Expectation-maximization of the potential of mean force and diffusion coefficient in Langevin dynamics from single molecule FRET data photon by photon

Efficient, nonparametric removal of noise and recovery of probability distributions from time series using nonlinear-correlation functions: Photon and photon-counting noise

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.

Trajectory Statistical Learning of the Potential Mean of Force and Diffusion Coefficient from Molecular Dynamics Simulations

Central to studying the conformational changes of a complex protein is understanding the dynamics and energetics involved. Phenomenologically, structural dynamics can be formulated using an overdamped Langevin model along an observable, e.g., the distance between two residues in the protein. The Langevin model is specified by the deterministic force (the potential of mean force, PMF) and stochastic force (characterized by the diffusion coefficient, D). It is therefore of great interest to be able to extract both PMF and D from an observable time series but under the same computational framework. Here, we approach this challenge in molecular dynamics (MD) simulations by treating it as a missing-data Bayesian estimation problem. An important distinction in our methodology is that the entire MD trajectory, as opposed to the individual data elements, is used as the statistical variable in Bayesian imputation. This idea is implemented through an eigen-decomposition procedure for a time-symmetrized Fokker-Planck equation, followed by maximizing the likelihood for parameter estimation. The mathematical expressions for the functional derivatives used in learning PMF and D also provide new physical insights for the manner by which the information on both the deterministic and stochastic forces is encoded in the dynamics data. An all-atom MD simulation of a nontrivial biomolecule case is used to illustrate the application of this approach. We show that, interestingly, the results of trajectory statistical learning can motivate new order parameters for an improved description of the kinetic bottlenecks in conformational changes. Complementing purely data-driven or black-box methods, this work underscores the advantages of physics-based machine learning in gaining chemical insights from quantitative parameter estimation.

Theory and Analysis of Single-Molecule FRET Experiments

Inter-dye distances and conformational dynamics can be studied using single-molecule FRET measurements. We consider two approaches to analyze sequences of photons with recorded photon colors and arrival times. The first approach is based on FRET efficiency histograms obtained from binned photon sequences. The experimental histograms are compared with the theoretical histograms obtained using the joint distribution of acceptor and donor photons or the Gaussian approximation. In the second approach, a photon sequence is analyzed without binning. The parameters of a model describing conformational dynamics are found by maximizing the appropriate likelihood function. The first approach is simpler, while the second one is more accurate, especially when the population of species is small and transition rates are fast. The likelihood-based analysis as well as the recoloring method has the advantage that diffusion of molecules through the laser focus can be rigorously handled.

Learning non-stationary Langevin dynamics from stochastic observations of latent trajectories

Many complex systems operating far from the equilibrium exhibit stochastic dynamics that can be described by a Langevin equation. Inferring Langevin equations from data can reveal how transient dynamics of such systems give rise to their function. However, dynamics are often inaccessible directly and can be only gleaned through a stochastic observation process, which makes the inference challenging. Here we present a non-parametric framework for inferring the Langevin equation, which explicitly models the stochastic observation process and non-stationary latent dynamics. The framework accounts for the non-equilibrium initial and final states of the observed system and for the possibility that the system's dynamics define the duration of observations. Omitting any of these non-stationary components results in incorrect inference, in which erroneous features arise in the dynamics due to non-stationary data distribution. We illustrate the framework using models of neural dynamics underlying decision making in the brain.

Prior-Apprised Unsupervised Learning of Subpixel Curvilinear Features in Low Signal/Noise Images

Many biophysical problems involve molecular and nanoscale targets moving next to a curvilinear track, e.g., a cytosolic cargo transported by motor proteins moving along a microtubule. For this type of problem, fluorescence imaging is usually the primary tool of choice. There is, however, an ∼20-fold mismatch between target-localization precision and track-imaging resolution such that questions requiring high-fidelity definition of the target's track remain inaccessible. On the other hand, if the contextual image of the tracks can be refined to a level comparable to that of the target, many intuitive yet mechanistically important issues can begin to be addressed. This work demonstrates that it is possible to statistically infer, to subpixel precision, curvilinear features in a low signal/noise image. This is achieved by a framework that consists of three stages: the Hessian-based feature enhancement, the subimage feature sampling and registration, and the statistical learning of the underlying curvilinear structure using a new, to our knowledge, method developed here for inferring the principal curves. In each stage, the descriptive prior information that the features come from curvilinear elements is explicitly taken into account. It is fully automated without user supervision, which is distinctly different from approaches that require user seeding or well-defined training data sets. Computer simulations of realistic images are used to investigate the performance of the framework and its implementation. The characterization results suggest that curvilinear features are refined to the same order of precision as that of the target and that the bootstrap confidence intervals from the analysis allow an estimate for the statistical bounds of the simulated "true" curve. Also shown are analyses of experimental images from three different microscopy modalities: two-photon laser-scanning microscopy, epifluorescence microscopy, and total internal reflection fluorescence microscopy. The practical application of this prior-apprised unsupervised learning framework as well as its potential outlook are discussed.

Statistical Learning of Discrete States in Time Series

Time series obtained from time-dependent experiments contain rich information on kinetics and dynamics of the system under investigation. This work describes an unsupervised learning framework, along with the derivation of the necessary analytical expressions, for the analysis of Gaussian-distributed time series that exhibit discrete states. After the time series has been partitioned into segments in a model-free manner using the previously developed change-point (CP) method, this protocol starts with an agglomerative hierarchical clustering algorithm to classify the detected segments into possible states. The initial state clustering is further refined using an expectation-maximization (EM) procedure, and the number of states is determined by a Bayesian information criterion (BIC). Also introduced here is an achievement scalarization function, usually seen in artificial intelligence literature, for quantitatively assessing the performance of state determination. The statistical learning framework, which is comprised of three stages, detection of signal change, clustering, and number-of-state determination, was thoroughly characterized using simulated trajectories with random intensity segments that have no underlying kinetics, and its performance was critically evaluated. The application to experimental data is also demonstrated. The results suggested that this general framework, the implementation of which is based on firm theoretical foundations and does not require the imposition of any kinetics model, is powerful in determining the number of states, the parameters contained in each state, as well as the associated statistical significance.

Compound Molecular Logic in Accessing the Active Site of Mycobacterium tuberculosis Protein Tyrosine Phosphatase B

Protein tyrosine phosphatase B (PtpB) from Mycobacterium tuberculosis (Mtb) extends the bacteria's survival in hosts and hence is a potential target for Mtb-specific drugs. To study how Mtb-specific sequence insertions in PtpB may regulate access to its active site through large-amplitude conformational changes, we performed free-energy calculations using an all-atom explicit solvent model. Corroborated by biochemical assays, the results show that PtpB's active site is controlled via an "either/or" compound conformational gating mechanism, an unexpected discovery that Mtb has evolved to bestow a single enzyme with such intricate logical operations. In addition to providing unprecedented insights for its active-site surroundings, the findings also suggest new ways of inactivating PtpB.

Linking time-series of single-molecule experiments with molecular dynamics simulations by machine learning

Single-molecule experiments and molecular dynamics (MD) simulations are indispensable tools for investigating protein conformational dynamics. The former provide time-series data, such as donor-acceptor distances, whereas the latter give atomistic information, although this information is often biased by model parameters. Here, we devise a machine-learning method to combine the complementary information from the two approaches and construct a consistent model of conformational dynamics. It is applied to the folding dynamics of the formin-binding protein WW domain. MD simulations over 400 μs led to an initial Markov state model (MSM), which was then "refined" using single-molecule Förster resonance energy transfer (FRET) data through hidden Markov modeling. The refined or data-assimilated MSM reproduces the FRET data and features hairpin one in the transition-state ensemble, consistent with mutation experiments. The folding pathway in the data-assimilated MSM suggests interplay between hydrophobic contacts and turn formation. Our method provides a general framework for investigating conformational transitions in other proteins.

Plasmon-Coupled Resonance Energy Transfer

In this study, we overview resonance energy transfer between molecules in the presence of plasmonic structures and derive an explicit Förster-type expression for the rate of plasmon-coupled resonance energy transfer (PC-RET). The proposed theory is general for energy transfer in the presence of materials with any space-dependent, frequency-dependent, or complex dielectric functions. Furthermore, the theory allows us to develop the concept of a generalized spectral overlap (GSO) J̃ (the integral of the molecular absorption coefficient, normalized emission spectrum, and the plasmon coupling factor) for understanding the wavelength dependence of PC-RET and to estimate the rate of PC-RET W_ET. Indeed, W_ET = (8.785 × 10^-25 mol) ϕ_Dτ_D^-1J̃, where ϕ_D is donor fluorescence quantum yield and τ_D is the emission lifetime. Simulations of the GSO for PC-RET show that the most important spectral region for PC-RET is not necessarily near the maximum overlap of donor emission and acceptor absorption. Instead a significant plasmonic contribution can involve a different spectral region from the extinction maximum of the plasmonic structure. This study opens a promising direction for exploring exciton transport in plasmonic nanostructures, with possible applications in spectroscopy, photonics, biosensing, and energy devices.

Extraction of Protein Conformational Modes from Distance Distributions Using Structurally Imputed Bayesian Data Augmentation

Protein conformational changes are known to play important roles in assorted biochemical and biological processes. Driven by thermal motions of surrounding solvent molecules, such a structural remodeling often occurs stochastically. Yet, regardless of how random the conformational reconfiguration may appear, it could in principle be described by a linear combination of a set of orthogonal modes which, in turn, are contained in the intramolecular distance distributions. The central challenge is how to obtain the distribution. This contribution proposes a Bayesian data-augmentation scheme to extract the predominant modes from only few distance distributions, be they from computational sampling or directly from experiments such as single-molecule Förster-type resonance energy transfer (smFRET). The inference of the complete protein structure from insufficient data was recognized as isomorphic to the missing-data problem in Bayesian statistical learning. Using smFRET data as an example, the missing coordinates were deduced, given protein structural constraints and multiple but limited number of smFRET distances; the Boltzmann weighing of each inferred protein structure was then evaluated using computational modeling to numerically construct the posterior density for the global protein conformation. The conformational modes were then determined from the iteratively converged overall conformational distribution using principal component analysis. Two examples were presented to illustrate these basic ideas as well as their practical implementation. The scheme described herein was based on the theory behind the powerful Tanner-Wang algorithm that guarantees convergence to the true posterior density. However, instead of assuming a mathematical model to calculate the likelihood as in conventional statistical inference, here the protein structure was treated as a statistical parameter and was imputed from the numerical likelihood function based on structural information, a probability model-free method. The framework put forth here is anticipated to be generally applicable, offering a new way to articulate protein conformational changes in a quantifiable manner.

Nonlinear reconstruction of single-molecule free-energy surfaces from univariate time series

The stable conformations and dynamical fluctuations of polymers and macromolecules are governed by the underlying single-molecule free energy surface. By integrating ideas from dynamical systems theory with nonlinear manifold learning, we have recovered single-molecule free energy surfaces from univariate time series in a single coarse-grained system observable. Using Takens' Delay Embedding Theorem, we expand the univariate time series into a high dimensional space in which the dynamics are equivalent to those of the molecular motions in real space. We then apply the diffusion map nonlinear manifold learning algorithm to extract a low-dimensional representation of the free energy surface that is diffeomorphic to that computed from a complete knowledge of all system degrees of freedom. We validate our approach in molecular dynamics simulations of a C(24)H(50) n-alkane chain to demonstrate that the two-dimensional free energy surface extracted from the atomistic simulation trajectory is - subject to spatial and temporal symmetries - geometrically and topologically equivalent to that recovered from a knowledge of only the head-to-tail distance of the chain. Our approach lays the foundations to extract empirical single-molecule free energy surfaces directly from experimental measurements.

Analysis of Fluorescence Lifetime and Energy Transfer Efficiency in Single-Molecule Photon Trajectories of Fast-Folding Proteins

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.

Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET

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.

A Method for Extracting the Free Energy Surface and Conformational Dynamics of Fast-Folding Proteins from Single Molecule Photon Trajectories

Single molecule fluorescence spectroscopy holds the promise of providing direct measurements of protein folding free energy landscapes and conformational motions. However, fulfilling this promise has been prevented by technical limitations, most notably, the difficulty in analyzing the small packets of photons per millisecond that are typically recorded from individual biomolecules. Such limitation impairs the ability to accurately determine conformational distributions and resolve sub-millisecond processes. Here we develop an analytical procedure for extracting the conformational distribution and dynamics of fast-folding proteins directly from time-stamped photon arrival trajectories produced by single molecule FRET experiments. Our procedure combines the maximum likelihood analysis originally developed by Gopich and Szabo with a statistical mechanical model that describes protein folding as diffusion on a one-dimensional free energy surface. Using stochastic kinetic simulations, we thoroughly tested the performance of the method in identifying diverse fast-folding scenarios, ranging from two-state to one-state downhill folding, as a function of relevant experimental variables such as photon count rate, amount of input data, and background noise. The tests demonstrate that the analysis can accurately retrieve the original one-dimensional free energy surface and microsecond folding dynamics in spite of the sub-megahertz photon count rates and significant background noise levels of current single molecule fluorescence experiments. Therefore, our approach provides a powerful tool for the quantitative analysis of single molecule FRET experiments of fast protein folding that is also potentially extensible to the analysis of any other biomolecular process governed by sub-millisecond conformational dynamics.

Fast single-molecule FRET spectroscopy: theory and experiment

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.

Fast Step Transition and State Identification (STaSI) for Discrete Single-Molecule Data Analysis

We introduce a step transition and state identification (STaSI) method for piecewise constant single-molecule data with a newly derived minimum description length equation as the objective function. We detect the step transitions using the Student's t test and group the segments into states by hierarchical clustering. The optimum number of states is determined based on the minimum description length equation. This method provides comprehensive, objective analysis of multiple traces requiring few user inputs about the underlying physical models and is faster and more precise in determining the number of states than established and cutting-edge methods for single-molecule data analysis. Perhaps most importantly, the method does not require either time-tagged photon counting or photon counting in general and thus can be applied to a broad range of experimental setups and analytes.