Seeing the forest for the trees: fluorescence studies of single enzymes in the context of ensemble experiments

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.

Dynamic disorder in simple enzymatic reactions induces stochastic amplification of substrate

A growing amount of evidence over the last two decades points to the fact that many enzymes exhibit fluctuations in their catalytic activity, which are associated with conformational changes on a broad range of timescales. The experimental study of this phenomenon, termed dynamic disorder, has become possible thanks to advances in single-molecule enzymology measurement techniques, through which the catalytic activity of individual enzyme molecules can be tracked in time. The biological role and importance of these fluctuations in a system with a small number of enzymes, such as a living cell, have only recently started being explored. In this work, we examine a simple stochastic reaction system consisting of an inflowing substrate and an enzyme with a randomly fluctuating catalytic reaction rate that converts the substrate into an outflowing product. To describe analytically the effect of rate fluctuations on the average substrate abundance at steady state, we derive an explicit formula that connects the relative speed of enzymatic fluctuations with the mean substrate level. Under fairly general modelling assumptions, we demonstrate that the relative speed of rate fluctuations can have a dramatic effect on the mean substrate, and lead to large positive deviations from predictions based on the assumption of deterministic enzyme activity. Our results also establish an interesting connection between the amplification effect and the mixing properties of the Markov process describing the enzymatic activity fluctuations, which can be used to easily predict the fluctuation speed above which such deviations become negligible. As the techniques of single-molecule enzymology continuously evolve, it may soon be possible to study the stochastic phenomena due to enzymatic activity fluctuations within living cells. Our work can be used to formulate experimentally testable hypotheses regarding the nature and magnitude of these fluctuations, as well as their phenotypic consequences.

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.

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.

The recent developments and applications of the traceless-Staudinger reaction in chemical biology study

Bioorthogonal reactions are one of the most important topics in chemical biology. Traceless-Staudinger reaction/ligation has been investigated and widely applied in life science. Herein, the current developments, mechanism studies, and biological applications are summarized.

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.

12-Channel Peltier array temperature control unit for single molecule enzymology studies using capillary electrophoresis

Capillary electrophoresis has been used to demonstrate that individual molecules of a given enzyme support different catalytic rates. In order to determine how rate varies with temperature, and determine activation energies for individual β-galactosidase molecules, a 12-channel Peltier array temperature control device was constructed where the temperature of each cell was separately controlled. This array was used to control the temperature of the central 30 cm of a 50 cm long capillary, producing a temperature gradient along its length. Continuous flow single β-galactosidase molecule assays were performed allowing measurement of the catalytic rates at different temperatures. Arrhenius plots were produced and the distribution of activation energies for individual β-galactosidase molecules was found to be 56 ± 10 kJ/mol with a range of 34-72 kJ/mol.

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

The dynamics of a protein along a well-defined coordinate can be formally projected onto the form of an overdamped Lagevin equation. Here, we present a comprehensive statistical-learning framework for simultaneously quantifying the deterministic force (the potential of mean force, PMF) and the stochastic force (characterized by the diffusion coefficient, D) from single-molecule Förster-type resonance energy transfer (smFRET) experiments. The likelihood functional of the Langevin parameters, PMF and D, is expressed by a path integral of the latent smFRET distance that follows Langevin dynamics and realized by the donor and the acceptor photon emissions. The solution is made possible by an eigen decomposition of the time-symmetrized form of the corresponding Fokker-Planck equation coupled with photon statistics. To extract the Langevin parameters from photon arrival time data, we advance the expectation-maximization algorithm in statistical learning, originally developed for and mostly used in discrete-state systems, to a general form in the continuous space that allows for a variational calculus on the continuous PMF function. We also introduce the regularization of the solution space in this Bayesian inference based on a maximum trajectory-entropy principle. We use a highly nontrivial example with realistically simulated smFRET data to illustrate the application of this new method.

Numerical construction of estimators for single-molecule fluorescence measurements

A novel scheme to estimate the values of the underlying physical quantity and those of any functions of the quantity from measured observable(s) contaminated with stochastic noise is presented for any arbitrary probability distribution. The constructed estimators can either maximize the unbiasedness (i.e., minimize the amount of the deviation of the expectation value from the true value buried in the measurement) or minimize the risk (the average deviation from the true value) depending on the relative priority of unbiasedness and risk in the data analysis. The performance of the constructed estimators is demonstrated with computer simulations of Förster-type resonance energy transfer (FRET) measurements and also with FRET experimental data of the agonist-binding domain of the GluA2 subunit of AMPA receptors with agonists chloro- and iodo-willardiines and with adenylate kinase both in the apo form and with substrates AMP-PNP and AMP. It is shown that the estimators constructed by the present method can quantify faithfully not only the physical quantity to be monitored but also the functions of that quantity for a wide range of values.

Importance of protein dynamics during enzymatic C-H bond cleavage catalysis

Quantum tunneling and protein dynamics have emerged as important components of enzyme function. This review focuses on soybean lipoxygenase-1, to illustrate how the properties of enzymatic C-H bond activation link protein motions to the fundamental bond making-breaking processes.

Hydrogen tunneling links protein dynamics to enzyme catalysis

The relationship between protein dynamics and function is a subject of considerable contemporary interest. Although protein motions are frequently observed during ligand binding and release steps, the contribution of protein motions to the catalysis of bond making/breaking processes is more difficult to probe and verify. Here, we show how the quantum mechanical hydrogen tunneling associated with enzymatic C-H bond cleavage provides a unique window into the necessity of protein dynamics for achieving optimal catalysis. Experimental findings support a hierarchy of thermodynamically equilibrated motions that control the H-donor and -acceptor distance and active-site electrostatics, creating an ensemble of conformations suitable for H-tunneling. A possible extension of this view to methyl transfer and other catalyzed reactions is also presented. The impact of understanding these dynamics on the conceptual framework for enzyme activity, inhibitor/drug design, and biomimetic catalyst design is likely to be substantial.

Analyzing Förster resonance energy transfer with fluctuation algorithms

Fluorescence correlation spectroscopy (FCS) in combination with Förster resonance energy transfer (FRET) has been developed to a powerful statistical tool, which allows for the analysis of FRET fluctuations in the huge time of nanoseconds to seconds. FRET-FCS utilizes the strong distance dependence of the FRET efficiency on the donor (D)-acceptor (A) distance so that it developed to a perfect method for studying structural fluctuation in biomolecules involved in conformational flexibility, structural dynamics, complex formation, folding, and catalysis. Structural fluctuations thereby result in anticorrelated donor and acceptor signals, which are analyzed by FRET-FCS in order to characterize underlying structural dynamics. Simulated and experimental examples are discussed. First, we review experimental implementations of FRET-FCS and present theory for a two-state interconverting system. Additionally, we consider a very common case of FRET dynamics in the presence of donor-only labeled species. We demonstrate that the mean relaxation time for the structural dynamics can be easily obtained in most of cases, whereas extracting meaningful information from correlation amplitudes can be challenging. We present a strategy to avoid a fit with an underdetermined model function by restraining the D and A brightnesses of the at least one involved state, so that both FRET efficiencies and both rate constants (i.e., the equilibrium constant) can be determined. For samples containing several fluorescent species, the use of pulsed polarized excitation with multiparameter fluorescence detection allows for filtered FCS (fFCS), where species-specific correlation functions can be obtained, which can be directly interpreted. The species selection is achieved by filtering using fluorescence decays of individual species. Analytical functions for species auto- and cross-correlation functions are given. Moreover, fFCS is less affected by photophysical artifacts and often offers higher contrast, which effectively increases its time resolution and significantly enhances its capability to resolve multistate kinetics. fFCS can also differentiate between species even when their brightnesses are the same and thus opens up new possibilities to characterize complex dynamics. Alternative fluctuation algorithms to study FRET dynamics are also briefly reviewed.

Interpreting single turnover catalysis measurements with constrained mean dwell times

Observation of a chemical transformation at the single-molecule level yields a detailed view of kinetic pathways contributing to the averaged results obtained in a bulk measurement. Studies of a fluorogenic reaction catalyzed by gold nanoparticles have revealed heterogeneous reaction dynamics for these catalysts. Measurements on single nanoparticles yield binary trajectories with stochastic transitions between a dark state in which no product molecules are adsorbed and a fluorescent state in which one product molecule is present. The mean dwell time in either state gives information corresponding to a bulk measurement. Quantifying fluctuations from mean kinetics requires identifying properties of the fluorescence trajectory that are selective in emphasizing certain dynamic processes according to their time scales. We propose the use of constrained mean dwell times, defined as the mean dwell time in a state with the constraint that the immediately preceding dwell time in the other state is, for example, less than a variable time. Calculations of constrained mean dwell times for a kinetic model with dynamic disorder demonstrate that these quantities reveal correlations among dynamic fluctuations at different active sites on a multisite catalyst. Constrained mean dwell times are determined from measurements of single nanoparticle catalysis. The results indicate that dynamical fluctuations at different active sites are correlated, and that especially rapid reaction events produce particularly slowly desorbing product molecules.