Protein Conformational Dynamics Probed by Single-Molecule Electron Transfer

Anomalous lineshapes and aging effects in two-dimensional correlation spectroscopy

Multitime correlation functions provide useful probes for the ensembles of trajectories underlying the stochastic dynamics of complex systems. These can be obtained by measuring their optical response to sequences of ultrashort optical pulse. Using the continuous time random walk model for spectral diffusion, we analyze the signatures of anomalous relaxation in two-dimensional four wave mixing signals. Different models which share the same two point joint probability distribution show markedly different lineshapes and may be distinguished. Aging random walks corresponding to waiting time distributions with diverging first moment show dependence of 2D lineshapes on initial observation time, which persist for long times.

A hyperspectral fluorescence lifetime probe for skin cancer diagnosis

The autofluorescence of biological tissue can be exploited for the detection and diagnosis of disease but, to date, its complex nature and relatively weak signal levels have impeded its widespread application in biology and medicine. We present here a portable instrument designed for the in situ simultaneous measurement of autofluorescence emission spectra and temporal decay profiles, permitting the analysis of complex fluorescence signals. This hyperspectral fluorescence lifetime probe utilizes two ultrafast lasers operating at 355 and 440 nm that can excite autofluorescence from many different biomolecules present in skin tissue including keratin, collagen, nicotinamide adenine dinucleotide (phosphate), and flavins. The instrument incorporates an optical fiber probe to provide sample illumination and fluorescence collection over a millimeter-sized area. We present a description of the system, including spectral and temporal characterizations, and report the preliminary application of this instrument to a study of recently resected (<2 h) ex vivo skin lesions, illustrating its potential for skin cancer detection and diagnosis.

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.

Anomalous escape governed by thermal 1/f noise

We present an analytic study for subdiffusive escape of overdamped particles out of a cusp-shaped parabolic potential well which are driven by thermal, fractional Gaussian noise with a 1/omega 1-alpha power spectrum. This long-standing challenge becomes mathematically tractable by use of a generalized Langevin dynamics via its corresponding non-Markovian, time-convolutionless master equation: We find that the escape is governed asymptotically by a power-law whose exponent depends exponentially on the ratio of barrier height and temperature. This result is in distinct contrast to a description with a corresponding subdiffusive fractional Fokker-Planck approach, thus providing experimentalists an amenable testbed to differentiate between the two escape scenarios.

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.

Illuminating the mechanistic roles of enzyme conformational dynamics

Many enzymes mold their structures to enclose substrates in their active sites such that conformational remodeling may be required during each catalytic cycle. In adenylate kinase (AK), this involves a large-amplitude rearrangement of the enzyme's lid domain. Using our method of high-resolution single-molecule FRET, we directly followed AK's domain movements on its catalytic time scale. To quantitatively measure the enzyme's entire conformational distribution, we have applied maximum entropy-based methods to remove photon-counting noise from single-molecule data. This analysis shows unambiguously that AK is capable of dynamically sampling two distinct states, which correlate well with those observed by x-ray crystallography. Unexpectedly, the equilibrium favors the closed, active-site-forming configurations even in the absence of substrates. Our experiments further showed that interaction with substrates, rather than locking the enzyme into a compact state, restricts the spatial extent of conformational fluctuations and shifts the enzyme's conformational equilibrium toward the closed form by increasing the closing rate of the lid. Integrating these microscopic dynamics into macroscopic kinetics allows us to model lid opening-coupled product release as the enzyme's rate-limiting step.

Probing polyproline structure and dynamics by photoinduced electron transfer provides evidence for deviations from a regular polyproline type II helix

Polyprolines are well known for adopting a regular polyproline type II helix in aqueous solution, rendering them a popular standard as molecular ruler in structural molecular biology. However, single-molecule spectroscopy studies based on Förster resonance energy transfer (FRET) have revealed deviations of experimentally observed end-to-end distances of polyprolines from theoretical predictions, and it was proposed that the discrepancy resulted from dynamic flexibility of the polyproline helix. Here, we probe end-to-end distances and conformational dynamics of poly-l-prolines with 1-10 residues using fluorescence quenching by photoinduced-electron transfer (PET). A single fluorophore and a tryptophan residue, introduced at the termini of polyproline peptides, serve as sensitive probes for distance changes on the subnanometer length scale. Using a combination of ensemble fluorescence and fluorescence correlation spectroscopy, we demonstrate that polyproline samples exhibit static structural heterogeneity with subpopulations of distinct end-to-end distances that do not interconvert on time scales from nano- to milliseconds. By observing prolyl isomerization through changes in PET quenching interactions, we provide experimental evidence that the observed heterogeneity can be explained by interspersed cis isomers. Computer simulations elucidate the influence of trans/cis isomerization on polyproline structures in terms of end-to-end distance and provide a structural justification for the experimentally observed effects. Our results demonstrate that structural heterogeneity inherent in polyprolines, which to date are commonly applied as a molecular ruler, disqualifies them as appropriate tool for an accurate determination of absolute distances at a molecular scale.

Dielectric dispersion interpretation of single enzyme dynamic disorder, spectral diffusion, and radiative fluorescence lifetime

A formulation based on measurable dielectric dispersion of enzymes is developed to estimate fluctuations in electrostatic interaction energy on time scales as long as milliseconds to seconds at a local site in enzymes. Several single molecule experimental obsevations occur on this time scale, currently unreachable by real time computational trajectory simulations. We compare the experimental results on the autocorrelation function of the fluctuations of catalysis rate with the calculations using the dielectric dispersion formulation. We also discuss the autocorrelation functions of the fluorescence lifetime and of spectral diffusion. We use a previously derived relation between the observables and the electric field fluctuations and calculate the latter using dielectric dispersion data for the proteins and the Onsager regression hypothesis.

Diffusive hidden Markov model characterization of DNA looping dynamics in tethered particle experiments

In many biochemical processes, proteins bound to DNA at distant sites are brought into close proximity by loops in the underlying DNA. For example, the function of some gene-regulatory proteins depends on such 'DNA looping' interactions. We present a new technique for characterizing the kinetics of loop formation in vitro, as observed using the tethered particle method, and apply it to experimental data on looping induced by lambda repressor. Our method uses a modified ('diffusive') hidden Markov analysis that directly incorporates the Brownian motion of the observed tethered bead. We compare looping lifetimes found with our method (which we find are consistent over a range of sampling frequencies) to those obtained via the traditional threshold-crossing analysis (which can vary depending on how the raw data are filtered in the time domain). Our method does not involve any time filtering and can detect sudden changes in looping behavior. For example, we show how our method can identify transitions between long-lived, kinetically distinct states that would otherwise be difficult to discern.

Estimating the influence of finite instrumental resolution on elastic neutron scattering intensities from proteins

Recent experimental and simulation studies show that the fractional Ornstein-Uhlenbeck process describes well the single particle motions in internal protein dynamics. Here the authors use this model to estimate the influence of finite instrumental resolution on elastic neutron scattering intensities from hydrated protein powders. They give, in particular, an estimation of the attenuation factor for the observed atomic position fluctuations, assuming a Gaussian and a triangular resolution function.

An interpretation of fluctuations in enzyme catalysis rate, spectral diffusion, and radiative component of lifetimes in terms of electric field fluctuations

Time-dependent fluctuations in the catalysis rate (deltak(t)) observed in single-enzyme experiments were found in a particular study to have an autocorrelation function decaying on the same time scale as that of spectral diffusion deltaomega(0)(t). To interpret this similarity, the present analysis focuses on a factor in enzyme catalysis, the local electrostatic interaction energy (E) at the active site and its effect on the activation free energy barrier. We consider the slow fluctuations of the electrostatic interaction energy (deltaE(t)) as a contributor to deltak(t) and relate the latter to deltaomega(0)(t). The resulting relation between deltak(t) and deltaomega(0)(t) is a dynamic analog of the solvatochromism used in interpreting solvent effects on organic reaction rates. The effect of the postulated deltaE(t) on fluctuations in the radiative component (deltagamma(r)(-1)(t)) of the fluorescence decay of chromophores in proteins also is examined, and a relation between deltagamma(r)(-1)(t) and deltaomega(0)(t) is obtained. Experimental tests will determine whether the correlation functions for deltak(t), deltaomega(0)(t), and deltagamma(r)(-1) are indeed similar for any enzyme. Measurements of dielectric dispersion, epsilon(omega), for the enzyme discussed elsewhere will provide further insight into the correlation function for deltaE(t). They also will determine whether fluctuations in the nonradiative component gamma(nr)(-1) of the lifetime decay has a different origin, fluctuations in distance for example.

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.

MCM forked substrate specificity involves dynamic interaction with the 5'-tail

The archaeal minichromosome maintenance protein MCM forms a homohexameric complex that functions as the DNA replicative helicase and serves as a model system for its eukaryotic counterpart. Here, we applied single molecule fluorescence resonance energy transfer methods to probe the substrate specificity and binding mechanism of MCM from the hyperthermophilic Archaea Sulfolobus solfataricus on various DNA substrates. S. solfataricus MCM displays a binding preference for forked substrates relative to partial or full duplex substrates. Moreover, the nature of MCM binding to Y-shaped substrates is distinct in that MCM loads on the 3'-tail while interacting with the 5'-tail likely via the MCM surface. These results provide the first elucidation of a dynamic nature of interaction between a ring-shaped helicase interacting with an opposing single-stranded DNA tail. This interaction contributes to substrate selectivity and increases the stability of the forked DNA-MCM complex, with possible implications for the MCM unwinding mechanism.

Studying individual events in biology

Studying the properties of individual events and molecules offers a host of advantages over taking only macroscopic measurements of populations. Here we review such advantages, as well as some pitfalls, focusing on examples from biological imaging. Examples include single proteins, their interactions in cells, organelles, and their interactions both with each other and with parts of the cell. Additionally, we discuss constraints that limit the study of single events, along with the criteria that must be fulfilled to determine whether single molecules or events are being detected.

Dynamic disorder in single-molecule Michaelis-Menten kinetics: The reaction-diffusion formalism in the Wilemski-Fixman approximation

Single-molecule equations for the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] mechanism of enzyme action are analyzed within the Wilemski-Fixman [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)] approximation after the effects of dynamic disorder--modeled by the anomalous diffusion of a particle in a harmonic well--are incorporated into the catalytic step of the reaction. The solution of the Michaelis-Menten equations is used to calculate the distribution of waiting times between successive catalytic turnovers in the enzyme beta-galactosidase. The calculated distribution is found to agree qualitatively with experimental results on this enzyme obtained at four different substrate concentrations. The calculations are also consistent with measurements of correlations in the fluctuations of the fluorescent light emitted during the course of catalysis, and with measurements of the concentration dependence of the randomness parameter.

Evaluation of intermolecular interactions of self-etch dentin adhesive primer molecules with type 1 collagen: computer modeling and in vitro binding analysis

The objective of this investigation was to study adhesion of self-etch primer systems to dentin by computer-modeled docking simulations and in vitro binding assay methods. Computer modeling employed analysis of docking simulations of a self-etch primer molecule 10-methacryloxydecamethylene phosphoric acid (MDP) and its calcium salt (MDPCa) as ligands. Typical type 1 collagen segments were selected as targets to reflect potential differences in the amino acid residues in dentinal type 1 collagen triple helix motif. The binding assay involved immunochemical analysis of the modification of anti-collagen binding to collagen by prior exposure of the demineralized dentin to MDP. The estimated mean docking energy values ranged between -4.5 and -8.9kcalmol(-1). The results revealed significant differences in the docking energy estimates as a function of ligand and target structures (p<0.01). Van der Waals and electrostatic contributions were also significantly influenced by ligand selection and collagen structure. Both MDP and MDPCa appear to be important in the overall interactions. Binding assay studies also lend evidence of collagen-ligand intermolecular interactions. It is suggested that the ability of self-etch dentin primer systems to bond effectively to dentin is not limited to the interaction of the primer with the hydroxyapatite of dentin, but also due to the ability to prime dentin efficiently through intermolecular interactions between the primer and its calcium salt with the collagen matrix. Virtual screening methods may be very valuable to select primer molecules for dentin bonding.

Shot-noise limited single-molecule FRET histograms: comparison between theory and experiments

We describe a simple approach and present a straightforward numerical algorithm to compute the best fit shot-noise limited proximity ratio histogram (PRH) in single-molecule fluorescence resonant energy transfer diffusion experiments. The key ingredient is the use of the experimental burst size distribution, as obtained after burst search through the photon data streams. We show how the use of an alternated laser excitation scheme and a correspondingly optimized burst search algorithm eliminates several potential artifacts affecting the calculation of the best fit shot-noise limited PRH. This algorithm is tested extensively on simulations and simple experimental systems. We find that dsDNA data exhibit a wider PRH than expected from shot noise only and hypothetically account for it by assuming a small Gaussian distribution of distances with an average standard deviation of 1.6 A. Finally, we briefly mention the results of a future publication and illustrate them with a simple two-state model system (DNA hairpin), for which the kinetic transition rates between the open and closed conformations are extracted.

Closed-form expressions of quantum electron transfer rate based on the stationary-phase approximation

Closed-form rate expressions are derived on the basis of the stationary-phase approximation for the Fermi golden rule expression of the quantum electron-transfer (ET) rate. First, on the basis of approximate solutions of the stationary-phase points near DeltaG = 0, -lambda, and lambda, where DeltaG is the reaction free energy and lambda is the reorganization energy, three closed-form rate expressions are derived, which are respectively valid near each value of DeltaG. Numerical tests for a model Ohmic spectral density with an exponential cutoff demonstrate good performance of the derived expressions in the respective regions of their validity. In particular, the expression near DeltaG = -lambda, which differs from the semiclassical approximation only by a prefactor quadratic in DeltaG, works substantially better than the latter. Then, a unified formula is suggested, which interpolates the three approximate expressions and serves as a good approximation in all three regions. We have also demonstrated that the interpolation formula can serve as a good quantitative means for understanding the temperature dependence of the quantum ET rate.

When does the Michaelis-Menten equation hold for fluctuating enzymes?

Enzymes are dynamic entities: both their conformation and catalytic activity fluctuate over time. When such fluctuations are relatively fast, it is not surprising that the classical Michaelis-Menten (MM) relationship between the steady-state enzymatic velocity and the substrate concentration still holds. However, recent single-molecule experiments have shown that this is the case even for an enzyme whose catalytic activity fluctuates on the 10(-4)-10 s range. The purpose of this paper is to examine various scenarios in which slowly fluctuating enzymes would still obey the MM relationship. Specifically, we consider (1) the quasi-static condition (e.g., the conformational fluctuation of the enzyme-substrate complex is much slower than binding, catalysis, and the conformational fluctuations of the free enzyme), (2) the quasi-equilibrium condition (when the substrate dissociation is much faster than catalysis, irrespective of the time scales or amplitudes of conformational fluctuations), and (3) the conformational-equilibrium condition (when the dissociation and catalytic rates depend on the conformational coordinate in the same way). For each of these scenarios, the physical meaning of the apparent Michaelis constant and catalytic rate constant is provided. Finally, as an example, the theoretical analysis of a recent single-molecule enzyme assay is considered in light of the perspectives presented in this paper.

Ab Initio Prediction of Tryptophan Fluorescence Quenching by Protein Electric Field Enabled Electron Transfer

We report quantum mechanical-molecular mechanical (QM-MM) predictions of fluorescence quantum yields for 20 tryptophans in 17 proteins, whose yields span the range from 0.01 to 0.3, using ab initio computed coupling matrix elements for photoinduced electron transfer from the 1La excited indole ring to a local backbone amide. The average coupling elements span the range 140-1000 cm-1, depending on tryptophan rotamer conformation. The matrix elements were from the singles configuration interaction matrix, and were largely insensitive to which of the three basis sets was used. Large fluctuations were seen on the time scale of tens of femtoseconds, caused primarily by side chain and backbone torsional variations for 150 ps of dynamics at 300 K. The largest coupling occurs for the chi1 = -60 degrees rotamer and is purely through-bond. There is no apparent correlation between the coupling magnitude and quantum yield, which is still dominated by energy gap and reorganization energy. The source of error bars for predicted quenching rates using the weak coupling golden rule may be due to inaccurate averaged Franck-Condon weighted densities because of inadequate simulation times and parameters and/or to failure of the weak coupled golden rule used in these predictions because of the broad distribution of Landau-Zener probabilities arising from the large variable coupling.

High-resolution, single-molecule measurements of biomolecular motion

Many biologically important macromolecules undergo motions that are essential to their function. Biophysical techniques can now resolve the motions of single molecules down to the nanometer scale or even below, providing new insights into the mechanisms that drive molecular movements. This review outlines the principal approaches that have been used for high-resolution measurements of single-molecule motion, including centroid tracking, fluorescence resonance energy transfer, magnetic tweezers, atomic force microscopy, and optical traps. For each technique, the principles of operation are outlined, the capabilities and typical applications are examined, and various practical issues for implementation are considered. Extensions to these methods are also discussed, with an eye toward future application to outstanding biological problems.

Self-suppression of reset induced triggering in picosecond SPAD timing circuits

We present a new photon timing circuit that achieves a time resolution of 35 ps full width at half maximum with single photon avalanche diodes having active area diameters up to 200 microm. The timing circuit is based on a double avalanche current sensing network that makes it particularly suited to operation at high photon counting rates. Thanks to its self-adjusting capabilities, no trimming is needed even when changing the photodetector operating conditions over a wide range.

New directions in single-molecule imaging and analysis

Optical imaging and analysis of single molecules continue to unfold as powerful ways to study the individual behavior of biological systems, unobscured by ensemble averaging. Current expansion of interest in this field is great, as evidenced by new meetings, journal special issues, and the large number of new investigators. Selected recent advances in biomolecular analysis are described, and two new research directions are summarized: superresolution imaging using single-molecule fluorescence and trapping of single molecules in solution by direct suppression of Brownian motion.

Single-molecule fluorescence spectroscopy in (bio)catalysis

The ever-improving time and space resolution and molecular detection sensitivity of fluorescence microscopy offer unique opportunities to deepen our insights into the function of chemical and biological catalysts. Because single-molecule microscopy allows for counting the turnover events one by one, one can map the distribution of the catalytic activities of different sites in solid heterogeneous catalysts, or one can study time-dependent activity fluctuations of individual sites in enzymes or chemical catalysts. By experimentally monitoring individuals rather than populations, the origin of complex behavior, e.g., in kinetics or in deactivation processes, can be successfully elucidated. Recent progress of temporal and spatial resolution in single-molecule fluorescence microscopy is discussed in light of its impact on catalytic assays. Key concepts are illustrated regarding the use of fluorescent reporters in catalytic reactions. Future challenges comprising the integration of other techniques, such as diffraction, scanning probe, or vibrational methods in single-molecule fluorescence spectroscopy are suggested.

Cooperative effects in photon statistics of molecular dimers with spectral diffusion

The two-point fluorescence intensity correlation function g(2)t and the Mandel parameter Mt are calculated for a strongly pumped dimer of two-level molecules undergoing Gaussian-Markovian frequency fluctuations. The effects of detuning and saturation are examined. All fluctuation time scale regimes are explored using a continued fraction solution of the stochastic Liouville equation for the generating function. Bunching and antibunching are observed for slow and fast fluctuations, respectively. The short-time antibunching dip in g(2) and its variation with intermolecular coupling, the exciton annihilation rate, and laser detuning are studied.

Single-molecule and ensemble fluorescence assays for a functionally important conformational change in T7 DNA polymerase

We report fluorescence assays for a functionally important conformational change in bacteriophage T7 DNA polymerase (T7 pol) that use the environmental sensitivity of a Cy3 dye attached to a DNA substrate. An increase in fluorescence intensity of Cy3 is observed at the single-molecule level, reflecting a conformational change within the T7 pol ternary complex upon binding of a dNTP substrate. This fluorescence change is believed to reflect the closing of the T7 pol fingers domain, which is crucial for polymerase function. The rate of the conformational change induced by a complementary dNTP substrate was determined by both conventional stopped-flow and high-time-resolution continuous-flow fluorescence measurements at the ensemble-averaged level. The rate of this conformational change is much faster than that of DNA synthesis but is significantly reduced for noncomplementary dNTPs, as revealed by single-molecule measurements. The high level of selectivity of incoming dNTPs pertinent to this conformational change is a major contributor to replicative fidelity.

Sensitive bioanalysis—combining single-molecule spectroscopy with mono-labeled self-quenching probes

Fluorescence single-molecule spectroscopy is an appropriate tool for modern bioanalysis. This technique enables the development of ultra sensitive assays, especially when combined with self-quenching probes. In this review we report novel DNA, enzyme, and antibody assays based on mono-labeled fluorescent probes that are quenched by photoinduced electron transfer (PET).

Development of a technique for the investigation of folding dynamics of single proteins for extended time periods

A technique was developed for the detection of fluorescence signals from free single molecules for extended time periods and was applied to the characterization of the unfolded states of iso-1-cytochrome c (cyt c). Protein molecules labeled with fluorescent dye were slowly injected into a capillary at concentrations that allow for the observation of one molecule at a time. A laser was introduced into the capillary coaxially, and the fluorescence was imaged as traces by using a lens with a large focal depth and wide field of view. Thus, the traces reflect the time-dependent changes in the fluorescence signals from single proteins. Cyt c was labeled with Alexa Fluor 532 at the C-terminal cysteine (cyt c-Alexa). In bulk experiments, cyt c-Alexa was shown to possess different fluorescence intensity for the native state, the unfolded state (U), and the intermediate state. Single-molecule traces of cyt c-Alexa were recorded by using the device. Intensity histograms of the traces revealed two distributions with broad and narrow widths, which were interpreted to correspond to the U and intermediate state, respectively, observed in the bulk measurements. The broad width of the U suggested the existence of a relatively slow conformational dynamics, which might be consistent with the correlation time ( approximately 15 ms) estimated from the traces assignable to the U. The technique was expected to reveal dynamics of proteins along the folding processes without artifacts caused by immobilization.

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.

Evolution equation of phenotype distribution: general formulation and application to error catastrophe

An equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics. The equation is solved by using the singular perturbation method, and assuming that the number of bases in the genetic sequence is large. Applying the equation to the mutation-selection model by Eigen provides the critical mutation rate for the error catastrophe. Phenotypic fluctuation of clones (individuals sharing the same gene) is introduced into this evolution equation. With this formalism, it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations, i.e., noise can enhance robustness of a fitted state to mutation. Our formalism is systematic and general, while approximations to derive more tractable evolution equations are also discussed.

Network analysis of protein dynamics

The network paradigm is increasingly used to describe the topology and dynamics of complex systems. Here, we review the results of the topological analysis of protein structures as molecular networks describing their small-world character, and the role of hubs and central network elements in governing enzyme activity, allosteric regulation, protein motor function, signal transduction and protein stability. We summarize available data how central network elements are enriched in active centers and ligand binding sites directing the dynamics of the entire protein. We assess the feasibility of conformational and energy networks to simplify the vast complexity of rugged energy landscapes and to predict protein folding and dynamics. Finally, we suggest that modular analysis, novel centrality measures, hierarchical representation of networks and the analysis of network dynamics will soon lead to an expansion of this field.

Fluorescence probe of Trp-cage protein conformation in solution and in gas phase

Measurements of protein unfolding in the absence of solvent, when combined with unfolding studies in solution, offer a unique opportunity to measure the effects of solvent on protein structure and dynamics. The experiments presented here rely on the fluorescence of an attached dye to probe the local conformational dynamics through interactions with a Trp residue and fields originating on charge sites. We present fluorescence measurements of thermal fluctuations accompanying conformational change of a miniprotein, Trp-cage, in solution and in gas phase. Molecular dynamics (MD) simulations are performed as a function of temperature, charge state, and charge location to elucidate the dye-protein conformational dynamics leading to the changes in measured fluorescence. The results indicate that the stability of the unsolvated protein is dominated by hydrogen bonds. Substituting asparagine for aspartic acid at position 9 results in a dramatic alteration of the solution unfolding curve, indicating that the salt bridge involving Lys8, Asp9, and Arg16 (+ - +) is essential for Trp-cage stability in solution. In contrast, this substitution results in minor changes in the unfolding curve of the unsolvated protein, showing that hydrogen bonds are the major contributor to the stability of Trp-cage in gas phase. Consistent with this hypothesis, the decrease in the number of hydrogen bonds with increasing temperature indicated by MD simulations agrees reasonably well with the experimentally derived enthalpies of conformational change. The simulation results display relatively compact conformations compared with NMR structures that are generally consistent with experimental results. The measured unfolding curves of unsolvated Trp-cage ions are invariant with the acetonitrile content of the solution from which they are formed, possibly as a result of conformational relaxation during or after desolvation. This work demonstrates the power of combined solution and gas-phase studies and of single-point mutations to identify specific noncovalent interactions which contribute to protein-fold stability. The combination of experiment and simulation is particularly useful because these approaches yield complementary information which can be used to deduce the details of structural changes of proteins in the gas phase.

Biochemistry. Photosynthesis from the protein's perspective

Slow motions of proteins modulate electron-transfer rates during the early stages of photosynthesis.

A single-quantum methyl 13C-relaxation dispersion experiment with improved sensitivity

A pulse sequence is described for recording single-quantum (13)C-methyl relaxation dispersion profiles of (13)C-selectively labeled methyl groups in proteins that offers significant improvements in sensitivity relative to existing approaches where initial magnetization derives from (13)C polarization. Sensitivity gains in the new experiment are achieved by making use of polarization from (1)H spins and (1)H --> (13)C --> (1)H type magnetization transfers. Its utility has been established by applications involving three different protein systems ranging in molecular weight from 8 to 28 kDa, produced using a number of different selective labeling approaches. In all cases exchange parameters from both (13)C-->(1)H and (1)H --> (13)C --> (1)H classes of experiment are in good agreement, with gains in sensitivity of between 1.7 and 4-fold realized using the new scheme.

Single molecule photon emission statistics for non-Markovian blinking models

The statistics of photon emission from a single molecule under continuous wave excitation are considered. In particular, we study stochastic model systems where photon emission rates evolve in time with non-Markovian dynamics. Our calculations are based on the recently introduced generalized optical Bloch equation (GBE) formalism, but with numerical complications beyond those seen in previous Markovian stochastic models. A spectral representation is introduced to facilitate the numerical solution of the GBE equations for these more challenging cases.

Exact analytical evaluation of time dependent transmission coefficient from the method of reactive flux for an inverted parabolic barrier

The paper demonstrates an elegant way of combining the normal mode analysis and the method of reactive flux to evaluate the time dependent transmission coefficient for a classical particle coupled to a set of harmonic oscillators, surmounting a one dimensional barrier. The author's analysis reproduces the results of Kohen and Tannor [J. Chem. Phys. 103, 6013 (1995)] and Bao [J. Chem. Phys. 124, 114103 (2006)]. Moreover the use of normal mode analysis has a better physical meaning.

Computational Study of a Single Surface-Immobilized Two-Stranded Coiled-Coil Polypeptide

The equilibrium structure and dynamics of a two-stranded coiled-coil polypeptide are investigated via Langevin dynamics simulations. An off-lattice model of the polypeptide chain is employed, which gives rise to a well-defined helical dimer native state and two-state folding kinetics. The behavior of the freely diffusing and surface-immobilized polypeptide is studied under different surface and denaturation conditions. The effect of surface immobilization on the distributions of structural and dynamical properties is considered in detail. The relationship between the simulation results and recent single-molecule fluorescence resonance energy transfer experiments performed on the two-stranded coiled-coil from the yeast transcription factor GCN4 (Jia et al. Chem. Phys. 1999, 247, 69; Talaga et al. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 13021) is discussed.

Extracting the Time Scales of Conformational Dynamics from Single-Molecule Single-Photon Fluorescence Statistics

The quenching rate of a fluorophore attached to a macromolecule can be rather sensitive to its conformational state. The decay of the corresponding fluorescence lifetime autocorrelation function can therefore provide unique information on the time scales of conformational dynamics. The conventional way of measuring the fluorescence lifetime autocorrelation function involves evaluating it from the distribution of delay times between photoexcitation and photon emission. However, the time resolution of this procedure is limited by the time window required for collecting enough photons in order to establish this distribution with sufficient signal-to-noise ratio. Yang and Xie have recently proposed an approach for improving the time resolution, which is based on the argument that the autocorrelation function of the delay time between photoexcitation and photon emission is proportional to the autocorrelation function of the square of the fluorescence lifetime [Yang, H.; Xie, X. S. J. Chem. Phys. 2002, 117, 10965]. In this paper, we show that the delay-time autocorrelation function is equal to the autocorrelation function of the square of the fluorescence lifetime divided by the autocorrelation function of the fluorescence lifetime. We examine the conditions under which the delay-time autocorrelation function is approximately proportional to the autocorrelation function of the square of the fluorescence lifetime. We also investigate the correlation between the decay of the delay-time autocorrelation function and the time scales of conformational dynamics. The results are demonstrated via applications to a two-state model and an off-lattice model of a polypeptide.

Picosecond Multidimensional Fluorescence Spectroscopy: A Tool to Measure Real-time Protein Dynamics During Function†

Advanced multidimensional time-correlated single photon counting (mdTCSPC) and picosecond time-resolved fluorescence in combination with site-directed fluorescence labeling are valuable tools to study the properties of membrane protein surface segments on the pico- to nanoseconds time scale. Time-resolved fluorescence anisotropy changes of protein bound fluorescent probes reveal changes in protein dynamics and steric restriction. In addition, the change in fluorescence lifetime and intensity of the covalently bound fluorescent dye is indicative of environmental changes at the protein surface. In this study, we have measured the changes in fluorescence lifetime traces of the fluorescent dye fluorescein covalently bound to the first cytoplasmic loop of bacteriorhodopsin (bR) after light activation of protein function. The fluorescence is excited by a picosecond laser pulse. The retinylidene chromophore of bR is light-activated by a 10 ns laser pulse, which in turn triggers recording of a sequence of fluorescence lifetime traces in the mdTCSPC-module. The fluorescence decay changes upon protein function occur predominantly in the 100 ps time range. The kinetics of these changes shows two transitions between three intermediate states in the second part of the bR photocycle. Correlation with photocycle kinetics allows for the determination of reaction intermediates at the proteins surface which are coupled to changes in the retinal binding pocket.

A close look at fluorescence quenching of organic dyes by tryptophan

Understanding fluorescence quenching processes of organic dyes by biomolecular compounds is of fundamental importance for in-vitro and in-vivo fluorescence studies. It has been reported that the excited singlet state of some oxazine and rhodamine derivatives is efficiently and almost exclusively quenched by the amino acid tryptophan (Trp) and the DNA base guanine via photoinduced electron transfer (PET). We present a detailed analysis of the quenching interactions between the oxazine dye MR121 and Trp in aqueous buffer. Steady-state and time-resolved fluorescence spectroscopy, together with fluorescence correlation spectroscopy (FCS), reveal three contributing quenching mechanisms: 1) diffusion-limited dynamic quenching with a bimolecular quenching rate constant k(d) of 4.0 x 10(9) s(-1) M(-1), 2) static quenching with a bimolecular association constant K(s) of 61 M(-1), and 3) a sphere-of-action contribution to static quenching described by an exponential factor with a quenching constant lambda of 22 M(-1). The latter two are characterized as nonfluorescent complexes, formed with approximately 30 % efficiency upon encounter, that are stable for tens of nanoseconds. The measured binding energy of 20-30 kJ mol(-1) is consistent with previous estimates from molecular dynamics simulations that proposed stacked complexes due to hydrophobic forces. We further evaluate the influence of glycerol and denaturant (guanidine hydrochloride) on the formation and stability of quenched complexes. Comparative measurements performed with two other dyes, ATTO 655 and Rhodamine 6G show similar results and thus demonstrate the general applicability of utilizing PET between organic dyes and Trp for the study of conformational dynamics of biopolymers on sub-nanometer length and nanosecond time-scales.

A theoretical analysis of rate constants and kinetic isotope effects corresponding to different reactant valleys in lactate dehydrogenase

In some enzymatic systems large conformational changes are coupled to the chemical step, in such a way that dispersion of rate constants can be observed in single-molecule experiments, each corresponding to the reaction from a different reactant valley. Under this perspective here we present a computational study of pyruvate to lactate transformation catalyzed by lactate dehydrogenase. The reaction consists of a hydride transfer and a proton transfer that seem to take place concertedly. The degree of asynchronicity and the energy barrier depend on the particular starting reactant valley. In order to estimate rate constants we used a free energy perturbation technique adapted to follow the intrinsic reaction coordinate for several possible reaction paths. Tunneling effects are also obtained with a slightly modified version of the ensemble-averaged variational transition state theory with multidimensional tunneling contributions. According to our results the closure of the active site by means of a flexible loop can lead to the formation of different reactant complexes displaying different features in the disposition of some key residues (such as Arg109), interactions with the substrate and number of water molecules in the active site. The chemical step of the reaction takes place with a different reaction rate from each of these complexes. Finally, primary kinetic isotope effects for replacement of the transferring hydrogen of the cofactor with a deuteride are in good agreement with experimental observations, thus validating our methodology.

Model of Fluorescence Intermittency in Single Enzymes

The intermittent emission of fluorescent light from single enzymes, quantum dots, and other nanoscale systems is often characterized by statistical correlations in the emitted signal. A one-dimensional model of such correlations in enzymes, based on a model of protein conformational fluctuations developed by Kou and Xie (Phys. Rev. Lett. 2004, 93, 180603), is formulated in the present paper in terms of the dynamics of a particle moving stochastically between "on" and "off" states under the action of fractional Gaussian noise. The model yields predictions for the short and long time behavior of the following quantities: the time correlation function, C(t), of the fluctuations of the signal intensity, the distribution, f(t), of time intervals between intensity fluctuations, and the Mandel parameter, Q(t), describing the extent of bunching or anti-bunching in the signal. At short times, C(t) and f(t) are found to decay exponentially, while, at long times, they are found to decay as power laws, the exponents being functions solely of the nature of the temporal correlations in the noise. The results are in good qualitative agreement with results from single-molecule experiments on fluorescence intermittency in the enzyme cholesterol oxidase carried out by Xie and co-workers (Science 1998, 282, 1877). The Mandel parameter, Q(t), for this model is positive at short and long times, indicating super-Poisson statistics in these limits, consistent with bunching of the fluorescent signal.

Dwell-time distribution analysis of polyprotein unfolding using force-clamp spectroscopy

Using the recently developed single molecule force-clamp technique we quantitatively measure the kinetics of conformational changes of polyprotein molecules at a constant force. In response to an applied force of 110 pN, we measure the dwell times of 1647 unfolding events of individual ubiquitin modules within each protein chain. We then establish a rigorous method for analyzing force-clamp data using order statistics. This allows us to test the success of a history-independent, two-state model in describing the kinetics of the unfolding process. We find that the average unfolding trajectory is independent of the number of protein modules N in each trajectory, which varies between 3 and 12 (the engineered protein length), suggesting that the unfolding events in each chain are uncorrelated. We then derive a binomial distribution of dwell times to describe the stochastic dynamics of protein unfolding. This distribution successfully describes 81% of the data with a single rate constant of alpha = 0.6 s(-1) for all N. The remainder of the data that cannot be accounted for suggests alternative unfolding barriers in the energy landscape of the protein. This method investigates the statistical features of unfolding beyond the average measurement of a single rate constant, thus providing an attractive alternative for measuring kinetics by force-clamp spectroscopy.

Docking and molecular dynamics simulation of the Azurin–Cytochrome c551 electron transfer complex

We coupled protein-protein docking procedure with molecular dynamics (MD) simulation to investigate the electron transfer (ET) complex Azurin-Cytochrome c551 whose transient character makes difficult a direct experimental investigation. The ensemble of complexes generated by the docking algorithm are filtered according to both the distance between the metal ions in the redox centres of the two proteins and to the involvement of suitable residues at the interface. The resulting best complex (BC) is characterized by a distance of 1.59 nm and involves Val23 and Ile59 of Cytochrome c551. The ET properties have been evaluated in the framework of the Pathways model and compared with experimental data. A 60 ns long MD simulation, carried on at full hydration, evidenced that the two protein molecules retain their mutual spatial positions upon forming the complex. An analysis of the ET properties of the complex, monitored at regular time intervals, has revealed that several different ET paths are possible, with the occasional intervening of water molecules. Furthermore, the temporal evolution of the geometric distance between the two redox centres is characterized by very fast fluctuations around an average value of 1.6 nm, with periodic jumps at 2 nm with a frequency of about 70 MHz. Such a behaviour is discussed in connection with a nonlinear dynamics of protein systems and its possible implications in the ET process are explored.

Detectors for single-molecule fluorescence imaging and spectroscopy

Single-molecule observation, characterization and manipulation techniques have recently come to the forefront of several research domains spanning chemistry, biology and physics. Due to the exquisite sensitivity, specificity, and unmasking of ensemble averaging, single-molecule fluorescence imaging and spectroscopy have become, in a short period of time, important tools in cell biology, biochemistry and biophysics. These methods led to new ways of thinking about biological processes such as viral infection, receptor diffusion and oligomerization, cellular signaling, protein-protein or protein-nucleic acid interactions, and molecular machines. Such achievements require a combination of several factors to be met, among which detector sensitivity and bandwidth are crucial. We examine here the needed performance of photodetectors used in these types of experiments, the current state of the art for different categories of detectors, and actual and future developments of single-photon counting detectors for single-molecule imaging and spectroscopy.