When is a single molecule heterogeneous? A multidimensional answer and its application to dynamics near the glass transition

Efficient, nonparametric removal of noise and recovery of probability distributions from time series using nonlinear-correlation functions: Photon and photon-counting noise

A preceding paper [M. Dhar, J. A. Dickinson, and M. A. Berg, J. Chem. Phys. 159, 054110 (2023)] shows how to remove additive noise from an experimental time series, allowing both the equilibrium distribution of the system and its Green's function to be recovered. The approach is based on nonlinear-correlation functions and is fully nonparametric: no initial model of the system or of the noise is needed. However, single-molecule spectroscopy often produces time series with either photon or photon-counting noise. Unlike additive noise, photon noise is signal-size correlated and quantized. Photon counting adds the potential for bias. This paper extends noise-corrected-correlation methods to these cases and tests them on synthetic datasets. Neither signal-size correlation nor quantization is a significant complication. Analysis of the sampling error yields guidelines for the data quality needed to recover the properties of a system with a given complexity. We show that bias in photon-counting data can be corrected, even at the high count rates needed to optimize the time resolution. Using all these results, we discuss the factors that limit the time resolution of single-molecule spectroscopy and the conditions that would be needed to push measurements into the submicrosecond region.

Two-Dimensional Fluctuation Correlation Spectroscopy (2D-FlucCS): A Method to Determine the Origin of Relaxation Rate Dispersion

Relaxation rate dispersion, i.e., nonexponential or multicomponent kinetics, is observed in complex systems when measuring relaxation kinetics. Often, the origin of rate dispersion is associated with the heterogeneity in the system. However, both homogeneous (where all molecules experience the same rate but inherently nonexponential) and heterogeneous (where all molecules experience different rates) systems can exhibit rate dispersion. A multidimensional correlation analysis method has been demonstrated to detect and quantify rate dispersion observed in molecular rotation, diffusion, solvation, and reaction kinetics. One-dimensional (1D) autocorrelation function detects rate dispersion and measures its extent. Two-dimensional (2D) autocorrelation function measures the origin of rate dispersion and distinguishes homogeneous from heterogeneous. In a heterogeneous system, implicitly there exist subensembles of molecules experiencing different rates. A three-dimensional (3D) autocorrelation function measures subensemble exchange if present and reveals if the system possesses static or dynamic heterogeneity. This perspective discusses the principles, applications, and potential and also presents a future outlook of two-dimensional fluctuation correlation spectroscopy (2D-FlucCS). The method is applicable to any experiment or simulation where a time series of fluctuation in an observable (emission, scattering, current, etc.) around a mean value can be obtained in steady state (equilibrium or nonequilibrium), provided the system is ergodic.

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.

Nonlinear measurements of kinetics and generalized dynamical modes. II. Application to a simulation of solvation dynamics in an ionic liquid

Solvation dynamics in ionic liquids show features that are often associated with supercooled liquids, including "stretched" nonexponential relaxation. To better understand the mechanism behind the stretching, the nonlinear mode-correlation methods proposed in Paper I [S. R. Hodge and M. A. Berg, J. Chem. Phys. 155, 024122 (2021)] are applied to a simulation of a prototypical ionic liquid. A full Green's function is recovered. In addition, specific tests for non-Gaussian dynamics are made. No deviations from Gaussian dynamics are found. This finding is incompatible with rate heterogeneity as a cause of the nonexponential relaxation and appears to be in conflict with an earlier multidimensional analysis of the same data. Although this conflict is not resolved here, this work does demonstrate the practicality of mode-correlation analysis in the face of finite datasets and calculations.

Jump-precursor state emerges below the crossover temperature in supercooled o-terphenyl

In a supercooled liquid, the crossover temperature T_{c} separates a high-temperature region of diffusive dynamics from a low-temperature region of activated dynamics. A molecular-dynamics simulation of all-atom, flexible o-terphenyl [Eastwood et al., J. Phys. Chem. B 117, 12898 (2013)10.1021/jp402102w] is analyzed with advanced statistical methods to reveal the molecular features associated with this crossover. The simulations extend to an α-relaxation time of 14 μs (272.5 K), two orders of magnitude slower than at T_{c} (290 K). At T_{c} and below, a distinct state emerges that immediately precedes an orientational jump. Compared to the initial, tightly caged state, this jump-precursor state has a looser cage, with solid-angular excursions of 0.054-0.0125 × 4π sr. At T_{c} (290 K), rate heterogeneity is already the dominant cause of stretched relaxation. Exchange within the distribution of rates is faster than α relaxation at T_{c}, but becomes equal to it at the lowest temperature simulated (272.5 K). The results trend toward a recent experimental observation near the glass transition (243 K) [Kaur et al., Phys. Rev. E 98, 040603(R) (2018)10.1103/PhysRevE.98.040603], which saw exchange substantially slower than α relaxation. Overall, the dynamic crossover comprises multiple phenomena: the development of heterogeneity, an increasing jump size, an emerging jump-precursor state, and a lengthening exchange time. The crossover is neither sharp, nor a simple superposition of the high- and low-temperature regimes; it is a broad region that contains unique and complex phenomena.

Micelle Heterogeneity from the 2D Kinetics of Solute Rotation

The chemical and physical properties of microstructured materials vary with position. The photophysics of solute molecules can measure these local properties, but they often show multiple rates (rate dispersion), which complicates the interpretation. In the case of micelles, rate dispersion in a solute's anisotropy decay has been assigned to either local anisotropy or heterogeneity in the local viscosity. To resolve this conflict, the rotation of PM597 molecules in SDS micelles has been measured by polarized MUPPETS (multiple population-period transient spectroscopy). This 2D technique shows that heterogeneity is strong and that local anisotropy is minimal. The results suggest that on a subnanosecond time scale, the solute sees only one strong fluctuation of the micelle structure. The anisotropic, average structure emerges on longer time scales.

Using microsecond single-molecule FRET to determine the assembly pathways of T4 ssDNA binding protein onto model DNA replication forks

DNA replication is a core biological process that occurs in prokaryotic cells at high speeds (∼1 nucleotide residue added per millisecond) and with high fidelity (fewer than one misincorporation event per 10⁷ nucleotide additions). The ssDNA binding protein [gene product 32 (gp32)] of the T4 bacteriophage is a central integrating component of the replication complex that must continuously bind to and unbind from transiently exposed template strands during DNA synthesis. We here report microsecond single-molecule FRET (smFRET) measurements on Cy3/Cy5-labeled primer-template (p/t) DNA constructs in the presence of gp32. These measurements probe the distance between Cy3/Cy5 fluorophores that label the ends of a short (15-nt) segment of ssDNA attached to a model p/t DNA construct and permit us to track the stochastic interconversion between various protein bound and unbound states. The length of the 15-nt ssDNA lattice is sufficient to accommodate up to two cooperatively bound gp32 proteins in either of two positions. We apply a unique multipoint time correlation function analysis to the microsecond-resolved smFRET data obtained to determine and compare the kinetics of various possible reaction pathways for the assembly of cooperatively bound gp32 protein onto ssDNA sequences located at the replication fork. The results of our analysis reveal the presence and translocation mechanisms of short-lived intermediate bound states that are likely to play a critical role in the assembly mechanisms of ssDNA binding proteins at replication forks and other ss duplex junctions.

Using Multiorder Time-Correlation Functions (TCFs) To Elucidate Biomolecular Reaction Pathways from Microsecond Single-Molecule Fluorescence Experiments

Recent advances in single-molecule fluorescence imaging have made it possible to perform measurements on microsecond time scales. Such experiments have the potential to reveal detailed information about the conformational changes in biological macromolecules, including the reaction pathways and dynamics of the rearrangements involved in processes, such as sequence-specific DNA "breathing" and the assembly of protein-nucleic acid complexes. Because microsecond-resolved single-molecule trajectories often involve "sparse" data, that is, they contain relatively few data points per unit time, they cannot be easily analyzed using the standard protocols that were developed for single-molecule experiments carried out with tens-of-millisecond time resolution and high "data density." Here, we describe a generalized approach, based on time-correlation functions, to obtain kinetic information from microsecond-resolved single-molecule fluorescence measurements. This approach can be used to identify short-lived intermediates that lie on reaction pathways connecting relatively long-lived reactant and product states. As a concrete illustration of the potential of this methodology for analyzing specific macromolecular systems, we accompany the theoretical presentation with the description of a specific biologically relevant example drawn from studies of reaction mechanisms of the assembly of the single-stranded DNA binding protein of the T4 bacteriophage replication complex onto a model DNA replication fork.

Rate and Amplitude Heterogeneity in the Solvation Response of an Ionic Liquid

In contrast with conventional liquids, ionic liquids have solvation dynamics with more rate dispersion and with average times that do not agree with dielectric measurements. A kinetic analog of multidimensional spectroscopy is introduced and used to look for heterogeneity in simulations of coumarin 153 in [Im12][BF4]. Strong heterogeneity is found in the diffusive solvation rate. An unanticipated heterogeneity in the amplitude of the inertial solvation is also seen. Both heterogeneities exchange at the same rate. This rate is similar to the mean diffusive solvation time, putting it in the intermediate-exchange region. Overall, there are multiple violations of the assumptions usually invoked in the theory of reaction dynamics.