scholarly journals A Continuous Time Representation of smFRET for the Extraction of Rapid Kinetics

2020 ◽  
Author(s):  
Zeliha Kilic ◽  
Ioannis Sgouralis ◽  
Wooseok Heo ◽  
Kunihiko Ishii ◽  
Tahei Tahara ◽  
...  
Keyword(s):  
Single Molecule ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Hidden Markov ◽  
Photon Emission ◽  
Simulated Data ◽  
Emission Rates ◽  
Markov Jump ◽  
Conformational States ◽  
Donor And Acceptor

AbstractOur 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.

scholarly journals A Model of Indel Evolution by Finite-State, Continuous-Time Machines

Genetics ◽  
2020 ◽  
Vol 216 (4) ◽  
pp. 1187-1204
Author(s):  
Ian Holmes
Keyword(s):  
Differential Equations ◽  
Continuous Time ◽  
Markov Models ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Coarse Graining ◽  
Automata Theory ◽  
State Spaces ◽  
Finite State ◽  
Related Approach

We introduce a systematic method of approximating finite-time transition probabilities for continuous-time insertion-deletion models on sequences. The method uses automata theory to describe the action of an infinitesimal evolutionary generator on a probability distribution over alignments, where both the generator and the alignment distribution can be represented by pair hidden Markov models (HMMs). In general, combining HMMs in this way induces a multiplication of their state spaces; to control this, we introduce a coarse-graining operation to keep the state space at a constant size. This leads naturally to ordinary differential equations for the evolution of the transition probabilities of the approximating pair HMM. The TKF91 model emerges as an exact solution to these equations for the special case of single-residue indels. For the more general case of multiple-residue indels, the equations can be solved by numerical integration. Using simulated data, we show that the resulting distribution over alignments, when compared to previous approximations, is a better fit over a broader range of parameters. We also propose a related approach to develop differential equations for sufficient statistics to estimate the underlying instantaneous indel rates by expectation maximization. Our code and data are available at https://github.com/ihh/trajectory-likelihood.

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

2020 ◽  
Author(s):  
Zeliha Kilic ◽  
Ioannis Sgouralis ◽  
Steve Pressé
Keyword(s):  
Data Acquisition ◽  
Discrete Time ◽  
Continuous Time ◽  
Hidden Markov ◽  
Jump Processes ◽  
Transition Rates ◽  
Markov Jump ◽  
Acquisition Rate ◽  
Physical Systems ◽  
Markov Jump Processes

AbstractThe hidden Markov model (HMM) is a framework for time series analysis widely applied to single molecule experiments. It has traditionally been used to interpret signals generated by 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, originally developed for applications outside the Natural Sciences, such as speech recognition, transitions between states, such as molecular conformational states, are modeled as occurring at the end of each data acquisition period and are described using transition probabilities. Yet, while 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, as 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, as 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 (HMJP). We explicitly show in what limit the HMJP 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.

scholarly journals Finite-Time Asynchronous Stabilization for Nonlinear Hidden Markov Jump Systems with Parameter Varying in Continuous-Time Case

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Lianjun Xiao ◽  
Xiaofeng Wang ◽  
Lingling Gao
Keyword(s):  
Continuous Time ◽  
Finite Time ◽  
Hidden Markov ◽  
Sufficient Conditions ◽  
Linear Parameter ◽  
Markov Jump ◽  
Linear Parameter Varying ◽  
Markov Jump Systems ◽  
Parameter Varying ◽  
Jump Systems

The finite-time asynchronous stabilization problem has received great attention because of the wide application of actual engineering. In this paper, we consider the problem of finite-time asynchronous stabilization for nonlinear hidden Markov jump systems (HMJSs) with linear parameter varying. Compared with the existing research results on Markov jump systems, this paper considers the HMJSs which contain both the hidden state and the observed state in continuous-time case. Moreover, we consider the parameters of the systems are time varying. The aim of the paper is to design a proper observation-mode-based asynchronous controller such that the closed-loop HMJSs with linear parameter varying be stochastically finite-time bounded with H ∞ performance (SFTB- H ∞ ). Then, we give some sufficient conditions to solve the SFTB- H ∞ asynchronous controller by considering the stochastic Lyapunov–Krasovskii functional (SLKF) methods. Finally, a numerical example is used to show the validity of the main results.

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