Protein Conformational Dynamics Probed by Single-Molecule Electron Transfer

Khinchin theorem and anomalous diffusion

A recent Letter [M. H. Lee, Phys. Rev. Lett. 98, 190601 (2007)] has called attention to the fact that irreversibility is a broader concept than ergodicity, and that therefore the Khinchin theorem [A. I. Khinchin, (Dover, New York, 1949)] may fail in some systems. In this Letter we show that for all ranges of normal and anomalous diffusion described by a generalized Langevin equation the Khinchin theorem holds.

Stochastic analysis of stepwise fluorescence quenching reactions on single-walled carbon nanotubes: single molecule sensors

The 1D quantum confinement of photogenerated excitons in single-walled carbon nanotubes (SWNT) can amplify the detection of molecular adsorption to where single-molecule discrimination is realizable, even from within living cells and tissues. Toward this aim, we present a type 1 collagen film, similar to those used as 3D cell scaffolds for tissue engineering, containing embedded SWNT capable of reporting single-molecule adsorption of quenching molecules. We utilize hidden Markov modeling to link single-molecule adsorption events to rate constants for H2O2, H+, and Fe(CN)6(3-). Among the three kinds of reactant molecules studied, H2O2 has the highest quenching equilibrium constant of 1.59 at 20 microM, whereas H+ is so insensitive that a similar equilibrium constant is achieved only with a concentration of 0.1 M (pH 1). The results were self-consistent because reverse (unquenching) rate constants (600 micros(-1) for H2O2, 1130 micros(-1) for H+ and 4000 micros(-1) for Fe(CN)6(3-)) were observed to be concentration-independent and the forward (quenching) rate constants varied monotonically with concentration. The quenching rate constants also increased with an increase in the redox potential of the quencher, indicating that electron transfer increases the adsorption equilibrium constant on the nanotube surface and, hence, the dwell time of the quencher. These developments provide the material, analytical, and mechanistic groundwork for SWNT to function as single-molecule stochastic biosensors.

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.

Principal component analysis for protein folding dynamics

Protein folding is considered here by studying the dynamics of the folding of the triple beta-strand WW domain from the Formin-binding protein 28. Starting from the unfolded state and ending either in the native or nonnative conformational states, trajectories are generated with the coarse-grained united residue (UNRES) force field. The effectiveness of principal components analysis (PCA), an already established mathematical technique for finding global, correlated motions in atomic simulations of proteins, is evaluated here for coarse-grained trajectories. The problems related to PCA and their solutions are discussed. The folding and nonfolding of proteins are examined with free-energy landscapes. Detailed analyses of many folding and nonfolding trajectories at different temperatures show that PCA is very efficient for characterizing the general folding and nonfolding features of proteins. It is shown that the first principal component captures and describes in detail the dynamics of a system. Anomalous diffusion in the folding/nonfolding dynamics is examined by the mean-square displacement (MSD) and the fractional diffusion and fractional kinetic equations. The collisionless (or ballistic) behavior of a polypeptide undergoing Brownian motion along the first few principal components is accounted for.

Correlation between square of electron tunneling matrix element and donor-acceptor distance in fluctuating protein media

Correlation between fluctuations of the square of electron tunneling matrix element T_DA² and the donor-acceptor distance R_DA in the electron transfer (ET) reaction from bacteriopheophytin anion to the primary quinone of the reaction center in the photosynthetic bacteria Rhodobacter sphaeroides is investigated by a combined study of molecular dynamics simulations of the protein conformation fluctuation and quantum chemical calculations. We adopted two kinds of R_DA ; edge-to-edge distance R_EE and center-to-center distance R_CC . The value of T_DA² distributed over more than 5 orders of magnitude and the fluctuation of the value of R_DA distributed over more than 1.8 Å for the 10⁶ instantaneous conformations of 1 ns simulation. We made analysis of the time-averaged correlation step by step as follows. We divide the 10⁶ simulation data into 1000/t parts of small data set to obtain the averaged data points of <T_DA²> _t and <R_EE > _t or <R_CC > _t . Plotting the 1000/t sets of log₁₀ <T_DA²> _t as a function of <R_EE > _t or <R_CC > _t , we made a principal coordinate analysis for these distributions. The slopes <β_E > _t and <β_C > _t of the primary axis are very large at small value of t and they are decreased considerably as t becomes large. The ellipticity for the distribution of <T_DA²> _tvs <R_EE > _t which can be a measure for the degree of correlation became very small when t is large, while it does not hold for the distribution of <T_DA²> _tvs <R_CC > _t . These results indicate that only the correlation between <T_DA²> _t and <R_EE > _t for large t satisfies the well-known linear relation ("Dutton law"), although the slope is larger than the original value 1.4 Å^-1. Based on the present result, we examined the analysis of the dynamic disorder by means of the single-molecule spectroscopy by Xie and co-workers with use of the "Dutton law".

Fractional Langevin equation: overdamped, underdamped, and critical behaviors

The dynamical phase diagram of the fractional Langevin equation is investigated for a harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) alpha_{c}=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase, (ii) alpha_{R}=0.441... marks a transition to a resonance phase when an external oscillating field drives the system, and (iii) alpha_{chi_{1}}=0.527... and (iv) alpha_{chi_{2}}=0.707... mark transitions to a double-peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over and underdamped regimes, with or without resonances, show behaviors different from normal.

Monte carlo simulations of protein assembly, disassembly, and linear motion on DNA

We use Monte Carlo simulations to analyze the simultaneous interactions of multiple proteins to a long DNA molecule. We study the time dependence of protein organization on DNA for different regimes that comprise (non)cooperative sequence-independent protein assembly, dissociation, and linear motion. A range of different behaviors is observed for the dynamics, final coverage, and cluster size distributions. We observe that the DNA substrate is almost never completely covered by protein when taking into account only (non)cooperative binding, because gaps remain on the substrate that are smaller than the binding site size of the protein. Due to these gaps, the apparent binding size of a protein during noncooperative binding can be overestimated by up to 30%. During dissociation of cooperatively bound proteins, the dissociation curve can be exponentially shaped even when allowing only end-dependent dissociation. We discuss the potential of our method for the analysis of a number of single-molecule experiments, for example, the binding of the DNA-repair proteins RecA and Rad51 to DNA.

A theoretical study of the catalytic mechanism of formate dehydrogenase

A theoretical study of the hydride transfer between formate anion and nicotinamide adenine dinucleotide (NAD(+)) catalyzed by the enzyme formate dehydrogenase (FDH) has been carried out by a combination of two hybrid quantum mechanics/molecular mechanics techniques: statistical simulation methods and internal energy minimizations. Free energy profiles, obtained for the reaction in the enzyme active site and in solution, allow obtaining a comparative analysis of the behavior of both condensed media. Moreover, calculations of the reaction in aqueous media can be used to probe the dramatic differences between reactants state in the enzyme active site and in solution. The results suggest that the enzyme compresses the substrate and the cofactor into a conformation close to the transition structure by means of favorable interactions with the amino acid residues of the active site, thus facilitating the relative orientation of donor and acceptor atoms to favor the hydride transfer. Moreover, a permanent field created by the protein reduces the work required to reach the transition state (TS) with a concomitant polarization of the cofactor that would favor the hydride transfer. In contrast, in water the TS is destabilized with respect to the reactant species because the polarity of the solute diminishes as the reaction proceeds, and consequently the reaction field, which is created as a response to the change in the solute polarity, is also decreased. Therefore protein structure is responsible of both effects; substrate preorganization and TS stabilization thus diminishing the activation barrier. Because of the electrostatic features of the catalyzed reaction, both media preferentially stabilize the ground-state, thus explaining the small rate constant enhancement of this enzyme, but FDH does so to a much lower extent than aqueous solution. Finally, a good agreement between experimental and theoretical kinetic isotope effects is found, thus giving some credit to our results.

Average electron tunneling route of the electron transfer in protein media

We present a new theoretical method to determine and visualize the average tunneling route of the electron transfer (ET) in protein media. In this, we properly took into account the fluctuation of the tunneling currents and the quantum-interference effect. The route was correlated with the electronic factor <TDA(2)> in the case of ET by the elastic tunneling mechanism. We expanded <TDA(2)> by the interatomic tunneling currents <Jab(2)>'s. Incorporating the quantum-interference effect into the mean-square interatomic tunneling currents, denoted as <Jab(2)>, we could express <TDA(2)> as a sum of variant Planck's over 2pi(2)<Jab(2)>. Drawing the distribution of <Jab(2)> on the protein structure, we obtain the <Jab(2)> map which visually represents which parts of bonds and spaces most significantly contribute to <TDA(2)>. We applied this method to the ET from the bacteriopheophytin anion to the primary quinone in the bacterial photosynthetic reaction center of Rhodobacter sphaeroides. We obtained <Jab(2)>'s by a combined method of molecular dynamics simulations and quantum chemical calculations. In calculating <Jab(2)>, we found that much destructive interference works among the interatomic tunneling currents even after taking the average. We drew the <Jab(2)> map by a pipe model where atoms a and b are connected by a pipe with width proportional to the magnitude of <Jab(2)>. We found that two groups of <Jab(2)>'s, which are mutually coupled with high correlation in each group, have broad pipes and form the average tunneling routes, called Trp route and Met route. Each of the two average tunneling routes is composed of a few major pathways in the Pathways model which are fused at considerable part to each other. We also analyzed the average tunneling route for the ET by the inelastic tunneling mechanism.

Protein modification for single molecule fluorescence microscopy

Single molecule methods have emerged as a powerful new tool for exploring biological phenomena. We provide a brief overview of the scope of current experiments and assess the limitations of both fluorescent labels and the means to achieve protein modification for single molecule microscopy.

Protein dynamics from a NMR perspective: networks of coupled rotators and fractional Brownian dynamics

Nuclear magnetic resonance (NMR) has proven to be the most valuable tool for investigating internal dynamics of proteins. In this perspective, the interpretation of NMR relaxation data eventually relies on a model of the motions. In this article, we propose to compare two radically different approaches that aim at describing internal dynamics in proteins. It is shown that the correlation functions predicted by a network of coupled rotators can be interpreted in terms of a heuristic approach based on fractional Brownian dynamics for each of the vectors in the network. Our results are interpreted in terms of the probability distributions of relaxation modes in both processes, the median of which turns out to be the relevant quantity for the comparison of both models.

Do-it-yourself guide: how to use the modern single-molecule toolkit

Single-molecule microscopy has evolved into the ultimate-sensitivity toolkit to study systems from small molecules to living cells, with the prospect of revolutionizing the modern biosciences. Here we survey the current state of the art in single-molecule tools including fluorescence spectroscopy, tethered particle microscopy, optical and magnetic tweezers, and atomic force microscopy. We also provide guidelines for choosing the right approach from the available single-molecule toolkit for applications as diverse as structural biology, enzymology, nanotechnology and systems biology.

High-linearity analog-to-digital acquisition board for photon-timing applications

Nowadays a wide range of scientific applications require the detection of very weak and fast luminescence signals in the time domain. Single molecule spectroscopy, diffuse optical tomography, time-resolved emission spectra are only a few examples among them. Advanced time-correlated single photon counting, that relies on multidetector and multidimensional acquisition, reveals itself as a powerful technique to gather deeper insight into various photochemical and biological processes. In this paper we present a high-speed, high-linearity A/D acquisition board that is designed as a building block for a compact, low-cost, multidetector acquisition system for photon-timing experiments. The prototype has been experimentally tested and the results fully comply the goal design specifications.

Subdiffusion in peptides originates from the fractal-like structure of configuration space

Molecular dynamics simulation of oligopeptide chains reveals configurational subdiffusion at equilibrium extending from 10(-12) to 10(-8) s. Trap models, involving a random walk with a distribution of waiting times, cannot account for the subdiffusion, which is found rather to arise from the fractal-like structure of the accessible configuration space.

Continuous-time random walk for open systems: fluctuation theorems and counting statistics

We consider continuous-time random walks (CTRW) for open systems that exchange energy and matter with multiple reservoirs. Each waiting time distribution (WTD) for times between steps is characterized by a positive parameter alpha , which is set to alpha=1 if it decays at least as fast as t{-2} at long times and therefore has a finite first moment. A WTD with alpha<1 decays as t{-alpha-1} . A fluctuation theorem for the trajectory quantity R , defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory, holds for any CTRW. However, R can be identified as a trajectory entropy change only if the WTDs have alpha=1 and satisfy separability (also called "direction time independence"). For nonseparable WTDs with alpha=1 , R can only be identified as a trajectory entropy change at long times, and a fluctuation theorem for the entropy change then only holds at long times. For WTDs with 0<alpha<1 no meaningful fluctuation theorem can be derived. We also show that the (experimentally accessible) nth moments of the energy and matter transfers between the system and a given reservoir grow as t{nalpha} at long times.

Tunable blinking kinetics of cy5 for precise DNA quantification and single-nucleotide difference detection

Fluorescence correlation spectroscopy (FCS) can resolve the intrinsic fast-blinking kinetics (FBKs) of fluorescent molecules that occur on the order of microseconds. These FBKs can be heavily influenced by the microenvironments in which the fluorescent molecules are contained. In this work, FCS is used to monitor the dynamics of fluorescence emission from Cy5 labeled on DNA probes. We found that the FBKs of Cy5 can be tuned by having more or less unpaired guanines (upG) and thymines (upT) around the Cy5 dye. The observed FBKs of Cy5 are found to predominantly originate from the isomerization and back-isomerization processes of Cy5, and Cy5-nucleobase interactions are shown to slow down these processes. These findings lead to a more precise quantification of DNA hybridization using FCS analysis, in which the FBKs play a major role rather than the diffusion kinetics. We further show that the alterations of the FBKs of Cy5 on probe hybridization can be used to differentiate DNA targets with single-nucleotide differences. This discrimination relies on the design of a probe-target-probe DNA three-way-junction, whose basepairing configuration can be altered as a consequence of a single-nucleotide substitution on the target. Reconfiguration of the three-way-junction alters the Cy5-upG or Cy5-upT interactions, therefore resulting in a measurable change in Cy5 FBKs. Detection of single-nucleotide variations within a sequence selected from the Kras gene is carried out to validate the concept of this new method.

DNA-ligand binding mode discrimination by characterizing fluorescence resonance energy transfer through lifetime measurements with picosecond resolution

We describe a method for distinguishing between minor groove binders and base intercalators that is based on measurements of the fluorescence lifetime of a donor (D) in the presence of an acceptor (A). The D-A pair is separated by a short double helix DNA with which the ligands interact. By plotting the D fluorescence lifetime as a function of the ligand-to-base pair concentration ratio we find a clear signature that distinguishes between the two binding mechanisms: minor groove binding induces an asymptotic decrease of the D fluorescence lifetime, while intercalation gives a monotonically increasing lifetime and the appearance of an additional short lifetime. We assayed Quinacrine, Hoechst and 4'-6'diamidine-2-phenyl indole, which in control experiments performed on oligodeoxyribonucleotides (oligos) lacking the A are demonstrated not to interfere with the D fluorescence. The changes in fluorescence lifetimes measured in the case of dual-labeled oligos are thus caused by structural changes in the DNA that modify the D-A distance. The appearance of the short-lived transient in the fluorescence decay of Ds attached to dual-labeled oligos upon binding of an intercalator can be interpreted as denaturation.

Distinct and long-lived activity states of single enzyme molecules

Individual enzyme molecules have been observed to possess discrete and different turnover rates due to the presence of long-lived activity states. These stable activity states are thought to result from different molecular conformations or post-translational modifications. The distributions in kinetic activity observed in previous studies were obtained from small numbers of single enzyme molecules. Due to this limitation, it has not been possible to fully characterize the different kinetic and equilibrium binding parameters of single enzyme molecules. In this paper, we analyze hundreds of single beta-galactosidase molecules simultaneously; using a high-density array of 50,000 fL-reaction chambers, we confirm the presence of long-lived kinetic states within a population of enzyme molecules. Our analysis has isolated the source of kinetic variability to kcat. The results explain the kinetic variability within enzyme molecule populations and offer a deeper understanding of the unique properties of single enzyme molecules. Gaining a more fundamental understanding of how individual enzyme molecules work within a population should provide insight into how they affect downstream biochemical processes. If the results reported here can be generalized to other enzymes, then the stochastic nature of individual enzyme molecule kinetics should have a substantial impact on the overall metabolic activity within a cell.

Single-molecule biophysics: at the interface of biology, physics and chemistry

Single-molecule methods have matured into powerful and popular tools to probe the complex behaviour of biological molecules, due to their unique abilities to probe molecular structure, dynamics and function, unhindered by the averaging inherent in ensemble experiments. This review presents an overview of the burgeoning field of single-molecule biophysics, discussing key highlights and selected examples from its genesis to our projections for its future. Following brief introductions to a few popular single-molecule fluorescence and manipulation methods, we discuss novel insights gained from single-molecule studies in key biological areas ranging from biological folding to experiments performed in vivo.

The enzyme mechanism of nitrite reductase studied at single-molecule level

A generic method is described for the fluorescence "readout" of the activity of single redox enzyme molecules based on Förster resonance energy transfer from a fluorescent label to the enzyme cofactor. The method is applied to the study of copper-containing nitrite reductase from Alcaligenes faecalis S-6 immobilized on a glass surface. The parameters extracted from the single-molecule fluorescence time traces can be connected to and agree with the macroscopic ensemble averaged kinetic constants. The rates of the electron transfer from the type 1 to the type 2 center and back during turnover exhibit a distribution related to disorder in the catalytic site. The described approach opens the door to single-molecule mechanistic studies of a wide range of redox enzymes and the precise investigation of their internal workings.

Critical exponent of the fractional Langevin equation

We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent alpha(c)=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase. The critical exponent alpha(R)=0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the underdamped, the overdamped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.

A network model to correlate conformational change and the impedance spectrum of single proteins

Integrated nanodevices based on proteins or biomolecules are attracting increasing interest in today's research. In fact, it has been shown that proteins such as azurin and bacteriorhodopsin manifest some electrical properties that are promising for the development of active components of molecular electronic devices. Here we focus on two relevant kinds of protein: bovine rhodopsin, prototype of G-protein-coupled-receptor (GPCR) proteins, and the enzyme acetylcholinesterase (AChE), whose inhibition is one of the most qualified treatments of Alzheimer's disease. Both these proteins exert their function starting with a conformational change of their native structure. Our guess is that such a change should be accompanied with a detectable variation of their electrical properties. To investigate this conjecture, we present an impedance network model of proteins, able to estimate the different impedance spectra associated with the different configurations. The distinct types of conformational change of rhodopsin and AChE agree with their dissimilar electrical responses. In particular, for rhodopsin the model predicts variations of the impedance spectra up to about 30%, while for AChE the same variations are limited to about 10%, which supports the existence of a dynamical equilibrium between its native and complexed states.

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.

Two-dimensional reaction free energy surfaces of catalytic reaction: effects of protein conformational dynamics on enzyme catalysis

We introduce a two-dimensional (2D) multisurface reaction free energy description of the catalytic cycle that explicitly connects the recently observed multi-time-scale conformational dynamics as well as dispersed enzymatic kinetics to the classical Michaelis-Menten equation. A slow conformational motion on a collective enzyme coordinate Q facilitates the catalytic reaction along the intrinsic reaction coordinate X, providing a dynamic realization of Pauling's well-known idea of transition-state stabilization. The catalytic cycle is modeled as transitions between multiple displaced harmonic wells in the XQ space representing different states of the cycle, which is constructed according to the free energy driving force of the cycle. Subsequent to substrate association with the enzyme, the enzyme-substrate complex under strain exhibits a nonequilibrium relaxation toward a new conformation that lowers the activation energy of the reaction, as first proposed by Haldane. The chemical reaction in X is thus enslaved to the down hill slow motion on the Q surface. One consequence of the present theory is that, in spite of the existence of dispersive kinetics, the Michaelis-Menten expression of the catalysis rate remains valid under certain conditions, as observed in recent single-molecule experiments. This dynamic theory builds the relationship between the protein conformational dynamics and the enzymatic reaction kinetics and offers a unified description of enzyme fluctuation-assisted catalysis.

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.