set  $V_{\text{bias}}({\vec {s}}\,):=0$ 

every MD step:
    compute CV values:
         ${\vec {s}}_{t}:={\vec {s}}(\{{\vec {r}}\})$ 
    
    every  $n$  MD steps:
        update bias potential:
             $V_{\text{bias}}({\vec {s}}\,):=V_{\text{bias}}({\vec {s}}\,)+\tau \omega \exp \!\!\left(\!-{\frac {1}{2}}\left|{\frac {{\vec {s}}-{\vec {s}}_{t}}{\vec {\sigma }}}\right|^{2}\right)$ 
    
    compute atomic forces:
         ${\vec {F}}_{i}:=-{\frac {\partial V(\{{\vec {r}}\,\})}{\partial {\vec {r}}_{i}}}\overbrace {\left.-{\frac {\partial V_{\text{bias}}({\vec {s}}\,)}{\partial {\vec {s}}}}\right|_{{\vec {s}}_{t}}\!\!\!{\frac {\partial {\vec {s}}(\{{\vec {r}}\,\})}{\partial {\vec {r}}_{i}}}} ^{{\vec {F}}_{{\text{bias}},i}}$ 
    
    propagate  $\{{\vec {r}}\}$  and  $\{{\vec {v}}\}$  by  $\Delta t$

Applications

Metadynamics has been used to study:

Implementations

PLUMED

PLUMED^[31] is an open-source library implementing many MTD algorithms and collective variables. It has a flexible object-oriented design^[32]^[33] and can interfaced with several MD programs (AMBER, GROMACS, LAMMPS, NAMD, Quantum ESPRESSO, and CP2K).^[34]^[35]

Other

Other MTD implementations exist in LAMMPS, NAMD, ORAC, CP2K,^[36] and Desmond.

External links

References

1 2 3 Laio, A.; Parrinello, M. (2002). "Escaping free-energy minima". Proceedings of the National Academy of Sciences of the United States of America. 99 (20): 12562–12566. arXiv:cond-mat/0208352. Bibcode:2002PNAS...9912562L. doi:10.1073/pnas.202427399. PMC 130499. PMID 12271136.
↑ Babin, V.; Roland, C.; Sagui, C. (2008). "Stabilization of resonance states by an asymptotic Coulomb potential". J. Chem. Phys. 128 (2): 134101/1–134101/7. Bibcode:2008JChPh.128b4101A. doi:10.1063/1.2821102.
↑ Barnett, C.B.; Naidoo, K.J. (2009). "Free Energies from Adaptive Reaction Coordinate Forces (FEARCF): An application to ring puckering". Mol. Phys. 107 (8): 1243–1250. Bibcode:2009MolPh.107.1243B. doi:10.1080/00268970902852608.
↑ Hansen, H.S.; Hünenberger, P.H. (2010). "Using the local elevation method to construct optimized umbrella sampling potentials: Calculation of the relative free energies and interconversion barriers of glucopyranose ring conformers in water". J. Comput. Chem. 31 (1): 1–23. doi:10.1002/jcc.21253. PMID 19412904.
1 2 Barducci, A.; Bussi, G.; Parrinello, M. (2008). "Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method". Physical Review Letters. 100 (2): 020603. arXiv:0803.3861. Bibcode:2008PhRvL.100b0603B. doi:10.1103/PhysRevLett.100.020603. PMID 18232845.
↑ Dickson, B.M. (2011). "Approaching a parameter-free metadynamics". Phys. Rev. E. 84: 037701–037703. arXiv:1106.4994. Bibcode:2011PhRvE..84c7701D. doi:10.1103/PhysRevE.84.037701.
↑ Junghans, Christoph, Danny Perez, and Thomas Vogel. "Molecular Dynamics in the Multicanonical Ensemble: Equivalence of Wang–Landau Sampling, Statistical Temperature Molecular Dynamics, and Metadynamics." Journal of Chemical Theory and Computation 10.5 (2014): 1843-1847.
↑ Crippen, Gordon M.; Scheraga, Harold A. (1969). "Minimization of polypeptide energy. 8. Application of the deflation technique to a dipeptide". Proceedings of the National Academy of Sciences. 64 (1): 42–49. Bibcode:1969PNAS...64...42C. doi:10.1073/pnas.64.1.42. PMC 286123. PMID 5263023.
↑ Levy, A.V.; Montalvo, A. (1985). "The Tunneling Algorithm for the Global Minimization of Functions". SIAM J. Sci. Stat. Comput. 6: 15–29. doi:10.1137/0906002.
↑ Glover, Fred (1989). "Tabu Search—Part I". ORSA Journal on Computing. 1 (3): 190–206. doi:10.1287/ijoc.1.3.190.
↑ Huber, T.; Torda, A.E.; van Gunsteren, W.F. (1994). "Local elevation: A method for improving the searching properties of molecular dynamics simulation". J. Comput. -Aided. Mol. Des. 8 (6): 695–708. Bibcode:1994JCAMD...8..695H. doi:10.1007/BF00124016. PMID 7738605.
↑ Grubmüller, H. (1995). "Predicting slow structural transitions in macromolecular systems: Conformational flooding". Phys. Rev. E. 52 (3): 2893–2906. Bibcode:1995PhRvE..52.2893G. doi:10.1103/PhysRevE.52.2893.
↑ Engkvist, O.; Karlström, G. (1996). "A method to calculate the probability distribution for systems with large energy barriers". Chem. Phys. 213: 63–76. Bibcode:1996CP....213...63E. doi:10.1016/S0301-0104(96)00247-9.
↑ Darve, E.; Pohorille, A. (2001). "Calculating free energies using average force". J. Chem. Phys. 115 (20): 9169. Bibcode:2001JChPh.115.9169D. doi:10.1063/1.1410978.
↑ http://www.grs-sim.de/cms/upload/Carloni/Presentations/Marinelli.ppt
↑ Branduardi, Davide; Bussi, Giovanni; Parrinello, Michele (2012-06-04). "Metadynamics with Adaptive Gaussians". Journal of Chemical Theory and Computation. 8 (7): 2247–2254. doi:10.1021/ct3002464.
↑ Spiwok, V.; Lipovová, P.; Králová, B. (2007). "Metadynamics in essential coordinates: free energy simulation of conformational changes". The Journal of Physical Chemistry B. 111 (12): 3073–3076. doi:10.1021/jp068587c. PMID 17388445.
↑ Ceriotti, Michele; Tribello, Gareth A.; Parrinello, Michele (2013-02-22). "Demonstrating the Transferability and the Descriptive Power of Sketch-Map". Journal of Chemical Theory and Computation. 9 (3): 1521–1532. doi:10.1021/ct3010563.
↑ Hashemian, Behrooz; Millán, Daniel; Arroyo, Marino (2013-12-07). "Modeling and enhanced sampling of molecular systems with smooth and nonlinear data-driven collective variables". The Journal of Chemical Physics. 139 (21): 214101. Bibcode:2013JChPh.139u4101H. doi:10.1063/1.4830403. ISSN 0021-9606.
↑ Raiteri, Paolo; Laio, Alessandro; Gervasio, Francesco Luigi; Micheletti, Cristian; Parrinello, Michele (2005-10-28). "Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics †". The Journal of Physical Chemistry B. 110 (8): 3533–3539. doi:10.1021/jp054359r.
↑ Bussi, Giovanni; Gervasio, Francesco Luigi; Laio, Alessandro; Parrinello, Michele (October 2006). "Free-Energy Landscape for β Hairpin Folding from Combined Parallel Tempering and Metadynamics". Journal of the American Chemical Society. 128 (41): 13435–13441. doi:10.1021/ja062463w.
1 2 Piana, S.; Laio, A. (2007). "A bias-exchange approach to protein folding". The Journal of Physical Chemistry B. 111 (17): 4553–4559. doi:10.1021/jp067873l. PMID 17419610.
↑ Gil-Ley, Alejandro; Bussi, Giovanni (2015-02-19). "Enhanced Conformational Sampling Using Replica Exchange with Collective-Variable Tempering". Journal of Chemical Theory and Computation. 11 (3): 1077–1085. doi:10.1021/ct5009087. PMC 4364913. PMID 25838811.
↑ Plattner, Nuria; Doll, J. D.; Dupuis, Paul; Wang, Hui; Liu, Yufei; Gubernatis, J. E. (2011-10-07). "An infinite swapping approach to the rare-event sampling problem". The Journal of Chemical Physics. 135 (13): 134111. arXiv:1106.6305. Bibcode:2011JChPh.135m4111P. doi:10.1063/1.3643325. ISSN 0021-9606.
↑ Suwa, Hidemaro (2010-01-01). "Markov Chain Monte Carlo Method without Detailed Balance". Physical Review Letters. 105 (12). arXiv:1007.2262. Bibcode:2010PhRvL.105l0603S. doi:10.1103/PhysRevLett.105.120603.
↑ Galvelis, Raimondas; Sugita, Yuji (2015-07-15). "Replica state exchange metadynamics for improving the convergence of free energy estimates". Journal of Computational Chemistry. 36 (19): 1446–1455. doi:10.1002/jcc.23945. ISSN 1096-987X.
↑ Ensing, B.; De Vivo, M.; Liu, Z.; Moore, P.; Klein, M. (2006). "Metadynamics as a tool for exploring free energy landscapes of chemical reactions". Accounts of Chemical Research. 39 (2): 73–81. doi:10.1021/ar040198i. PMID 16489726.
↑ Gervasio, F.; Laio, A.; Parrinello, M. (2005). "Flexible docking in solution using metadynamics". Journal of the American Chemical Society. 127 (8): 2600–2607. doi:10.1021/ja0445950. PMID 15725015.
↑ Vargiu, A. V.; Ruggerone, P.; Magistrato, A.; Carloni, P. (2008). "Dissociation of minor groove binders from DNA: insights from metadynamics simulations". Nucleic Acids Research. 36 (18): 5910–5921. doi:10.1093/nar/gkn561. PMC 2566863. PMID 18801848.
↑ Martoňák, R.; Laio, A.; Bernasconi, M.; Ceriani, C.; Raiteri, P.; Zipoli, F.; Parrinello, M. (2005). "Simulation of structural phase transitions by metadynamics". Zeitschrift für Kristallographie. 220 (5–6): 489. arXiv:cond-mat/0411559. Bibcode:2005ZK....220..489M. doi:10.1524/zkri.220.5.489.65078.
↑ "PLUMED". www.plumed.org. Retrieved 2016-01-26.
↑ Bonomi, Massimiliano; Branduardi, Davide; Bussi, Giovanni; Camilloni, Carlo; Provasi, Davide; Raiteri, Paolo; Donadio, Davide; Marinelli, Fabrizio; Pietrucci, Fabio (2009-10-01). "PLUMED: A portable plugin for free-energy calculations with molecular dynamics". Computer Physics Communications. 180 (10): 1961–1972. arXiv:0902.0874. Bibcode:2009CoPhC.180.1961B. doi:10.1016/j.cpc.2009.05.011.
↑ Tribello, Gareth A.; Bonomi, Massimiliano; Branduardi, Davide; Camilloni, Carlo; Bussi, Giovanni (2014-02-01). "PLUMED 2: New feathers for an old bird". Computer Physics Communications. 185 (2): 604–613. arXiv:1310.0980. Bibcode:2014CoPhC.185..604T. doi:10.1016/j.cpc.2013.09.018.
↑ "MD engines - PLUMED". www.plumed.org. Retrieved 2016-01-26.
↑ "howto:install_with_plumed [CP2K Open Source Molecular Dynamics ]". www.cp2k.org. Retrieved 2016-01-26.
↑ http://manual.cp2k.org/trunk/CP2K_INPUT/MOTION/FREE_ENERGY/METADYN.html

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