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

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

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

Interpreting single turnover catalysis measurements with constrained mean dwell times

Observation of a chemical transformation at the single-molecule level yields a detailed view of kinetic pathways contributing to the averaged results obtained in a bulk measurement. Studies of a fluorogenic reaction catalyzed by gold nanoparticles have revealed heterogeneous reaction dynamics for these catalysts. Measurements on single nanoparticles yield binary trajectories with stochastic transitions between a dark state in which no product molecules are adsorbed and a fluorescent state in which one product molecule is present. The mean dwell time in either state gives information corresponding to a bulk measurement. Quantifying fluctuations from mean kinetics requires identifying properties of the fluorescence trajectory that are selective in emphasizing certain dynamic processes according to their time scales. We propose the use of constrained mean dwell times, defined as the mean dwell time in a state with the constraint that the immediately preceding dwell time in the other state is, for example, less than a variable time. Calculations of constrained mean dwell times for a kinetic model with dynamic disorder demonstrate that these quantities reveal correlations among dynamic fluctuations at different active sites on a multisite catalyst. Constrained mean dwell times are determined from measurements of single nanoparticle catalysis. The results indicate that dynamical fluctuations at different active sites are correlated, and that especially rapid reaction events produce particularly slowly desorbing product molecules.

Communications: Can one identify nonequilibrium in a three-state system by analyzing two-state trajectories?

For a three-state Markov system in a stationary state, we discuss whether, on the basis of data obtained from effective two-state (or on-off) trajectories, it is possible to discriminate between an equilibrium state and a nonequilibrium steady state. By calculating the full phase diagram we identify a large region where such data will be consistent only with nonequilibrium conditions. This regime is considerably larger than the region with oscillatory relaxation, which has previously been identified as a sufficient criterion for nonequilibrium.

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.

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

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

Path-probability density functions for semi-Markovian random walks

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

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

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

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.

Utilizing the information content in two-state trajectories

The signal from many single-molecule experiments monitoring molecular processes, such as enzyme turnover by means of fluorescence and opening and closing of ion channel through the flux of ions, consists of a time series of stochastic "on" and "off" (or open and closed) periods, termed a two-state trajectory. This signal reflects the dynamics in the underlying multisubstate on-off kinetic scheme (KS) of the process. The determination of the underlying KS is difficult and sometimes even impossible because of the loss of information in the mapping of the multidimensional KS onto two dimensions. Here we introduce a previously undescribed procedure that efficiently and optimally relates the signal to all equivalent underlying KS. This procedure partitions the space of KS into canonical (unique) forms that can handle any KS and obtains the topology and other details of the canonical form from the data without the need for fitting. Also established are relationships between the data and the topology of the canonical form to the on-off connectivity of a KS. The suggested canonical forms constitute a powerful tool in discriminating between KS. Based on our approach, the upper bound on the information content in two-state trajectories is determined.

Closed-form solutions for continuous time random walks on finite chains

Continuous time random walks (CTRWs) on finite arbitrarily inhomogeneous chains are studied. By introducing a technique of counting all possible trajectories, we derive closed-form solutions in Laplace space for the Green's function (propagator) and for the first passage time probability density function (PDF) for nearest neighbor CTRWs in terms of the input waiting time PDFs. These solutions are also the Laplace space solutions of the generalized master equation. Moreover, based on our counting technique, we introduce the adaptor function for expressing higher order propagators (joint PDFs of time-position variables) for CTRWs in terms of Green's functions. Using the derived formula, an escape problem from a biased chain is considered.