Evaluating multi-state free energy profiles from splitting probability

Single molecule experiments monitor the structural transitions of biomolecules under a constant mechanical force to study their fold-unfold transitions. The activation barrier for such transitions is obtained by inverting the observed committor, which is the probability that the molecule starting from a given extension reaches the folded state before the unfolded state. This work proposes an analytical model for committor analysis of the multi-state conformational dynamics of a DNA hairpin in a complex cellular environment, within the framework of the generalized Langevin equation using a general asymmetric bistable potential with a power-law frictional memory kernel. We obtained exact analytical expressions for the probability density function, first passage time distribution, and the committor. The results are compared with those obtained from steered molecular dynamics simulation of a three-state DNA hairpin, and earlier experimental data. We investigated the dependence of the committor and the corresponding committor-inverted profiles on the linker stiffness, barrier height, and degree of asymmetry in the bistable potential. This model successfully captures the fold-unfold dynamics, reproducing the multi-state free energy profile with asymmetric energy barriers.