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Dhar M, Berg MA. Efficient, nonparametric removal of noise and recovery of probability distributions from time series using nonlinear-correlation functions: Photon and photon-counting noise. J Chem Phys 2024; 161:034116. [PMID: 39028845 DOI: 10.1063/5.0212157] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2024] [Accepted: 06/28/2024] [Indexed: 07/21/2024] Open
Abstract
A preceding paper [M. Dhar, J. A. Dickinson, and M. A. Berg, J. Chem. Phys. 159, 054110 (2023)] shows how to remove additive noise from an experimental time series, allowing both the equilibrium distribution of the system and its Green's function to be recovered. The approach is based on nonlinear-correlation functions and is fully nonparametric: no initial model of the system or of the noise is needed. However, single-molecule spectroscopy often produces time series with either photon or photon-counting noise. Unlike additive noise, photon noise is signal-size correlated and quantized. Photon counting adds the potential for bias. This paper extends noise-corrected-correlation methods to these cases and tests them on synthetic datasets. Neither signal-size correlation nor quantization is a significant complication. Analysis of the sampling error yields guidelines for the data quality needed to recover the properties of a system with a given complexity. We show that bias in photon-counting data can be corrected, even at the high count rates needed to optimize the time resolution. Using all these results, we discuss the factors that limit the time resolution of single-molecule spectroscopy and the conditions that would be needed to push measurements into the submicrosecond region.
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Affiliation(s)
- Mainak Dhar
- Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208, USA
| | - Mark A Berg
- Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208, USA
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Kilic Z, Sgouralis I, Heo W, Ishii K, Tahara T, Pressé S. Extraction of rapid kinetics from smFRET measurements using integrative detectors. CELL REPORTS. PHYSICAL SCIENCE 2021; 2:100409. [PMID: 34142102 PMCID: PMC8208598 DOI: 10.1016/j.xcrp.2021.100409] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/09/2023]
Abstract
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.
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Affiliation(s)
- Zeliha Kilic
- Center for Biological Physics, Department of Physics, Arizona State University, Tempe, AZ 85287, USA
| | - Ioannis Sgouralis
- Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
| | - Wooseok Heo
- Molecular Spectroscopy Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Kunihiko Ishii
- Molecular Spectroscopy Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
- Ultrafast Spectroscopy Research Team, RIKEN Center for Advanced Photonics (RAP), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Tahei Tahara
- Molecular Spectroscopy Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
- Ultrafast Spectroscopy Research Team, RIKEN Center for Advanced Photonics (RAP), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Steve Pressé
- Center for Biological Physics, Department of Physics, Arizona State University, Tempe, AZ 85287, USA
- School of Molecular Sciences, Arizona State University, Tempe, AZ 85287, USA
- Lead contact
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Tavakoli M, Jazani S, Sgouralis I, Shafraz OM, Sivasankar S, Donaphon B, Levitus M, Pressé S. Pitching single-focus confocal data analysis one photon at a time with Bayesian nonparametrics. PHYSICAL REVIEW. X 2020; 10:011021. [PMID: 34540355 PMCID: PMC8445401 DOI: 10.1103/physrevx.10.011021] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
Fluorescence time traces are used to report on dynamical properties of molecules. The basic unit of information in these traces is the arrival time of individual photons, which carry instantaneous information from the molecule, from which they are emitted, to the detector on timescales as fast as microseconds. Thus, it is theoretically possible to monitor molecular dynamics at such timescales from traces containing only a sufficient number of photon arrivals. In practice, however, traces are stochastic and in order to deduce dynamical information through traditional means-such as fluorescence correlation spectroscopy (FCS) and related techniques-they are collected and temporally autocorrelated over several minutes. So far, it has been impossible to analyze dynamical properties of molecules on timescales approaching data acquisition without collecting long traces under the strong assumption of stationarity of the process under observation or assumptions required for the analytic derivation of a correlation function. To avoid these assumptions, we would otherwise need to estimate the instantaneous number of molecules emitting photons and their positions within the confocal volume. As the number of molecules in a typical experiment is unknown, this problem demands that we abandon the conventional analysis paradigm. Here, we exploit Bayesian nonparametrics that allow us to obtain, in a principled fashion, estimates of the same quantities as FCS but from the direct analysis of traces of photon arrivals that are significantly smaller in size, or total duration, than those required by FCS.
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Affiliation(s)
- Meysam Tavakoli
- Department of Physics, Indiana University-Purdue University Indianapolis, IN 46202
| | - Sina Jazani
- Center for Biological Physics, Department of Physics, Arizona State University, Tempe, AZ 85287
| | - Ioannis Sgouralis
- Center for Biological Physics, Department of Physics, Arizona State University, Tempe, AZ 85287
| | - Omer M. Shafraz
- Department of Biomedical Engineering, University of California, Davis, CA 95616
| | - Sanjeevi Sivasankar
- Department of Biomedical Engineering, University of California, Davis, CA 95616
| | - Bryan Donaphon
- Biodesign Institute, Arizona State University, Tempe, AZ 85287
| | - Marcia Levitus
- Center for Biological Physics, Department of Physics, Arizona State University, Tempe, AZ 85287
- Biodesign Institute, Arizona State University, Tempe, AZ 85287 and School of Molecular Sciences, Arizona State University, Tempe, AZ 85287
| | - Steve Pressé
- Corresponding author. ; Website: http://statphysbio.physics.asu.edu
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Tavakoli M, Taylor JN, Li CB, Komatsuzaki T, Pressé S. Single Molecule Data Analysis: An Introduction. ADVANCES IN CHEMICAL PHYSICS 2017. [DOI: 10.1002/9781119324560.ch4] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
Affiliation(s)
- Meysam Tavakoli
- Physics Department; Indiana University-Purdue University Indianapolis; Indianapolis IN 46202 USA
| | - J. Nicholas Taylor
- Research Institute for Electronic Science; Hokkaido University; Kita 20 Nishi 10 Kita-Ku Sapporo 001-0020 Japan
| | - Chun-Biu Li
- Research Institute for Electronic Science; Hokkaido University; Kita 20 Nishi 10 Kita-Ku Sapporo 001-0020 Japan
- Department of Mathematics; Stockholm University; 106 91 Stockholm Sweden
| | - Tamiki Komatsuzaki
- Research Institute for Electronic Science; Hokkaido University; Kita 20 Nishi 10 Kita-Ku Sapporo 001-0020 Japan
| | - Steve Pressé
- Physics Department; Indiana University-Purdue University Indianapolis; Indianapolis IN 46202 USA
- Department of Chemistry and Chemical Biology; Indiana University-Purdue University Indianapolis; Indianapolis IN 46202 USA
- Department of Cell and Integrative Physiology; Indiana University School of Medicine; Indianapolis IN 46202 USA
- Department of Physics and School of Molecular Sciences; Arizona State University; Tempe AZ 85287 USA
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Chung HS, Louis JM, Gopich IV. Analysis of Fluorescence Lifetime and Energy Transfer Efficiency in Single-Molecule Photon Trajectories of Fast-Folding Proteins. J Phys Chem B 2016; 120:680-99. [PMID: 26812046 DOI: 10.1021/acs.jpcb.5b11351] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
Abstract
In single-molecule Förster resonance energy transfer (FRET) spectroscopy, the dynamics of molecular processes are usually determined by analyzing the fluorescence intensity of donor and acceptor dyes. Since FRET efficiency is related to fluorescence lifetimes, additional information can be extracted by analyzing fluorescence intensity and lifetime together. For fast processes where individual states are not well separated in a trajectory, it is not easy to obtain the lifetime information. Here, we present analysis methods to utilize fluorescence lifetime information from single-molecule FRET experiments, and apply these methods to three fast-folding, two-state proteins. By constructing 2D FRET efficiency-lifetime histograms, the correlation can be visualized between the FRET efficiency and fluorescence lifetimes in the presence of the submicrosecond to millisecond dynamics. We extend the previously developed method for analyzing delay times of donor photons to include acceptor delay times. To determine the kinetics and lifetime parameters accurately, we used a maximum likelihood method. We found that acceptor blinking can lead to inaccurate parameters in the donor delay time analysis. This problem can be solved by incorporating acceptor blinking into a model. While the analysis of acceptor delay times is not affected by acceptor blinking, it is more sensitive to the shape of the delay time distribution resulting from a broad conformational distribution in the unfolded state.
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Affiliation(s)
- Hoi Sung Chung
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health , Bethesda, Maryland 20892-0520, United States
| | - John M Louis
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health , Bethesda, Maryland 20892-0520, United States
| | - Irina V Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health , Bethesda, Maryland 20892-0520, United States
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Gopich IV. Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET. J Chem Phys 2015; 142:034110. [PMID: 25612692 DOI: 10.1063/1.4904381] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Photon sequences from single-molecule Förster resonance energy transfer (FRET) experiments can be analyzed using a maximum likelihood method. Parameters of the underlying kinetic model (FRET efficiencies of the states and transition rates between conformational states) are obtained by maximizing the appropriate likelihood function. In addition, the errors (uncertainties) of the extracted parameters can be obtained from the curvature of the likelihood function at the maximum. We study the standard deviations of the parameters of a two-state model obtained from photon sequences with recorded colors and arrival times. The standard deviations can be obtained analytically in a special case when the FRET efficiencies of the states are 0 and 1 and in the limiting cases of fast and slow conformational dynamics. These results are compared with the results of numerical simulations. The accuracy and, therefore, the ability to predict model parameters depend on how fast the transition rates are compared to the photon count rate. In the limit of slow transitions, the key parameters that determine the accuracy are the number of transitions between the states and the number of independent photon sequences. In the fast transition limit, the accuracy is determined by the small fraction of photons that are correlated with their neighbors. The relative standard deviation of the relaxation rate has a "chevron" shape as a function of the transition rate in the log-log scale. The location of the minimum of this function dramatically depends on how well the FRET efficiencies of the states are separated.
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Affiliation(s)
- Irina V Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892, USA
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Abstract
Single-molecule spectroscopy is widely used to study macromolecular dynamics. Although this technique provides unique information that cannot be obtained at the ensemble level, the possibility of studying fast molecular dynamics is limited by the number of photons detected per unit time (photon count rate), which is proportional to the illumination intensity. However, simply increasing the illumination intensity often does not help because of various photophysical and photochemical problems. In this Perspective, we show how to improve the dynamic range of single-molecule fluorescence spectroscopy at a given photon count rate by considering each and every photon and using a maximum likelihood method. For a photon trajectory with recorded photon colors and inter-photon times, the parameters of a model describing molecular dynamics are obtained by maximizing the appropriate likelihood function. We discuss various likelihood functions, their applicability, and the accuracy of the extracted parameters. The maximum likelihood method has been applied to analyze the experiments on fast two-state protein folding and to measure transition path times. Utilizing other information such as fluorescence lifetimes is discussed in the framework of two-dimensional FRET efficiency-lifetime histograms.
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Affiliation(s)
- Hoi Sung Chung
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health (NIH), Bethesda, MD, 20892-0520
| | - Irina V. Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health (NIH), Bethesda, MD, 20892-0520
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Waligórska M, Molski A. Maximum likelihood-based analysis of photon arrival trajectories in single-molecule FRET. Chem Phys 2012. [DOI: 10.1016/j.chemphys.2012.05.009] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Abstract
We consider the analysis of a class of experiments in which the number of photons in consecutive time intervals is recorded. Sequence of photon counts or, alternatively, of FRET efficiencies can be studied using likelihood-based methods. For a kinetic model of the conformational dynamics and state-dependent Poisson photon statistics, the formalism to calculate the exact likelihood that this model describes such sequences of photons or FRET efficiencies is developed. Explicit analytic expressions for the likelihood function for a two-state kinetic model are provided. The important special case when conformational dynamics are so slow that at most a single transition occurs in a time bin is considered. By making a series of approximations, we eventually recover the likelihood function used in hidden Markov models. In this way, not only is insight gained into the range of validity of this procedure, but also an improved likelihood function can be obtained.
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Affiliation(s)
- Irina V Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892, USA
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