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Classical rate theories often fail in cases where the observable(s) or order parameter(s) used are poor reaction coordinates or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling (MSM) approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality (RCQ). The RCQ can be bounded from below by the directly computable observation quality (OQ), thus providing a measure allowing the RCQ to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, here focusing on the case of two-state kinetics.
Fragment-size distributions have been studied experimentally in masticated viscoelastic food (fish sausage).The mastication experiment in seven subjects was examined. We classified the obtained results into two groups, namely, a single lognormal dist
The projected discovery and exclusion capabilities of particle physics and astrophysics/cosmology experiments are often quantified using the median expected $p$-value or its corresponding significance. We argue that this criterion leads to flawed res
We introduce and study random bipartite networks with hidden variables. Nodes in these networks are characterized by hidden variables which control the appearance of links between node pairs. We derive analytic expressions for the degree distribution
By exerting mechanical force it is possible to unfold/refold RNA molecules one at a time. In a small range of forces, an RNA molecule can hop between the folded and the unfolded state with force-dependent kinetic rates. Here, we introduce a mesoscopi
We introduce a filter-construction method for pulse processing that differs in two respects from that in standard optimal filtering, in which the average pulse shape and noise-power spectral density are combined to create a convolution filter for est