Abstract
In calculations of relative free energy differences, the number of atoms of the initial and final states is rarely the same. This necessitates the introduction of dummy atoms. These placeholders interact with the physical system only by bonded energy terms. We investigate the conditions necessary so that the presence of dummy atoms does not influence the result of a relative free energy calculation. On the one hand, one has to ensure that dummy atoms only give a multiplicative contribution to the partition function so that their contribution cancels from double-free energy differences. On the other hand, the bonded terms used to attach a dummy atom (or group of dummy atoms) to the physical system have to maintain it in a well-defined position and orientation relative to the physical system. A detailed theoretical analysis of both aspects is provided, illustrated by 24 calculations of relative solvation free energy differences, for which all four legs of the underlying thermodynamic cycle were computed. Cycle closure (or lack thereof) was used as a sensitive indicator to probing the effects of dummy atom treatment on the resulting free energy differences. We find that a naive (but often practiced) treatment of dummy atoms results in errors of up to kBT when calculating the relative solvation free energy difference between two small solutes, such as methane and ammonia. While our analysis focuses on the so-called single topology approach to set up alchemical transformations, similar considerations apply to dual topology, at least many widely used variants thereof.
Highlights
-called alchemical free energy simulations (FES) have become a standard tool of computational chemists, both in academia as well as in pharmaceutical research in industry, in particular for lead optimization.[1−3] While the methodology can be used to compute “absolute” free energies,[4−6] in many cases knowledge of relative free energy differences, for example, the binding free energy difference between two ligands, is sufficient.[1−3] These successful applications and the beginning widespread use by nonexperts make it important to keep an eye on remaining methodological challenges and pitfalls.Calculations of relative free energy differences rely on a thermodynamic cycle as shown in Figure 1(a).[7]
We report the results along the alchemical path (ΔΔGsolv), as well as the difference of the respective absolute solvation free energy differences (ΔΔGsaoblsv), together with the difference ΔΔGsolv − ΔΔGsaoblsv, which ideally should be identically zero
Our model transformations involve up to 10 dummy atoms (TOL2MET); overall, our results indicate that any contributions from dummy atoms cancel from the double free energy difference of interest if best practice is followed
Summary
Calculations of relative free energy differences rely on a thermodynamic cycle as shown in Figure 1(a).[7] Rather than computing the double free energy difference of interest according to ΔΔA = ΔA4 − ΔA3, as would be done in an experiment, it is obtained along the “alchemical” paths, ΔΔA = ΔA2 − ΔA1. In the so-called single topology paradigm,[8] these placeholders are usually referred to as dummy atoms. This is made explicit, where the presence of dummy atoms is indicated by the superscripts D. One really computes ΔΔA′ = A2′ − A1′ as shown, relying on that ΔΔA′ equals ΔΔA from the idealized cycle Figure 1(a)
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