Generalizing HMMs to Continuous Time for Fast Kinetics: Hidden Markov Jump Processes

Abstract

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.

Inferring kinetic rate constants from single-molecule FRET trajectories – a blind benchmark of kinetic analysis tools

“…This may be attributed to the inevitable effect of time discretization and related intensity averaging: when a transition between the high- and low-FRET states happens during a time bin, time-weighted averaging (camera blurring) of the FRET efficiencies occurs, leading, in some cases, to mid-FRET observations that are indistinguishable from those caused by a bona fide biomolecular conformation. While, at the single datapoint level this discretization artefact cannot be prevented, the inference accuracy may be improved by treating discretization-induced averaging explicitly in the analysis 31,41 ; or using pulsed illumination to reduce blurring 42,43 . Overall, postFRET and Tracy inferred the most accurate rate constants with average GT deviations of 9 % and 14 %, respectively.…”

“…While, at the single datapoint level this discretization artefact cannot be prevented, the inference accuracy may be improved by treating discretization-induced averaging explicitly in the analysis 31,41 ; or using pulsed illumination to reduce blurring 42,43 . Overall, postFRET and Tracy inferred the most accurate rate constants with average GT deviations of 9 % and 14 %, respectively.…”

Extraction of rapid kinetics from smFRET measurements using integrative detectors

“…By taking advantage of algorithmic advancements in inference on continuous-time processes that are scarcely half a decade old, 40 the HMJP scales as O ( MK ), in which M is the number of jumps, and K is the number of states (see Robert and Casella 60 and Kilic et al 61 ). This scaling is, in itself, a testament to the powerful method of uniformization that only introduces jumps if they are warranted based on the data.…”

Memory effects and static disorder reduce information in single-molecule signals

“…[7][8][9][10][11][12][13] ), but the challenge, then, is to choose the right dynamical model out of the multitude of possibilities 5 . Data-driven Bayesian inference models of single-molecule time series have enjoyed considerable success in recent years [14][15][16][17][18][19] , but they usually require physical insight in order to constrain the space of possible models, and they, too, often assume that the observed dynamics is a one-dimensional random walk even if the number of discrete states is not specified a priori.…”