Two-dimensional fluorescence resonance energy transfer as a probe for protein folding: a theoretical study

We describe a two-dimensional (2D), four-color fluorescence resonance energy transfer (FRET) scheme, in which the conformational dynamics of a protein is followed by simultaneously observing the FRET signal from two different donor-acceptor pairs. For a general class of models that assume Markovian conformational dynamics, we relate the properties of the emission correlation functions to the rates of elementary kinetic steps in the model. We further use a toy folding model that treats proteins as chains with breakable cross-links to examine the relationship between the cooperativity of folding and FRET data and to establish what additional information about the folding dynamics can be gleaned from 2D, as opposed to one-dimensional FRET experiments. We finally discuss the potential advantages of the four-color FRET over the three-color FRET technique.     

Protein structure and dynamics from single-molecule fluorescence resonance energy transfer

The pros and cons of single-molecule vs ensemble-averaged fluorescence resonance energy transfer (FRET) experiments, performed on proteins, are explored with the help of Langevin dynamics simulations. An off-lattice model of the polypeptide chain is employed, which gives rise to a well-defined native state and two-state folding kinetics. A detailed analysis of the distribution of the donor-acceptor distance is presented at different points along the denaturation curve, along with its dependence on the averaging time window. We show that unique information on the correlation between structure and dynamics, which can only be obtained from single-molecule experiments, is contained in the correlation between the donor-acceptor distance and its displacement. The latter is shown to provide useful information on the free energy landscape of the protein, which is complementary to that obtained from the distribution of donor-acceptor distances.     

Identifiability of models for intramolecular two-state excited-state processes with added quencher and coupled species-dependent rotational diffusion

The parameters describing the kinetics of excited-state processes can possibly be recovered by analysis of the fluorescence decay surface measured as a function of the experimental variables. The identifiability analysis of a photophysical model assuming errorless time-resolved fluorescence data can verify whether the model parameters can be determined and may lead to the minimal experimental conditions under which this is possible. In this work, we used the method of similarity transformation to investigate the identifiability of three kinetic models utilized to describe the time-resolved fluorescence of reversible intramolecular two-state excited-state processes in isotropic environments: (1) model without added quencher, (2) model with added quencher, (3) model with added quencher coupled with species-dependent rotational diffusion described by Brownian reorientation. Without a priori information, model 1 is not identifiable. For model 2, two sets of quenching rate constants and combinations of excited-state deactivation/exchange rate constants are possible, but they cannot be allocated to a specific excited-state species. For both sets, upper and lower limits on the excited-state deactivation/exchange rate constants can be obtained. For model 3, both spherically and cylindrically symmetric rotors, with no change in the principal axes of rotation in the latter, are considered. The fluorescence delta-response functions I(parallel)(t) and I(perpendicular)(t), for fluorescence polarized parallel and perpendicular, respectively, to the electric vector of linearly polarized excitation, are used to define the sum S(t) identically equal to I(parallel)(t) + 2 I(perpendicular)(t) and the difference D(t) identically equal to I(parallel)(t) - I(perpendicular)(t). The identifiability analysis is performed using the S(t) and D(t) functions. Also for model 3, two sets of kinetic parameters (i.e., quenching rate constants, combinations of deactivation/exchange rate constants, and rotational diffusion coefficients) exist, but these parameters cannot be assigned unequivocally to a specific species. For the three models, an infinite number of alternative spectroscopic parameters associated with excitation and emission are found.     

Extraction of kinetic information from single-molecule experiments.

We discuss two possible approaches for extracting kinetic information from single-molecule experiments. The first approach is based on computing correlation functions from measured fluorescence signals, and the second on studying the statistics of on and off times of the same fluorescence signal. We show that in both cases it is possible to extract kinetic information about the nature of intramolecular fluctuations of the single molecule. We show that for single-molecule kinetics the intramolecular fluctuations produce stochastic memory effects which lead to new dynamic features that do not exist in traditional chemical kinetics. In particular, we investigate a new type of chemical oscillations in correlation functions observed experimentally by Edman and Rigler (Proc. Natl. Acad. Sci. USA 2000, 97, 8266).     

On the relationships between kinetic schemes and two-state single molecule trajectories

Trajectories of a signal that fluctuates between two states which originate from single molecule activities have become ubiquitous. Common examples are trajectories of ionic flux through individual membrane channels and of photon counts collected from diffusion, activity, and conformational changes of biopolymers. By analyzing the trajectory, one wishes to deduce the underlying mechanism, which is usually described by a multisubstate kinetic scheme. In previous works [O. Flomenborn, J. Klafter, and A. Szabo, Biophys. J. 88, 3780 (2005); O. Flomenbom and J. Klafter, Acta Phys. Pol. B 36, 1527 (2005)], we divided kinetic schemes that generate two-state trajectories into two types: reducible schemes and irreducible schemes. A full characterization of the reducible ones was given. We showed that all the information in trajectories generated from reducible schemes is contained in the waiting time probability density functions (PDFs) of the two states. It follows that reducible schemes with the same waiting time PDFs are not distinguishable; namely, such schemes lead to identical two-state trajectories in the statistical sense. In this work, we further characterize the topologies of kinetic schemes, now of irreducible schemes, and further study two-state trajectories from the two types of scheme. We suggest various methods for extracting information about the underlying kinetic scheme from the trajectory (e.g., calculate the binned successive waiting times PDFs and analyze the ordered waiting time trajectories), and point out the advantages and disadvantages of each. We show that the binned successive waiting times PDFs are not only more robust than other functions when analyzing finite trajectories, but contain, in most cases, more information about the underlying kinetic scheme than other functions in the limit of infinitely long trajectories. For some cases, however, analyzing the ordered waiting times trajectory may supply unique information about the underlying kinetic scheme.     

Proton-coupled electron transfer in soybean lipoxygenase

The proton-coupled electron transfer reaction catalyzed by soybean lipoxygenase-1 is studied with a multistate continuum theory that represents the transferring hydrogen nucleus as a quantum mechanical wave function. The inner-sphere reorganization energy of the iron cofactor is calculated with density functional theory, and the outer-sphere reorganization energy of the protein is calculated with the frequency-resolved cavity model for conformations obtained with docking simulations. Both classical and quantum mechanical treatments of the proton donor-acceptor vibrational motion are presented. The temperature dependence of the calculated rates and kinetic isotope effects is in agreement with the experimental data. The weak temperature dependence of the rates is due to the relatively small free energy barrier arising from a balance between the reorganization energy and the reaction free energy. The unusually high deuterium kinetic isotope effect of 81 is due to the small overlap of the reactant and product proton vibrational wave functions and the dominance of the lowest energy reactant and product vibronic states in the tunneling process. The temperature dependence of the kinetic isotope effect is strongly influenced by the proton donor-acceptor distance with the dominant contribution to the overall rate. This dominant proton donor-acceptor distance is significantly smaller than the equilibrium donor-acceptor distance and is determined by a balance between the larger coupling and the smaller Boltzmann probability as the distance decreases. Thus, the proton donor-acceptor vibrational motion plays a vital role in decreasing the dominant donor-acceptor distance relative to its equilibrium value to facilitate the proton-coupled electron transfer reaction.     

Millisecond association kinetics of K+ with triazacryptand-based K+ indicators measured by fluorescence correlation spectroscopy

We recently introduced a water-soluble, long-wavelength K(+)-sensing indicator, TAC-Red, consisting of a triazacryptand K(+)-selective ionophore coupled to a xanthylium chromophore (Nat. Methods 2005, 2, 825-827). Stopped-flow kinetic analysis indicated that in response to changes in K(+) concentration TAC-Red fluorescence enhancement occurs in milliseconds or less. Here, we use fluorescence correlation spectroscopy to quantify the binding kinetics of K(+) with TAC-Red and a new, longer-wavelength sensor, TAC-Crimson. Autocorrelation functions, G(tau), were similar at 0 and high (150 mM) K(+) concentrations, with the appearance of a prominent kinetic process with a correlation time in the millisecond range for K(+) concentrations between approximately 20 and 60 mM. Control experiments with increased illumination volume and solution viscosity indicated that the millisecond component represented K(+)/TAC-Red association. K(+)-dependent G(tau) data, modeled using a global regression to a binding/diffusion model, gave association and dissociation rate constants of 0.0020 +/- 0.0003 mM(-1) ms(-1) and 0.12 +/- 0.02 ms(-1), respectively, for TAC-Red. Similar results were obtained for TAC-Crimson. The rapid K(+) binding kinetics with triazacryptand-based sensors support their utility for measuring changes in K(+) concentrations during rapid neural signaling and ion channel gating.     

Compartmental modeling of reversible intermolecular two-state excited-state processes coupled with rotational diffusion or with added quencher

Analysis of related time-resolved fluorescence measurements can possibly lead to the determination of the kinetic parameters of excited-state processes. A deterministic identifiability analysis on an error-free fluorescence decay data surface has to be executed to verify whether the parameters of a particular model can be determined and may point to the minimal experimental conditions under which this will become possible. In this work, similarity transformation is chosen as an identifiability analysis approach because it also gives the explicit relationships between the true and alternative model parameters. Results are presented for two kinetic models of a reversible intermolecular two-state excited-state process in isotropic environments: (a) with coupled species-dependent rotational diffusion described by Brownian reorientation and (b) with added quencher. For model a, both spherically and cylindrically symmetric rotors, with no change in the principal axes of rotation in the latter, are considered. The fluorescence delta-response functions I(parallel)(t) and I(perpendicular)(t), for fluorescence polarized respectively parallel and perpendicular to the electric vector of linearly polarized excitation, are used to define the sum S(t) = I( parallel)(t) + 2 I( perpendicular)(t) and the difference D(t) = I(parallel)(t) - I(perpendicular)(t) function. The identifiability analysis is carried out using the S(t) and D(t) functions. The analysis involving S(t) shows that two physically acceptable possible solutions for the overall rate constants of the excited-state process exist. Inclusion of information from polarized fluorescence measurements on the rotational kinetic behavior contained in D(t) results in the unique set of rate constants and rotational diffusion coefficients when the rotational diffusion coefficients are different. For model b, it is shown that addition of quencher plays formally the same role as rotational diffusion as far as the identification is concerned. When the quenching rate constants are different, the rate constants of a reversible intermolecular two-state excited-state process with added quencher can be uniquely determined.     

Reversal of Solvatochromism: A New Strategy to Construct Activatable Two-photon Fluorescent Probes for Sensing

Two-photon (TP) imaging with a donor-acceptor (D?A) type fluorophore is an emerging tool for bioimaging and sensing. However, current TP probes suffer from serious solvatochromic quenching in aqueous solution due to their strong intramolecular charge transfer (ICT) in excited states. In this work, based on solvatochromism reversal, we report a novel strategy to develop TP probes for bioimaging. Specifically, compared with the normal two-photon probes that showed a fluorescence off with ICT suppressed, the novel probes exhibited strong fluorescence in the aqueous solution when their ICT was inhibited. This strategy not only provides a new way for the design of high-performance TP probes, but also expands the biological analysis toolbox for use in living systems.     

Kinetic energy density study of some representative semilocal kinetic energy functionals

There is a number of explicit kinetic energy density functionals for noninteracting electron systems that are obtained in terms of the electron density and its derivatives. These semilocal functionals have been widely used in the literature. In this work, we present a comparative study of the kinetic energy density of these semilocal functionals, stressing the importance of the local behavior to assess the quality of the functionals. We propose a quality factor that measures the local differences between the usual orbital-based kinetic energy density distributions and the approximated ones, allowing us to ensure if the good results obtained for the total kinetic energies with these semilocal functionals are due to their correct local performance or to error cancellations. We have also included contributions coming from the Laplacian of the electron density to work with an infinite set of kinetic energy densities. For all but one of the functionals, we have found that their success in the evaluation of the total kinetic energy is due to global error cancellations, whereas the local behavior of their kinetic energy density becomes worse than that corresponding to the Thomas-Fermi functional.     

Calibration and Limits of Camera‐Based Fluorescence Correlation Spectroscopy: A Supported Lipid Bilayer Study

Camera‐based fluorescence correlation spectroscopy (FCS) approaches allow the measurement of thousands of contiguous points yielding excellent statistics and details of sample structure. Imaging total internal reflection FCS (ITIR‐FCS) provides these measurements on lipid membranes. Herein, we determine the influence of the point spread function (PSF) of the optical system, the laser power used, and the time resolution of the camera on the accuracy of diffusion coefficient and concentration measurements. We demonstrate that the PSF can be accurately determined by ITIR‐FCS and that the laser power and time resolution can be varied over a wide range with limited influence on the measurement of the diffusion coefficient whereas the concentration measurements are sensitive to changes in the measurement parameters. One advantage of ITIR‐FCS is that the measurement of the PSF has to be performed only once for a given optical setup, in contrast to confocal FCS in which calibrations have to be performed at least once per measurement day. Using optimized experimental conditions we provide diffusion coefficients for over ten different lipid membranes consisting of one, two and three constituents, measured in over 200000 individual correlation functions. Using software binning and thus the inherent advantage of ITIR‐FCS of providing multiple observation areas in a single measurement we test the FCS diffusion law and show how they can be complemented by the local information provided by the difference in cross‐correlation functions (ΔCCF). With the determination of the PSF by ITIR‐FCS and the optimization of measurement conditions ITIR‐FCS becomes a calibration‐free method. This allows us to provide measurements of absolute diffusion coefficients for bilayers with different compositions, which were stable over many different bilayer preparations over a time of at least one year, using a single PSF calibration.     

Application of a microcomputer for the cross-spectral analysis with pseudorandom signal in electroanalytical chemistry

A digital memory device and a microcomputer are introduced to calculate correlation functions and Fourier transform in voltammetric measurements. Typical cross-spectra for simple diffusion and kinetic processes are obtained. The fast Fourier transform (FFT) modified for microcomputers was used for speed-up of the calculation. The cross-spectra obtained gave information about the electrode process such as the pattern of the accompanied chemical reaction and the kinetic parameter of the chemical reaction.     

Proton-coupled electron transfer in soybean lipoxygenase: dynamical behavior and temperature dependence of kinetic isotope effects

The dynamical behavior and the temperature dependence of the kinetic isotope effects (KIEs) are examined for the proton-coupled electron transfer reaction catalyzed by the enzyme soybean lipoxygenase. The calculations are based on a vibronically nonadiabatic formulation that includes the quantum mechanical effects of the active electrons and the transferring proton, as well as the motions of all atoms in the complete solvated enzyme system. The rate constant is represented by the time integral of a probability flux correlation function that depends on the vibronic coupling and on time correlation functions of the energy gap and the proton donor-acceptor mode, which can be calculated from classical molecular dynamics simulations of the entire system. The dynamical behavior of the probability flux correlation function is dominated by the equilibrium protein and solvent motions and is not significantly influenced by the proton donor-acceptor motion. The magnitude of the overall rate is strongly influenced by the proton donor-acceptor frequency, the vibronic coupling, and the protein/solvent reorganization energy. The calculations reproduce the experimentally observed magnitude and temperature dependence of the KIE for the soybean lipoxygenase reaction without fitting any parameters directly to the experimental kinetic data. The temperature dependence of the KIE is determined predominantly by the proton donor-acceptor frequency and the distance dependence of the vibronic couplings for hydrogen and deuterium. The ratio of the overlaps of the hydrogen and deuterium vibrational wavefunctions strongly impacts the magnitude of the KIE but does not significantly influence its temperature dependence. For this enzyme reaction, the large magnitude of the KIE arises mainly from the dominance of tunneling between the ground vibronic states and the relatively large ratio of the overlaps between the corresponding hydrogen and deuterium vibrational wavefunctions. The weak temperature dependence of the KIE is due in part to the dominance of the local component of the proton donor-acceptor motion.     

Theoretical analysis of proton relays in electrochemical proton-coupled electron transfer

The coupling of long-range electron transfer to proton transport over multiple sites plays a vital role in many biological and chemical processes. Recently the concerted proton-coupled electron transfer (PCET) reaction in a molecule with a hydrogen-bond relay inserted between the proton donor and acceptor sites was studied electrochemically. The standard rate constants and kinetic isotope effects (KIEs) were measured experimentally for this double proton transfer system and a related single proton transfer system. In the present paper, these systems are studied theoretically using vibronically nonadiabatic rate constant expressions for electrochemical PCET. Application of this approach to proton relays requires the calculation of multidimensional proton vibrational wave functions and the incorporation of multiple proton donor-acceptor motions. The decrease in proton donor-acceptor distances due to thermal fluctuations and the contributions from excited electron-proton vibronic states play important roles in these systems. The calculated KIEs and the ratio of the standard rate constants for the single and double proton transfer systems are in agreement with the experimental data. The calculations indicate that the standard PCET rate constant is lower for the double proton transfer system because of the smaller overlap integral between the ground state reduced and oxidized proton vibrational wave functions, resulting in greater contributions from excited electron-proton vibronic states with higher free energy barriers. The theory predicts that this rate constant may be increased by modifying the molecule in a manner that decreases the equilibrium proton donor-acceptor distances or alters the molecular thermal motions to facilitate the concurrent decrease of these distances. These insights may guide the design of more efficient catalysts for energy conversion devices.     

Surface sticking and lateral diffusion of lipids in supported bilayers

The diffusion of fluorescently labeled lipids in supported bilayers is studied using two different methods: Z-scan fluorescence correlation spectroscopy (z-scan FCS) and two-focus fluorescence correlation spectroscopy (2f-FCS). It is found that the data can be fitted consistently only when taking into account partial sticking of the labeled lipids to the supporting glass surface. A kinetic reaction-diffusion model is developed and applied to the data. We find a very slow sticking rate which, however, when neglected, leads to strongly varying estimates of the free diffusion coefficient. The study reveals a strong sensitivity of FCS on even slight binding/unbinding kinetics of the labeled molecules, which has significance for related diffusion measurements in cellular lipid membranes.     

pH-induced intramolecular folding dynamics of i-motif DNA

Using the combination of fluorescence resonance energy transfer (FRET) and fluorescence correlation spectroscopy (FCS) technique, we investigate the mechanism and dynamics of the pH-induced conformational change of i-motif DNA in the bulk phases and at the single-molecule level. Despite numerous studies on i-motif that is formed from cytosine (C)-rich strand at slightly acidic pH, its detailed conformational dynamics have been rarely reported. Using the FRET technique to provide valuable information on the structure of biomolecules such as a protein and DNA, we clearly show that the partially folded species as well as the single-stranded structure coexist at neutral pH, supporting that the partially folded species may exist substantially in vivo and play an important role in a process of gene expression. By measuring the FCS curves of i-motif, we observed the gradual decrease of the diffusion coefficient of i-motif with increasing pH. The quantitative analysis of FCS curves supports that the gradual decrease of diffusion coefficient (D) associated with the conformational change of i-motif is not only due to the change in the intermolecular interaction between i-motif and solvent accompanied by the increase of pH but also due to the change of the shape of DNA. Furthermore, FCS analysis showed that the intrachain contact formation and dissociation for i-motif are 5-10 times faster than that for the open form. The fast dynamics of i-motif with a compact tetraplex is due to the intrinsic conformational changes at the fluorescent site including the motion of alkyl chain connecting the dye to DNA, whereas the slow intrachain contact formation observed from the open form is due to the DNA motion corresponding to an early stage interaction in the folding process of the unstructured open form.     

Hydrogen tunneling in an enzyme active site: a quantum wavepacket dynamical perspective

We study the hydrogen tunneling problem in a model system that represents the active site of the biological enzyme, soybean lipoxygenase-1. Toward this, we utilize quantum wavepacket dynamics performed on potential surfaces obtained by using hybrid density functional theory under the influence of a dynamical active site. The kinetic isotope effect is computed by using the transmission amplitude of the wavepacket, and the experimental value is reproduced. By computing the hydrogen nuclear orbitals (eigenstates) along the reaction coordinate, we note that tunneling for both hydrogen and deuterium occurs through the existence of distorted, spherical s-type proton wave functions and p-type polarized proton wave functions for transfer along the donor-acceptor axis. In addition, there is also a significant population transfer through distorted p-type proton wave functions directed perpendicular to the donor-acceptor axis (via intervening pi-type proton eigenstate interactions) which underlines the three-dimensional nature of the tunneling process. The quantum dynamical evolution indicates a significant contribution from tunneling processes both along the donor-acceptor axis and along directions perpendicular to the donor-acceptor axis. Furthermore, the tunneling process is facilitated by the occurrence of curve crossings and avoided crossings along the proton eigenstate adiabats.     

Octacoordinate carbons encaged inside carborane clusters: a density functional theory investigation

This work focuses on the computational design and characterization of a novel series of endohedral carborane clusters containing octacoordinate carbon centers. The structural and bonding features and the thermodynamic and kinetic stabilities are discussed extensively based on density functional theory calculations. These nonclassical carboranes are fascinating in structure not only for the octacoordinate carbon center but also for the surrounding carbon and boron ligands with inverted bonding configuration. These endohedral carboranes are higher in energy than the corresponding exohedral isomers due to the high strain in the system. A new stability rule based on the donor-acceptor model is proposed to predict the stability ordering for these carborane isomers. In addition, some of these octacoordinate carboranes might have relatively high kinetic stabilities, which is rather hopeful for the experimental syntheses.     

Shedding light on biomolecule conformational dynamics using fluorescence measurements of trapped ions

Biomolecule conformational change has been widely investigated in solution using several methods; however, much less experimental data about structural changes are available for completely isolated, gas-phase biomolecules. Studies of conformational change in unsolvated biomolecules are required to complement the interpretation of mass spectrometry measurements and in addition, can provide a means to directly test theoretical simulations of biomolecule structure and dynamics independent of a simulated solvent. In this Feature Article, we review our recent introduction of a fluorescence-based method for probing local conformational dynamics in unsolvated biomolecules through interactions of an attached dye with tryptophan (Trp) residues and fields originating on charge sites. Dye-derivatized biomolecule ions are formed by electrospray ionization and are trapped in a variable-temperature quadrupole ion trap in which they are irradiated with either continuous or short pulse lasers to excite fluorescence. Fluorescence is measured as a function of temperature for different charge states. Optical measurements of the dye fluorescence include average intensity changes, changes in the emission spectrum, and time-resolved measurements of the fluorescence decay. These measurements have been applied to the miniprotein, Trp-cage, polyproline peptides and to a beta-hairpin-forming peptide, and the results are presented as examples of the broad applicability and utility of these methods. Model fits to Trp-cage fluorescence data measured as a function of temperature provide quantitative information on the thermodynamics of conformational changes, which are reproduced well by molecular dynamics. Time-resolved measurements of the fluorescence decays of Trp-cage and small polyproline peptides definitively demonstrate the occurrence of fluorescence quenching by the amino acid Trp in unsolvated biomolecules.     

Kinetic approach to heavy metal mobilization assessment in sediments: choose of kinetic equations and models to achieve maximum information

Studies of trace metal mobilization in sediments are generally performed using sequential extraction schemes at equilibrium. In the present work, a kinetic fractionation of trace metals in sediments has been developed to assess that information. The extraction rate data have been obtained using a single extraction scheme with EDTA and following a protocol previously optimized. Two kinetic equations and two kinetic models were used to fit the experimental data. The two constants equation fits well the extraction rate data used in this work but does not present any physico-chemical meaning. The diffusion model and the two first-order reactions model allow determining which parameter (the reaction between the metal M and the EDTA or the diffusion of the complex M/EDTA) is rate limiting in the trace metal extraction by EDTA. It appears that the two first-order reactions model is more efficient than the diffusion model to fit the present extraction rate data so it can be deduced that the diffusion of the complex M/EDTA is not the limiting step of the trace metal extraction by EDTA in estuarine sediments. In a second part, relationships between the fraction of metals determined with the two first-order reactions model and the sediments composition were established.