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

Transcription

1 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

2 TyOutline What is free energy and why is important Traditional QC approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 2

3 TyOutline What is free energy and why is important Traditional QC approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 3

4 Ty What is free energy and why is important F (s) = k B T ln P (s) P (s) = e V (x) k bt δ(s (x) s)dx Q conformational transitions and equilibria s= state (bound/unbound, reactant/products, phasea/phaseb) ligand binding mechanism of transporters or channels chemical reactions and catalysis phase transitions... P (out) P (near) P (in) 4

5 TyOutline What is free energy and why is important Traditional QC approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 5

6 TyTraditional QC approach Rigid rotor/ harmonic approximation is well known [1] F (s) =H(s) TS(s) S(s) =R + R ln (q t (s)q e (s)q r (s)q v (s)) + RT ln qt (s)q e (s)q r (s)q v (s) T trasl, elect, rot, vib partition functions where the partition functions are obtained from the calculated vibrational spectrum (Hessian), moments of inertia, electronic structure H(s) =H el (s)+h t (s)+h v (s)+h r (s) It requires electronic structure and Hessian in the local minima and transition states: 1 single conformation per state. [1] McQuarrie, Simon Physical Chemistry, a molecular approach 6

7 Ty Finding the RC (I) Static approach: look for saddle points by using chemical intuition and TS optimization. m1 m2 7910

8 Ty Finding the RC Static approach: look for saddle points by using chemical intuition and TS optimization. m1 Verify then by IRC: move forward and backward to verify to end in minima

9 Ty Finding the RC Static approach: look for saddle points by using chemical intuition and TS optimization. m1 m2 Verify then by IRC: move forward and backward to verify to end in minima

10 Ty Finding the RC Static approach: look for saddle points by using chemical intuition and TS optimization. m1 m2 Verify then by IRC: move forward and backward to verify to end in minima. G = H T S Rates may be derived from Eyring equation [2] : k(t )= k bt hc 0 e G/RT [2] Eyring, Chem. Rev vol. 17 pp

11 TyTraditional QC approach PRO: one calculation per point (3 single points calculations per rate and free energy differences) CONS: optimizing TS is not trivial (redundant coordinates), the landscape must be very simple (Free) energy ortho coor Reaction coor 11

12 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12

13 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness A TS B Perfectly parallel basins, Not too bad (small correction over static approaches). Typical case of symmetry in a reaction 12

14 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12

15 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness Perfectly parallel basins but energetically asymmetric, Can be done with static QC but need to sample all the paths 12

16 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12

17 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness Rough paths: vibrations are meaningless, paths from IRC are not representative of reactions: USE MD BASED METHODS 12

18 TyProblems with traditional QC approach Many problems are not smooth as before: many parallel valleys (conformers), solvent induced roughness 12

19 TyOutline What is free energy and why is important Traditional QM approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 13

20 TyThe ingredients of the probability picture Free energy is connected to the probability of a given state to occur in a single measurement F (s) = k B T ln P (s) P (s) = Probability e V (x) k B T δ(s (x) s)dx e V (x) k B T dx State (bond length, folded/ unfolded descr., liquid/ crystal/amorphous) 14

21 TyExplicit sampling via molecular dynamics a real system (cuvette) a a F (s) = k B T ln P (s) b a a a a b a a a a a set of independent objects (ensemble) a a a a b b MD: evolve in time with Newton s eq. of motion use ergodic hypothesis a b a a a a b a a t 2 t 3 t... all the available states can be sampled with t->infinite P (s) 15

22 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K Free Energy React V 1 Prod 2 k BT RC ν = ν 0 exp ( β V ) ν 0 = s 1 16

23 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K Free Energy React V 1 Prod 2 k BT RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! ν = ν 0 exp ( β V ) ν 0 = s 1 16

24 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K Free Energy React V 1 Prod 2 k BT RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! ν = ν 0 exp ( β V ) property ν 0 = s 1 time 16

25 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K Free Energy 100 ps React V 1 Prod 2 k BT RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! ν = ν 0 exp ( β V ) property ν 0 = s 1 time 16

26 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K Free Energy 100 ps React V 1 Prod 2 k BT RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! ν = ν 0 exp ( β V ) property ν 0 = s 1 time If you want thermodynamics you must average over events 16

27 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K Free Energy React 100 ps V 1 Prod 2 k BT RC MD (DFT) can reach hundreds of ps. Lucky if you see one single event! ν = ν 0 exp ( β V ) property ν 0 = s 1 time If you want thermodynamics you must average over events property time 16

28 TyLimitations Transition of 8 kcal/mol can take up to milliseconds at 300K Free Energy 100 ps V 1 MD (DFT) can reach hundreds of ps. Lucky if you see one single event! ν = ν 0 exp ( β V ) ν 0 = s 1 This 2 k BT React is not Prod your solution RC property time If you want thermodynamics you must average over events property time 16

29 TyOutline What is free energy and why is important Traditional QM approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 17

30 TyThe ancestor: 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

31 TyLessons from Torrie & Valleau Biasing potentials: Free Energy React Prod 1 2 k BT RC [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp

32 TyLessons from Torrie & Valleau Biasing potentials: Free Energy React biasing pot Prod 1 2 k BT RC [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp

33 TyLessons from Torrie & Valleau Free Energy Biasing potentials: React biasing pot 1 2 k BT resulting potential Prod 1 2 k BT RC Free Energy Metastabilities are React removed Prod RC [3] Torrie, Valleau J. Comp. Phys, 1977 vol. 23, pp

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

35 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 MetadynamicsLaio & Parrinello, PNAS 2002, vol. 20, pp Conformational Flooding Grubmüller, Phys Rev E 1995, vol. 52, pp

36 Ty Adaptive sampling methods (e.g. metadynamics) without ES with ES Φ Ψ from milli secs... free energy is faster not black box to nano secs! 21

37 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 s0 = k B T 1 e V (x) k B T δ(s (x) s)dx s k B T δ(s (x) s)dx e V (x) s0 Dirac delta function! Gaussian function (I like this trick!) F (s) s s0 k B T e V (x) k B T e 1 k 2k B T (s (x) s) 2 dx It is an average of a biased simulation! F (s) s s0 1 e V (x)+k/2(s (x) s 0 ) 2 k B T dx s e V (x) k B T e k 2k B T (s (x) s) 2 dx k(s (x) s 0 )e V (x)+k/2(s (x) s0 ) 2 k B T dx s0 = ks (x) s 0 bias,s0 (opposite of the mean force ) 22

38 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

39 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

40 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

41 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

42 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

43 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

44 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

45 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

46 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

47 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

48 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

49 TyThermodynamic Integration " 101 Make a constrained simulation (go over barrier!) Acquire mean force Integrate probability +WHAM etc... 23

50 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 SN2 reaction: if you like this movie come to the tutorial to see how to make it! 24

51 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 in DFT, rbf fitting schemes) very effective Sometimes trivially parallel (many umbrellas at same time) 25

52 TyThe problem of the s F (s) = k B T ln P (s) A multidimensional landscape: turn into monodimensional 26

53 TyThe problem of the s F (s) = k B T ln P (s) A multidimensional landscape: turn into monodimensional CV2 b a CV1 26

54 TyThe problem of the s F (s) = k B T ln P (s) A multidimensional landscape: turn into monodimensional CV2 b a CV1 26

55 TyThe problem of the s F (s) = k B T ln P (s) A multidimensional landscape: turn into monodimensional CV2 b a CV1 Lodola, Branduardi, De Vivo, Capoferri, Mor, Piomelli, Cavalli, PlosOne, 2012, vol 7, pp. e

56 TyThe problem of the s Use machine learning approaches Tribello et. al. PNAS 2010, vol. 107, pp

57 TyOutline What is free energy and why is important Traditional QM approach Explicit sampling Enhanced sampling PLUMED plugin for free energy calculations 28

58 TyThe PLUMED plugin: Bonomi, Branduardi et al., Comp. Phys. Comm vol. 180 pp.1961 small connection with MD code slow fast Already in many classical MD codes (LAMMPS, DLPOLY, SANDER, GROMACS, NAMD) now in FHI-aims! 29

59 TyPLUMED: what it contains Methods Constrained relaxation Umbrella sampling Metadynamics (welltempered) d-afed Steered MD Harmonic walls Ratcheting Reweighting schemes String method Descriptors (with der.) Distances, angles, dihedrals Coordination numbers Contact maps Path collective variables Energy Function of variables SPRINT variables... + MetaGUI analysis interface and PLUMED preparation GUI in VMD+analysis tools! 30

60 TyIn FHI-aims compile separately: Makefile.meta in control.in plumed.dat this enables PLUMED set a distance as set anotherdistance as set repulsive boundaries set metadynamics 31

61 TyAcknowledgements Luca Ghiringhelli (FHI, Berlin) The PLUMED developers (ETHZ, USILU, Cambridge, UCSF, SISSA, Mount Sinai, Curtin, EPFL, MPI-BP) Giovanni Bussi (SISSA, Triest) Michele Parrinello (ETHZ/USILU, Lugano) José Faraldo-Gómez (MPI-BP, Frankfurt/Main) 32

62 Ty Thank you! 33