Hybrid neMD/MC lipid swapping algorithm to equilibrate membrane simulation with thermodynamic reservoir

Molecular dynamics (MD) simulations based on detailed all-atom models offer a powerful approach to study the structure and dynamics of biological membranes. However, the complexity of biological membranes in terms of chemical diversity presents an outstanding challenge. Particularly, difficulties are encountered when a given lipid type is present at very low abundance. While considering a very large simulation system with a small number of the low abundance lipid may offer a practical solution in some cases, resorting to increasingly large system rapidly becomes computationally costly and impractical. More fundamentally, an additional issue may be encountered if the low abundance lipid displays a high affinity for some protein in the simulation system. What is needed is to treat the simulation box as an open system in which the number of lipids can naturally fluctuate, as in the Grand Canonical Monte Carlo (MC) algorithm. However, this approach, in which a whole lipid molecule needs to be inserted or annihilated, is essentially impractical in the context of an all-atom simulation. To enforce equilibrium between a simulated system and an infinite surrounding bath, we propose a hybrid non-equilibrium (neMD)-MC algorithm, in which a randomly chosen lipid molecule in the simulated system is swapped with a lipid picked in a separate system standing as a thermodynamic "reservoir" with the desired mole fraction for all lipid components. The neMD/MC algorithm consists in driving the system via short non-equilibrium trajectories to generate a new state of the system that are subsequently accepted or rejected via a Metropolis MC step. The probability of exchanges in the context of an infinite reservoir with the desired mole fraction for all lipid components is derived and tested with a few illustrative systems for phosphatidylcholine and phosphatidylglycerol lipid mixtures.