Andrzej Molski

Statistics of the bleaching number and the bleaching time in single-molecule fluorescence spectroscopy

The kinetics of bleaching in single-molecule fluorescence spectroscopy (SMS) is studied using renewal theory. A five-state model of a dye molecule is considered where bleaching occurs from the excited triplet states. An exact formalism is developed to calculate the distributions of the bleaching number (i.e., the number of photon counts) and bleaching time (i.e., the time before photobleaching). For photostable dyes those distributions are well approximated by exponential distributions determined by the average bleaching number ν and the average bleaching time τ respectively. Exact formulation is developed to calculate ν and τ in terms of the transition rate constants. For photostable dyes the exact ν and τ are well approximated by expressions derived from a steady-state solution to the kinetic rate equations describing the molecule. The theory implies that experimental multiexponential fits to the distributions of the bleaching number and bleaching time are an indication that the SMS system is heterogeneous.

Monomer–excimer kinetics in solution. II. Statistical nonequilibrium thermodynamic approach

The statistical nonequilibrium thermodynamic theory of diffusion‐influenced reactions is employed to study reversible excimer formation in solution. Three types of rate coefficients for bimolecular excimer formation are discussed: (i) molecular rate coefficients defined by one‐way reactive fluxes, (ii) phenomenological rate constants based on form of the macroscopic rate equations, and (iii) modified, time‐dependent phenomenological rate coefficients. Relations are derived linking: (i) the molecular rate coefficients for reversible and irreversible excimer formation rates, and (ii) the steady‐state molecular rate constant and the Laplace transform of the time resolved irreversible rate coefficient. The relationship between the present approach and the microscopic–stochastic theory of excimer formation is discussed.

Molecular and phenomenological rate coefficients of fast reactions in solutions

The paper is concerned with the problem of formulating chemical rate equations for reversible reactions in solution in terms of concentration‐independent, phenomenological rate coefficients. These time‐dependent rate coefficients approach, after an initial transient, the rate constants that can be obtained in a relaxation experiment. We start with the coupled evolution equations for the macroscopic concentrations, and for the two‐particle distribution functions describing association–dissociation (A+B=C), bimolecular isomerization (A+B=B+C), and double decomposition (A+B=C+D). The effects of interparticle forces and long‐ranged reactivity are included. We derive general identities linking the reactants and products radial distribution functions. For association–dissociation this leads to relations among the molecular rate coefficients which are valid for both contact and long ranged reactivities. For the other two reaction types, we were able to derive analogous relations only for contact reactivities. We demonstrate how the phenomenological rate coefficients can be defined via the solutions of the corresponding diffusional boundary‐value problems. This approach is quite general, and valid for both contact and long‐ranged reactivities and if interaction forces are included.

Monomer–excimer kinetics in solution. I. Stochastic many‐particle approach

A stochastic many‐particle approach is applied to study the kinetics of reversible excimer formation in solution. Coupled dynamic equations for the macroscopic concentrations and for the radial distribution function are derived, and applied to analyze (i) time resolved kinetics after a short pulse, and (ii) steady‐state kinetics. Renormalization of the phenomenological excimer dissociation rate coefficient due to nonequilibrium effects is discussed. A relation is demonstrated between steady‐state, reversible monomer–excimer kinetics and irreversible fluorescence kinetics. Explicit results are given for the excimer fluorescence yield, assuming the Smoluchowski–Collins–Kimball reactivity model.

Deterministic identifiability of photophysical kinetic models with transients via the method of similarity transformation

The method of similarity transformation is shown to be applicable to the problem of deterministic identifiability of photophysical kinetic models with transients. A two-state excited-state model of irreversible intermolecular association with transients is demonstrated to be identifiable from two decay traces at two observation wavelengths and one co-reactant concentration, when either only the species that associates with the co-reactant is directly excited or the excited species spectra are separated such that each species can be monitored individually. When transients are absent, two decays at one co-reactant concentration do not guarantee the recovery of the model parameters, which indicates that transient effects change identifiability criteria compared to the case with time-invariant rate constants.

Generalization of the Smoluchowski theory to include the effect of unimolecular processes: fluorescence quenching

Abstract A generalization of the Smoluchowski theory to include the effect of particle generation and decay is discussed. First, the standard Smoluchowski theory of the irreversible bimolecular reaction A + B → products is formulated in terms of the radial distribution function g AB rather than in terms of the local concentration of one sort of reactants around a sink. Then, the effect of unimolecular processes on g AB is evaluated via an analysis of the time evolution of the two-particle distribution function ƒ AB = ϱ A ϱ B g AB . A generalized Smoluchowski theory describing fluorescence quenching is developed, and employed to derive relationships among the steady-state, time-dependent, and frequency-dependent quenching rates. Applications of these relations for analysis of quenching data are discussed.

Maximum likelihood-based analysis of single-molecule photon arrival trajectories

In this work we explore the statistical properties of the maximum likelihood-based analysis of one-color photon arrival trajectories. This approach does not involve binning and, therefore, all of the information contained in an observed photon strajectory is used. We study the accuracy and precision of parameter estimates and the efficiency of the Akaike information criterion and the Bayesian information criterion (BIC) in selecting the true kinetic model. We focus on the low excitation regime where photon trajectories can be modeled as realizations of Markov modulated Poisson processes. The number of observed photons is the key parameter in determining model selection and parameter estimation. For example, the BIC can select the true three-state model from competing two-, three-, and four-state kinetic models even for relatively short trajectories made up of 2 × 10(3) photons. When the intensity levels are well-separated and 10(4) photons are observed, the two-state model parameters can be estimated with about 10% precision and those for a three-state model with about 20% precision.

Deterministic identifiability of two-state excited-state models with transients: Recovery of deactivation rate constants

A deterministic identifiability analysis of kinetic models of two-state excited-state processes in the presence of transient effects is performed to establish the conditions for the unique determination of the excited-state species deactivation rate constants with no assumption regarding the kinetics of interconversion in the excited state. It is necessary that decays of at least one excited-state species can be monitored separately to uniquely determine the excited-state species deactivation rate constants. This conclusion holds for reversible and irreversible intermolecular as well as intramolecular excited-state processes in the presence of transient effects. Sufficient conditions for the recovery of the deactivation rate constants are established for several cases of practical relevance.

Monomer‐excimer kinetics in solution. III. Generalized Smoluchowski approach

A generalized Smoluchowski approach developed in A. Molski, Chem. Phys. 182, 203 (1994) is employed to study reversible excimer formation in solution. For contact excimer formation, relations among the rate coefficients are analyzed for three modes of excitation; initially pulsed, steady‐state, and periodic. A new kinetic Laplace transform relation for the frequency domain is demonstrated in the linear harmonic regime. The Laplace transform relations between the time domain and steady states, derived in W. Naumann and A. Molski, J. Chem. Phys. 100, 1511, 1520 (1994) for interaction‐free models, are shown to be also valid when the interaction forces are included.

A coarse-grained MARTINI-like force field for DNA unzipping in nanopores

In nanopore force spectroscopy (NFS) a charged polymer is threaded through a channel of molecular dimensions. When an electric field is applied across the insulating membrane, the ionic current through the nanopore reports on polymer translocation, unzipping, dissociation, and so forth. We present a new model that can be applied in molecular dynamics simulations of NFS. Although simplified, it does reproduce experimental trends and all‐atom simulations. The scaled conductivities in bulk solution are consistent with experimental results for NaCl for a wide range of electrolyte concentrations and temperatures. The dependence of the ionic current through a nanopore on the applied voltage is symmetric and, in the voltage range used in experiments (up to 2 V), linear and in good agreement with experimental data. The thermal stability and geometry of DNA is well represented. The model was applied to simulations of DNA hairpin unzipping in nanopores. The results are in good agreement with all‐atom simulations: the scaled translocation times and unzipping sequence are similar.