Transition path times in asymmetric barriers

Active Brownian particles in a circular disk with an absorbing boundary

We solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle exploring a circular region with an absorbing boundary. Using the passive Brownian particle as basis states and dealing with the activity as a perturbation, we provide a matrix representation of the Fokker-Planck operator and we express the propagator in terms of the perturbed eigenvalues and eigenfunctions. Alternatively, we show that the propagator can be expressed as a combination of the equilibrium eigenstates with weights depending only on time and on the initial conditions, and obeying exact iterative relations. Our solution allows also obtaining the survival probability and the first-passage time distribution. These latter quantities exhibit peculiarities induced by the nonequilibrium character of the dynamics; in particular, they display a strong dependence on the activity of the particle and, to a less extent, also on its rotational diffusivity.

Microscopic origin of diffusive dynamics in the context of transition path time distributions for protein folding and unfolding

Experimentally measured transition path time distributions are usually analyzed theoretically in terms of a diffusion equation over a free energy barrier. It is though well understood that the free energy profile separating the folded and unfolded states of a protein is characterized as a transition through many stable micro-states which exist between the folded and unfolded states. Why is it then justified to model the transition path dynamics in terms of a diffusion equation, namely the Smoluchowski equation (SE)? In principle, van Kampen has shown that a nearest neighbor Markov chain of thermal jumps between neighboring microstates will lead in a continuum limit to the SE, such that the friction coefficient is proportional to the mean residence time in each micro-state. However, the practical question of how many microstates are needed to justify modeling the transition path dynamics in terms of an SE has not been addressed. This is a central topic of this paper where we compare numerical results for transition paths based on the diffusion equation on the one hand and the nearest neighbor Markov jump model on the other. Comparison of the transition path time distributions shows that one needs at least a few dozen microstates to obtain reasonable agreement between the two approaches. Using the Markov nearest neighbor model one also obtains good agreement with the experimentally measured transition path time distributions for a DNA hairpin and PrP protein. As found previously when using the diffusion equation, the Markov chain model used here also reproduces the experimentally measured long time tail and confirms that the transition path barrier height is ∼3k_BT. This study indicates that in the future, when attempting to model experimentally measured transition path time distributions, one should perhaps prefer a nearest neighbor Markov model which is well defined also for rough energy landscapes. Such studies can also shed light on the minimal number of microstates needed to unravel the experimental data.

What can we learn from transition path time distributions for protein folding and unfolding?

Recent advances in experimental measurements of transition path time distributions have raised intriguing theoretical questions. The present interpretation of the experimental data indicates a small value of the fitted transition path barrier height as compared to the barrier height of the unfolded to folded transition. Secondly, as shown in this paper, it is essential to analyse the experimental data using absorbing boundary conditions at the end points used to determine the transition paths. Such an analysis reveals long time tails that have thus far eluded quantitative theoretical interpretation. Is this due to uncertainty in the experimental data or does it call for a rethinking of the theoretical interpretation? A detailed study of the transition path time distribution using a diffusive model leads to the following conclusions. a. The present experimental data is not accurate enough to discern between functional forms of bell shaped free energy barriers. b. Long time tails indicate the possible existence of a "trap" in the transition path region. c. The "trap" may be considered as a well in the free energy surface. d. The long time tail is quite sensitive to the form of the trap so that future measurements of the long time tail as a function of the location of the end points of the transition path may make it possible to not only determine the well depth but also to distinguish between different functional forms for the free energy surface. e. Introduction of a well along the transition path leads to good fits with the experimental data provided that the transition path barrier height is ∼3k_BT, substantially higher than the estimates of ∼1k_BT based on bell shaped functions. The results presented here negate the need of introducing multi-dimensional effects, free energy barrier asymmetry, sub-diffusive memory kernels or systematic ruggedness to explain the experimentally measured data.

Optimal navigation strategy of active Brownian particles in target-search problems

We investigate exploration patterns of a microswimmer, modeled as an active Brownian particle, searching for a target region located in a well of an energy landscape and separated from the initial position of the particle by high barriers. We find that the microswimmer can enhance its success rate in finding the target by tuning its activity and its persistence in response to features of the environment. The target-search patterns of active Brownian particles are counterintuitive and display characteristics robust to changes in the energy landscape. On the contrary, the transition rates and transition-path times are sensitive to the details of the specific energy landscape. In striking contrast to the passive case, the presence of additional local minima does not significantly slow down the active-target-search dynamics.

Nonreciprocality of a micromachine driven by a catalytic chemical reaction

We propose a model that describes cyclic state transitions of a micromachine driven by a catalytic chemical reaction. We consider a mechanochemical coupling of variables representing the degree of a chemical reaction and the internal state of a micromachine. The total free energy consists of a tilted periodic potential and a mechanochemical coupling energy. We assume that the reaction variable obeys a deterministic stepwise dynamics characterized by two typical timescales, i.e., the mean first passage time and the mean first transition path time. To estimate the functionality of a micromachine, we focus on the quantity called "nonreciprocality" and further discuss its dependence on the properties of catalytic reaction. For example, we show that the nonreciprocality is proportional to the square of the mean first transition path time. The explicit calculation of the two timescales within the decoupling approximation model reveals that the nonreciprocality is inversely proportional to the square of the energy barrier of catalytic reaction.

Target Search of Active Agents Crossing High Energy Barriers

Target search by active agents in rugged energy landscapes has remained a challenge because standard enhanced sampling methods do not apply to irreversible dynamics. We overcome this nonequilibrium rare-event problem by developing an algorithm generalizing transition-path sampling to active Brownian dynamics. This method is exemplified and benchmarked for a paradigmatic two-dimensional potential with a high barrier. We find that even in such a simple landscape the structure and kinetics of the ensemble of transition paths changes drastically in the presence of activity. Indeed, active Brownian particles reach the target more frequently than passive Brownian particles, following longer and counterintuitive search patterns.