Computing free energies with PLUMED 2.0

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1 Computing free energies with PLUMED 2.0 Davide Branduardi Formerly at MPI Biophysics, Frankfurt a.m.

2 TyOutline of the talk Relevance of free energy computation Free-energy from histograms: issues Tackling the sampling problem Tackling the order-parameter problem 2

TyRelevance of free energy calculations Free

React Drug binding A A Prod s (RC) A(s 0 )=

3 TyRelevance of free energy calculations Free energy is a measure of probability Chemical reactions Phase transitions Free Energy find elusive structures and mechanisms! React Drug binding A A Prod s (RC) A(s 0 )= k B T ln P (s 0 ) Biophysics Molecular recognition 3

4 TyRare event: troubles with ergodic hypothesis Free Energy React A stop? Prod 1 2 k BT RC k(t )= k BT hc 0 e A /RT Now I sampled all the two states. Is it sufficient? property Free energy requires statistics! A = 1 ln N 1 N 2 property [3] Eyring, Chem. Rev vol. 17 pp 65 time time 4

5 Ty A test on ATP-Mg 2+ in water ATP 4- -Mg 2+ is of fundamental importance in bioenergetics and regulation Free energy of α β γ to β γ α β γ β γ OW3 O2α OW2 + OW4 O1α OW3 Mg OW4 OW1 + OW2 O2α O1α [1] Liao, Sun, Chandler, Oster Eur. Biophys. J. 2004, vol. 33, p. 29 5

6 TyErgodic hypothesis: am I visiting all the states? Classical MD, CHARMM27 force field, 915 TIP3P water molecules, GROMACS code 6

7 TyPossible solutions Replica exchange methods: multiple simulations at different temperatures or Hamiltonians Biased dynamics: add a potential function along the degree(s) of freedom you want to accelerate Biased dynamics + Replica exchange 7

8 Ty Thermodynamic-Integration-like approaches The ancestor is mean force calculation through harmonic potential F (s) = k B T ln P (s) F (s) s s 0 = k B T 1 R V (x) e k B T (s 0 (x) s)dx s Z e V (x) k B T (s 0 (x) s)dx s 0 Dirac delta function Gaussian function (I like this trick!) F (s) s s 0 ' k B T F (s) s s 0 ' R e V (x) k B T e 1 R e V (x)+k/2(s 0 (x) s 0 ) 2 k B T dx 1 k 2k B T (s0 (x) s) 2 dx s It is an average of a biased simulation! Z k(s 0 (x) Z e V (x) k B T e k 2k B T (s0 (x) s) 2 dx s 0 )e V (x)+k/2(s 0 (x) s0 ) 2 k B T dx s 0 = khs 0 (x) s 0 i bias,s0 (opposite of the mean force ) 8

9 TyThermodynamic Integration 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability descriptor descriptor or +WHAM etc... descriptor 9

10 TyAn example with ATP-Mg Free energy (kcal/mol) O2α -6 Mg Distance (Å) O2α Mg 10

TyThermodynamic-Integration-like approaches Thermodynamic integration WHAM Roux, Comp Phys Comm 1995, vol. 91, pp. 275.

11 TyThermodynamic-Integration-like approaches Thermodynamic integration WHAM Roux, Comp Phys Comm 1995, vol. 91, pp Free energy perturbation Beveridge, DiCapua, Annu Rev Biophys Biophys Chem 1989, vol 18, pp Beveridge, DiCapua, Annu Rev Biophys Biophys Chem 1989, vol 18, pp Jarzynski-equation based approaches (steered-md) Jarzynski Phys Rev Lett 1997, vol. 78, pp Crooks-equation based approaches (two directions steered-md) Crooks J Stat Phys 1998, vol. 90, pp

12 Ty Avoid recoding with MD positions time=t forces time=t evolution time=t+ t Enhanced sampling calculate additional (history dependent) forces Each single engine had only few enhanced sampling techniques (according developers interests) in NAMD, GROMACS, AMBER, LAMMPS, QUANTUM-ESPRESSO Bonomi, Branduardi, Bussi, Camilloni, Provasi, Raiteri, Donadio, Marinelli, Pietrucci, Broglia, Parrinello, M. Comp. Phys. Commun. 2009, 180, Tribello, Bonomi, Branduardi, Camilloni, Bussi Comp. Phys. Commun. (2013) in press. 12

13 TyNow in ESPResSo: 13

14 TyHow to set this up in PLUMED? Mg O2α m(x,t)= k 2 (d(x) x 0(t)) 2 c(x) = SOD X i PHO X j 1 1 NN xi x j d 0 r 0 MM xi x j d 0 r 0 c(x) > 0.2 : U(X) =k (c(x) 0.2) 4 c(x) < 0.2 : U(X) =0. 14

TyAn example with ATP-Mg 2+ Free energy (kcal/mol) 12 9 6 3 0-3 Hysteresis!

15 TyAn example with ATP-Mg 2+ Free energy (kcal/mol) Hysteresis! Forward Backward O2α -6 Mg Distance (Å) O2α Mg 15

16 TyAdaptive methods: Torrie & Valleau [3] Free energy: Free energy for a biased system: use biasing potential to overcome barriers and retrieve free energies [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp

17 TyLessons from Torrie & Valleau Free Energy Biasing potentials: React biasing pot 1 2 k BT resulting potential Prod 1 2 k BT RC allow to measure free energies: useful allow to overcome barriers: cheap React Prod need only an additional energy term to standard force field: easy Free Energy Metastabilities are removed RC [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp

18 TyAdaptive sampling methods Adaptive Biasing Force Darve & Pohorille, J. Chem. Phys. 2001, vol. 115, pp Adaptive Umbrella Sampling Mezei, M. J Comput Phys 1987, vol. 68, pp Self Healing Umbrella Sampling Marsili et. al J. Chem. Phys. B vol. 110, pp Metadynamics Laio & Parrinello, PNAS 2002, vol. 20, pp Conformational Flooding Grubmüller, Phys Rev E 1995, vol. 52, pp

Ty Enhanced sampling: metadynamics Apply a repulsive gaussian potential up to compensate the underlying free energy: V g (t, s(x)) = X N hills i=1 We (s(x) s i ) 2 2 eh(t, x) =H(x)+V g (t, s(x))

19 Ty Enhanced sampling: metadynamics Apply a repulsive gaussian potential up to compensate the underlying free energy: V g (t, s(x)) = X N hills i=1 We (s(x) s i ) 2 2 eh(t, x) =H(x)+V g (t, s(x)) Produces a non markovian dynamics. The negative of the potential deposited is proven to converge to the correct free energy surface [6] Transition between metastable states are enhanced by artificial removal of metastable states [4] Laio & Parrinello, PNAS 2002, vol. 20, p [5] Bussi & Laio, Parrinello Phys. Rev. Lett. 2006, vol 96, p [6] Branduardi, Bussi, Parrinello, J. Chem. Theory Comput. 2012, vol 8, p

20 TyHow to set this up in PLUMED? V g (t, s(x)) = NX hills i=1 We (s(x) s i ) 2 2 eh(t, x) =H(x)+V g (t, s(x)) d(x) 0.65 c(x) > 0.65 : U(X) = c(x) < 0.65 : U(X) =

21 TyA test on ATP-Mg 2+ in water: meta with distance Choose the CV(s): maximum 3! Free energy estimate React. Prod. AReact.-AProd. 21

22 TyAdaptive sampling vs TI-like methods Adaptive sampling: The problem is multidimensional but probably the accessible phase space is limited: they explore only what you need The free energy landscape has competitive, parallel reactive paths (solvent degrees of freedom, rotations) that can be overcome at some point TI-like: In 1-d (max 2-d, via WHAM or, better with rbf fitting schemes) very effective Sometimes trivially parallel (many umbrellas at same time) 22

23 TyMisconceptions in free energy calculations Technique (TI, WHAM, SMD, MetaD, ABF?) Choice of order parameter 23

24 TyMisconceptions in free energy calculations Technique (TI, WHAM, SMD, MetaD, ABF?) Choice of order parameter 24

25 Ty The importance of the order parameter Different order parameters give different landscapes HighE LowE hidden var random Transition state is mixed pre post Poor convergence of free energy free energy enhanced A CV CV Unreliable transition rates k(t )= k BT hc 0 e A /RT Hard to compare with Expt! [7] Eyring, Chem. Rev vol. 17 p 65 25

26 Ty How to choose the order parameter Different order parameters give different landscapes HighE LowE hidden var Combination of CVs hidden var free energy A CV CV free energy enhanced PRO: the best perspective has good convergence and rates! CONS: it is a trial and error procedure! enhanced A CV CV 26

TyPath free energy calculations Monodimensional nonlinear mapping of the transition in terms of many coordinates Progress Distance along from the path hidden var selected path path snapshots in a in

27 TyPath free energy calculations Monodimensional nonlinear mapping of the transition in terms of many coordinates Progress Distance along from the path hidden var selected path path snapshots in a in given a given representation z(x) s(x) = actual MD snapshot actual MD snapshot P P i ie x x i P 1 PX ln P i e e x x i metrics (cartesian? MSD?) CV i x x i CV CV Adaptive (it uses information from a previous transition) Preserves low dimensional description for efficiency Allows extensions (distance from path) Beyond metadynamics: Steered molecular dynamics and others [8] Branduardi, Gervasio, Parrinello,J. Chem. Phys. 2007, vol.126, p

28 Ty Path collective variables: ATP-Mg 2+ Free energy estimate AReact.-AProd. React. Prod. the shape is conserved, the free energy difference has smaller fluctuations! 28

29 TyPLUMED provides many CVs: are they enough? 29

30 Ty Plumed provides many CVs : combine with functions 30

31 Ty Plumed provides many CVs : matheval c(t) =cos(0.5t 1.028) s(t) =sin(0.5t 1.028) y x U(X,t) = 100 [(s(t) sin(x)) 2 +(c(t) cos(x)) 2 ] 31

32 Ty Plumed provides many CVs : matheval 32

33 TyPlumed is also a set of tools 33

34 TySummary Enhanced sampling method makes you cross barriers Free energies can be calculated from biased dynamics Free energies depends on the technique and from the order parameter Order parameters are as important as the accelerated sampling method PLUMED offers a flexible tool to access a variety of methods and collective variables Beyond what I shown here: mixed replica exchange/biased dynamics (PT-WTE, PT-MetaD, Bias Exchange) Even more: String methods 34

(Queen s University, Belfast) Carlo Camilloni

35 TyAcknowledgements Giovanni Bussi (SISSA, Trieste) Max Bonomi (UCSF, San Francisco) Gareth Tribello (Queen s University, Belfast) Carlo Camilloni (Cambridge University) Michele Parrinello (ETH, Zurich) 35

The PLUMED plugin and free energy methods in electronic-structure-based molecular dynamics

The PLUMED plugin and free energy methods in electronic-structure-based molecular dynamics Davide Branduardi, Theoretical Molecular Biophysics Group Max Planck for Biophysics, Frankfurt am Main, Germany