What can one learn from two-state single-molecule trajectories?

Information-theoretical limit on the estimates of dissipation by molecular machines using single-molecule fluorescence resonance energy transfer experiments

Single-molecule fluorescence resonance energy transfer (FRET) experiments are commonly used to study the dynamics of molecular machines. While in vivo molecular processes often break time-reversal symmetry, the temporal directionality of cyclically operating molecular machines is often not evident from single-molecule FRET trajectories, especially in the most common two-color FRET studies. Solving a more quantitative problem of estimating the energy dissipation/entropy production by a molecular machine from single-molecule data is even more challenging. Here, we present a critical assessment of several practical methods of doing so, including Markov-model-based methods and a model-free approach based on an information-theoretical measure of entropy production that quantifies how (statistically) dissimilar observed photon sequences are from their time reverses. The Markov model approach is computationally feasible and may outperform model free approaches, but its performance strongly depends on how well the assumed model approximates the true microscopic dynamics. Markov models are also not guaranteed to give a lower bound on dissipation. Meanwhile, model-free, information-theoretical methods systematically underestimate entropy production at low photoemission rates, and long memory effects in the photon sequences make these methods demanding computationally. There is no clear winner among the approaches studied here, and all methods deserve to belong to a comprehensive data analysis toolkit.

Identifying the generator matrix of a stationary Markov chain using partially observable data

Given that most states in real-world systems are inaccessible, it is critical to study the inverse problem of an irreversibly stationary Markov chain regarding how a generator matrix can be identified using minimal observations. The hitting-time distribution of an irreversibly stationary Markov chain is first generalized from a reversible case. The hitting-time distribution is then decoded via the taboo rate, and the results show remarkably that under mild conditions, the generator matrix of a reversible Markov chain or a specific case of irreversibly stationary ones can be identified by utilizing observations from all leaves and two adjacent states in each cycle. Several algorithms are proposed for calculating the generator matrix accurately, and numerical examples are presented to confirm their validity and efficiency. An application to neurophysiology is provided to demonstrate the applicability of such statistics to real-world data. This means that partially observable data can be used to identify the generator matrix of a stationary Markov chain.

Increasing the accuracy of single-molecule data analysis using tMAVEN

Time-dependent single-molecule experiments contain rich kinetic information about the functional dynamics of biomolecules. A key step in extracting this information is the application of kinetic models, such as hidden Markov models (HMMs), which characterize the molecular mechanism governing the experimental system. Unfortunately, researchers rarely know the physicochemical details of this molecular mechanism a priori, which raises questions about how to select the most appropriate kinetic model for a given single-molecule data set and what consequences arise if the wrong model is chosen. To address these questions, we have developed and used time-series modeling, analysis, and visualization environment (tMAVEN), a comprehensive, open-source, and extensible software platform. tMAVEN can perform each step of the single-molecule analysis pipeline, from preprocessing to kinetic modeling to plotting, and has been designed to enable the analysis of a single-molecule data set with multiple types of kinetic models. Using tMAVEN, we have systematically investigated mismatches between kinetic models and molecular mechanisms by analyzing simulated examples of prototypical single-molecule data sets exhibiting common experimental complications, such as molecular heterogeneity, with a series of different types of HMMs. Our results show that no single kinetic modeling strategy is mathematically appropriate for all experimental contexts. Indeed, HMMs only correctly capture the underlying molecular mechanism in the simplest of cases. As such, researchers must modify HMMs using physicochemical principles to avoid the risk of missing the significant biological and biophysical insights into molecular heterogeneity that their experiments provide. By enabling the facile, side-by-side application of multiple types of kinetic models to individual single-molecule data sets, tMAVEN allows researchers to carefully tailor their modeling approach to match the complexity of the underlying biomolecular dynamics and increase the accuracy of their single-molecule data analyses.

Increasing the accuracy of single-molecule data analysis using tMAVEN

Time-dependent single-molecule experiments contain rich kinetic information about the functional dynamics of biomolecules. A key step in extracting this information is the application of kinetic models, such as hidden Markov models (HMMs), which characterize the molecular mechanism governing the experimental system. Unfortunately, researchers rarely know the physico-chemical details of this molecular mechanism a priori, which raises questions about how to select the most appropriate kinetic model for a given single-molecule dataset and what consequences arise if the wrong model is chosen. To address these questions, we have developed and used time-series Modeling, Analysis, and Visualization ENvironment (tMAVEN), a comprehensive, open-source, and extensible software platform. tMAVEN can perform each step of the single-molecule analysis pipeline, from pre-processing to kinetic modeling to plotting, and has been designed to enable the analysis of a single-molecule dataset with multiple types of kinetic models. Using tMAVEN, we have systematically investigated mismatches between kinetic models and molecular mechanisms by analyzing simulated examples of prototypical single-molecule datasets exhibiting common experimental complications, such as molecular heterogeneity, with a series of different types of HMMs. Our results show that no single kinetic modeling strategy is mathematically appropriate for all experimental contexts. Indeed, HMMs only correctly capture the underlying molecular mechanism in the simplest of cases. As such, researchers must modify HMMs using physico-chemical principles to avoid the risk of missing the significant biological and biophysical insights into molecular heterogeneity that their experiments provide. By enabling the facile, side-by-side application of multiple types of kinetic models to individual single-molecule datasets, tMAVEN allows researchers to carefully tailor their modeling approach to match the complexity of the underlying biomolecular dynamics and increase the accuracy of their single-molecule data analyses.

Free-energy landscape of two-state protein acylphosphatase with large contact order revealed by force-dependent folding and unfolding dynamics

Acylphosphatase (AcP) is a small protein with 98 amino acid residues that catalyzes the hydrolysis of carboxyl-phosphate bonds. AcP is a typical two-state protein with slow folding rate due to its relatively large contact order in the native structure. The mechanical properties and unfolding behavior of AcP has been studied by atomic force microscope. Here using stable magnetic tweezers, we measured the force-dependent folding rates within a force range 1-3 pN, and unfolding rates 15-40 pN. The obtained unfolding rates show different force sensitivities at forces below and above ∼27 pN, which determines a free-energy landscape with two energy barriers. Our results indicate that the free-energy landscape of small globule proteins have general Bactrian camel shape, and large contact order of the native state produces a high barrier dominate at low forces.

Hidden Intermediate State and Second Pathway Determining Folding and Unfolding Dynamics of GB1 Protein at Low Forces

Atomic force microscopy experiments found that GB1, a typical two-state model protein used for study of folding and unfolding dynamics, can sustain forces of more than 100 pN, but its response to low forces still remains unclear. Using ultrastable magnetic tweezers, we discovered that GB1 has an unexpected nonmonotonic force-dependent unfolding rate at 5-160 pN, from which a free energy landscape with two main barriers and a hidden intermediate state was constructed. A model combining two separate models by Dudko et al. with two pathways between the native state and this intermediate state is proposed to rebuild the unfolding dynamics over the full experimental force range. One candidate of this transient intermediate state is the theoretically proposed molten globule state with a loosely collapsed conformation, which might exist universally in the folding and unfolding processes of two-state proteins.

Efficient use of single molecule time traces to resolve kinetic rates, models and uncertainties

Single molecule time traces reveal the time evolution of unsynchronized kinetic systems. Especially single molecule Förster resonance energy transfer (smFRET) provides access to enzymatically important time scales, combined with molecular distance resolution and minimal interference with the sample. Yet the kinetic analysis of smFRET time traces is complicated by experimental shortcomings-such as photo-bleaching and noise. Here we recapitulate the fundamental limits of single molecule fluorescence that render the classic, dwell-time based kinetic analysis unsuitable. In contrast, our Single Molecule Analysis of Complex Kinetic Sequences (SMACKS) considers every data point and combines the information of many short traces in one global kinetic rate model. We demonstrate the potential of SMACKS by resolving the small kinetic effects caused by different ionic strengths in the chaperone protein Hsp90. These results show an unexpected interrelation between conformational dynamics and ATPase activity in Hsp90.

Single-Molecule Analysis beyond Dwell Times: Demonstration and Assessment in and out of Equilibrium

We present a simple and robust technique for extracting kinetic rate models and thermodynamic quantities from single-molecule time traces. Single-molecule analysis of complex kinetic sequences (SMACKS) is a maximum-likelihood approach that resolves all statistically relevant rates and also their uncertainties. This is achieved by optimizing one global kinetic model based on the complete data set while allowing for experimental variations between individual trajectories. In contrast to dwell-time analysis, which is the current standard method, SMACKS includes every experimental data point, not only dwell times. As a result, it works as well for long trajectories as for an equivalent set of short ones. In addition, the previous systematic overestimation of fast over slow rates is solved. We demonstrate the power of SMACKS on the kinetics of the multidomain protein Hsp90 measured by single-molecule Förster resonance energy transfer. Experiments in and out of equilibrium are analyzed and compared to simulations, shedding new light on the role of Hsp90's ATPase function. SMACKS resolves accurate rate models even if states cause indistinguishable signals. Thereby, it pushes the boundaries of single-molecule kinetics beyond those of current methods.

Hidden Markov Modeling with Detailed Balance and Its Application to Single Protein Folding

Hidden Markov modeling (HMM) has revolutionized kinetic studies of macromolecules. However, results from HMM often violate detailed balance when applied to the transitions under thermodynamic equilibrium, and the consequence of such violation has not been well understood. Here, to our knowledge, we developed a new HMM method that satisfies detailed balance (HMM-DB) and optimizes model parameters by gradient search. We used free energy of stable and transition states as independent fitting parameters and considered both normal and skew normal distributions of the measurement noise. We validated our method by analyzing simulated extension trajectories that mimicked experimental data of single protein folding from optical tweezers. We then applied HMM-DB to elucidate kinetics of regulated SNARE zippering containing degenerate states. For both simulated and measured trajectories, we found that HMM-DB significantly reduced overfitting of short trajectories compared to the standard HMM based on an expectation-maximization algorithm, leading to more accurate and reliable model fitting by HMM-DB. We revealed how HMM-DB could be conveniently used to derive a simplified energy landscape of protein folding. Finally, we extended HMM-DB to correct the baseline drift in single-molecule trajectories. Together, we demonstrated an efficient, versatile, and reliable method of HMM for kinetics studies of macromolecules under thermodynamic equilibrium.

Theory of single molecule emission spectroscopy

A general theory and calculation framework for the prediction of frequency-resolved single molecule photon counting statistics is presented. Expressions for the generating function of photon counts are derived, both for the case of naive "detection" based solely on photon emission from the molecule and also for experimentally realizable detection of emitted photons, and are used to explicitly calculate low-order photon-counting moments. The two cases of naive detection versus physical detection are compared to one another and it is demonstrated that the physical detection scheme resolves certain inconsistencies predicted via the naive detection approach. Applications to two different models for molecular dynamics are considered: a simple two-level system and a two-level absorber subject to spectral diffusion.

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

Output-input ratio in thermally fluctuating biomolecular machines

Biological molecular machines are proteins that operate under isothermal conditions and hence are referred to as free energy transducers. They can be formally considered as enzymes that simultaneously catalyze two chemical reactions: the free energy-donating (input) reaction and the free energy-accepting (output) one. Most if not all biologically active proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. This makes the description of the enzymatic reaction kinetics in terms of conventional rate constants insufficient. In the steady state, upon taking advantage of the assumption that each reaction proceeds through a single pair (the gate) of transition conformational substates of the enzyme-substrates complex, the degree of coupling between the output and the input reaction fluxes has been expressed in terms of the mean first-passage times on a conformational transition network between the distinguished substates. The theory is confronted with the results of random-walk simulations on the five-dimensional hypercube. The formal proof is given that, for single input and output gates, the output-input degree of coupling cannot exceed unity. As some experiments suggest such exceeding, looking for the conditions for increasing the degree of coupling value over unity challenges the theory. Performed simulations of random walks on several model networks involving more extended gates indicate that the case of the degree of coupling value higher than 1 is realized in a natural way on critical branching trees extended by long-range shortcuts. Such networks are scale-free and display the property of the small world. For short-range shortcuts, the networks are scale-free and fractal, representing a reasonable model for biomolecular machines displaying tight coupling, i.e., the degree of coupling equal exactly to unity. A hypothesis is stated that the protein conformational transition networks, as just as higher-level biological networks, the protein interaction network, and the metabolic network, have evolved in the process of self-organized criticality.

Single molecule counting statistics for systems with periodic driving

We extend the generating function approach for calculation of event statistics observed in single molecule spectroscopy to cases where the single molecule evolves under explicitly time-dependent and periodic perturbation. Floquet theory is used to recast the generating function equations for the periodically driven system into effective equations devoid of explicit time-dependence. Two examples are considered, one employing simple stochastic dynamics and the other quantum dynamics, to demonstrate the versatility and numerical accuracy of the methodology.

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.

Probing single-molecule enzyme active-site conformational state intermittent coherence

The relationship between protein conformational dynamics and enzymatic reactions has been a fundamental focus in modern enzymology. Using single-molecule fluorescence resonance energy transfer (FRET) with a combined statistical data analysis approach, we have identified the intermittently appearing coherence of the enzymatic conformational state from the recorded single-molecule intensity-time trajectories of enzyme 6-hydroxymethyl-7,8-dihydropterin pyrophosphokinase (HPPK) in catalytic reaction. The coherent conformational state dynamics suggests that the enzymatic catalysis involves a multistep conformational motion along the coordinates of substrate-enzyme complex formation and product releasing, presenting as an extreme dynamic behavior intrinsically related to the time bunching effect that we have reported previously. The coherence frequency, identified by statistical results of the correlation function analysis from single-molecule FRET trajectories, increases with the increasing substrate concentrations. The intermittent coherence in conformational state changes at the enzymatic reaction active site is likely to be common and exist in other conformation regulated enzymatic reactions. Our results of HPPK interaction with substrate support a multiple-conformational state model, being consistent with a complementary conformation selection and induced-fit enzymatic loop-gated conformational change mechanism in substrate-enzyme active complex formation.

Modeling stochastic dynamics in biochemical systems with feedback using maximum caliber

Complex feedback systems are ubiquitous in biology. Modeling such systems with mass action laws or master equations requires information rarely measured directly. Thus rates and reaction topologies are often treated as adjustable parameters. Here we present a general stochastic modeling method for small chemical and biochemical systems with emphasis on feedback systems. The method, Maximum Caliber (MaxCal), is more parsimonious than others in constructing dynamical models requiring fewer model assumptions and parameters to capture the effects of feedback. MaxCal is the dynamical analogue of Maximum Entropy. It uses average rate quantities and correlations obtained from short experimental trajectories to construct dynamical models. We illustrate the method on the bistable genetic toggle switch. To test our method, we generate synthetic data from an underlying stochastic model. MaxCal reliably infers the statistics of the stochastic bistability and other full dynamical distributions of the simulated data, without having to invoke complex reaction schemes. The method should be broadly applicable to other systems.

Probing and controlling fluorescence blinking of single semiconductor nanoparticles

In this review we present an overview of the experimental and theoretical development on fluorescence intermittency (blinking) and the roles of electron transfer in semiconductor crystalline nanoparticles. Blinking is a very interesting phenomenon commonly observed in single molecule/particle experiments. Under continuous laser illumination, the fluorescence time trace of these single nanoparticles exhibit random light and dark periods. Since its first observation in the mid-1990s, this intriguing phenomenon has attracted wide attention among researchers from many disciplines. We will first present the historical background of the discovery and the observation of unusual inverse power-law dependence for the waiting time distributions of light and dark periods. Then, we will describe our theoretical modeling efforts to elucidate the causes for the power-law behavior, to probe the roles of electron transfer in blinking, and eventually to control blinking and to achieve complete suppression of the blinking, which is an annoying feature in many applications of quantum dots as light sources and fluorescence labels for biomedical imaging.

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.

Single-biomolecule kinetics: the art of studying a single enzyme

The potential of single-enzyme studies to unravel the complex energy landscape of these polymeric catalysts is the next critical step in enzymology. From its inception in Rotman's emulsion experiments in the 1960s, the field of single-molecule enzymology has now advanced into the time-resolved age. Technological advances have enabled individual enzymatic turnover reactions to be observed with a millisecond time resolution. A number of initial studies have revealed the underlying static and dynamic disorder in the catalytic rates originating from conformational fluctuations. Although these experiments are still in their infancy, they may be able to relate the topography of the energy landscape to the biological function and regulation of enzymes. This review summarizes some of the experimental techniques and data-analysis methods that have been used to study individual enzyme molecules in search of a deeper understanding of their kinetics.

Single-molecule photon emission statistics for systems with explicit time dependence: Generating function approach

Generating function calculations are extended to allow for laser pulse envelopes of arbitrary shape in numerical applications. We investigate photon emission statistics for two-level and V- and Lambda-type three-level systems under time-dependent excitation. Applications relevant to electromagnetically induced transparency and photon emission from single quantum dots are presented.

The effects of anomalous diffusion on power-law blinking statistics of CdSe nanorods

In this study of fluorescence blinking statistics for nanorods, we present a diffusion-controlled reaction model that leads to a more general formula: t(-m) exp[-(Gammat)(n)]. This formula describes a short-time power law with a crossover to a stretched exponential decay at later times. Based on quantum Brownian motion for a coupled central harmonic oscillator coupled to heat bath oscillators, we show that the position distribution follows anomalous diffusion with time-dependent diffusion coefficient and drift coefficient. The first and the second moments of the energy fluctuations are shown to be related to the exponent m and n for the blinking statistics.

Single-molecule fluorescence studies of protein folding

The structural and dynamic details of protein folding are still widely unexplored due to the enormous level of heterogeneity intrinsic to this process. The unfolded polypeptide chain can assume a vast number of possible conformations, and many complex pathways lead from the ensemble of unfolded conformations to the ensemble of native conformations in an overall funnel-shaped energy landscape. Classical experimental methods involve measurements on bulk samples and usually yield only average values characteristic of the entire molecular ensemble under study. The observation of individual molecules avoids this averaging and allows, in principle, microscopic distributions of conformations and folding trajectories to be revealed. Fluorescence-based techniques are arguably the most versatile single-molecule methods at present, and Förster resonance energy transfer (FRET) between two dye molecules specifically attached to the protein of interest provides a means of studying the inter-dye distance and, thereby, the conformation of folding polypeptide chains in real time. This chapter focuses on practical aspects and different experimental realizations for protein folding investigations by using single-molecule fluorescence.

Toolbox for analyzing finite two-state trajectories

In many experiments, the aim is to deduce an underlying multisubstate on-off kinetic scheme (KS) from the statistical properties of a two-state trajectory. However, a two-state trajectory that is generated from an on-off KS contains only partial information about the KS, and so, in many cases, more than one KS can be associated with the data. We recently showed that the optimal way to solve this problem is to use canonical forms of reduced dimensions (RDs). RD forms are on-off networks with connections only between substates of different states, where the connections can have nonexponential waiting time probability density functions (WT-PDFs). In theory, only a single RD form can be associated with the data. To utilize RD forms in the analysis of the data, a RD form should be associated with the data. Here, we give a toolbox for building a RD form from a finite time, noiseless, two-state trajectory. The methods in the toolbox are based on known statistical methods in data analysis, combined with statistical methods and numerical algorithms designed specifically for the current problem. Our toolbox is self-contained-it builds a mechanism based only on the information it extracts from the data, and its implementation is fast (analyzing a 10;{6}cycle trajectory from a 30-parameter mechanism takes a couple of hours on a PC with a 2.66GHz processor). The toolbox is automated and is freely available for academic research upon electronic request.

Analysis of the entire sequence of a single photon experiment on a flavin protein

The large amount of statistical data collected by single biomolecule experiments often demonstrates complex and non-Markovian relaxation over many time scales. Analyzing and interpreting these data is a major challenge because of the inherently statistical noise and the lack of definite theoretical descriptions or computer simulations on biologically relevant time scales. This paper reports one of the first complete sequence analyses of a single photon experiment on the flavin protein to determine an underlying physical picture for protein motions on the millisecond to second regimes. The robustness of Bayesian information analysis combined with the nonparametric maximum entropy method (MEM) incorporates all available information of the single-molecule data sequence and maximizes our ability to test the legitimacy of possible models. Our analysis of the experimental data is consistent with the stochastic Gaussian diffusion model where the slow protein motions are modeled as a collection of over-damped diffusive normal modes and reveals non-universal and distinct dynamic features that are specific for protein functions.

Multiscale complex network of protein conformational fluctuations in single-molecule time series

Conformational dynamics of proteins can be interpreted as itinerant motions as the protein traverses from one state to another on a complex network in conformational space or, more generally, in state space. Here we present a scheme to extract a multiscale state space network (SSN) from a single-molecule time series. Analysis by this method enables us to lift degeneracy--different physical states having the same value for a measured observable--as much as possible. A state or node in the network is defined not by the value of the observable at each time but by a set of subsequences of the observable over time. The length of the subsequence can tell us the extent to which the memory of the system is able to predict the next state. As an illustration, we investigate the conformational fluctuation dynamics probed by single-molecule electron transfer (ET), detected on a photon-by-photon basis. We show that the topographical features of the SSNs depend on the time scale of observation; the longer the time scale, the simpler the underlying SSN becomes, leading to a transition of the dynamics from anomalous diffusion to normal Brownian diffusion.

COCIS: Markov processes in single molecule fluorescence

This article examines the current status of Markov processes in single molecule fluorescence. For molecular dynamics to be described by a Markov process, the Markov process must include all states involved in the dynamics and the FPT distributions out of those states must be describable by a simple exponential law. The observation of non-exponential first-passage time distributions or other evidence of non-Markovian dynamics is common in single molecule studies and offers an opportunity to expand the Markov model to include new dynamics or states that improve understanding of the system.

Construction of effective free energy landscape from single-molecule time series

A scheme for extracting an effective free energy landscape from single-molecule time series is presented. This procedure uniquely identifies a non-Gaussian distribution of the observable associated with each local equilibrium state (LES). Both the number of LESs and the shape of the non-Gaussian distributions depend on the time scale of observation. By assessing how often the system visits and resides in a chosen LES and escapes from one LES to another (with checking whether the local detailed balance is satisfied), our scheme naturally leads to an effective free energy landscape whose topography depends on in which time scale the system experiences the underlying landscape. For example, two metastable states are unified as one if the time scale of observation is longer than the escape time scale for which the system can visit mutually these two states. As an illustrative example, we present the application of extracting the effective free energy landscapes from time series of the end-to-end distance of a three-color, 46-bead model protein. It indicates that the time scales to attain the local equilibrium tend to be longer in the unfolded state than those in the compact collapsed state.

Kinetic analysis of sequential multistep reactions

Many processes in biology and chemistry involve multistep reactions or transitions. The kinetic data associated with these reactions are manifested by superpositions of exponential decays that are often difficult to dissect. Two major challenges have hampered the kinetic analysis of multistep chemical reactions: (1) reliable and unbiased determination of the number of reaction steps, and (2) stable reconstruction of the distribution of kinetic rate constants. Here, we introduce two numerically stable integral transformations to solve these two challenges. The first transformation enables us to deduce the number of rate-limiting steps from kinetic measurements, even when each step has arbitrarily distributed rate constants. The second transformation allows us to reconstruct the distribution of rate constants in the multistep reaction using the phase function approach, without fitting the data. We demonstrate the stability of the two integral transformations by both analytic proofs and numerical tests. These new methods will help provide robust and unbiased kinetic analysis for many complex chemical and biochemical reactions.

Probing lipid vesicles by bimolecular association and dissociation trajectories of single molecules

Vesicles prepared by DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine) and SOPC (1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine) lipid molecules having sizes smaller than the diffraction-limited focused laser beam have been used to confine single molecules in the laser focus. The confinement of single molecules in a volume smaller than the focused laser beam leads to a Gaussian distribution of single molecule fluorescence intensity. The interactions of single Nile Red molecules with DMPC and SOPC lipid bilayers were studied by single molecule fluorescence confocal microscopy. Nile Red molecules were observed to associate with and dissociate from individual DMPC and SOPC vesicles adsorbed on a glass surface, generating on-and-off fluctuations in a fluorescence signal representing a very low noise two-state trajectory. Off-time statistics were used to investigate the mean radius of the vesicles and the size distribution functions. The means of the on-time distributions of Nile Red in DMPC and SOPC vesicles were significantly different. The association and dissociation reactions of single Nile Red molecules with a vesicle have been studied. Features of the bimolecular interaction between the probe Nile Red and the vesicle were evaluated from the uncorrelated mean on-time and vesicle radius distributions, and the linear Nile Red concentration dependence of the mean off-time. Nile Red is shown to be a useful probe of the structural fluctuations and heterogeneity of these membrane structures, and it is a useful model with which to directly study a diffusion-influenced reversible bimolecular reaction.

Path-probability density functions for semi-Markovian random walks

In random walks, the path representation of the Green's function is an infinite sum over the length of path probability density functions (PDFs). Recently, a closed-form expression for the Green's function of an arbitrarily inhomogeneous semi-Markovian random walk in a one-dimensional (1D) chain of L states was obtained by utilizing path-PDFs calculations. Here we derive and solve, in Laplace space, the recursion relation for the n order path PDF for the same system. The recursion relation relates the n order path PDF to L/2 (round towards zero for an odd L) shorter path PDFs and has n independent coefficients that obey a universal formula. The z transform of the recursion relation straightforwardly gives the generating function for path PDFs, from which we recover the Green's function of the random walk, but, moreover, derive an explicit expression for any path PDF of the random walk. These expressions give the most detailed description of arbitrarily inhomogeneous semi-Markovian random walks in 1D.

Properties of the generalized master equation: Green's functions and probability density functions in the path representation

The Green's function for the master equation and the generalized master equation in path representation is an infinite sum over the length of path probability density functions (PDFs). In this paper, the properties of path PDFs are studied both qualitatively and quantitatively. The results are used in building efficient approximations for Green's function in 1D, and are relevant in modeling and in data analysis.

Hidden Markov model analysis of multichromophore photobleaching

The interpretation of single-molecule measurements is greatly complicated by the presence of multiple fluorescent labels. However, many molecular systems of interest consist of multiple interacting components. We investigate this issue using multiply labeled dextran polymers that we intentionally photobleach to the background on a single-molecule basis. Hidden Markov models allow for unsupervised analysis of the data to determine the number of fluorescent subunits involved in the fluorescence intermittency of the 6-carboxy-tetramethylrhodamine labels by counting the discrete steps in fluorescence intensity. The Bayes information criterion allows us to distinguish between hidden Markov models that differ by the number of states, that is, the number of fluorescent molecules. We determine information-theoretical limits and show via Monte Carlo simulations that the hidden Markov model analysis approaches these theoretical limits. This technique has resolving power of one fluorescing unit up to as many as 30 fluorescent dyes with the appropriate choice of dye and adequate detection capability. We discuss the general utility of this method for determining aggregation-state distributions as could appear in many biologically important systems and its adaptability to general photometric experiments.

Size Effects and Breakdown of the Power-Law Blinking Statistics of CdSe Nanorods

In this study, the dependence of sample size and light intensity on the fluorescence intermittency of semiconductor nanorods is investigated. We present a model with diffusion-controlled electron-transfer reactions involving anomalous diffusion in energy configuration space. This model leads to a general formula t(-m) exp[-(Gammat)n] for the temporal behavior of blinking statistics, where m and n are related to the time dependence of the spectral diffusion. We reanalyze the experimental data of the long-time bending tail of CdSe nanorods and elucidate the size effects of the bending rates and activation energy.

Theory of the statistics of kinetic transitions with application to single-molecule enzyme catalysis

Single-molecule spectroscopy can monitor transitions between two microscopic states when these transitions are associated with the emission of photons. A general formalism is developed for obtaining the statistics of such transitions from a microscopic model when the dynamics is described by master or rate equations or their continuum analog, multidimensional reaction-diffusion equations. The focus is on the distribution of the number of transitions during a fixed observation time, the distribution of times between transitions, and the corresponding correlation functions. It is shown how these quantities are related to each other and how they can be explicitly calculated in a straightforward way for both immobile and diffusing molecules. Our formalism reduces to renewal theory when the monitored transitions either go to or originate from a single state. The influence of dynamics slow compared with the time between monitored transitions is treated in a simple way, and the probability distributions are expressed in terms of Mandel-type formulas. The formalism is illustrated by a detailed analysis of the statistics of catalytic turnovers of enzymes. When the rates of conformational changes are slower than the catalytic rates which are in turn slower than the binding relaxation rate, (1) the mean number of turnovers is shown to have the classical Michaelis-Menten form, (2) the correlation function of the number of turnovers is a direct measure of the time scale of catalytic rate fluctuations, and (3) the distribution of the time between consecutive turnovers is determined by the steady-state distribution.

Analytical theory of hysteresis in ion channels: two-state model

Channel-forming proteins in a lipid bilayer of a biological membrane usually respond to variation of external voltage by changing their conformations. Periodic voltages with frequency comparable with the inverse relaxation time of the protein produce hysteresis in the occupancies of the protein conformations. If the channel conductance changes when the protein jumps between these conformations, hysteresis in occupancies is observed as hysteresis in ion current through the channel. We develop an analytical theory of this phenomenon assuming that the channel conformational dynamics can be described in terms of a two-state model. The theory describes transient behavior of the channel after the periodic voltage is switched on as well as the shape and area of the hysteretic loop as functions of the frequency and amplitude of the applied voltage. The area vanishes as the voltage period T tends to zero and infinity. Asymptotic behaviors of the loop area A in the high- and low-frequency regimes, respectively, are A approximately T and A approximately T(-1).

Transformations that preserve detailed balance in Markov models

Aggregated Markov processes related by similarity transformation are equivalent in that they cannot be distinguished by steady-state experiments. We derive an explicit formula for the set of all detailed-balance preserving similarity transformations between such continuous time Markov chains with N states. The matrices that define the allowed similarity transformations are found to be a simple non-linear function applied to almost any element of the special orthogonal group in N dimensions. Since a model is identifiable only if there is no similarity transformations to an equivalent model, we expect this result to prove useful in the theory of identification of aggregated Markov chains, an enterprise of growing importance as more and more single molecules yield to observation.

Single Molecule Photon Counting Statistics for Quantum Mechanical Chromophore Dynamics

We extend the generating function technique for calculation of single molecule photon emission statistics (Zheng, Y.; Brown, F. L. H. Phys. Rev. Lett. 2003, 90, 238305) to systems governed by multi-level quantum dynamics. This opens up the possibility to study phenomena that are outside the realm of purely stochastic and mixed quantum-stochastic models. In particular, the present methodology allows for calculation of photon statistics that are spectrally resolved and subject to quantum coherence. Several model calculations illustrate the generality of the technique and highlight quantitative and qualitative differences between quantum mechanical models and related stochastic approximations when they arise. Calculations suggest that studying photon statistics as a function of photon frequency has the potential to reveal more about system dynamics than the usual broadband detection schemes.