where ∆U is defined as above and the angled brackets indicate an ensemble average over the system with the separation fixed at q. The first term is the ordinary free energy perturbation expression, and the second arises from the unbiasing of the velocities. An equivalent formulation can be given that uses the constraint forces (the forces needed to maintain the constraints), which are readily available in many simulation programs [8,53].

If additional, auxiliary constraints are present that are not part of the reaction coordinate (e.g., constraints on covalent bond lengths), the formulas are much more complicated, and the algebra becomes rapidly prohibitive. The same is true when q is a multidimensional coordinate (e.g., a set of dihedrals). Umbrella sampling approaches (discussed in previous sections) are vastly simpler in such cases and appear to be the method of choice for all but the simplest reaction coordinates.

V.ELECTROSTATIC FREE ENERGIES

Many important biochemical processes involve charge separation or transfer. Examples include proton and electron transfer, ion binding, and point mutations that replace a neutral residue with a charged one. To study such processes, alchemical free energy calculations are frequently performed in which a net charge is created, deleted, or displaced (e.g., see Figs. 2b and 3). This poses specific problems, because long-range interactions usually make a significant contribution to the free energy change. Two main families of approximations have been used to treat long-range interactions in electrostatic free energy calculations. The first treats distant regions as a simple dielectric medium, i.e., either as a continuum or as a lattice of polarizable dipoles [9,14]. The second does not introduce a solvent continuum; rather, it assumes periodic boundary conditions and calculates electrostatic interactions over an infinite distance range through lattice summation methods such as the Ewald or particle mesh methods [15]. We discuss these two approaches in turn.

A. Dielectric Reaction Field Approaches

Consider an alchemical transformation of a particle in water, where the particle’s charge is changed from 0 to q (e.g., neon → sodium; q 1). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant εw of water:

This estimate should be accurate if the droplet is sufficiently large (a few tens of angstroms).

The idea of a finite simulation model subsequently transferred into bulk solvent can be applied to a macromolecule, as shown in Figure 5a. The alchemical transformation is introduced with a molecular dynamics or Monte Carlo simulation for the macromolecule, which is solvated by a limited number of explicit water molecules and otherwise surrounded by vacuum. Then the finite model is transferred into a bulk solvent continuum

and the transfer free energy is obtained from continuum electrostatics. This involves performing finite-difference Poisson–Boltzmann calculations that take into account the detailed shape of the macromolecule and an infinite bulk solvent continuum [25,26]. This protocol is computationally very efficient, because the macromolecule does not have to be fully solvated in the alchemical simulation step. It includes all electrostatic interactions (provided the alchemical simulation of the finite model is done without a cutoff). The approximation for the long-range interactions is well-defined (in contrast to cutoff treatments), and it can be systematically improved by increasing the size of the explicit solvent region.

A very simple version of this approach was used in early applications. An alchemical charging calculation was done using a distance-based cutoff Rc for electrostatic interactions, either with a finite or a periodic model. Then a cut-off correction equal to the Born free energy, Eq. (38), was added, with the spherical radius taken to be R Rc. This is a convenient but ill-defined approximation, because the system with a cutoff is not equivalent to a spherical charge of radius Rc. A more rigorous cutoff correction was derived recently that is applicable to sufficiently homogeneous systems [54] but appears to be impractical for macromolecules in solution.

An approach widely used in liquid simulations is to include the bulk solvent medium in the alchemical simulation step, calculating the reaction field it produces on-the-fly [15], thus eliminating the need for a subsequent transfer step. This approach can be implemented for a macromolecule in several ways. If the biomolecule is fully solvated with periodic boundary conditions, then a standard liquid simulation approach can be used, in principle. The reaction field on each charge qi (belonging either to the biomolecule or to a water molecule) due to charges beyond a certain cutoff distance rRF is calculated with a continuum approximation, which assumes that the medium beyond rRF is a homogeneous dielectric polarized by the inner medium [55]. The reaction field has a simple analytical form, but the homogeneity assumption will be accurate only if the system is predominantly made up of solvent and if rRF is greater than the biomolecule’s diameter (see Fig. 5b); this implies a large and costly model. Nevertheless, with rapidly increasing computer power, such protocols will become increasingly feasible. The homogeneous medium assumption can be dropped, but this implies a much more complicated and expensive reaction field calculation, which takes into account the exact distribution of solvent and solute in the simulation cell (see Fig. 5b).

A variant of this approach (Fig. 5c) uses a finite (usually spherical) simulation region surrounded by an infinite dielectric continuum [56,57]. The reaction field calculation, again, remains simple as long as the medium outside the simulation region is homogeneous. This implies, again, a large explicit solvent region completely surrounding the biomolecule. The surrounding dielectric continuum has sometimes been replaced by a large lattice of polarizable dipoles that follow a simplified Brownian dynamics [9].

Another variant that may turn out to be the method of choice performs the alchemical free energy simulation with a spherical model surrounded by continuum solvent, neglecting portions of the macromolecule that lie outside the spherical region. The reaction field due to the outer continuum is easily included, because the model is spherical. Additional steps are used to change the dielectric constant of that portion of the macromolecule that lies in the outer region from its usual low value to the bulk solvent value (before the alchemical simulation) and back to its usual low value (after the alchemical simulation); the free energy for these steps can be obtained from continuum electrostatics [58].

B. Lattice Summation Methods

Lattice summation methods, particularly the Ewald summation, are used increasingly in biomolecular simulations, as reviewed by Darden [15] and elsewhere [12]. They have been used recently in free energy simulations of charge creation in liquid water [57,59]. The Ewald summation method assumes periodic boundary conditions; for each partial charge qi in the unit cell, a Gaussian charge distribution centered on qi but with opposite sign is added to the system, screening the charge–charge interactions and making them short-ranged. The same Gaussian distribution is subtracted to recover the original system; interactions due to the subtracted Gaussians are handled using Fourier transforms.

For systems with a net charge, it is necessary to include a uniform background charge to neutralize the system. Such a uniform neutralizing charge does not affect the forces in the system and contributes a constant term to the potential energy. It is also trivial to implement, so it is usually not a technical concern. However, it must be correctly accounted for when the net charge of the system is modified, as in a charging free energy calculation. The electrostatic part of the potential energy has the form

where the sums are over all charges or pairs of charges qi, qj; rij is the vector connecting charges i and j; φEw is the Ewald interaction potential, and ξEw is the ‘‘self-potential.’’ The Ewald potential φEw includes short-range interactions between the screened partial charges and long-range interactions between the partial charges and the array of subtracted Gaussians. The self-potential represents the interaction of a unit charge with both the periodic images of its two associated Gaussians and the neutralizing background charge [59]. Therefore, to correctly account for the free energy associated with the neutralizing background, it is necessary and sufficient to take into account the self-term in either thermodynamic perturbation or thermodynamic integration formulas. For example, if qi is

Figure 5 Continuum reaction field approaches for electrostatic free energies. (a) A two-step approach. The mutation introduces a positive charge near the center of a protein (shown in tube representation). The mutation in the fully solvated protein (left) is decomposed into two steps. Step I: The mutation is performed with a finite cap of explicit water molecules (shown in stick representation); the system is otherwise surrounded by vacuum. Step II: The two finite models (before and after mutation) are transfered into bulk solvent, treated as a dielectric continuum. The transfer free energy is obtained from continuum electrostatics. (From Ref. 25.) (b) Molecular dynamics with periodic boundary conditions: on-the-fly reaction field calculation. One simulation cell is shown. For each charge qi, interactions with groups within rRF are calculated in microscopic detail; everything beyond rRF is viewed as a homogeneous dielectric medium, producing a reaction field on qi [55]. The mutation is introduced using MD or MC simulations. As shown, for many of the charges the medium beyond rRF is not truly homogeneous, being made up of both solvent and solute groups.

(c) Spherical boundary conditions with continuum reaction field [56]. The region within the sphere (large circle) is simulated with MD or MC and explicit solvent; the region outside is treated as a dielectric continuum, which produces a reaction field on each charge within the sphere. If the sphere is smaller than the protein (as here), the outer region is heterogeneous and the reaction field calculation is rather difficult.