THERMODYNAMICS OF PROTEIN CONFORMATIONAL CHANGES BY UMBRELLA SAMPLING AND FREE ENERGY CALCULATION

Spontaneous transitions between the native and non-native protein conformations are normally rare events that hardly take place in typical unbiased molecular dynamics simulations. It was recently demonstrated that such transitions could be well described by a reaction coordinate, Q, that represents the collective fraction of the native contacts between the protein atoms. Here we attempt to use this reaction coordinate to enhance the conformational sampling. We perform umbrella sampling simulations with biasing potentials on for two model proteins, Trp-Cage and BBA, using the CHARMM force field. Hamiltonian replica exchange is implemented in these simulations to further facilitate the sampling. The simulations appear to have reached satisfactory convergence, resulting in unbiased, free energies as a function of . In addition to the native structure, multiple folded conformations are identified in the reconstructed equilibrium ensemble. Some conformations without any native contacts nonetheless have rather compact geometries and are stabilized by hydrogen bonds not present in the native structure. Whereas the enhanced sampling along with reasonably reproduces the equilibrium conformational space, we also find that the folding of an α-helix in Trp-Cage is a slow degree of freedom orthogonal to and therefore cannot be accelerated by biasing the reaction coordinate. Overall, we conclude that whereas is an excellent parameter to analyze the simulations, it is not necessarily a perfect reaction coordinate for enhanced sampling, and better incorporation of other slow degrees of freedom may further improve this reaction coordinate.

To analyze such behavior like slow degrees of freedom, we conducted another research study. Proteins may adopt multiple conformations, and they undergo various transitions from one conformation to another. A well-defined reaction coordinate can describe these transitions. However, there is no efficient way to define the entire conformational space of a complex biological system by only one reaction coordinate. In the two-state system, a protein can adopt two different conformations, A and B. We implemented a stepwise transition model. The targeted protein starts from metastable A, and it will undergo a transition to another intermediate state, and from that intermediate state, the protein undergoes another transition to and so on. Therefore, by N transitions, we can get to the metastable state B. During each step transition, we apply a boundary potential over other degrees of freedom to keep them unchanged. With this strategy, along with a simple definition of the reaction coordinate, we have high accuracy in our thermodynamics and protein dynamics measurements. As a case study, we implemented all-atom Umbrella Sampling simulations to characterize the conformational changes between outward-facing open (OF) and outward-facing occluded (OC) states of transmembrane protein Mhp1. For each step transition, the reaction coordinate was defined by a simple dihedral angle or a bond length. We could obtain six transition steps with five intermediate states that connect the two OF and OC stable states. We measured each step transition free energy profile from WHAM equations. We performed two independent sampling simulations with different initial structures: the transition initiates from OF state indicated InitOF, and the transition initiates from the OC state indicated as InitOC transition. By comparing the obtained free energy profiles with the stepwise model, we implied the extent of convergence in our calculations. The energy difference between OF and OC states in our study is -1.02 kcal/mol for InitOF and InitOC transition, respectively.