successori+1 can be calculated, and the binding free energy can be
recovered as the sum of all theseΔGi,i + 1.
ΔGb¼Xi¼N 2
i¼ 0 ΔGi,iþ^1 ð^9 Þ
There are six states along the absolute binding free energy cycle
that are conceptually helpful to think of: the two physically mean-
ingful end states (i.e., the bound and unbound states), and four
alchemical intermediate states where the ligand isdecoupledfrom
the environment, i.e., it does not interact with any other molecular
species in the simulation. In general, we use the termdecoupledto
indicate a state in which the intermolecular interactions of the
ligand have been removed, while the intramolecular interactions
are still present; i.e., the atoms in the ligand feel the forces resulting
from electrostatic and van der Waals (vdW) interactions with the
other atoms in the same molecule. On the other hand, we use the
termannihilatedwhen also the intramolecular interactions have
been removed. Figure1 shows these intermediate states visually, to
help understand their nature and how they connect the two end
states together. In the unbound end state (state A), we are consid-
ering a ligand that is free in solution. Simulating a box containing
only the ligand is computationally efficient and ensures there are no
interactions with the protein. This fully interacting ligand in solu-
tion (Fig.1. State A; in orange) is then transformed into a nonin-
teracting solute (Fig.1. State B; in white) by scaling its electrostatic
and vdW interactions to zero through several nonphysical states
that can be simulated independently. The ligand is then restrained
to limit its accessible sampling volume while still not interacting
with the environment (Fig.1, State C; in white with a paper clip).
Restraining the ligand substantially aids the convergence of the
calculations. In fact, if the ligand was left unrestrained when
decoupled, it could leave the binding pocket and float around the
whole simulation box. Then, once its interactions with the environ-
ment were turned back on, it would have to go through a physical
binding process in order to find its position in the protein again.
State C is equivalent to having the noninteracting and restrained
ligand within the protein cavity (Fig.1, State D), since no work is
needed to change the relative positions of the completely noninter-
acting protein and ligand. The decoupled and restrained ligand in
complex with the protein has then its electrostatic and vdW inter-
actions turned back on again (Fig.1, State E). The restraints
between ligand and protein are then finally removed, closing the
cycle, and reaching the other physical end state, that is, the bound
protein–ligand state (Fig.1, State F).
If the cycle just described is discretized intoNintermediate
states, it is then possible to recover the binding free energyΔGb.
Following from the discussion above, the cycle can be split into four
main steps, each of them corresponding to the free energy
Absolute Alchemical Free Energy 207