Multipoint Fluorescence Quenching-Time Statistics for Single Molecules with Anomalous Diffusion

Nonlinear measurements of kinetics and generalized dynamical modes. I. Extracting the one-dimensional Green's function from a time series

Often, a single correlation function is used to measure the kinetics of a complex system. In contrast, a large set of k-vector modes and their correlation functions are commonly defined for motion in free space. This set can be transformed to the van Hove correlation function, which is the Green's function for molecular diffusion. Here, these ideas are generalized to other observables. A set of correlation functions of nonlinear functions of an observable is used to extract the corresponding Green's function. Although this paper focuses on nonlinear correlation functions of an equilibrium time series, the results are directly connected to other types of nonlinear kinetics, including perturbation-response experiments with strong fields. Generalized modes are defined as the orthogonal polynomials associated with the equilibrium distribution. A matrix of mode-correlation functions can be transformed to the complete, single-time-interval (1D) Green's function. Diagonalizing this matrix finds the eigendecays. To understand the advantages and limitation of this approach, Green's functions are calculated for a number of models of complex dynamics within a Gaussian probability distribution. Examples of non-diffusive motion, rate heterogeneity, and range heterogeneity are examined. General arguments are made that a full set of nonlinear 1D measurements is necessary to extract all the information available in a time series. However, when a process is neither dynamically Gaussian nor Markovian, they are not sufficient. In those cases, additional multidimensional measurements are needed.

Theory of single molecule emission spectroscopy

A general theory and calculation framework for the prediction of frequency-resolved single molecule photon counting statistics is presented. Expressions for the generating function of photon counts are derived, both for the case of naive "detection" based solely on photon emission from the molecule and also for experimentally realizable detection of emitted photons, and are used to explicitly calculate low-order photon-counting moments. The two cases of naive detection versus physical detection are compared to one another and it is demonstrated that the physical detection scheme resolves certain inconsistencies predicted via the naive detection approach. Applications to two different models for molecular dynamics are considered: a simple two-level system and a two-level absorber subject to spectral diffusion.

Single molecule counting statistics for systems with periodic driving

We extend the generating function approach for calculation of event statistics observed in single molecule spectroscopy to cases where the single molecule evolves under explicitly time-dependent and periodic perturbation. Floquet theory is used to recast the generating function equations for the periodically driven system into effective equations devoid of explicit time-dependence. Two examples are considered, one employing simple stochastic dynamics and the other quantum dynamics, to demonstrate the versatility and numerical accuracy of the methodology.

Aging and nonergodicity beyond the Khinchin theorem

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.

Single-molecule photon emission statistics for systems with explicit time dependence: Generating function approach

Generating function calculations are extended to allow for laser pulse envelopes of arbitrary shape in numerical applications. We investigate photon emission statistics for two-level and V- and Lambda-type three-level systems under time-dependent excitation. Applications relevant to electromagnetically induced transparency and photon emission from single quantum dots are presented.

Joint probability distributions and multipoint correlations of the continuous-time random walk

We present an efficient method to determine the Fourier-Laplace transform of the joint n-point probability distribution of a continuous-time random walk for arbitrary finite n. Additionally, we devise a recursive procedure with which it is possible to calculate the Laplace transforms of the multipoint correlation functions without having to determine the joint probability distributions first. The methods are used on several examples with both independent and dependent distributions for the waiting time and the spatial step size.

Aging correlation functions of the interrupted fractional Fokker-Planck propagator

The authors explore aging in a general semi-Markov process with arbitrary waiting time distributions and discuss the role of trapping phenomenon in glasses in determining the waiting time distributions. In certain limits, they obtain a two-time propagator for the fractional Fokker-Planck equation through coordination, but one can generalize the construction to incorporate cutoffs in the aging of the system, i.e., the interrupted aging. The construction allows the incorporation of cutoffs in the aging of the system, and the exploration of signatures of aging effects in systems that eventually achieve equilibrium. Cutoffs in aging effects are illustrated for interrupted fractional diffusion in a harmonic potential and for the on-time probability of a quantum dot system.

Super- and sub-Poissonian photon statistics for single molecule spectroscopy

We investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave laser field. The spectral diffusion is modeled based on a stochastic approach, in the spirit of the Anderson-Kubo line shape theory. Using a generating function formalism we solve the generalized optical Bloch equations and obtain an exact analytical formula for the line shape and Mandel's Q parameter. The line shape exhibits well-known behaviors, including motional narrowing when the stochastic modulation is fast and power broadening. The Mandel parameter, describing the line shape fluctuations, exhibits a transition from a quantum sub-Poissonian behavior in the fast modulation limit to a classical super-Poissonian behavior found in the slow modulation limit. Our result is applicable for weak and strong laser fields, namely, for arbitrary Rabi frequency. We show how to choose the Rabi frequency in such a way so that the quantum sub-Poissonian nature of the emission process becomes strongest. A lower bound on Q is found and simple limiting behaviors are investigated. A nontrivial behavior is obtained in the intermediate modulation limit, when the time scales for spectral diffusion and the lifetime of the excited state become similar. A comparison is made between our results and previous ones derived, based on the semiclassical generalized Wiener-Khintchine formula.

Single Molecule Photon Counting Statistics for Quantum Mechanical Chromophore Dynamics

We extend the generating function technique for calculation of single molecule photon emission statistics (Zheng, Y.; Brown, F. L. H. Phys. Rev. Lett. 2003, 90, 238305) to systems governed by multi-level quantum dynamics. This opens up the possibility to study phenomena that are outside the realm of purely stochastic and mixed quantum-stochastic models. In particular, the present methodology allows for calculation of photon statistics that are spectrally resolved and subject to quantum coherence. Several model calculations illustrate the generality of the technique and highlight quantitative and qualitative differences between quantum mechanical models and related stochastic approximations when they arise. Calculations suggest that studying photon statistics as a function of photon frequency has the potential to reveal more about system dynamics than the usual broadband detection schemes.

Mapping the energy landscape of biomolecules using single molecule force correlation spectroscopy: theory and applications

We present, to our knowledge, a new theory that takes internal dynamics of proteins into account to describe forced-unfolding and force-quench refolding in single molecule experiments. In the current experimental setup (using either atomic force microscopy or laser optical tweezers) the distribution of unfolding times, P(t), is measured by applying a constant stretching force f(S) from which the apparent f(S)-dependent unfolding rate is obtained. To describe the complexity of the underlying energy landscape requires additional probes that can incorporate the dynamics of tension propagation and relaxation of the polypeptide chain upon force quench. We introduce a theory of force correlation spectroscopy to map the parameters of the energy landscape of proteins. In force correlation spectroscopy, the joint distribution P(T, t) of folding and unfolding times is constructed by repeated application of cycles of stretching at constant f(S) separated by release periods T during which the force is quenched to f(Q) < f(S). During the release period, the protein can collapse to a manifold of compact states or refold. We show that P(T, t) at various f(S) and f(Q) values can be used to resolve the kinetics of unfolding as well as formation of native contacts. We also present methods to extract the parameters of the energy landscape using chain extension as the reaction coordinate and P(T, t). The theory and a wormlike chain model for the unfolded states allows us to obtain the persistence length l(p) and the f(Q)-dependent relaxation time, giving us an estimate of collapse timescale at the single molecular level, in the coil states of the polypeptide chain. Thus, a more complete description of landscape of protein native interactions can be mapped out if unfolding time data are collected at several values of f(S) and f(Q). We illustrate the utility of the proposed formalism by analyzing simulations of unfolding-refolding trajectories of a coarse-grained protein (S1) with beta-sheet architecture for several values of f(S), T, and f(Q) = 0. The simulations of stretch-relax trajectories are used to map many of the parameters that characterize the energy landscape of S1.

Anomalous continuous-time random-walk spectral diffusion in coherent third-order optical response

Recursive relations are derived for nonlinear optical response functions of a two-level chromophore with stochastic frequency fluctuations described by a continuous-time random walk. Stationary ensembles are constructed and signatures of anomalous relaxation in the photon echo signal are discussed for a two state jump modulation with a power law waiting time density function psi(t) approximately t(-alpha-1). Stretched exponential decay of the photon echo signal is predicted for 0<alpha<1 and power law asymptotics for 1<alpha<2.

Probing protein-protein interactions by dynamic force correlation spectroscopy

We develop a formalism for single molecule dynamic force spectroscopy to map the energy landscape of protein-protein complex (P(1)P(2)). The joint distribution P(tau(1),tau(2)) of unbinding lifetimes tau(1) and tau(2), measurable in a compression-tension cycle, which accounts for the internal relaxation dynamics of the proteins under tension, shows that the histogram of tau(1) is not Poissonian. The theory is applied to the forced unbinding of protein P1, modeled as a wormlike chain, from P(1)P(2). We propose a new class of experiments which can resolve the effect of internal protein dynamics on the unbinding lifetimes.

Multipoint correlation functions for continuous-time random walk models of anomalous diffusion

Recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous-time random walk (CTRW) in an external potential. Two- and three-point correlation functions are calculated for waiting-time distributions with an anomalous power-law profile t(-alpha-1), 0 < alpha < 1, on intermediate time scales with a crossover to an exponential long time decay. Comparison of the CTRW with the Brownian harmonic oscillator model (Gaussian process) illustrates how higher-order correlation functions may be used to distinguish between dynamical models that have the same two-point correlation function.

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.

Gadolinium(III)-Loaded Nanoparticulate Zeolites as Potential High-Field MRI Contrast Agents: Relationship Between Structure and Relaxivity

The effects of dealumination, pore size, and calcination on the efficiency (as expressed in the relaxivity) of Gd3+-loaded zeolites for potential application as magnetic resonance imaging (MRI) contrast agents were studied. Partial dealumination of zeolites NaY or NaA by treatment with (NH4)2SiF6 or diluted HCl resulted in materials that, upon loading with Gd3+, had a much higher relaxivity than the corresponding non-dealuminated materials. Analysis of the 1H NMR dispersion profiles of the various zeolites showed that this can be mainly ascribed to an increase of the amount of water inside the zeolite cavities as a result of the destruction of walls between cavities. However, the average residence time of water inside the Gd3+-loaded cavities did not change significantly, which suggests that the windows of the Gd3+-loaded cavities are not affected by the dealumination. Upon calcination, the Gd3+ ions moved to the small sodalite cavities and became less accessible for water, resulting in a decrease in relaxivity. The important role of diffusion for the relaxivity was demonstrated by a comparison of the relaxivity of Gd3+-loaded zeolite NaY and NaA samples. NaA had much lower relaxivities due to the smaller pore sizes. The transversal relaxivities of the Gd3+-doped zeolites are comparable in magnitude to the longitudinal ones at low magnetic fields (<60 MHz). However at higher fields, the transversal relaxivities steeply increased, whereas the longitudinal relaxivities decreased as field strength increased. Therefore, these materials have potential as T1 MRI contrast agents at low field, and as T2 agents at higher fields.

Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule

By introducing fractional Gaussian noise into the generalized Langevin equation, the subdiffusion of a particle can be described as a stationary Gaussian process with analytical tractability. This model is capable of explaining the equilibrium fluctuation of the distance between an electron transfer donor and acceptor pair within a protein that spans a broad range of time scales, and is in excellent agreement with a single-molecule experiment.

Single molecule photon emission statistics in the slow modulation limit

A framework for calculating photon emission statistics for single chromophores perturbed by slow environmental fluctuations is introduced. When internal chromophore dynamics are significantly faster than time scales for environmental modulation it becomes possible to invoke a type of adiabatic approximation, allowing for straightforward calculation of photon counting moments including explicitly quantum effects. Unlike previous exact treatments, the present methodology involves calculation of dynamics reflecting only the modulation characteristics of the environment and quantum dynamics of an isolated chromophore separately, i.e., the complicated intermingling of chromophore quantum dynamics and the environmental modulation are suppressed via the adiabatic approximation. This leads to significant conceptual and computational simplifications. Within its regime of applicability, the present approximation reproduces exact calculations quantitatively. We demonstrate this accuracy explicitly for the case of a two-level chromophore modulated by a number of different stochastic models.