Estimating the Relative Free Energy of Different Molecular States with Respect to a Single Reference State

Abstract

We have investigated the feasibility of predicting free energy differences between a manifold of molecular states from a single simulation or ensemble representing one reference state. Two formulas that are based on the so-called λ- coupling parameter approach are analyzed and compared: (i) expansion of the free energy F(λ) into a Taylor series around a reference state (λ = 0), and (ii) the so-called free energy perturbation formula. The results obtained by these extrapolation methods are compared to exact (target) values calculated by thermodynamic integration for mutations in two molecular systems: a model dipolar diatomic molecule in water, and a series of para-substituted phenols in water. For moderate charge redistribution (≈0.5 e), both extrapolation methods reproduce the exact free energy differences. For free energy changes due to a change of atom type or size, the Taylor expansion method fails completely, while the perturbation formula yields moderately accurate predictions. Both extrapolation methods fail when a mutation involves the creation or deletion of atoms, due to the poor sampling in the reference state simulation of the configurations that are important in the end states of interest. To overcome this sampling difficulty, a procedure based on the perturbation formula and on biasing the sampling in the reference state is proposed, in which soft-core interaction sites are incorporated into the Hamiltonian of the reference state at positions where atoms are to be created or deleted. For mutations going from p-methylphenol to the other five differently para-substituted phenols, the differences in free energy are correctly predicted using extrapolation based on a single simulation of a biased, non-physical reference state. Since a large number of mutations can be investigated using a recorded trajectory of a single simulation, the proposed method is potentially viable in practical applications such as drug design.

Comparison of enveloping distribution sampling and thermodynamic integration to calculate binding free energies of phenylethanolamine N-methyltransferase inhibitors

“…10-13) requires one simulation, just as the onestep perturbation technique. 14 Pathway methods may use different approaches to estimate the free enthalpy difference between two end states A and B. In TI, the system is perturbed along a userdefined pathway between the end states by introducing a λ-dependence into the Hamiltonian H ( r N , p N ; λ) with r N = ( r 1 , r 2 ,..., r N ) denoting the Cartesian coordinates of all N particles in the system and p N = ( p 1 , p 2 ,..., p N ) denoting the corresponding conjugate momenta.…”

Improving the accuracy of molecular dynamics simulations: parameterisation of interaction potentials for small molecules

“…This means that using SSP the free energy for a wide range of alternative parameter combinations can be estimated from a single trajectory of a given reference state. 168 This has the advantage that such a trajectory can be precalculated and the change in free energy associated with any change in the Hamiltonian determined by resampling this stored trajectory. Reference ensembles were generated for all molecules in each solvent and in vacuum.…”

“…Indeed, Smith et al 167 extended this approach to incorporate higher order derivatives and was able to predict the effect of changing the charge distribution in a diatomic molecule on the free energy of hydration. In principle, a complete Taylor expansion could be used to predict the change in free energy associated with any combination of parameters from a single simulation, but as Liu et al 168 later showed this is formally equivalent to free energy perturbation 43 assuming a single reference state, Eq. 35.…”

“…35 is only over the reference state ensemble, the free energy for a wide range of alternative end states (X) can be estimated from a single trajectory of this reference state. 168 This has the advantage that a trajectory at a given reference state can be pre-calculated and the change in free energy associated with any change in the Hamiltonian determined by resampling the stored trajectory. However, the accuracy of the results relies on a fundamental assumption: that the trajectory at the reference state (R) contains a representative sample of configurations appropriate for the alternative end states (X).…”

“…However, the accuracy of the results relies on a fundamental assumption: that the trajectory at the reference state (R) contains a representative sample of configurations appropriate for the alternative end states (X). Single-step perturbation (SSP) 45,160,168 (also referred to as one-step perturbation 44,169 and single-step free energy perturbation 170 ) has predominantly been used for the prediction of differences in binding free energies in drug design applications. It has also been used to examine the sensitivity of peptide conformations to changes in torsional parameters [171][172] however its utility in terms of facilitating the incorporation of thermodynamic data in the parametrization of empirical force fields more generally has not been investigated extensively.…”