Calculating Watson-Crick to Hoogsteen Transition Kinetics in DNA with Langevin Dynamics and Fokker-Planck Diffusion in Reduced Configuration Space

Abstract

Previous studies have shown that an important aspect of designing drugs for RNA drug targets, in particular, the transactivation response element (TAR) from HIV type 1 (HIV-1), is the determination of the thermodynamics and kinetics for WC to HG (Watson-Crick to Hoogsteen) conformation transitions. Previous work by Eunae Kim and Ioan Andricioaei successfully implemented umbrella sampling to produce a two-dimensional free energy surface for WC to HG transitions, where the two independent variables are the “flip-over” angle and “flip-out” angle of the adenine moiety undergoing the transition. The current study builds upon the work of Kim and Andricioaei by exploring the kinetics that arise from the time evolution of the probability density function of configurations, subjected to this potential energy surface. A position-dependent diffusion matrix was first calculated from the umbrella sampling data, in order to accurately describe diffusive motions in this reduced space. Next, a 2D polynomial of order 20 was fit to the energy surface, in order to obtain an analytical expression for the 2D surface. The diffusion matrix and analytical function describing the 2D surface were then used to carry out Langevin dynamics. Numerous Langevin trajectories between states were then analyzed in multiple ways, including calculating first passage time distributions between WC and HG configurations, and determination of regions in configuration space of maximum flux. In order to make the system numerically tractable for a Fokker-Planck approach to diffusion, the energy surface was also fit to a simpler 2D analytical function with a 20 term Fourier series solely in the “flip-over” dimension and a separate harmonic term approximating the “flip-out” dimension. Both approaches were utilized to determine favorable pathways for WC/HG transitions, and their respective time scales.