Free Energy Simulation Methods

Transcription

1 Free Energy Simulation Methods

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3 Free energy simulation methods Many methods have been developed to compute (relative) free energies on the basis of statistical mechanics Free energy perturbation Thermodynamic integration Umbrella sampling Slow growth Fast growth L. Landau R. W. Zwanzig J. G. Kirkwood

4 Recap of statistical mechanics: Partition function, free energy, enthalpy, & entropy Partition function of canonical (NVT) ensemble Z = exp( E / kt) dpdx E = E + E kinetic potential In classical mechanics, potential energy is independent of kinetic energy

5 Free energy, enthalpy, & entropy Free energy (Helmholtz) Internal energy A = kt ln exp( E / kt ) dx <>: Ensemble average Entropy S = ( U A) T

6 Difficulty for absolute free energy simulation Absolute free energy requires converged integration on 3N dimensions Z = exp( E / kt) dx 3N k B T No experimental absolute free energies available Relative free energies are what really matter

7 How computational chemists compute free energy? 1. Minimize geometry 2. Frequency calculation 3N-6 internal vibrations 3. Ideal gas model for translation 4. Ideal rotor model for rotation Severe problems for macromolecules Harmonic approximation Multiple conformation states Solvated states are difficult to model

8 Free energy perturbation Free energy difference between two states State 1; energy E1 State 2; energy E2

9 Free energy perturbation Free energy difference between two states ΔA1 2 Zwanzig, R. W., J. Chem. Phys. 1954, 22:

10 Free energy perturbation by Functional Calculus A( E( X)) = kt ln exp( E/ kt) dx δa( E( X)) = AE ( ) δedx E Homework: Derive Free Energy Perturbation equation using functional calculus.

11 Free energy perturbation Symmetry of FEP equations Δ A = A A ( E ) = kt ln exp( E / kt ) ( E ) = kt ln exp( E / kt ) = ( A A ) = ΔA Forward and backward simulations to reduce the error [ ] Δ A = ΔA ΔA /2 kt = ln exp( ( E E )/ kt) + ln exp( ( E E )/ kt)

12 Free energy perturbation Accuracy of FEP ρ(δe 1 2 )

13 Free energy perturbation Accuracy of FEP Δ A = kt ln exp( βδe ) ρ ( ΔE ) dδe Δ A = ktln exp( βδe ) ρ ( ΔE ) dδe

14 Free energy perturbation Technical complications Phase-space overlapping between two states Energy

15 Free energy perturbation Implementation λ 9 ΔA = ΔA 1 2 λ λ i i i+ 1 λ 1 λ 2 λ 3 λ 0 In general, free energy differences between discrete points can be determined and summed to give the total free energy difference

16 Approximated free energy perturbation Δ A = kt ln exp( βδe ) ρ ( ΔE ) dδe ( βδe ) i ln ρ1( 1 2) 1 2 i= 0 i! Δ A = kt ΔE dδe ( βδe ) 2 = kt ln 1 + ( βδ E ) ρ ( ΔE ) dδe = kt + Δ βδe1 2 βδe1 2 ln 1 β E i = kt Ci ( β ) / i! i= 1

17 Approximated free energy perturbation i Δ A1 2 = kt Ci ( β ) / i! i= 1 C = ΔE C = ΔE ΔE C = ΔE ΔE = Δ 1 2 Δ C E E C C 5 =... When distribution of ΔE is gaussian ( ) Δ A = ΔE βσ ΔE

18 Thermodynamic integration A virtual path linking two states 2 1 Kirkwood, J. G., J. Chem. Phys. 1935, 3:

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20 Thermodynamic integration Sometimes we have a real reaction path Δ A = 0 λ ' = λ ' 0 λ ' 0 A( λ) dλ λ E( λ) dλ λ λ the area under the curve corresponds to the free energy change This is often used in simulating free energy profile and activation free energies for enzyme-catalyzed reactions.

21 Free Energy Perturbation vs Thermodynamic Integration Δ A = ktln exp( βδe ) Δ A = E( λ) λ λ dλ 1. Requirement for gradient 2. Error analysis 3. Easiness for implementation 4. Can be combined together

22 Umbrella sampling for modeling reaction processes ClCH 3 + OH- HOCH 3 + Cl-

23 Umbrella sampling E(R) E (R) P(R) Torrie, G. M., and Valleau, J. P., J. Comput. Phys. 1977, 23:

24 Umbrella sampling Modified energy function E '( R) = E( R) + V( R) The potential of mean force (PMF) of R is defined as [ ] A( R) = ktln( exp E( R)/ kt δ ( r R) dx) = kt ln ρ( R) + C PMF of biased energy function is [ ] A'( R) = ktln( exp E'( R) / kt δ ( r R) dx) = AR ( ) ktln = AR ( ) + V( R) + C [ ER ( )/ kt] δ ( r R) dx exp[ ] = kt ln( exp V( R)/ kt δ ( r R) dx) [ ] exp V( R)/ kt δ ( r R) dx

25 Umbrella sampling For two neighboring windows A'( R) = A( R) + V ( R) + C A'( R) = A( R) + V ( R) + C 2 2 2

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28 A' ( R) = A( R) + V ( R) + C 1 1 1

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30 Weighted histogram analysis method ρ Minimizing overall sampling uncertainties from linear combination of samples of different windows S β Vi( R) fi b ( u) ( R) = e ρi ( R) ρ ( R) = c ( R) ρ ( R) ci ( R) ( u) ( ) i 0 i= 1 i i S i= 1 = 1 2 ( σ ρ0 ( R) ) c ( R) = i 0 S S 2 2 ( ) i i i i= 1 i= 1 u c ( R) σ ρ ( R) μ c ( R) 1 2 ( σ ρ0 ( R) ) ci ( R) 2 ( u) ( ) ( ) 0 = = 2ci R σ ρi R μ ( i ) 2 ( u) ( ) ( ) 1 ci R = σ ρ R μ ( ) ( R) ( ) ( R) ( R) ( R) ( R) 2 ( b) i ( b) i i NiΔ ( R) ( ) ( ) 2 ( u) 2β Vi R fi 2 ( b) σ ρi R = e σ ρi R g σ ρ = ρ = g N ( ) i β Vi R fi ρ0 R e iδ

31 Umbrella sampling Overlapping between different sampling windows

32 Slow growth Replace discrete sampling configuration points to continuous quasi-equilibrium process ΔA = - WORK (done by a reversible / quasi-static isothermal process) Example: gas in a piston B Δ A = PdV = dw A B A Let s drive the system from one state to another in a very slow (quasistatic) way ΔA 1 2 N E( λi ) δλ i λ i= 0 i

33 Slow growth Noise and hysteresis in slow growth simulations ΔA + ΔA

34 Fast growth and Jarzynski s equality ΔA limw For a quasi-static slow process 0 1 = τ 0 N For a non-quasi-static process ΔA W Thermodynamic second law W For a non-quasi-static process =ΔA

35 Fast growth method J. Am. Chem. Soc., 2005, 127:

36 Thermodynamic cycle Thermodynamic cycle can be constructed to avoid the direct simulation of computationally difficult processes. Example: which of the two ligands, L1 and L2, binds stronger to the enzyme E? Approach 1: direct simulation of the two binding processes L(aq) 1 + E(aq) L-E(aq) 1 ΔG1 L(aq) + E(aq) L-E(aq) ΔG ΔΔ G = ΔG ΔG 1 2 Approach 2: simulating other two processes as Mutate free L1 into free L2 in solution Mutate L1 into L2 within the enzyme s active site L (aq) L (aq) ΔG L-E(aq) L-E(aq) ΔG

37 Thermodynamic cycle L 1 (aq) + E(aq) ΔG 1 L 1 -E(aq) ΔG 3 ΔG 4 L 2 (aq) + E(aq) ΔG 2 L 2 -E(aq) ΔG 1 ΔG + ΔG 2 4 ΔG ΔΔG = ΔG 1 = ΔG 4 2 ΔG ΔG 4 ΔG 3 3 ΔG = 3 0 Thermodynamic cycle can be constructed to avoid the direct simulation of computationally difficult processes.

38 QM/MM Methods for Complex Reaction Processes

39 Quantum mechanics In computational chemistry, the most immediate task is to compute energy and gradient for any given molecular conformational states! Computational cost Hartree-Fock: N 4 MP2: N 5 MP4: N 6 Coupled Cluster: N 7 DFT: N 3 ~N 4 Challenge developing cheap methods for the simulation of macromolecules!

40 General MM force field forms Five basic terms in all force fields V= bond stretching + angle bending + torsions + + bonds angles torsions atom pairs atom pairs electrostatic interaction van der Waals interaction Covalent interactions Non-bonded interactions

41 General MM force field forms Non-bonded interactions (electrostatic and van der Waals interactions) are calculated only between atoms not involved in direct bonding (1-2 interaction), or angle connections (1-3 interaction)

42 Example case: CO vibration ΔE 1412*(b-1.13) 2 Think about perturbation method in quantum chemistry!

43 Example case: H 2 O vibration ΔE *(θ-105) 2

44 Electrostatic potential based atomic charges Molecular electrostatic potential (ESP) is well defined: ϕ () r = ϕ () r + ϕ () r QM nuc ele nuclei Z A ρ( r ') = dr ' r r r r' A A

45 Electrostatic potential based atomic charges Fitting molecular ESP for point charges ϕ () r = MM atom A qa r r A min w( r) ϕmm ( r) ϕqm ( r) dr 2 w(r): weighting functions

46 Development of MM force fields Two fundamental assumptions: Born-Oppenheimer approximation The fast-moving electrons are adjusting instantaneously to the positions of the (relatively) slow-moving nuclei. The contributions from the electronic degrees of freedom are thus implicitly determined by the positions of nuclei. Transferable functional groups in chemistry Chemically identical function groups (for example OH, -C=O, -NH 2, ) can be described by the same set of interaction terms. Only limited number of functional groups need to be considered.

47 Development of MM force fields Unique features of MM force fields Electron-less Transferability Concept of atom type Requires extensive/tedious parameterization process Cannot describe chemical reactions in general!!!

48 Why QM/MM? QM MM Accuracy Computational cost Long-MD Polarization Bond formation / breaking Long-range vdw High High No Included Yes No Medium to low low Yes Ignored/ approximated No Yes

49 General QM/MM scheme QM MM E = E + E + E QM QM / MM MM 1. QM is used to describe the site where reactions occur, including those atoms make important and direct interactions to atoms undergoing valence change in the reactions process. 2. MM is used to describe the rest of the system. Presumably atoms in these regions contribute to the reaction moieties through a static and classical electrostatic fashion.

50 A simple approach: ONIOM method Our own n-layered Integrated molecular Orbital + molecular mechanics Method + = E QM (1) E (1+ 2) MM = E (1) + E () 2 + E (1/ 2) MM MM MM Mechanical embedding model E / (1+ 2) QM MM = E (1) + E (2) + E QM MM MM (1/ 2) = E (1) + E (1+ 2) E (1) QM MM MM

51 General QM/MM scheme MM QM H + = H + H + H QM MM QM QM/ MM MM = H + H + H QM QM/ MM, ele QM / MM, nuc l QM/ MM, vdw QM/ MM,cov a MM + H + + H + H E = E + E + E QM QM / MM MM

52 Electronic structure at QM/MM boundary Dangling QM/MM bond

53 Electronic structure at QM/MM boundary Linked hydrogen MM QM Pros: easy to implement Cons: different ensembles due to the additional hydrogen atom

54 Electronic structure at QM/MM boundary Frozen local orbital / Local SCF A strictly localized bond orbital (SLBO) for the bond. The SLBO is assumed to be transferable. SLBO is excluded from the SCF optimization and does not mix with other orbitals. To compensate for the additional electron introduced with the doubly occupied SLBO, an extra charge of 1e is placed on M1, which interacts with all other MM charges. More charges can be placed on MM atoms around to improve.

55 Electronic structure at QM/MM boundary Pseudo-bond approach A monovalent, fluorine-like boundary atom with seven valence electrons, Z = 7, was used to replace the boundary atom. An angular-momentum dependent effective core potential (ECP) was used to mimic the original C-C bonds. ECP is therefore specific to the type of the bond on the boundary and is obtained by fitting. But fitted parameters show weak dependence on the QM method.

56 Electrostatic embedding model MM QM HQM+ MM = H + H + H QM QM/ MM MM = H + H + H QM QM/ MM, ele QM / MM, nucl + H + H + H ψ H ψ = ψ H QM/ MM, vdw QM/ MM,cova MM QM+ MM QM + H + E + E QM/ MM, ele QM/ MM, nucl QM/ MM, vd W + E + E QM/ MM,cova MM When MM atoms are represented as point charges H + H = H + QM QM / MM, ele QM ψ q i i MM r r i

57 QM/MM electrostatic interactions Point charge Point MM charge has singular point at distance 0 Over-polarization of QM electrons Gaussian charge or damping of the QM/MM electrostatic interactions will improve

58 QM/MM elec. interactions: lack of Pauli interactions Pauli repulsion ensures two nuclei won t collapse into each other Lack of Pauli repulsion leads to two effects: incorrect energetics incorrect electron densities

59 QM/MM electrostatic interaction Long range electrostatic interactions Do QM images see each other? Limits the size of the system and the QM basis sets Contribution of image charge distributions to the central QM region Ewald type of method Multiple grids

60 QM/MM vdw Simple Lennard-Jones 12-6 form is often used E C C = r r 12 6 QM / MM, vdw 12 6 But dispersions of QM atoms depend on the QM electron distribution which further depends on the surrounding MM point charges T.J. Giese & D. M. York, JCP, 127:194101

61 Hierarchy of Hamiltonians Ab initio quantum chemistry Divide-and-conquer methods Fragment based approaches Effective fragment potential Molecular fractionation with conjugate caps Density fragment interaction QM/MM methods MM methods Coarse-grained methods Mesoscopic and continuum methods

62 A matrix interaction scheme

63 QM/MM: direct-sampling based free energy calculation QM SCF calculation at every MD/MC step E( rqm, rmm ) E( rqm, rmm ) E( rqm, rmm ),, and r r ψ H + H QM QM QM The forces on QM atoms r The forces on MM atoms ψ + = ψ r MM, i MM, i MM ( H H /, ) QM / MM, ele QM QM MM ele QM ψ HQM + HQM/ MM, ele ψ ψ qi/ r ri ψ = r r ψ

64 QM/MM: direct-sampling based free energy calculation Reaction coordinate R is known, computes P( R), probability distribution of R or E( rqm, rmm ), acting force on R R Umbrella sampling Thermodynamic integration Too costly for ab initio QM in general! Choice of R strongly affects the convergence

65 Example: electron-transfer reaction [ Fe( H O) ] [ Fe( H O) ] Δ A = E( η) η η dη Janak theorem Removing electron Adding electron E( η) η E( η) η = ε = ε HOMO LUMO

66 QM/MM: free-energy perturbation based free energy calculation Discrete representation of a reaction path by the QM conformations {r QM } i, i=1,,n Free energy difference between two adjacent QM conformations is computed by free energy perturbation 1 Δ A = ln exp( β E { } E { } ) β r r d r ( ) { } i + i 1 QM i+ 1 QM i MM i Zhang, Liu, & Yang, J. Chem. Phys., 2000

67 QM/MM: two computational challenges QM calculation 50 ~ 100 atoms, 500 ~ 1000 basis functions, 10 ~ 20 energy+gradient/day MM conformation Ensemble of reaction paths Convergence of MM sampling

68 QM-PMF: beyond single potential energy surface PMF of QM coordinates Integration yields free energy PMF Gradient

69 QM/MM: zero-order ESP charge approximation Decompose the QM/MM total electrostatic energy QM/MM electrostatic energy E QM internal energy ( r, r ) ESP QM / MM QM MM = q Q ( r, r ) j r - r j MMi QM QM, i MM, j i QM MM ESP-fitted QM charges E ( r, r ) = Ψ H Ψ E ( r, r ) ESP 1 QM MM eff QM / MM QM MM E 0 0 ( rqm, rmm ) E1 ( rqm, rmm ) + qq 0 0 j i( rqm, rmm) r - r j MMi QM QM, i MM, j + E ( r, r ) + E ( r, r ) + E ( r ) QM / MM, vdw QM MM QM / MM,cov QM MM MM MM

70 QM/MM: first-order polarizable QM ESP charge approximation First-order polarization of ESP charge Q ( r, v ) = Q, + χ ij vmm ( rqm j ) vmm ( r QM j ) + κ r r i QM MM QM i,, ij QM, j QM, j j QM j QM χ ij Q VMM ( QM, j ) r QM, i = N κ ij Q QM, i = rqm, j N E ( r, v ) = Q ( r, v ) v r ( ) ESP QM / MM QM MM i QM MM MM QM, i i QM = Q + χ v ( r ) v ( r ) v r i QM ( ) 0 0 QM, i ij MM QM, j MM QM, j MM QM, i j QM ( ) ( ) ( ) E ( r, v ) = E ( r, v ) χ v r v r v r QM MM 1 QM MM ij MM QM, i MM QM, j MM QM, j i QM j QM vmm ( QM, i ) v (, ) MM QM i χ ij vmm ( QM, j ) vmm ( QM, j ) 2 r r r r i QM j QM Lu & Yang, J. Chem. Phys., 2004

71 QM/MM: first-order polarizable QM ESP charge approximation First-order polarization of ESP charge E ( r, r ) = E ( r, v ) + E ( r, v ) ESP QM MM 1 QM MM QM / MM QM MM + E ( r, r ) + E ( r, r ) + E ( r ) QM / MM, vdw QM MM QM / MM,cov QM MM MM MM = E ( r, v ) + Q v ( r ) QM MM i MM QM, i i QM vmm ( QM, i ) v (, ) MM QM i χ ij vmm ( QM, j ) vmm ( QM, j ) 2 r r r r + E i, j QM QM / MM, vdw ( r, r ) + E ( r, r ) + E ( r ) QM MM QM / MM,cov QM MM MM MM E ( r, r ) E ( r, v ) v ( r ) = QM MM 1 QM MM 0 MM QM, i QQM, i rqm, i rqm, i rqm, i j QM κ 0 ij vmm ( rqm, j ) vm M( rqm, j) vmm( rqm, i) + v ( r ) v ( r ) χ v ( r ) v ( r ) r MM QM, j MM QM, j ji MM QM, i MM QM, i j QM ( EQM / MM, vdw( rqm, rm M ) + EQM / MM,cov( rqm, rmm )) r QM, i QM, i Lu & Yang, J. Chem. Phys., 2004

72 QM/MM-MFEP: straightforward optimization 1) Set initial QM structure r (0) QM, set counter n=0; 2) Increase counter n=n+1; a) Carry out MD simulation of the MM environment with QM geometry fixed at r (n-1) QM n { r τ τ = N} ( ) ( ), 1,..., MD sampling based on E ( r ) MM ref MM b) Carry out one step of QM optimization based on PMF and PMF gradient ( n-1) A( r ) ( n) ( n-1) QM QM one step in the QM optimization based on A( rqm ) and rqm r c) Update QM geometry 3) Go to step (2) until converged Hu, Lu, & Yang, J. Chem. Theory, & Comput., 2007

73 QM/MM-MFEP: path optimization NEB, Ayala-Schlegel MEP, QSM General procedures: 1. Optimize the structures of the reactant and product states, separately 2. General initial guess of the reaction path, by coordinate driving or interpolation 3. Fix the structures of the RS and PS, carry out MFEP optimization 4. Optimize the TS structure Hu, Lu, & Yang, J. Chem. Theory, & Comput., 2007

74 QM/MM Free energy perturbation Δ E = E ( r, r ) + E ( r, r ) + E ( r, r ) i j QM QM, j MM, i QM / MM, ele QM, j MM, i QM / MM, nuc QM, j MM, i - E ( r, r )- E ( r, r )- E ( r, r ) QM QM, i MM, i QM / MM, ele QM, i MM, i QM / MM, nuc QM, i MM, i + E ( r, r )- E ( r, r ) E QM / MM, vdw QM, j MM, i QM / MM, vdw QM, i MM, i ( r, r )- E ( r, r ) + QM / MM,cova QM, j MM, i QM/ MM,cov a QM, i MM, i

75 QM/MM-MFEP: bottleneck Time of MD simulations 50,000 atoms, ~ 150 ps /day (1 fs stepsize) 130,000 atoms, ~ 50 ps/day (1 fs stepsize) Time for QM calculations 800 basis functions, 10~20 energy+gradient/day Hundreds of QM optimization steps are needed Half day for one MD + QM calculation Too many repetitive MD simulations whose samplings are not efficiently utilized

76 QM/MM-FE: efficient iterative, sequential optimizations 1) Set initial QM structure r (0) QM, set counter n=0; 2) Increase counter n=n+1; a) Carry out MM optimization with QM geometry fixed at r QM (n-1) b) Carry out QM optimization with the MM conformation fixed at r QM (n-1) c) Update QM geometry 3) Go to step (2) until converged n ( ) r = arg min E r, r ( n) ( 1) MM QM MM r MM (n) ( ) r = arg min E r, r ( n) QM r QM MM QM In certain cases, sequential optimization may have unique advantages. In the QM/MM-FE method, sequential algorithm significantly reduces the number of QM calculations. Zhang, Liu, & Yang, J. Chem. Phys., 2000

77 Ab Initio QM/MM Minimum Free-Energy Path MD t

78 Iterative, sequential optimization QM MM

79 Iterative, sequential QM/MM-MFEP: algorithm 1) Set initial QM structure r (0,0) QM, set counter n=0; 2) Increase counter n=n+1; a) Carry out MD simulation of the MM environment with QM geometry fixed at r (n-1,0) QM n { r τ τ = N} ( ) ( ), 1,..., MD sampling based on E ( r ) MM ref MM b) Carry out QM optimization with the MM ensemble fixed at c) Update QM geometry 3) Go to step (2) until converged { r ( ( ) MM τ } { β ( τ ) ( τ ) } N ( ni, ) 1 1 ( ni, ) ( n,0) A ( rqm ) = Aref ln exp E QM, MM ( ) Eref MM ( ) β N r r r τ = 1 N E A ( r ) = rqm ( n) ( rqm, rmm ( τ )) ( ni, ) ( n,0) exp { β E (, ( τ) ) E ( ( τ) ) r r r r } QM, i N ( ni, ) ( n,0) exp { β E ( rqm, rmm ( τ) ) Eref ( rmm ( τ) ) } ( ni, ) QM MM ref MM QM τ = 1 τ = 1

80 Ab initio QM/MM minimum free energy path Advantages Complicated solution and enzyme reactions become gas-phase-like Removes the path-dependence of initial conformations Can be applied to solution reactions Adequate statistical sampling Reaction path optimization without explicitly defining the reaction coordinate Hu & Yang, Annu. Rev. Phys. Chem., 2008