Toolbox for analyzing finite two-state trajectories

Shedding light on protein folding, structural and functional dynamics by single molecule studies

The advent of advanced single molecule measurements unveiled a great wealth of dynamic information revolutionizing our understanding of protein dynamics and behavior in ways unattainable by conventional bulk assays. Equipped with the ability to record distribution of behaviors rather than the mean property of a population, single molecule measurements offer observation and quantification of the abundance, lifetime and function of multiple protein states. They also permit the direct observation of the transient and rarely populated intermediates in the energy landscape that are typically averaged out in non-synchronized ensemble measurements. Single molecule studies have thus provided novel insights about how the dynamic sampling of the free energy landscape dictates all aspects of protein behavior; from its folding to function. Here we will survey some of the state of the art contributions in deciphering mechanisms that underlie protein folding, structural and functional dynamics by single molecule fluorescence microscopy techniques. We will discuss a few selected examples highlighting the power of the emerging techniques and finally discuss the future improvements and directions.

SOLVING ION CHANNEL KINETICS WITH THE QuB SOFTWARE

The ability to record the currents from single ion channels led to the need to extract the underlying kinetic model from such data. This inverse hidden Markov problem is difficult but led to the creation of a software suite called QuB utilizing likelihood optimization. This review presents the software. The software is open source and, in addition to solving kinetic models, has many generic database operations including report generation with publishable graphics, function fitting and scripting for new and repeated processing and AD/DA I/O. The core algorithms allow for constraints such as fixed rates or maintaining detailed balance in the model. All rate constants can be driven by a stimulus and the system can analyze nonstationary data. QuB also can analyze the kinetics of multichannel data where individual events cannot be discriminated, but the fitting algorithms utilize the signal variance as well as the mean to fit models. QuB can be applied to any data appropriately modeled with Markov kinetics and has been utilized to solve ion channels but also the movement of motor proteins, the sleep cycles in mice, and physics processes.

[Formula: see text]Special Issue Comment: This is a review about the software QuB that can extract a model from the trajectory. It is connected with the review about treatments when solving single molecules,60 and the reviews about enzymes.61,62

INSIGHTS IN ENZYME FUNCTIONAL DYNAMICS AND ACTIVITY REGULATION BY SINGLE MOLECULE STUDIES

The advent of advanced single molecule measurements heralded the arrival of a wealth of dynamic information revolutionizing our understanding of protein dynamics and behavior in ways not deducible by conventional bulk assays. They offered the direct observation and quantification of the abundance and life time of multiple states and transient intermediates in the energy landscape that are typically averaged out in non-synchronized ensemble measurements, thus providing unprecedented insights into complex biological processes. Here we survey the current state of the art in single-molecule fluorescence microscopy methodology for studying the mechanism of enzymatic activity and the insights on protein functional dynamics. We will initially discuss the strategies employed to date, their limitations and possible ways to overcome them, and finally how single enzyme kinetics can advance our understanding on mechanisms underlying function and regulation of proteins.

[Formula: see text]Special Issue Comment: This review focuses on functional dynamics of individual enzymes and is related to the review on ion channels by Lu,44 the reviews on mathematical treatment of Flomenbom45 and Sach et al.,46 and review on FRET by Ruedas-Rama et al.41

CORE-SIZE DEPENDENT PHOTOLUMINESCENCE BLINKING OF ISOLATED QUANTUM DOT-FULLERENE HYBRIDS

We demonstrate the possibility to bias the photoluminescence blinking of isolated colloidal quantum dots coupled to fullerenes by varying the quantum dot core size. Changing the core size affects the energy offset between the donor and acceptor and directly affects the driving force for electron transfer between the two moieties. Single particle photoluminescence measurements reveal dramatic reduction in the on-time associated with the photoluminescence blinking in quantum dot-fullerene hybrids when the quantum dot core size decreases, a manifestation associated with enhanced electron transfer.

[Formula: see text]Special Issue Comment: This project is about the blink properties of QDs in the presence of electron acceptor moieties and it connects with two articles from this Special Issue involving treatments when solving single molecules.1,2

MATHEMATICAL TREATMENTS THAT SOLVE SINGLE MOLECULES

In this article, we talk about the ways that scientists can solve single molecule trajectories. Solving single molecules, that is, finding the model from the data, is complicated at least as much as measuring single molecules. We must filter the noise and take care of every step in the analysis when constructing the most accurate model from the data. Here, we present valuable solutions. Ways that solve clean discrete data are first presented. We review here our reduced dimensions forms (RDFs): unique models that are canonical forms of discrete data, and the statistical and numerical toolbox that builds a RDF from finite, clean, two-state data. We then review our most recent filter that "tackles" the noise when measuring two state noisy photon trajectories. The filter is a numerical algorithm with various special statistical treatments that is based on a general likelihood function that we have developed recently. We show the strengths of the filter (also over other approaches) and talk about its various new variants. This filter (with minor adjustments) can solve the noise in any discrete state trajectories, yet, extensions are needed in "tackling" the noise from other data, e.g. continuous data. Only the combined procedures enable creating the most accurate model from noisy discrete trajectories from single molecules. These concepts and methods (with adjustments) are valuable also when solving continuous trajectories and fluorescence resonance energy transfer trajectories. We also present a set of simple methods that can help any scientist with treating the trajectory perhaps encouraging applying the involved methods. The involved methods will appear in software that we are developing now, helping therefore the experimentalists utilizing these methods on real data. Comparisons with other known methods in this field are made.

[Formula: see text]Special Issue Comment: This article about mathematical treatments when solving single molecules is related to the reviews in this Special Issue about measuring enzymes67 and about FRET experiments2 and about the software QUB.6

SOLVING SINGLE BIOMOLECULES BY ADVANCED FRET-BASED SINGLE-MOLECULE FLUORESCENCE TECHNIQUES

The use of Förster resonance energy transfer (FRET) has undergone a renaissance in the last two decades, especially in the study of structure of biomolecules, biomolecular interactions, and dynamics. Thanks to powerful advances in single-molecule fluorescence (SMF) techniques, seeing molecules at work is a reality, which has helped to build up the mindset of molecular machines. In the last few years, many technical developments have broadened the applications of SMF-FRET, expanding the amount of information that can be recovered from individual molecules. Here, we focus on the non-standard SMF-FRET techniques, such as two-color coincidence detection (TCCD), alternating laser excitation (ALEX), multiparameter fluorescence detection (MFD); the addition of fluorescence lifetime as an orthogonal dimension in single-molecule experiments; or the development of novel and improved methods of analysis constituting to a set of advanced methodologies that may become routine tools in a close future.

[Formula: see text]Special Issue Comment: This review about advanced single-molecule FRET techniques is specially related to the review by Jørgensen and Hatzakis,6 who detail experimetal strategies to solve the activity of single enzymes. The advanced techniques described in our paper may serve as interesting alternatives when applied to enzyme studies. Our manuscript is also related to the reviews in this Special Issue that deal with model solving.22,130

Dynamic disorder in single-enzyme experiments: facts and artifacts

Using a single-molecule fluorescence approach, the time series of catalytic events of an enzymatic reaction can be monitored, yielding a sequence of fluorescent "on"- and "off"-states. An accurate on/off-assignment is complicated by the intrinsic and extrinsic noise in every single-molecule fluorescence experiment. Using simulated data, the performance of the most widely employed binning and thresholding approach was systematically compared to change point analysis. It is shown that the underlying on- and off-histograms as well as the off-autocorrelation are not necessarily extracted from the "signal'' buried in noise. The shapes of the on- and off-histograms are affected by artifacts introduced by the analysis procedure and depend on the signal-to-noise ratio and the overall fluorescence intensity. For experimental data where the background intensity is not constant over time we consider change point analysis to be more accurate. When using change point analysis for data of the enzyme α-chymotrypsin, no characteristics of dynamic disorder was found. In light of these results, dynamic disorder might not be a general sign of enzymatic reactions.

Statistics of reversible bond dynamics observed in force-clamp spectroscopy

We present a detailed analysis of two-state trajectories obtained from force-clamp spectroscopy (FCS) of reversibly bonded systems. FCS offers the unique possibility to vary the equilibrium constant in two-state kinetics, for instance, the unfolding and refolding of biomolecules, over many orders of magnitude due to the force dependence of the respective rates. We discuss two different kinds of counting statistics, the event counting usually employed in the statistical analysis of two-state kinetics and additionally the so-called cycle counting. While in the former case all transitions are counted, cycle counting means that we focus on one type of transitions. This might be advantageous in particular if the equilibrium constant is much larger or much smaller than unity because in these situations the temporal resolution of the experimental setup might not allow to capture all transitions of an event-counting analysis. We discuss how an analysis of FCS data for complex systems exhibiting dynamic disorder might be performed yielding information about the detailed force dependence of the transition rates and about the time scale of the dynamic disorder. In addition, the question as to which extent the kinetic scheme can be viewed as a Markovian two-state model is discussed.

Communications: Can one identify nonequilibrium in a three-state system by analyzing two-state trajectories?

For a three-state Markov system in a stationary state, we discuss whether, on the basis of data obtained from effective two-state (or on-off) trajectories, it is possible to discriminate between an equilibrium state and a nonequilibrium steady state. By calculating the full phase diagram we identify a large region where such data will be consistent only with nonequilibrium conditions. This regime is considerably larger than the region with oscillatory relaxation, which has previously been identified as a sufficient criterion for nonequilibrium.

Anomalous diffusion as a stochastic component in the dynamics of complex processes

We propose an interpolation expression using the difference moment (Kolmogorov transient structural function) of the second order as the average characteristic of displacements for identifying the anomalous diffusion in complex processes when the stochastic (the term "stochastic" in this paper refers to random variability in the signals of complex systems characterized by nonlinear interactions, dissipation, and inertia) dynamics of the system under study reaches a steady state (large time intervals). Our procedure based on this expression for identifying anomalous diffusion and calculating its parameters in complex processes is applied to the analysis of the dynamics of blinking fluorescence of quantum dots, x-ray emission from accreting objects, fluid velocity in Rayleigh-Bénard convection, and geoelectrical signal for a seismic area. For all four examples, the proposed interpolation is able to adequately describe the stochastic part of the experimental difference moment, which implies that anomalous diffusion manifests itself in these complex processes. The results of this study make it possible to broaden the range of complex natural processes in which anomalous diffusion can be identified.

Analysis of single-molecule fluorescence spectroscopic data with a Markov-modulated Poisson process

We present a photon-by-photon analysis framework for the evaluation of data from single-molecule fluorescence spectroscopy (SMFS) experiments using a Markov-modulated Poisson process (MMPP). A MMPP combines a discrete (and hidden) Markov process with an additional Poisson process reflecting the observation of individual photons. The algorithmic framework is used to automatically analyze the dynamics of the complex formation and dissociation of Cu2+ ions with the bidentate ligand 2,2'-bipyridine-4,4'dicarboxylic acid in aqueous media. The process of association and dissociation of Cu2+ ions is monitored with SMFS. The dcbpy-DNA conjugate can exist in two or more distinct states which influence the photon emission rates. The advantage of a photon-by-photon analysis is that no information is lost in preprocessing steps. Different model complexities are investigated in order to best describe the recorded data and to determine transition rates on a photon-by-photon basis. The main strength of the method is that it allows to detect intermittent phenomena which are masked by binning and that are difficult to find using correlation techniques when they are short-lived.