Markovian Milestoning for Computing Entry, Exit, and Internal Diffusion Rates of Ligands in Proteins

Abstract

Measuring diffusion rates of ligands plays a key role in understanding the kinetic processes inside proteins. For example, although many molecular simulation studies have reported free energy barriers to infer rates for CO diffusion in myoglobin (Mb), they typically do not include direct calculation of diffusion rates because of the long simulation times needed to infer these rates with statistical accuracy. We show in this talk how to apply Markovian milestoning along minimum free-energy pathways to calculate diffusion rates of both CO inside Mb and O2 inside monomeric sarcosine oxidase (MSOX). In Markovian milestoning, one partitions a suitable reaction coordinate space into regions and performs restrained molecular dynamics in each region to accumulate kinetic statistics that, when assembled across regions, provides an estimate of the mean first-passage time between states. We show here that the milestones can be chosen from a Voronoi tessellation defined by discrete centers taken from minimum free-energy pathways computed using the single-sweep reconstruction method. In the case of CO/Mb, we find that, although simulations predict the existence of many potential portals that connect the solvent phase with the distal pocket (DP, the site of CO attachment to the heme iron), the so-called “histidine gate” pathway is kinetically dominant for both CO entry and exit by approximately a factor of ten, in qualitative agreement with existing experimental data. Our calculations also semi-quantitatively agree with experiment, predicting a 60-ns mean exit time for CO from the DP and a mean entry time of 50 microsec under CO-saturating conditions. The Markovian milestoning approach seems from these results to be a promising way to estimate biomolecule-related transition rates, and ongoing work involves using it to time conformational changes.