Application of method of moments analysis to fluorescence decay lifetime distributions

In fluorescence decay work, distributions of exponential decay lifetimes are anticipated where complex systems are examined. We describe here methods of gaining information on such distributions using the method of moments analysis approach. The information obtained may be as simple as the average and deviation of the lifetime distribution, quantities which we show may be estimated directly from the results of a multiexponential analysis. An approximation to the actual distribution shape may also be obtained using a procedure we call the variable filter analysis (VFA) method without making any assumptions about the shape of the distribution. Tests of VFA using both simulated and experimental data are described. Limitations of this method and of distribution analysis methods in general are discussed. Results of analyses on experimental decays for ethidium intercalated in core particles and in free DNA are reported.     

Distance distributions in proteins recovered by using frequency-domain fluorometry. Applications to troponin I and its complex with troponin C

We used resonance energy transfer to examine the distribution of distances between two sites on troponin I (TnI). The donor (D) was the single tryptophan residue at site 158 (Trp 158), and the acceptor (A) was cysteine 133 (Cys 133) which was labeled with N-(iodoacetyl)-N'-(1-sulfo-5-naphthyl)ethylenediamine (IE). A distribution of D-A distances results in a distribution of donor decay times, which were resolved by using frequency-domain fluorometry. In the native state we recovered a relatively narrow distribution of D-A distances. The widths of the distance distributions were found to increase progressively and dramatically with increasing concentrations of guanidine hydrochloride. Binding of calcium-free troponin C (TnC) to troponin I did not alter the distance distribution. Addition of Ca2+ to the TnI.TnC complex resulted in a sharper distance distribution and protected against the guanidine hydrochloride induced increase in the width of the distance distribution. Additionally, the same distance distributions were recovered for native and denatured TnI when the Forster distance for energy transfer was decreased by acrylamide quenching. These results demonstrate that distance distributions can be recovered with good accuracy, to the extent of revealing modest changes due to binding of other components. This technique should have widespread applications in studies of protein folding.     

Resolvability of fluorescence lifetime distributions using phase fluorometry.

The analysis of the fluorescence decay using discrete exponential components assumes that a small number of species is present. In the absence of a definite kinetic model or when a large number of species is present, the exponential analysis underestimates the uncertainty of the recovered lifetime values. A different approach to determine the lifetime of a population of molecules is the use of probability density functions and lifetime distributions. Fluorescence decay data from continuous distributions of exponentially decaying components were generated. Different magnitudes of error were added to the data to simulate experimental conditions. The resolvability of the distributional model was studied by fitting the simulated data to one and two exponentials. The maximum width of symmetric distributions (uniform, gaussian, and lorentzian), which cannot be distinguished from single and double exponential fits for statistical errors of 1 and 0.1%, were determined. The width limits are determined by the statistical error of the data. It is also shown that, in the frequency domain, the discrete exponential analysis does not uniformly weights all the components of a distribution. This systematic error is less important when probability and distribution functions are used to recover the decay. Finally, it is shown that real lifetime distributions can be proved using multimodal probability density functions. In the companion paper that follows we propose a physical approach, which provides lifetime distribution functions for the tryptophan decay in proteins. In the third companion paper (Alcala, J.R., E. Gratton, and F.J. Prendergast, 1987, Biophys. J., in press) we use the distribution functions obtained to fit data from the fluorescence decay of single tryptophan proteins.     

Proflavin binding to poly[d(A-T)] and poly[d(A-br5U)]: triplet state and temperature-jump kinetics

The delayed fluorescence properties of proflavin have been exploited in studies of the excited-state binding kinetics of the dye to poly[d(A-T)] and its brominated analogue poly[d(A-br5U)] at room temperature and pH 7. The two analyzed luminescence decay times of the DNA-dye complex are dependent on the total nucleic acid concentration. This dependence is shown to reflect a temporal coupling of the intrinsic delayed emission decay rates with the dynamic chemical kinetic binding processes in the excited state. Temperature-jump kinetic studies conducted on the brominated polymer and corresponding information on poly[d(A-T)] from a previous study [Ramstein, J., Ehrenberg, M., & Rigler, R. (1980) Biochemistry 19, 3938-3948] provide complementary information about the ground state. In the ground state, the poly[d(A-T)]-proflavin complex has one chemical relaxation time, which reaches a plateau at high DNA concentrations. The brominated DNA-dye complex exhibits two relaxation times: a faster relaxation mode that behaves similarly to that for the unhalogenated DNA and a slower relaxation mode that is apparent at high DNA concentrations. The ground-state kinetic data are analyzed in terms of two alternative models incorporating series and parallel reaction schemes. The former consists of two sequential binding steps--a fast bimolecular process followed by a monomolecular step--while the latter consists of two coupled bimolecular steps. A similar analysis for the excited-state data yields reasonable kinetic constants only for the series model, which, in accordance with previous proposals for acridine intercalators, consists of a fast outside binding step followed by intercalation of the dye. A comparison of the ground- and excited-state kinetic parameters reveals that the external binding process is much stronger and the intercalation is much weaker in the excited state. That the excited-state data are only consistent with the series model suggests that delayed luminescence studies may provide a general tool for distinguishing between the two kinetic mechanisms. In particular, we demonstrate the use of delayed luminescence spectroscopy as a tool for probing dynamic DNA-ligand interactions in solution.     

Correction for incomplete labeling in the measurement of distance distributions by frequency-domain fluorometry.

Measurements of time-resolved fluorescence are now being used to recover conformational distributions of biological macromolecules. The fluorescence data of the donor are easily corrupted by incomplete labeling of the macromolecules by the acceptor. In the present paper we describe a general procedure to correct for incomplete acceptor labeling in the determination of distance distributions from frequency-domain measurements of the donor fluorescence decay kinetics. The method can also be used to determine the extent of acceptor labeling. Simulated data were used to determine the effect of incomplete labeling on resolution of the distance distribution and the effect on the recovered distributions if one fails to account for incomplete labeling by the acceptor. The expressions and implemented algorithm were verified using known mixtures of donor-control and donor-acceptor pair molecules, which simulated the presence of a donor population lacking the acceptor. Finally, we present data on the distance distributions between two labeled sites in myosin S1 (Cys-697 to Cys-707) where it was not possible to obtain complete labeling of the acceptor site.     

Particle size influences decay rates of environmental DNA in aquatic systems

Environmental DNA (eDNA) analysis is a powerful tool for remote detection of target organisms. However, obtaining quantitative and longitudinal information from eDNA data is challenging, requiring a deep understanding of eDNA ecology. Notably, if the various size components of eDNA decay at different rates, and we can separate them within a sample, their changing proportions could be used to obtain longitudinal dynamics information on targets. To test this possibility, we conducted an aquatic mesocosm experiment in which we separated fish-derived eDNA components using sequential filtration to evaluate the decay rate and changing proportion of various eDNA particle sizes over time. We then fit four alternative mathematical decay models to the data, building towards a predictive framework to interpret eDNA data from various particle sizes. We found that medium-sized particles (1–10 μm) decayed more slowly than other size classes (i.e., <1 and > 10 μm), and thus made up an increasing proportion of eDNA particles over time. We also observed distinct eDNA particle size distribution (PSD) between our Common carp and Rainbow trout samples, suggesting that target-specific assays are required to determine starting eDNA PSDs. Additionally, we found evidence that different sizes of eDNA particles do not decay independently, with particle size conversion replenishing smaller particles over time. Nonetheless, a parsimonious mathematical model where particle sizes decay independently best explained the data. Given these results, we suggest a framework to discern target distance and abundance with eDNA data by applying sequential filtration, which theoretically has both metabarcoding and single-target applications.     

Analysis of fluorescence decay kinetics from variable-frequency phase shift and modulation data.

Recently it has become possible to measure fluorescence phase-shift and modulation data over a wide range of modulation frequencies. In this paper we describe the analysis of these data by the method of nonlinear least squares to determine the values of the lifetimes and fractional intensities for a mixture of exponentially decaying fluorophores. Analyzing simulated data allowed us to determine those experimental factors that are most critical for successfully resolving the emissions from mixtures of fluorophores. The most critical factors are the accuracy of the experimental data, the relative difference of the individual decay times, and the inclusion of data measured at multiple emission wavelengths. After measuring at eight widely spaced modulation frequencies, additional measurements yielded only a modest increase in resolution. In particular, the uncertainty in the parameters decreased approximately as the reciprocal of the square root of the number of modulation frequencies. Our simulations showed that with presently available precision and data for one emission bandpass, two decay times could be accurately determined if their ratio were greater than or equal to 1.4. Three exponential decays could also be resolved, but only if the range of the lifetimes were fivefold or greater. To reliably determine closely-spaced decay times, the data were measured at multiple emission wavelengths so that the fractional intensities of the components could be varied. Also, independent knowledge of any of the parameters substantially increased the accuracy with which the remaining parameters could be determined. In the subsequent paper we present experimental results that broadly confirm the predicted resolving potential of variable-frequency phase-modulation fluorometry.     

Probing alpha-helical secondary structure at a specific site in model peptides via restriction of tryptophan side-chain rotamer conformation.

The relationship between alpha-helical secondary structure and the fluorescence properties of an intrinsic tryptophan residue were investigated. A monomeric alpha-helix forming peptide and a dimeric coiled-coil forming peptide containing a central tryptophan residue were synthesized. The fluorescence parameters of the tryptophan residue were determined for these model systems at a range of fractional alpha-helical contents. The steady-state emission maximum was independent of the fractional alpha-helical content. A minimum of three exponential decay times was required to fully describe the time-resolved fluorescence data. Changes were observed in the decay times and more significantly, in their relative contributions that could be correlated with alpha-helix content. The results were also shown to be consistent with a model in which the decay times were independent of both alpha-helix content and emission wavelength. In this model the relative contributions of the decay time components were directly proportional to the alpha-helix content. Data were also analyzed according to a continuous distribution of exponential decay time model, employing global analysis techniques. The recovered distributions had "widths" that were both poorly defined and independent of peptide conformation. We propose that the three decay times are associated with the three ground-state chi 1 rotamers of the tryptophan residue and that the changes in the relative contributions of the decay times are the result of conformational constraints, imposed by the alpha-helical main-chain, on the chi 1 rotamer populations.     

The lognormal distribution fits the decay profile of eukaryotic mRNA.

A general program was written which simulates radioactive labeling of RNA $\text{in vivo}$. The program was used to determine the effect that different distributions of half-lives would have on the composite decay curve observed in a pulse-chase experiment. Four biologically relevant points emerge: 1) The published, experimentally determined composite decay curves for eukaryotic mRNA are not compatible with a normal, uniform, or exponential distribution of decay times. 2) The experimental curves are compatible with a lognormal distribution of decay times as well as the two-component discrete distribution previously hypothesized. 3) If the lognormal or some similar distribution were correct, about half the mRNA species would decay faster than what is presently called the “fast component of decay”. This point is crucial to any argument about the fraction of poly (A) or other nuclear sequence that is transported to the cytoplasm. 4) If a $\text{particular}$ mRNA species is found to decay at a constant rate for 3 half-lives, that is not only consistent with 1 half-life for all the mRNA, but also consistent with 20 different half-lives which are normally or uniformly distributed.In addition to the decay of mRNA, the lognormal distribution is also compatible with data on the decay of poly(A)-containing nuclear RNA and total cellular protein.     

Emission wavelength-dependent decay of the 9-anthroyloxy-fatty acid membrane probes.

Using the phase-modulation technique, we have measured the fluorescence decay of 2- and 12-(9-anthroyloxy)-stearic acid (2- and 12-AS) and 16-(9-anthroyloxy)-palmitic acid (16-AP) bound to egg phosphatidylcholine vesicles or dissolved in nonpolar solvents. Heterogeneity analysis demonstrates that the decay is generally not monoexponential and exhibits large component variations across it emission spectrum. The mean decay time increases (and in parallel, the steady-state polarization decreases) monotonically with increasing wavelength from values at the blue end. The decay at the red side of the emission spectrum contains an exponential term with a negative amplitude, indicating that emission occurs from intermediates created in the excited-state. This behavior is interpreted as arising from intramolecular fluorophore relaxation occurring on the time scale of the fluorescence lifetime. We believe this to be the first study of wavelength-dependent fluorescent emission which is dominated by an intramolecular relaxation process. Although the three probes exhibit qualitatively similar effects, the emission band variations are greatest for 2-AS and smallest for 16-AP. The differences among the probes are not entirely due to environmental factors as demonstrated, for example, by the emission polarization differences observed in the isotropic solvent paraffin oil. In summary, while these findings point out some of the complexities in the 9-anthroyloxy-fatty acids as membrane probes, they also indicate how these complexities might be used as a sensitive measure of lipid-probe interaction.     

Resolution of the fluorescence decay of the two tryptophan residues of lac repressor using single tryptophan mutants.

We have studied the time-resolved intrinsic tryptophan fluorescence of the lac repressor (a symmetric tetramer containing two tryptophan residues per monomer) and two single-tryptophan mutant repressors obtained by site-directed mutagenesis, lac W201Y and lac W220Y. These mutant repressor proteins have tyrosine substituted for tryptophan at positions 201 and 220, respectively, leaving a single tryptophan residue per monomeric subunit at position 220 for the W201Y mutant and at position 201 in the W220Y mutant. It was found that the two decay rates recovered from the analysis of the wild type data do not correspond to the rates recovered from the analysis of the decays of the mutant proteins. Each of these residues in the mutant repressors displays at least two decay rates. Global analysis of the multiwavelength data from all three proteins, however, yielded results consistent with the fluorescence decay of the wild type lac repressor corresponding simply to the weighted linear combination of the decays from the mutant proteins. The effect of ligation by the antagonistic ligands, inducer and operator DNA, was similar for all three proteins. The binding of the inducer sugar resulted in a quenching of the long-lived species, while binding by the operator decreased the lifetime of the short components. Investigation of the time-resolved anisotropy of the intrinsic tryptophan fluorescence in these three proteins revealed that the depolarization of fluorescence resulted from a fast motion and the global tumbling of the macromolecule. Results from the simultaneous global analysis of the frequency domain data sets from the three proteins revealed anisotropic rotations for the macromolecule, consistent with the known elongated shape of the repressor tetramer. In addition, it appears that the excited-state dipole of tryptophan 220 is alighed with the long axis of the repressor.     

Protein fluorescence decay: discrete components or distribution of lifetimes? Really no way out of the dilemma?

A new methodology of fluorescence decay analysis by iterative reconvolution is presented. It is based on the recent finding that the statistics of single-photon time-correlated data are best described by a compound Poisson law and requires the recording of a sample of at least 20 decays. Application of multivariate statistical methods to the analysis of the recovered decay parameters results in improved accuracy and better estimation of the uncertainties of mono- and multiexponential decays. If it is, of course, not possible to distinguish unambiguously between discrete components and a continuous distribution of lifetimes, it is, however, possible to determine a higher limit of the width of such a distribution should it be present. With our methodology, the presence of a distribution of lifetimes with a width of approximately 20% of its center value inevitably leads to a failure in the deconvolution procedure, a fact of crucial importance in protein conformational studies, for example.     

Demonstration of an associated anisotropy decay by frequency-domain fluorometry

We used frequency-domain fluorometry to demonstrate the presence of an associated decay of fluorescence anisotropy. In such systems the individual correlation times are associated with distinct emitting species, each with its own characteristic lifetime and rotational correlation times. We obtained an associated system using 1-anilino-8-naphthalenesulfonic acid (ANS) in the presence of increasing amounts of apomyoglobin. When both free and apomyoglobin-bound ANS contributed to the emission the differential polarized phase angles become negative at particular frequencies, even though the fundamental anisotropy (r0) is greater than zero. Additionally, the modulated anisotropy decreases at high frequencies. Both observations appear to be the unique consequence of an associated anisotropy decay, and are not possible for a multiexponential anisotropy decay of a single species.     

Fluorescence lifetime spectroscopy in multiply scattering media with dyes exhibiting multiexponential decay kinetics

To investigate fluorescence lifetime spectroscopy in tissue-like scattering, measurements of phase modulation as a function of modulation frequency were made using two fluorescent dyes exhibiting single exponential decay kinetics in a 2% intralipid solution. To experimentally simulate fluorescence multiexponential decay kinetics, we varied the concentration ratios of the two dyes, 3,3-diethylthiatricarbocyanine iodide and indocynanine green (ICG), which exhibit distinctly different lifetimes of 1.33 and 0.57 ns, respectively. The experimental results were then compared with values predicted using the optical diffusion equation incorporating 1) biexponential decay, 2) average of the biexponential decay, as well as 3) stretched exponential decay kinetic models to describe kinetics owing to independent and quenched relaxation of the two dyes. Our results show that while all kinetic models could describe phase-modulation data in nonscattering solution, when incorporated into the diffusion equation, the kinetic parameters failed to likewise predict phase-modulation data in scattering solutions. We attribute the results to the insensitivity of phase-modulation measurements in nonscattering solutions and the inaccuracy of the derived kinetic parameters. Our results suggest the high sensitivity of phase-modulation measurements in scattering solutions may provide greater opportunities for fluorescence lifetime spectroscopy.     

Analysis of heterogeneous fluorescence decays in proteins. Using fluorescence lifetime of 8-anilino-1-naphthalenesulfonate to probe apomyoglobin unfolding at equilibrium

The solvatochromic fluorescent dye 8-anilino-1-naphthalenesulfonate (ANS) is one of the popular probes of protein folding. Folding kinetics is tracked with ANS fluorescence intensity, usually interpreted as a reflection of protein structure-the hydrophobicity of the binding environments. Such simplistic view overlooks the complicated nature of ANS-protein complexes: the fluorescence characteristics are convoluted results of the ground state populational distribution of the probe-protein complex, the structural changes in the protein and the excited state photophysics of the probe. Understanding of the interplay of these aspects is crucial in accurate interpretation of the protein dynamics. In this work, the fluorescence decay of ANS complexed with apomyoglobin at different conformations denatured by pH is modeled. The fluorescence decay of the ANS-apomyoglobin complex contains information on not only apomyoglobin structure but also molecular populational distributions. The challenge in modeling fluorescence decay profiles originates from the convolution of heterogeneous binding and excited-state relaxation of the fluorescent probe. We analyzed frequency-domain fluorescence lifetime data of ANS-apomyoglobin with both maximum entropy methods (MEM) and nonlinear least squares methods (NLLS). MEM recovers a model of two expanding-and-merging lifetime distributions for ANS-apomyoglobin in the equilibrium transition from the native (N) through an intermediate (I-1) to the acid-unfolded state U(A). At pH 6.5 and above, when apomyoglobin is mostly populated at the N-state, ANS-apomyoglobin emits a predominant long-lifetime fluorescence from a relaxed charge transfer state S(1,CT) of ANS, and a short-lifetime fluorescence that is mainly from a nascent excited-state S(1,np) of ANS stabilized by the strong ANS-apomyoglobin interaction. Lowering the pH diminishes the contribution from the S(1,np) state. Meanwhile, more protein molecules become populated at the U(A) state, which exhibits a short lifetime that is not distinguishable from the S(1,np) state. At pH 3.4, when the population of the U(A) becomes significant, the short-lifetime fluorescence comes predominantly from ANS binding to the U(A). Further lowering the pH leads to more exposure of the bound ANS. The long lifetime shifts toward and finally merges with the short lifetime and becomes one broad distribution that stands for ANS binding to the U(A) below pH 2.4. The above expanding-and-merging model is consistent with F-statistic analysis of NLLS models. The consistency of this model with the knowledge from the literature, as well as the continuity of the decay parameters changing upon experimental conditions are also crucial in drawing the conclusions.     

Dynamics of erabutoxin b as studied by nuclear magnetic resonance. Relaxation studies of methyl proton resonances

Longitudinal and transverse relaxation times were measured for well-resolved and assigned methyl proton resonances of erabutoxin b at 270 MHz, 300 MHz and 500 MHz. Both longitudinal and transverse magnetization decay curves are non-exponential due to cross-relaxation and cross-correlation effects. The longitudinal and transverse relaxation rates were obtained from the initial slope of both magnetization decay curves. The correlation times for the isotropic tumbling motion of the protein were determined to be 2.82 ns at 300 K and 1.62 ns at 330 K from the analysis of the relaxation data of some alpha protons. Using these values, the relaxation data of methyl protons were fitted to various theoretical models. Most of the methyl resonances could be fitted well to a model which allowed methyl rotation (in the range 0.01-0.05 ns) and an external contribution from protons assumed to be in positions derived from X-ray coordinates. The data for a few methyl groups, however, could not be fitted in this way. For these a smaller number of external protons than predicted by the X-ray coordinates was assumed. Additionally, a larger amplitude motion had to be introduced into the model for particular residues. This additional motion requires concerted protein motion close to these residues, since the X-ray structure suggests that steric hindrance would prevent local motion. These results are consistent with the idea of a flexible and dynamic structure for proteins.     

Identification of functional p53-binding motifs in the mouse wig-1 promoter

With selective excitation around BChl-B800 and BChl-B850 absorption bands, we observed the evolution of excited-state dynamics in LH2 from Rhodobacter sphaeroides 601. The dynamical traces demonstrate a dominant excited-state absorption (ESA) followed concomitantly by an ultrafast transmission increase and decay with pulse-width limited time scale at 818 nm and 828 nm excitation. The ESA occurring prior to excitonic thermalization or ground-state bleach was observed at 840 nm as well. These experimental results indicate the competition between the transition from excitonic states to higher-lying excited states and interexciton relaxation, which are of physical significance for understanding excitation transfer and related mechanisms in LH2.     

Lateral diffusion coefficients in membranes measured by resonance energy transfer and a new algorithm for diffusion in two dimensions

We describe measurements of lateral diffusion in membranes using resonance energy transfer. The donor was a rhenium (Re) metal-ligand complex lipid, which displays a donor decay time near 3 micros. The long donor lifetime resulted in an ability to measure lateral diffusion coefficient below 10(-8) cm(2)/s. The donor decay data were analyzed using a new numerical algorithm for calculation of resonance energy transfer for donors and acceptors randomly distributed in two dimensions. An analytical solution to the diffusion equation in two dimensions is not known, so the equation was solved by the relaxation method in Laplace space. This algorithm allows the donor decay in the absence of energy transfer to be multiexponential. The simulations show that mutual lateral diffusion coefficients of the donor and acceptor on the order of 10(-8) cm(2)/s are readily recovered from the frequency-domain data with donor decay times on the microsecond timescale. Importantly, the lateral diffusion coefficients and acceptor concentrations can be recovered independently despite correlation between these parameters. This algorithm was tested and verified using the donor decays of a long lifetime rhenium lipid donor and a Texas red-lipid acceptor. Lateral diffusion coefficients ranged from 4.4 x 10(-9) cm(2)/s in 1,2-dimyristoyl-sn-glycero-3-[phospho-rac-(1-glycerol)] (DMPG) at 10 degrees C to 1.7 x 10(-7) cm(2)/s in 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) at 35 degrees C. These results demonstrated the possibility of direct measurements of lateral diffusion coefficients using microsecond decay time luminophores.     

Spatio-temporal attributes of left ventricular pressure decay rate during isovolumic relaxation

Global left ventricular (LV) isovolumic relaxation rate has been characterized: 1) via the time constant of isovolumic relaxation τ or 2) via the logistic time constant τ(L). An alternate kinematic method, characterizes isovolumic relaxation (IVR) in accordance with Newton's Second Law. The model's parameters, stiffness E(k), and damping/relaxation μ result from best fit of model-predicted pressure to in vivo data. All three models (exponential, logistic, and kinematic) characterize global relaxation in terms of pressure decay rates. However, IVR is inhomogeneous and anisotropic. Apical and basal LV wall segments untwist at different times and rates, and transmural strain and strain rates differ due to the helically variable pitch of myocytes and sheets. Accordingly, we hypothesized that the exponential model (τ) or kinematic model (μ and E(k)) parameters will elucidate the spatiotemporal variation of IVR rate. Left ventricular pressures in 20 subjects were recorded using a high-fidelity, multipressure transducer (3 cm apart) catheter. Simultaneous, dual-channel pressure data was plotted in the pressure phase-plane (dP/dt vs. P) and τ, μ, and E(k) were computed in 1631 beats (average: 82 beats per subject). Tau differed significantly between the two channels (P < 0.05) in 16 of 20 subjects, whereas μ and E(k) differed significantly (P < 0.05) in all 20 subjects. These results show that quantifying the relaxation rate from data recorded at a single location has limitations. Moreover, kinematic model based analysis allows characterization of restoring (recoil) forces and resistive (crossbridge uncoupling) forces during IVR and their spatio-temporal dependence, thereby elucidating the relative roles of stiffness vs. relaxation as IVR rate determinants.