From a physical perspective, all water molecules are indistinguishable in stage 1, however in stage 2 (and in all subsequent stages of the thermodynamic cycle), the trapped water is distinguishable from solvent waters because of the restraint on it. Because of this, the harmonic restraint in stages 2 to 4 needs to be treated carefully in the MBAR calculation for stages 1 through 4. An MBAR calculation requires the evaluation of the Hamiltonians of each of the states involved in the free energy calculation with the trajectories of each of these states. Hence, to calculate the correct restraint potential on the trapped water, we need to evaluate the Hamiltonian of stage 1 with the trajectory of stage 2, even though no water is restrained in stage 1. If the trapped water exchanges with a water from bulk solvent in stage 1, the Hamiltonian of stage 1 would not change because the trapped water is indistinguishable from the bulk solvent water that occupies the trapped water's site. However, when we evaluate the Hamiltonian of stage 2 with the trajectory of stage 1, the restraint potential will be extremely high, because the Hamiltonian of stage 2 would consider the distinguishable trapped water, which is now in the bulk solvent in stage 1, to calculate the restraint potential. This would lead to extremely high restraint energies, making it difficult to converge the free energy calculation, but more importantly, also would not give us the free energy of interest. Specifically, we are interested in the free energy of restraining any (that is, indistinguishable) water to the target position, rather than the free energy of restraining this specific (distinguishable) water to the target position. Thus, we needed to change how our calculation handled the exchange of water. For indistinguishable water, upon water exchange, if the newly-arrived bulk solvent water was considered as the new trapped water and was selected for the calculation of the restraint potential, the restraint potential will be correct for the case of indistinguishable waters, which is what we want.