Free energy calculations and the potential of mean force IMA Workshop on Classical and Quantum Approaches in Molecular Modeling Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washington Square East New York University, New York, NY 10003
Free Energy Canonical Ensemble (Helmholtz free energy): 1 QNVT (,, ) = d p d r e = e N! h N N βh( pr, ) βa( N, V, T) 3 N DV ( ) A( NVT,, ) = ktln QNVT (,, ) Isothermal-Isobaric Ensemble (Gibbs free energy): 1 ( N, P, T) dv d d e = e VN! h = 0 N N β( H( pr, ) + PV) βg( N, P, T) p r 3N 0 DV ( ) GNPT (,, ) = ktln ( NPT,, )
Free Energy (cont d) State function: A21 = A2 A1 P 2 1 V T
Free energy and work If an amount of work W is required to change the thermodynamic state of the system from 1 to 2, then W A 21 21 Equality holds when the work is performed infinitely slowly or reversibly. Jarzynski s equality [PRL, 78 2690, (1997)] shows how to relate irreversible work to the free energy difference. Let W 21 (x) be a microscopic function whose ensemble average is the thermodynamic work W 21. e βw β A 21 21 1 = e
d 1 d 2 Free energy profiles A q= d1 d2 k = κe β A
Protein Folding Energetics From G. Bussi, et al. JACS 128, 13435 (2006)
Binding Free Energies Inhibition constant: U ( r, r ) = U ( r ) + U ( r ) 1 U ( r, r ) = U ( r ) + U ( r ) + U ( r, r ) 2 K i [ E][ I] [ EI ] = = E I E E I I G bind E I E E I I EI E I e β Thermodynamic state potentials: Meta-potential: U( r, r, λ) = f( λ) U ( r, r ) + g( λ) U ( r, r ) E I 1 E I 2 E I f(0) = 1 f(1) = 0 g(0) = 0 g(1) = 1 G = bind 1 0 dλ U λ Thermodynamic integration (Kirkwood, 1935) λ
Binding free energies: Thermodynamic perturbation 1 N N β H(, ) Z( NVT,, ) QNVT (,, ) = d pr p d r e = 3N ( ) 3N N! h DV N! λ Z N V T = d r e λ = βh πm ( ) 2 (,, ) N βu r / 2 DV ( ) Free energy difference related to partition function ratio: Perturbation formula: Q A21 = ktln = ktln Q 1 1 1 β ( U U ) 2 1 Z Z 2 2 1 1 Z2 1 1 = dr e = dr e e Z Z Z = e βu ( r) βu ( r) β( U ( r) U ( r)) 1 2 1 2 1 Need sufficient overlap between two ensembles
λ dynamics methods Use molecular dynamics to sample λ via a Hamiltonian: H λ 2 2 pλ pi = + + U( r1,..., rn, λ) 2m 2m λ i i 2 2 pλ pi = + + f( λ) U1( r1,..., rn ) + g( λ) U2( r1,..., rn ) 2m 2m λ i Free energy from probability distribution of λ: i P( λ) = dr e β λ U ( r, ) A( λ) = ktln P( λ) A = A(1) A(0) 21 Need to have best sampling at the endpoints of the λ-path, which are normally the most difficult to sample.
Aim for a profile with a barrier: A( λ) λ dynamics methods λ = 0 λ = 1 In order to generate such a profile, we need: 1. A high temperature T λ >> T to ensure barrier crossing 2. An adiabatic decoupling between λ and other degrees of freedom 3. Choose m λ >> m i.
λ dynamics methods Under adiabatic conditions, we generate a free energy profile at T λ β β λ λ βλ A( λ; β) βa( λ; β) β βu( r, λ) β e = e = dr e Free energy profile at temperature T from probability distribution generated under adiabatic conditions: A( λ; β) = kt ln P ( λ; β, β ) λ adb λ
Chemical Potential of Lennard-Jones Argon ( ) 4 2 λ = [ λ 1] 4 2 f g( λ) = [( λ 1) 1] m = 2000 m T = 200T λ λ TI
Nbins 1 ς () t = P( xi;) t Pexact ( xi) N bins i= 1 [ ]
0.06-0.683 H HO 0.06 0.418 H C 0.145 0.09 H Backbone (Serine) 0.09 H 0.09 H C -0.27 Backbone (Alanine) Solvation free energies of amino acid side-chain analogs 0.06-0.683 H HO 0.06 0.418 H C 0.085 0.09 H H 0.06 (Methanol) 0.09 H 0.09 H C -0.36 H 0.09 (Methane) 1 Solute (CHARMm22 Parameters) 256 TIP3P Water molecules Cubic Simulation Box (L = 19.066 A) Periodic Boundary Conditions Ewald Summation Technique for charges System Temperature: 298 K NVT via GGMT Thermostats (Liu,MET 2000) λ-afed Parameters: kt λ = 12,000 K = 50 kt m λ = 16,000 g.mol-1 a. Wolfenden, et al. Biochem. 20, 849 (1981) b. Shirts, et al. JCP 119, 5740 (2003); JCP 122, 134508 (2005) c. Yin and Mackerell J. Comp. Chem. 19, 334 (1998)
The Free Energy Profile Probability distribution function: 1 P q = d d e q q Q N N β H( pr, ) ( ) p r δ ( ( r1,..., rn ) ) Free energy profile in q: F( q) = ktln P( q)
Bluemoon Ensemble Approach E. A. Carter, et al.chem. Phys. Lett. 156. 472 (1989); M. Sprik and G. Ciccotti, J. Chem. Phys. 109, 7737 (1998). Impose a constraint of the form: q( r,..., r ) 1 N = q However, constraints also require: q q q ( r,..., r, p,..., p ) = ir = i = 0 p i 1 N 1 N i i ri i ri mi But in constrained MD, what we are actually computing is: 1 Pq = d p d r e δ qr r q δ q r r p p Q N N β H( pr, ) ( ) ( ( 1,..., N) ) ( ( 1,..., N, 1,..., N))
Bluemoon Ensemble df dq = Z 1/2 11 [ λ + kti] Z 1/2 11 Using the Lagrange multiplier to compute free energy: df dq = z 1/2 [ λ + kti] z 1/2 constr constr z 1 1 1 = i I= i i m r r z mm r r r r i 2 q q q q q 2 i i i i, j i j i i j j M. Sprik and G. Ciccotti, J. Chem. Phys. 109, 7737 (1998). **When using SHAKE/RATTLE, λ must be the SHAKE multiplier!!
d 1 d 2 Free energy profiles A δ = d d 1 2
From D. Marx and MET, PRL 86, 4946 (2001).
Variable transformations and statistical mechanics
Adiabatic dynamics and free energy profiles L. Rosso, P. Minary, Z. Zhu and MET, J. Chem. Phys. 116, 4389 (2000); Maragliano, Vanden Eijnden CPL 426, 168 (2006) Hamiltonian from transformation: n 2 3N 2 pi pi H = + + Uq ( 1,..., q3n ) 2m 2m i= 1 i i= n+ 1 i Adiabatic conditions: m T q k T m k Free energy surface: Aq (,..., q) = kt ln Pq (,..., q) 1 n q 1 n
Conformational sampling of the solvated alanine dipeptide [L Rosso, J. B. Abrams and MET J. Phys. Chem. B 109, 2099 (2005)] Time FF β α R C7 ax α L φ ψ Meta 1 5ns CHm27 1.0 0.0 4.8 7.4 US 2 400ns CHm22 0.0 1.41 3.9 4.4 AFED 5ns CHm22 0.2 0.0 4.5 7.6 AFED T φ,ψ = 5T, M φ,ψ = 50M C Umbrella Sampling 50 ns 4.7 ns 1. Ensing, et al. ACR 39, 73 (2005) 2. Smith, JCP 111, 5568 (1999) α R C7 ax α L CHARMm22 β
NATMA (gas phase) (N-acetyl-Tryptophan-methyl-amide) F(φ,ψ) kt(φ,ψ)=20kt, m(φ,ψ)=600m C t=2.5 ns N H H H CH 3 N ψ N φ O CH 3 O φ C7eq ψ C5(AΦ) Why NATMA? AFED Small Minima compared Predictions: to actual proteins and can be easily studied C5(AP) Ab initio energies 1 Conf. Tryptophan Energy side-chain gives Location rise to a (φ,ψ) free energy Conf. landscape endemic of Rel. actual Energy proteins C5(AP) 0.0 kcal/mol (-160, 160) C5 (AP) 0.0 kcal/mol C5(AΦ) Experimental +1.03 1 kcal/mol and DFT 1 Minimization (-140, 140) data available C5(AΦ) +0.65 kcal/mol C7eq +2.18 kcal/mol (-100, 100) C7eq +2.28 kcal/mol 1. Dian, B.C., et al. Science, 296, 2369 (2002); J. Chem. Phys. 117, 10688. (2002)
Add bias potential to Hamiltonian: Metadynamics A. Laio and M. Parrinello, PNAS 99, 12562 (2002); A. Laio, et al. JPCB 109, 6714 (2005) 2 i H = p + U( r1,..., rn) + UG( r1,..., rn, t) 2m i i UG( r,...,, t) W exp G G ( q () q ( () t ) 2 n k k G 1 rn = 2 t= τ,2 τ,... k= 1 2 r r Free energy is negative of bias potential: Aq (,..., q) W exp G G ( q q ( r ( t) ) 2 n 1 n = k k G 2 t= τ,2 τ,... k= 1 2
REPSWA (Reference Potential Spatial Warping Algorithm)
V No Transformation Transformation 5kT 10kT 10kT
How it works Forces: V =kt V =5kT ( V Vref ) x Fu ( ) = x u = F( x) F ( x) ( ) ref x u x / u becomes large in the barrier region! V =10kT
Barrier Crossing Transformations (cont d)
V ( φ) ref
P. Minary, G. J. Martyna and MET SIAM J. Sci. Comp. (accepted) V ( φ,{}) r = V ( φ) S ( φ) + αv ( r ( φ,{}) r r ) S ( r ( φ,{}) r r ) ref tors 1 inter 4 5 2 4 5
Comparison for 50-mer using TraPPE with all interactions PT replicas = 10; PT exchange prob. = 5%, REPSWA α = 0.8; Every 10 th dihedral not transformed
Comparison to parallel tempering and CBMC Siepmann and Frenkel, Mol. Phys. 75, 59 (1992) End-to-end distance fluctuations
Comparison for 50-mer using CHARMM22 all interactions
Comparison of 50-mer using CHARMM22 all interactions
Honeycutt and Thirumalai
No Transformation Parallel Tempering SDC-REPSWA PT replicas = 16; PT exchange prob. = 5% Model sheet protein β