Statistical Thermodynamics Lecture 19: Free Energies in Modern Computational Statistical Thermodynamics: FEP and Related Methods Dr. Ronald M. Levy ronlevy@temple.edu
Free energy calculations Free energy is the most important quantity that describes a thermodynamic process Two types of free energy calculations: 1. Path independent methods for calculation of free energies. e.g. Free Energy Perturbation (FEP), Thermodynamic Integration (TI) 2. Path dependent methods for calculations of free energies. e.g. Potential of Mean Force (PMF)
General perturbation theory Perturbation theory is one of the oldest and most useful, general techniques in applied mathematics. Its initial applications to physics were in celestial mechanics to explain how the presence of bodies other than the sun perturbed the elliptical orbits of planets. The main idea: One starts with an initial system, called the unperturbed or reference system. The system of interest, called the target system, is represented in terms of a perturbation to the reference system.
Statistical mechanics description of free energy In the canonical ensemble: A = k B T lnq N-particle reference system described by Hamiltonian H(x,p), function of 3N Cartesian coordinates x and conjugated momenta p. Partition function: Q = Free energy difference : ΔA = k B T ln 1 h 3N N! exp[ βh(x, p)]dx dp ΔA = A 1 A 0 = k B T ln Q 1 Q 0 exp[ βh 1 (x, p)] dxdp exp[ βh 0 (x, p)]dx dp
ΔA = k B T ln exp[ βh 1 (x, p)] dxdp exp[ βh 0 (x, p)]dx dp = k B T ln exp[ βh 1 (x, p)+ βh 0 (x, p)] exp[ βh 0 (x, p)]dxdp exp[ βh 0 (x, p)]dx dp = k B T ln exp[ βδh(x, p)] exp[ βh 0 (x, p)]dxdp exp[ βh 0 (x, p)]dx dp
Probability density function P 0 of finding the reference system in a state defined by x and p is: P 0 (x, p) = Substituting in expression of ΔA: ΔA = k B T ln exp[ βh 0 (x, p)] exp[ βh 0 (x, p)]dx dp exp[ βδh(x, p)] exp[ βh 0 (x, p)]dxdp exp[ βh 0 (x, p)]dx dp = k B T ln exp[ βδh(x, p)] P 0 (x, p)dxdp Fundamental FEP formula for the transformation 0 1: ΔA = k B T ln < exp[ βδh(x, p)] > 0 If the kinetic term cancels out : ΔA = k B T ln < exp[ βδu] > 0
Free energy perturbation method ΔA = k B T ln < exp[ βδu] > 0 In this formula, ΔU = U 1 (x) U 0 (x) is the difference in potential between target and reference system, and the average is over the ensemble of the initial state corresponding to reference system with potential U 0 (x). Similarly, the free energy difference can also be written in terms of an average over the ensemble of the final state corresponding to target system with potential U 1 (x): ΔA = A 1 A 0 = k B T ln Q 1 Q 0 = k B T ln Q 0 Q 1 = k B T ln < exp[βδu] > 1
Free energy perturbation calculations continue ΔA = k B T ln ΔA = k B T ln exp[ βδh(x, p)] P 0 (x, p)dxdp exp( βδu)p 0 (ΔU)dΔU According to the central limit theorem (CLT), ΔU would be Gaussian-distributed if U 1 and U 0 were the functions of a sufficient number of identically distributed random variables. Therefore: where: P 0 (ΔU) = 1 2πσ exp[ (ΔU < ΔU > 0 )2 2σ 2 ] σ 2 =< ΔU 2 > 0 < ΔU > 0 2 Analytical expression of ΔA: ΔA =< ΔU > 0 1 2 βσ 2 Notice: use of this analytical expression can be successful only if P 0 (ΔU) is a narrow function of ΔU; this condition does not imply that the ΔA must be small.
Example: hydration free energy of benzene reference state: solute has high probability to overlap with solvent in the gas phase target state: solute has low probability to overlap with solvent in the aqueous phase ΔA = -0.767 kcal/mol Although hydration free energy of benzene is small, this quantity cannot be successfully calculated directly using the FEP equation; because low-energy configurations in the target ensemble, which do not suffer from the overlap between the solute and solvent molecules, are not sampled in simulations of the reference state.
FEP calculation --- staging The difficulty in applying FEP formula can be circumvented through staging strategy. Construction of several intermediate states between reference and target state such that P(ΔU i,i+1 ) for two consecutive states i and i+1 sampled at state i is sufficiently narrow for the direct evaluation of the corresponding free energy difference ΔA i,i+1. With N-2 intermediate states, N 1 i=1 ΔA = ΔA i,i+1 = 1 β N 1 i=1 ln < exp( βδu i,i+1 ) > Intermediate states do not need to be physically meaningful, they do not have to correspond to systems that actually exist. i
FEP calculation --- formula The Hamiltonian can be considered to be a function of some parameter λ. λ can be defined between 0 and 1, such that λ = 0 for reference state and λ = 1 for target state. Dependence of hybrid Hamiltonian on λ : H(λ) = (1 λ)h 0 + λh 1 = H 0 + λδh ΔH is the perturbation Hamiltonian, equal to H 1 -H 0. If N-2 intermediate states are created to link the reference and target states, such that λ 1 = 0 and λ N = 1: ΔH i = H(λ i+1 ) H(λ i ) = (λ i+1 λ i )ΔH Total free energy difference: = Δλ i ΔH with Δλ i =λ i+1 -λ i ΔU i = Δλ i ΔU ΔA = 1 β N 1 i=1 ln < exp( βδλ i ΔU) > λ i
Important concept in FEP: Order parameter Order parameter: They are collective variables used to describe transformations from the reference system to the target one. order parameter: distance order parameter: dihedral An order parameter may or may not correspond to the path along which the transformation takes place in nature, and would be called the reaction coordinate if such were the case. There is more than one way to define an order parameter. The choice of order parameters may have a significant effect on the efficiency and accuracy of free energy calculations.
Application of FEP: binding free energy calculation The binding free energy of two molecules, ΔA binding, defined as the free energy difference between two molecules in the bound and free, unbound states, can be determined experimentally through the measurement of binding constants. For direct calculation of ΔA binding, it needs to define an order parameter that measures the separation between the ligand and the binding center of the protein. This may be difficult when the ligand is buried deep in the binding pocket. Alternative route: double annihilation of the ligand through a thermodynamic cycle.
Thermodynamic cycle and double annihilation ΔA binding = ΔA 0 annihilation ΔA1 annihilation + ΔA restrain Double annihilation involves the removal of the ligand molecule from aqueous solution in the unbound state and bound state, respectively. Through thermodynamic cycle, the double annihilation free energies ΔA 0 annihilation and ΔA 1 annihilation were related to ΔA binding The ligand in the binding pocket is annihilated from a strongly constrained position, whereas the unbound ligand can translate and rotate freely during annihilation. Proper corrections for the loss of translational, rotational and conformational entropies should be taken into account. ΔA binding protein + ligand protein ligand ΔA 0 annihilation ΔA1 annihilation ΔA restrain protein + nothing protein nothing
Example: Relative binding free energies for ligands In the case of seeking potential inhibitors of a target protein, determining relative binding free energies for a series of ligands is required. This can be handled by repeating the absolute binding free energy calculation for each ligand of interest. ΔA XLC = -6.93 kcal/mol Ligand XLC Protein FXa ΔA XLD = -9.98 kcal/mol Ligand XLD Protein FXa ΔΔA cal = ΔA XLD ΔA XLC = -3.05 kcal/mol. Match well with ΔΔA exp = -2.94 kcal/mol
Alchemical transformations ΔA A binding protein + ligand A protein ligand ΔA 0 mutation ΔA1 mutation ΔA B binding protein + ligand B protein ligand There is an alternate pathway for calculations of relative binding free energies. Mutation of ligand A into ligand B in both the bound and the free states, following a different thermodynamic cycle. Example: In the mutation of ethane into methanol, the former serves as the common topology. As the carbon atom is transformed into oxygen, two hydrogen atoms of the methyl moiety are turned into non-interacting, ghost particles by annihilating their point charges and van der Waals parameters.
Thermodynamic integration method If N-2 intermediate states are created to link the reference and target states, such that λ 1 = 0 and λ N = 1: ΔA = A 1 (λ N ) A 0 (λ 1 ) = λ da(λ) N dλ = dλ λ 1 1 0 da(λ) dλ dλ Q = da(λ) dλ exp[ βu(x, λ)] dx A = kt lnq = kt 1 Q d( exp[ βu(x, λ)]dx) dλ = kt β = du dλ λ du dλ exp( βu)dx Q TI: ΔA = 1 0 du dλ λ dλ FEP: ΔA = k B T ln < exp[ βδu] > 0
Potential of mean force method The reaction coordinate ξ, describes the path along which the transformation takes place in nature. Potential of mean force (PMF) is a path dependent method for calculations of free energies based on the reaction coordinate ξ. ΔA = A 1 (ξ N ) A 0 (ξ 1 ) = da(ξ) dξ = dξ du dξ du dξ ξ = F ξ ξ = constraint force Constraint force: force required to constrain the system at a fix reaction coordinate ξ; if this force can be extracted, free energy can be evaluated from integration of the mean force along the reaction coordinate. ξ dξ
Example: Single ion at liquid/vapor interface Z direction ξ Zero Position Gibbs Dividing Surface The reaction coordinate is the separation between ion and zero position along the z direction. Iodide shows more stability than chloride around interface from PMF profile.
Summary Different method for calculation free energy: free energy perturbation (FEP), thermodynamic integration (TI) and potential of mean force (PMF). In the next lab, we will use FEP method, implemented in the Schrodinger package, to calculate the relative binding free energy between a pair of ligand XLC and XLD binding to the receptor protein Fxa.