Photon-by-Photon Hidden Markov Model Analysis for Microsecond Single-Molecule FRET Kinetics

Abstract

The function of biological macromolecules involves large-scale conformational dynamics spanning multiple time scales, from microseconds to seconds. Such conformational motions, which may involve whole domains or subunits of a protein, play a key role in allosteric regulation. There is an urgent need for experimental methods to probe the fastest of these motions. Single-molecule fluorescence experiments can in principle be used for observing such dynamics, but there is a lack of analysis methods that can extract the maximum amount of information from the data, down to the microsecond time scale. To address this issue, we introduce HMM, a maximum likelihood estimation algorithm for photon-by-photon analysis of single-molecule fluorescence resonance energy transfer (FRET) experiments. HMM is based on analytical estimators for model parameters, derived using the Baum-Welch algorithm. An efficient and effective method for the calculation of these estimators is introduced. HMM is shown to accurately retrieve the reaction times from ∼1 s to ∼10 μs and even faster when applied to simulations of freely diffusing molecules. We further apply this algorithm to single-molecule FRET data collected from Holliday junction molecules and show that at low magnesium concentrations their kinetics are as fast as ∼10 s. The new algorithm is particularly suitable for experiments on freely diffusing individual molecules and is readily incorporated into existing analysis packages. It paves the way for the broad application of single-molecule fluorescence to study ultrafast functional dynamics of biomolecules.

Toward dynamic structural biology: Two decades of single-molecule Förster resonance energy transfer

“…Recently, Pirchi et al reported an analytical method to extract the values of these parameters by performing a photon-by-photon hidden Markov modeling analysis of smFRET experiments (H2MM) (93), as previously suggested by Gopich and Szabo (94). They were able to extract rate constants (ranging from ~10 μs to ~1 s) and the mean FRET efficiencies of the corresponding states.…”

Pitching single-focus confocal data analysis one photon at a time with Bayesian nonparametrics

“…Indeed, existing methods make assumptions that would render them inapplicable to diffusion in an inhomogenesouly illuminated volume. For example, they assume uniform illumination [38,76], bin the data and thereby reduce temporal resolution to ex-D=4. 26 ploit existing mathematical frameworks such as the Hidden Markov model [2,16,39,70,97], or focus on immobile biomolecules [23,24,37,39].…”

“…(7). The parameters not varied are held fixed at the following values: diffusion coefficient of 1 µm 2 /s which is typical of slower in vivo conditions [5,63,81,96], particle photon emission rates of 4 × 10 4 photons/s [57,76], and 4 as the number of labeled biomolecules contributing to the trace. We chose a small number of biomolecules (as opposed to a larger number of biomolecules) as this scenario presents the greatest analysis challenge (as very few photons, and thus thus little information, is gathered).…”

“…(5E, F)), diffusing at 1 µm 2 /s for a total time 30 ms. The particle photon emission rates per label of labeled biomolecules and background are taken to be 4 × 10 4 photons/s and 10 3 photons/s respectively, which are typical of confocal microscope [76]. Our method is general and can readily treat arbitrary confocal volume geometries such as a Lorentzians [10] or Airy patterns [105] though, for simplicity, in generating synthetic data we use a 3D Gaussian volume of size ω xy =0.3 µm and ω z =1.5 µm [54]).…”

“…Previously proposed methods to analyze single photon measurements [2,38,39,43,70,76,99] make assumptions that render them inappropriate for imaging biomolecules (or particles, more broadly speaking) moving through inhomogeneously illuminated volumes [38]. For example, they may assume that the biomolecules are fixed in space.…”