The hidden Markov model (HMM) is a framework for time series analysis widely applied to single-molecule experiments. Although initially developed for applications outside the natural sciences, the HMM has traditionally been used to interpret signals generated by physical systems, such as single molecules, evolving in a discrete state space observed at discrete time levels dictated by the data acquisition rate. Within the HMM framework, transitions between states are modeled as occurring at the end of each data acquisition period and are described using transition probabilities. Yet, whereas measurements are often performed at discrete time levels in the natural sciences, physical systems evolve in continuous time according to transition rates. It then follows that the modeling assumptions underlying the HMM are justified if the transition rates of a physical process from state to state are small as compared to the data acquisition rate. In other words, HMMs apply to slow kinetics. The problem is, because the transition rates are unknown in principle, it is unclear, a priori, whether the HMM applies to a particular system. For this reason, we must generalize HMMs for physical systems, such as single molecules, because these switch between discrete states in ''continuous time''. We do so by exploiting recent mathematical tools developed in the context of inferring Markov jump processes and propose the hidden Markov jump process. We explicitly show in what limit the hidden Markov jump process reduces to the HMM. Resolving the discrete time discrepancy of the HMM has clear implications: we no longer need to assume that processes, such as molecular events, must occur on timescales slower than data acquisition and can learn transition rates even if these are on the same timescale or otherwise exceed data acquisition rates.

The cystic fibrosis transmembrane conductance regulator (CFTR) is an anion channel that regulates salt and fluid homeostasis across epithelial membranes1. Alterations in CFTR cause cystic fibrosis, a fatal disease without a cure2,3. Electrophysiological properties of CFTR have been analysed for decades4–6. The structure of CFTR, determined in two globally distinct conformations, underscores its evolutionary relationship with other ATP-binding cassette transporters. However, direct correlations between the essential functions of CFTR and extant structures are lacking at present. Here we combine ensemble functional measurements, single-molecule fluorescence resonance energy transfer, electrophysiology and kinetic simulations to show that the two nucleotide-binding domains (NBDs) of human CFTR dimerize before channel opening. CFTR exhibits an allosteric gating mechanism in which conformational changes within the NBD-dimerized channel, governed by ATP hydrolysis, regulate chloride conductance. The potentiators ivacaftor and GLPG1837 enhance channel activity by increasing pore opening while NBDs are dimerized. Disease-causing substitutions proximal (G551D) or distal (L927P) to the ATPase site both reduce the efficiency of NBD dimerization. These findings collectively enable the framing of a gating mechanism that informs on the search for more efficacious clinical therapies.

We consider the evolution of a decaying passive scalar in the presence of a gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the short, intermediate, and long time symmetry properties of the evolving point wise probability measure (PDF) for the random passive scalar. We identify, for the cases of both point source and line source initial data, regions in the x-y plane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernel formulas for the scalars statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the available random Green's functions which allows for the fast construction of the PDF for single point statistics, as well as multi-point statistics including spatially integrated quantities natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on parallel planes and observe a dramatically different long time state resulting from the existence of the wall. In particular, the channel case collapsing invariant measure has negative skewness, with random states heavily leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state. arXiv:1802.03340v1 [physics.flu-dyn]

SUMMARY Hidden Markov models (HMMs) are used to learn single-molecule kinetics across a range of experimental techniques. By their construction, HMMs assume that single-molecule events occur on slower timescales than those of data acquisition. To move beyond that HMM limitation and allow for single-molecule events to occur on any timescale, we must treat single-molecule events in continuous time as they occur in nature. We propose a method to learn kinetic rates from single-molecule Förster resonance energy transfer (smFRET) data collected by integrative detectors, even if those rates exceed data acquisition rates. To achieve that, we exploit our recently proposed “hidden Markov jump process” (HMJP), with which we learn transition kinetics from parallel measurements in donor and acceptor channels. HMJPs generalize the HMM paradigm in two critical ways: (1) they deal with physical smFRET systems as they switch between conformational states in continuous time , and (2) they estimate transition rates between conformational states directly without having recourse to transition probabilities or assuming slow dynamics. Our continuous-time treatment learns the transition kinetics and photon emission rates for dynamic regimes that are inaccessible to HMMs, which treat system kinetics in discrete time. We validate our framework’s robustness on simulated data and demonstrate its performance on experimental data from FRET-labeled Holliday junctions.

An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Here, we observe and model mathematically an unexpected fundamental mechanism by which particles suspended within stratification may self-assemble and form large aggregates without adhesion. This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and aggregate geometry, which produces toroidal flows. We show that these flows yield attractive horizontal forces between particles at the same heights. We observe that many particles demonstrate a collective motion revealing a system which appears to solve jigsaw-like puzzles on its way to organizing into a large-scale disc-like shape, with the effective force increasing as the collective disc radius grows. Control experiments isolate the individual dynamics, which are quantitatively predicted by simulations. Numerical force calculations with two spheres are used to build many-body simulations which capture observed features of self-assembly.

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Our goal is to learn kinetic rates from single molecule FRET (smFRET) data even if these exceed the data acquisition rate. To achieve this, we develop a variant of our recently proposed hidden Markov jump process (HMJP) with which we learn transition kinetics from parallel measurements in donor and acceptor channels. Our HMJP generalizes the hidden Markov model (HMM) paradigm in two critical ways: (1) it deals with physical smFRET systems as they switch between conformational states in continuous time; (2) it estimates the transition rates between conformational states directly without having recourse to transition probabilities or assuming slow dynamics (as is necessary of the HMM). Our continuous time treatment learns transition kinetics and photon emission rates for dynamical regimes inaccessible to the HMM which treats system kinetics in discrete time. We validate the robustness of our framework on simulated data and demonstrate its performance on experimental data from FRET labeled Holliday junctions.