Reexamining experimental data of single-molecule fluorescence correlation spectroscopy for cholesterol oxidase, we find that the existing Michaelis-Menten models with dynamical disorder cannot explain strong correlations between subsequent turnover cycles revealed in the diagonal feature in the joint statistical distribution of adjacent ''on'' times of this enzyme. We suggest that functional conformational motions representing ordered sequences of transitions between a set of conformational substates are involved, along with equilibrium conformational fluctuations in the turnover cycle of cholesterol oxidase. A two-channel model of singleenzyme dynamics, including a slow functional conformational motion in one of the channels, is proposed that allows us to reproduce such strong correlations. Functionally important conformational motions are nonequilibrium processes that lead from one state of a protein to another and follow binding or dissociation of a ligand (1, 2). In enzymes, such ordered slow conformational motions can constitute an inherent part of a turnover cycle. Their functions may consist of transporting a substrate to the active center inside a macromolecule or bringing it into an appropriate arrangement with respect to this center. Generally, these are physical intramolecular motions that are needed for the occurrence of catalytic conversion and must precede it. Using time-resolved x-ray methods, functional conformational motions in Ha-ras p21 protein on GTP hydrolysis (3) and in a cytochrome P-450 enzyme (4) were recorded. In these experiments, enzyme molecules formed a crystal but could nonetheless perform their characteristic catalytic cycles. Today, many examples of ordered conformational motions in proteins, resolved by x-ray crystallography methods, are known (5-7) (see also the database at http:͞͞molmovdb.mbb.yale.edu͞molmovdb). Thermal conformational fluctuations for single macromolecules in solution were observed by using fluorescence correlation spectroscopy (FCS) (9, 10) and FRET methods (11). Conformational movements of enzymes in aqueous solvents could also be detected by NMR (12). Using FRET measurements, nonequilibrium conformational changes in single molecules of T4 lysozyme under reaction conditions were observed (13). Single-molecule FCS provides a powerful tool for monitoring chemical transitions during catalytic turnover cycles of individual molecules, and such experiments have been already performed for a number of enzymes, including cholesterol oxidase (14) and horseradish peroxidase (15,16). Simultaneous monitoring of chemical turnover cycles and physical processes of conformational changes was not, however, possible in the single-molecule experiments. Therefore, functional conformational motions during a turnover cycle for enzymic reactions in solution could not be identified. Nonetheless, the presence of such functional motions can be deduced by special statistical analysis of experimental data. In a previous publication, such statistical analysis, based on the memory functio...
The control of coherence and spectral properties of noise-induced oscillations by time-delayed feedback is studied in a FitzHugh–Nagumo system which serves as a paradigmatic model of excitable systems. A semianalytical approach based on a discrete model with waiting time densities is developed, which allows one to predict quantitatively the increase of coherence measured by the correlation time, and the modulation of the main frequencies of the stochastic dynamics in dependence on the delay time. The analytical mean-field approximation is in good agreement with numerical results for the full nonlinear model.
Fluorescent spectroscopy experiments with single-enzyme molecules yield a large volume of statistical data that can be analyzed and interpreted using stochastic models of enzyme action. Here, we present two models, each based on the mechanism that an enzyme molecule must pass through a sequence of conformational transformations to complete its catalytic turnover cycle. In the simplest model, only one path leading to the release of product is present. In contrast to this, two different catalytic paths are possible in the second considered model. If a cycle is started from an active state, immediately after the previous product release, it follows a different conformational route and is much shorter. Our numerical investigations show that both models generate nonMarkovian molecular statistics. However, their memory landscapes and distributions of cycle times are significantly different. The memory landscape of the double-path model bears strong similarity to the recent experimental data for horseradish peroxidase. L arge biomolecules may act as machines that generate motoric motions or, in the case of enzymes, catalyze individual reaction events. To understand the functions of enzymes at a molecular level, experiments with single molecules are important. Such experiments have been performed for lactate dehydrogenase (1, 2), alkaline phosphatase (3, 4), -D-galactosidase (5), cholesterol oxidase (6), horseradish peroxidase (7), staphylococcal nuclease (8), and RNA polymerase (9). Because realistic computer simulations of enzymic turnover cycles are not yet possible, interpretation of experimental data are based on phenomenological models of single-enzyme kinetics. Theoretical and experimental investigations reveal that proteins are characterized by rugged energy landscapes (10, 11). Intramolecular relaxation in proteins involves passing through a large number of metastable substates and may therefore be slow.Conformational changes are essential for the catalytic enzyme function. They are generally used to explain allosteric regulation and the phenomena of cooperativity in enzymes with several interacting subunits. Conformational memory has been found for the enzyme wheat germ hexokinase (12, 13). Experiments on peptide binding to class II MHC proteins have further suggested that conformational memory of previous functional states is present in such macromolecules (14,15). A strong memory effect has also been discovered in single molecules of the hairpin ribozyme (16). In the investigations of cholesterol oxidase by Lu et al. (6), slow conformational fluctuations in the equilibrium state of a single enzyme, leading to modulation of its affinity for a given substrate, were shown to explain correlations between subsequent catalytic cycles. In this case, conformational dynamics was not directly influenced by enzyme activity and played an external role with respect to its catalytic function. In contrast to this, experiments with horseradish peroxidase could be interpreted by assuming that the memory of earlier function...
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