Piecewise Constraints for Exact End Point Sampling with λ-Dynamics

λ-Dynamics simulations are used to explore differences between distinct chemical states through interpolation with a λ parameter. Existing algorithms for λ-dynamics treat the thermodynamic coupling parameter λ as an additional degree of freedom and integrate its motion along with all other physical degrees of freedom in the molecular system. This coupling parameter is constrained to 0 ≤ λ ≤ 1 to represent interpolation between two chemical end-states. In this paper, we present a mapping scheme which numerically satisfies this constraint and has desirable properties when generalized to the multiend-state constraint problem of satisfying both 0 ≤ λi ≤ 1 and ∑iλi = 1. Relative to the existing mapping schemes which implicitly satisfy these constraints, the introduced method samples the λ = {0, 1} end-states exactly and performs well as the number of end-states increases. The exact end-state sampling removes the need to approximate end-states with a finite-width histogram bin and produces unbiased free energy estimates. This new constraint was used within the framework of multisite λ-dynamics (MSλD) and led to improved sampling with many end-states. This was shown through calculations of single and multisite systems with a range of substituents at each site in relative small molecule solvation and protein side-chain mutations. This development makes MSλD calculations with many end-states more reliable and opens the door to more ambitious design projects in the future.