Development and Application of Free-energy Calculation Methods based on Molecular Dynamics Simulations

Abstract

Molecular dynamics (MD) simulations offer nowadays a valuable tool for studying phenomena in chemistry and biology. The history of models in chemistry, as well as the methodological and theoretical background of MD is presented in Chapter 1. Chapter 2 deals with the application of explicit-solvent molecular dynamics (MD) simulations to resorcin[4]arene cavitands, which can adopt a close/contracted (Vase) and an open/expanded (Kite) conformation. The Vase-Kite equilibria of a quinoxaline- and a dinitrobenzene-based resorcin[4]arene are investigated in three solvent environments (vacuum, chloroform and toluene) and at three temperatures (198.15, 248.15 and 298.15 K). The challenge of sampling the millisecond-timescale Vase-Kite transition is addressed by calculating relative free energies using ball-and-stick local elevation umbrella sampling (B&S-LEUS) to promote interconversion transitions. The calculated Vase-to-Kite free-energy changes ∆G are in qualitative agreement with the experimental magnitudes and trends. Chapters 3-5 present the development of the conveyor belt scheme, a method to calculate free-energy differences. The working principle relies on K coupled replicas of the system that are simulated at different values of a coupling parameter λ. The number K is taken to be even and the replicas are equally spaced on a forward-turn-backward-turn path, akin to a conveyor belt (CB) between the two end-states. As in λ-dynamics (λD), the λ-values associated with the individual systems evolve in time along the simulation. However, they do so in a concerted fashion, determined by the evolution of a single dynamical variable Λ of period 2π controlling the advance of the entire CB. Thus, a change of Λ is always associated with one half of the replicas moving forward and the other half moving backward along λ. As a result, the effective free-energy profile of the replica system along Λ is characterized with decrasing barriers upon increasing K, at least as K −1 in the limit of large K. When a sufficient number of replicas is used, these variations become small, which enables a complete and quasi-homogeneous coverage of the λ-range by the replica system. Chapter 3 introduces this scheme with respect to alchemical free-energy calculations. Therefore it is termed conveyor belt thermodynamic integration. It provides the mathematical/physical formulation of the scheme, along with an initial application of the method to the calculation of the hydration free energy of methanol. In Chapter 4, the conveyor belt thermodynamic integration is applied to a Lys-X-Lys tripeptide, involving a side-chain mutation in the central residue, and guanosine triphosphate, involving a hydrogen-to-bromine mutation. With both systems, sampling issues have been encountered, due to the large orthogonal barriers either along the backbone dihedral angles φ and ψ (tripeptide) or along ribose-base dihedral angle χ (guanosine). This relative merits of different sampling schemes, orthogonal biasing and estimators to improve the convergence are investigated. The thermodynamic integration scheme is shown to suffer from the constraint of simulations at fixed λ. There is no significant improvement upon changing from the Simpson’s quadrature to the MBAR estimator. Both Hamiltonian replica exchange and conveyor belt thermodynamic integration improve the results for the tripeptide. For the guanosine triphosphate, improvement is only achieved upon application of an orthogonal biasing potential, most efficiently in combination with Hamiltonian replica exchange or conveyor belt thermodynamic integration. The conveyor belt scheme is extended to the calculation of conformational free-energy differences in Chapter 5, resulting in a so-called conveyor belt umbrella sampling scheme (CBUS). CBUS is here initally applied to the calculation of 45 standard absolute binding free energies of five alkali cations to three crown ethers in three different solvents. Besides introducing and testing the new scheme, it is compared to other methods. Expectedly, the direct counting approach has convergence issues on the accessible simulation timescale, and the corresponding results are rather unreliable. On the other hand, the results obtained using traditional umbrella sampling and CBUS are very consistent. Additionally, comparison of the results to those of previous alchemical calculations via an alchemical pathway reveals excellent consistency while the trends of available experimental data are qualitatively reproduced. Conclusions and an outlook into future developments are given in Chapter 6.