Revival of high-order fluorescence correlation analysis: generalized theory and biochemical applications

Radial pair correlation of molecular brightness fluctuations maps protein diffusion as a function of oligomeric state within live-cell nuclear architecture

Nuclear proteins can modulate their DNA binding activity and the exploration volume available during DNA target search by self-associating into higher-order oligomers. Directly tracking this process in the nucleoplasm of a living cell is, however, a complex task. Thus, here we present a microscopy method based on radial pair correlation of molecular brightness fluctuations (radial pCOMB) that can extract the mobility of a fluorescently tagged nuclear protein as a function of its oligomeric state and spatiotemporally map the anisotropy of this parameter with respect to nuclear architecture. By simply performing a rapid frame scan acquisition, radial pCOMB has the capacity to detect, within each pixel, protein oligomer formation and the size-dependent obstruction nuclear architecture imparts on this complex's transport across sub-micrometer distances. From application of radial pCOMB to an oligomeric transcription factor and DNA repair protein, we demonstrate that homo-oligomer formation differentially regulates chromatin accessibility and interaction with the DNA template.

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.

Resolving fluorescent species by their brightness and diffusion using correlated photon-counting histograms

Fluorescence fluctuation spectroscopy (FFS) refers to techniques that analyze fluctuations in the fluorescence emitted by fluorophores diffusing in a small volume and can be used to distinguish between populations of molecules that exhibit differences in brightness or diffusion. For example, fluorescence correlation spectroscopy (FCS) resolves species through their diffusion by analyzing correlations in the fluorescence over time; photon counting histograms (PCH) and related methods based on moment analysis resolve species through their brightness by analyzing fluctuations in the photon counts. Here we introduce correlated photon counting histograms (cPCH), which uses both types of information to simultaneously resolve fluorescent species by their brightness and diffusion. We define the cPCH distribution by the probability to detect both a particular number of photons at the current time and another number at a later time. FCS and moment analysis are special cases of the moments of the cPCH distribution, and PCH is obtained by summing over the photon counts in either channel. cPCH is inherently a dual channel technique, and the expressions we develop apply to the dual colour case. Using simulations, we demonstrate that two species differing in both their diffusion and brightness can be better resolved with cPCH than with either FCS or PCH. Further, we show that cPCH can be extended both to longer dwell times to improve the signal-to-noise and to the analysis of images. By better exploiting the information available in fluorescence fluctuation spectroscopy, cPCH will be an enabling methodology for quantitative biology.

Two-Color Spatial Cumulant Analysis Detects Heteromeric Interactions between Membrane Proteins

Fluorescence fluctuation spectroscopy can be used to measure the aggregation of fluorescently labeled molecules and is typically performed using time series data. Spatial intensity distribution analysis and fluorescence moment image analysis are established tools for measuring molecular brightnesses from single-color images collected with laser scanning microscopes. We have extended these tools for analysis of two-color images to resolve heteromeric interactions between molecules labeled with spectrally distinct chromophores. We call these new methods two-color spatial intensity distribution analysis and two-color spatial cumulant analysis (2c-SpCA). To implement these techniques on a hyperspectral imaging system, we developed a spectral shift filtering technique to remove artifacts due to intrinsic cross talk between detector bins. We determined that 2c-SpCA provides better resolution from samples containing multiple fluorescent species; hence, this technique was carried forward to study images of living cells. We used fluorescent heterodimers labeled with enhanced green fluorescent protein and mApple to quantify the effects of resonance energy transfer and incomplete maturation of mApple on brightness measurements. We show that 2c-SpCA can detect the interaction between two components of trimeric G-protein complexes. Thus, 2c-SpCA presents a robust and computationally expedient means of measuring heteromeric interactions in cellular environments.

Determination of Equilibrium Constant and Relative Brightness in FRET-FCS by Including the Third-Order Correlations

Fluorescence correlation spectroscopy (FCS) encodes the information on the equilibrium constant (K), the relative fluorescence brightness of fluorophore (Q), and the forward and backward reaction rate constants (k₊ and k_-) on a physical or chemical relaxation. However, it has been a long-standing problem to completely resolve the FCS data to get the thermodynamic and kinetic information. Recently, we have solved the problem for fluorescence autocorrelation spectroscopy (FACS). Here, we extend the method to fluorescence cross-correlation spectroscopy (FCCS), which appears when FCS is coupled with fluorescence resonance energy transfer (FRET). Among 12 total second-order and third-order pre-exponential factors in a relaxation process probed by the FRET-FCS technique, 3 are independent. We presented and discussed 3 sets of explicit solutions to use these pre-exponential factors to calculate K and Q. Together with the relaxation time, the acquired K will allow people to obtain k₊ and k_-, so that the goal of deciphering the FRET-FCS data will be fully reached. The theory is verified by extensive computer simulations and tested experimentally on a system of oligonucleotide hybridization.

Analytical form of the autocorrelation function for the fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) can provide information about diffusion coefficients and rate constants of chemical reactions in small systems of interacting molecules. However, the interpretation of FCS experiments depends crucially on the model of the autocorrelation function for the fluorescence intensity fluctuations. In this theoretical work, we consider a system of fluorescent molecules that diffuse and interact with massive particles, e.g. surfactant micelles. Using the general formalism of FCS, we derive a new analytical approximation of the autocorrelation function for systems in which both diffusion and a binary reaction occur. This approximation provides a smooth interpolation between the limit of fast reaction (much faster than diffusion), and the opposite limit of slow reaction. Our studies of noncovalent interactions of micelles with dyes by FCS provided an experimental case to which the approximate autocorrelation function was successfully applied [X. Zhang, A. Poniewierski, A. Jelińska, A. Zagożdżon, A. Wisniewska, S. Hou and R. Hołyst, Soft Matter, 2016, 12, 8186-8194].

Quantifying the dynamics of the oligomeric transcription factor STAT3 by pair correlation of molecular brightness

Oligomerization of transcription factors controls their translocation into the nucleus and DNA-binding activity. Here we present a fluorescence microscopy analysis termed pCOMB (pair correlation of molecular brightness) that tracks the mobility of different oligomeric species within live cell nuclear architecture. pCOMB amplifies the signal from the brightest species present and filters the dynamics of the extracted oligomeric population based on arrival time between two locations. We use this method to demonstrate a dependence of signal transducer and activator of transcription 3 (STAT3) mobility on oligomeric state. We find that on entering the nucleus STAT3 dimers must first bind DNA to form STAT3 tetramers, which are also DNA-bound but exhibit a different mobility signature. Examining the dimer-to-tetramer transition by a cross-pair correlation analysis (cpCOMB) reveals that chromatin accessibility modulates STAT3 tetramer formation. Thus, the pCOMB approach is suitable for mapping the impact oligomerization on transcription factor dynamics.

The spectroscopic basis of fluorescence triple correlation spectroscopy

We have developed fluorescence triple correlation spectroscopy (F3CS) as an extension of the widely used fluorescence microscopy technique fluorescence correlation spectroscopy. F3CS correlates three signals at once and provides additional capabilities for the study of systems with complex stoichiometry, kinetic processes, and irreversible reactions. A general theory of F3CS was developed to describe the interplay of molecular dynamics and microscope optics, leading to an analytical function to predict experimental triple correlations of molecules that freely diffuse through the tight focus of the microscope. Experimental correlations were calculated from raw fluorescence data using triple correlation integrals that extend multiple-tau correlation theory to delay times in two dimensions. The quality of experimental data was improved by tuning specific spectroscopic parameters and employing multiple independent detectors to minimize optoelectronic artifacts. Experiments with the reversible system of freely diffusing 16S rRNA revealed that triple correlation functions contain symmetries predicted from time-reversal arguments. Irreversible systems are shown to break these symmetries, and correlation strategies were developed to detect time-reversal asymmetries in a comprehensive way with respect to two delay times, each spanning many orders of magnitude in time. The correlation strategies, experimental approaches, and theory developed here enable studies of the composition and dynamics of complex systems using F3CS.

Fluorescence correlation spectroscopy: past, present, future

In recent years fluorescence correlation spectroscopy (FCS) has become a routine method for determining diffusion coefficients, chemical rate constants, molecular concentrations, fluorescence brightness, triplet state lifetimes, and other molecular parameters. FCS measures the spatial and temporal correlation of individual molecules with themselves and so provides a bridge between classical ensemble and contemporary single-molecule measurements. It also provides information on concentration and molecular number fluctuations for nonlinear reaction systems that complement single-molecule measurements. Typically implemented on a fluorescence microscope, FCS samples femtoliter volumes and so is especially useful for characterizing small dynamic systems such as biological cells. In addition to its practical utility, however, FCS provides a window on mesoscopic systems in which fluctuations from steady states not only provide the basis for the measurement but also can have important consequences for the behavior and evolution of the system. For example, a new and potentially interesting field for FCS studies could be the study of nonequilibrium steady states, especially in living cells.