Constant pH Simulation with FMM Electrostatics in GROMACS. (B) GPU Accelerated Hamiltonian Interpolation.

The structural dynamics of biological macromolecules, such as proteins, DNA/RNA, or their complexes, are strongly influenced by protonation changes of their typically many titratable groups, which explains their pH sensitivity. Conversely, conformational and environmental changes in the biomolecule affect the protonation state of these groups. With a few exceptions, conventional force field-based molecular dynamics (MD) simulations do not account for these effects, nor do they allow for coupling to a pH buffer. The λ-dynamics method implements this coupling and thus allows for MD simulations at constant pH. It uses separate Hamiltonians for the protonated and deprotonated states of each titratable group, with a dynamic λ variable that continuously interpolates between them. However, rigorous implementations of Hamiltonian Interpolation (HI) λ-dynamics are prohibitively slow for typical numbers of sites when used with particle mesh Ewald (PME). To circumvent this problem, it has recently been proposed to interpolate the charges (QI) instead of the Hamiltonians. Here, in the second of two companion papers, we propose a rigorous yet efficient Multipole-Accelerated Hamiltonian Interpolation (MAHI) method to perform λ-dynamics in GROMACS. Starting from a charge-scaled Hamiltonian, precomputed with the Fast Multipole Method (FMM), the correct HI forces are calculated with negligible computational overhead. However, other electrostatic solvers, such as PME, can also be used for the precomputation. We compare Hamiltonian interpolation with charge interpolation and show that HI leads to more frequent transitions between protonation states, resulting in better sampling and accuracy. Our accuracy and performance benchmarks show that introducing, e.g., 512 titratable sites to a one million atom MD system increases runtime by less than 20% compared to a regular FMM-based simulation. We have integrated the scheme into our GPU-accelerated FMM code for the simulation software GROMACS, allowing easy and effortless transitions from standard force field simulations to constant pH simulations.