Computing transition rates for rare events: When Kramers theory meets the free-energy landscape

Abstract

Computing reactive trajectories and free energy (FE) landscapes associated to rare event kinetics is key to understanding the dynamics of complex systems. The analysis of the FE surface on which the underlying dynamics takes place has become central to compute transition rates. In the overdamped limit, most often encountered in biophysics and soft condensed matter, the Kramers' Theory (KT) has proved to be quite successful in recovering correct kinetics. However, the additional calculation to obtain rate constants in complex systems where configurational entropy is competing with energy is still challenging conceptually and computationally. Building on KT and the metadynamics framework, the rate is expressed in terms of the height of the FE barrier measured along the minimum FE path and an auxiliary measure of the configurational entropy. We apply the formalism to two different problems where our approach shows good agreement with simulations and experiments and can present significant improvement over the standard KT.

Thermodynamics and ordering kinetics in asymmetric PS-b-PMMA block copolymer thin films

“…Nucleation corresponds to a thermally activated transition from a less stable to a more stable state, moving through a transition state at high energy, which represents the nucleation barrier. [41], [42], [43], [44], [45] Indeed, the basic features of nucleation are a free energy variation, with competing gain and penalty components, along with a nucleation barrier that inversely depends on the free energy gain during nucleation. All these features match our experimental results.…”

Dynamical control of denaturation bubble nucleation in supercoiled DNA minicircles

“…3 where the free energy surfaces (FES) are reconstructed within the metaD framework along the two CVs, ρ max and φ min . The entropic contribution to the FES can be quantitatively assessed considering the definition of the free energy difference in terms of the joint probability distribution of the CVs [25,31],…”

“…To take into account the presence of slow non-reactive modes competing with enthalpic contribution, as characterized by the configurational entropic contribution along φ min , we extended the metadynamics scope discussed above to assess the characteristic times associated with the opening and closure of the long-lived thermally assisted denaturation bubbles when their direct numerical estimation was not feasible. To do so, we considered the convergence of the minimal free energy path (MFEP) and the convergence of the free energy of formation, F * 0 , defined in terms of the joint probability distribution of the CVs [25,31]:…”

“…We explore numerically specific design of DNA minicircles containing between 200 and 400 bps, also named microDNA, as they are representative of supercoiled DNA loops found in nature [21] and have a suitable size for exploring the relationship between twist and writhe [9]. To overcome the inherent limitations of atomistic simulations encountered at lengthand time-scales of interest [22], mesoscopic modeling [23] is combined with accelerated dynamics simulations [24,25] to study accurately the free energy landscape and the equilibrium rates associated with the nucleation and closure mechanisms of the long-lived thermally-assisted denaturation bubbles. We discuss how specific tuning of DNA structural parameters, such as the minicircle size and degree of supercoiling can lead to a large variety of equilibrium closure and nucleation rates that can be seen as a dynamical bandwidth which might enhance selectivity for DNA binding proteins.…”

Assessing Position-Dependent Diffusion from Biased Simulations and Markov State Model Analysis

“…As originally described by Hendrik A. Kramers in his seminal work [12], such quantities can be calculated by employing models of diffusive motion. In the simplest diffusive model, one can ignore inertia and memory effects and regard the diffusive motion as the random walk of a particle under a position-dependent potential [13,14]. The dynamics of the reaction coordinate, X(t), is determined by two functions: the potential of mean force, W (X), and the diffusivity, D(X), along this coordinate.…”