A simple method to optimize the spacing and number of alchemical intermediates in expanded ensemble free energy calculations

Alchemical free energy calculations are essential to modern structure-based drug design. Such calculations are usually performed at a series of discrete intermediates along a nonphysical thermodynamic pathway to estimate the free energy difference between two endpoints of an alchemical transformation. The efficiency and accuracy of the free energy estimate depends critically on the choice of alchemical intermediates. In this paper, we review the concept of thermodynamic length, and how it can be used as a principle to choose alchemical paths in free energy simulations. We then present an algorithm for optimizing the choice of alchemical intermediates in free energy simulations. Our method is similar to the thermodynamic trailblazing algorithm of Rizzi et al. (2020) but with several improvements for use with expanded ensemble (EE) simulations. Our method only requires a single initial round of EE simulation, and includes a method for optimizing the number of alchemical intermediates in an EE simulation, based on the predicted mixing time. We first show how the method performs in a simple toy model, and then demonstrate its use in a realistic example for an alchemical relative binding free energy calculation. We also show how our method can be used to optimize free energy estimates in other contexts, namely, calculating a score for model selection in the Bayesian Inference of Conformational Populations (BICePs) approach. We have implemented our optimization algorithm in a freely available Python package called pylambdaopt (https://github.com/vvoelz/pylambdaopt)