On the analysis of fluorescence decay kinetics by the method of least-squares

Analysis of fluorescence decay kinetics aims at the determination of the analytic expression and the numerical values of the pertinent parameters which describe the decay process. In the well-known method of least-squares, one assumes a plausible functional form for the decay data and adjusts the values of the parameters until the statistically best fit is obtained between the data and the calculated decay function, i.e., until the sum of the weighted squares of the residuals is at a minimum. It is shown that proper weighting of the squares of the residuals may markedly improve the quality of the analysis. Such weighting requires information about the character of the experimental noise, which is often available, e.g., when the noise is due to counting error in photon-counting techniques. Furthermore, dramatic improvements in the accuracy of the analysis may often be achieved by use of auxiliary information available about the system studied. For example, the preexponents in a multiexponential fluorescence decay of a mixture of chromophores (such as tryptophan residues in a protein molecule) may sometimes be estimated independently; much higher accuracy can then be attained for the decay lifetimes by analysis of the decay kinetics. It is proposed that the shape of the autocorrelation function of the weighted residuals may serve as a convenient criterion for the quality of fit between the experimental data and the decay function obtained by analysis. The above conclusions were reached by analysis of computer-simulated experiments, and the usefulness of this approach is illustrated. The importance of stating the uncertainties in the estimated parameters inherent in the analysis of decay kinetics is stressed.     

Stability of Radiation-Induced Organic Free Radicals: Decay by Heat

The rate of free radical decay was measured at various temperatures using electron paramagnetic resonance spectroscopy. Rate constants determined from first-order decay kinetics were used to determine the activation energy for the process of free radical decay. The similarity between the temperature dependence of free radical decay by heat and that of electrical conductivity has led us to consider the possibility that the two processes may be related. Mechanisms by which a population of electron-hole conducting states may lead to free radical decay are outlined and experimental data relating to these mechanisms are discussed.     

Spin trap determination of free radical burst kinetics in stimulated neutrophils

Electron spin resonance spin-trapping methods were used to investigate the free radical production kinetics of neutrophils stimulated with phorbol myristate acetate (PMA) and opsonized zymosan (OPZ). Using the spin trap 5,5-dimethyl-1-pyrroline-N-oxide, the principle spin adduct observed is DMPO-OH (trapped hydroxyl radical). The DMPO-OH ESR signal amplitude was observed to decay exponentially. In such cases a simple method may be used to analyze the raw kinetics amplitude data to yield true production rate and net production data. The method, pitfalls, and self-consistency criteria are illustrated with PMA and OPZ-stimulated neutrophils at 25 and 37 degrees C under varying oxygen tensions, and with noise-free simulated data. The simulations demonstrate that rate results are relatively insensitive to the precise choice of decay time constant, tc, while net production results are very sensitive to the choice of tc used to analyze the raw data. OPZ (0.6-2.4 mg/ml) yields a strong, sharp neutrophil burst which peaks in 2 min or less while PMA yields a slower burst which peaks in 3.4-14 min for PMA concentrations of 500-50 ng/ml, respectively. Increased oxygen tension during the PMA experiments increased the spin adduct lifetime. The methods presented are applicable to other cell systems or spin adducts which exhibit first order decay.     

Background suppression in frequency-domain fluorometry

Gated detection is often used in time-domain measurements of long-lived fluorophores for suppression of interfering short-lived autofluorescence. However, no direct method has been available for gated detection and background suppression when using frequency-domain fluorometry. We describe a direct method for real-time suppression of autofluorescence in frequency-domain fluorometry. The method uses a gated detector and the sample is excited by a pulsed train. The detector is gated on following each excitation pulse after a suitable time delay for decay of the prompt autofluorescence. Under the same experimental conditions a detectable reference signal is obtained by using a long lifetime standard with a known decay time. Because the sample and reference signals are measured under identical excitation, gating and instrumental conditions, the data can be analyzed as usual for frequency-domain data without further processing. We show by simulations that this method can be used to resolve single and multiexponential decays in the presence of short lifetime autofluorescence.     

Application of a reference convolution method to tryptophan fluorescence in proteins. A refined description of rotational dynamics

A reference method for the deconvolution of polarized fluorescence decay data is described. Fluorescence lifetime determinations for p-terphenyl, p-bis[2-(5-phenyloxazolyl)]benzene and N-acetyltryptophanamide (AcTrpNH2) show that with this method more reliable fits of the decays can be made than with the scatterer method, which is most frequently used. Analysis of the AcTrpNH2 decay with p-terphenyl as the reference compound yields an excellent fit with lifetimes of 2.985 ns for AcTrpNH2 and 1.099 ns for p-terphenyl (20 degrees C), whereas the AcTrpNH2 decay cannot be satisfactorily fitted when the scatterer method is used. The frequency of the detected photons is varied to determine the conditions where pulse pile-up starts to affect the measured decays. At detection frequencies of 5 kHz and 15 kHz, which corresponds to 1.7% and 5% respectively of the rate of the excitation photons no effects are found. Decays measured at 30 kHz (10%) are distorted, indicating that pile-up effects play a role at this frequency. The fluorescence and fluorescence anisotropy decays of the tryptophan residues in the proteins human serum albumin, horse liver alcohol dehydrogenase and lysozyme have been reanalysed with the reference method. The single tryptophan residue of the albumin is shown to be characterized by a triple-exponential fluorescence decay. The anisotropy decay of albumin was found to be mono-exponential with a rotational correlation time of 26 ns (20 degrees C). The alcohol dehydrogenase has two different tryptophan residues to which single lifetimes are assigned. It is found that the rotational correlation time for the dehydrogenase changes with excitation wavelength (33 ns for lambda ex = 295 nm and 36 ns for lambda ex = 300 nm at 20 degrees C), indicating a nonspherical protein molecule. Lysozyme has six tryptophan residues, which give rise to a triple-exponential fluorescence decay. A single-exponential decay with a rotational correlation time of 3.8 ns is found for the anisotropy. This correlation time is significantly shorter than that arising from the overall rotation and probably originates from intramolecular, segmental motion.     

F/F deconvolution of fluorescence decay data

An approach for the deconvolution of multiexponential fluorescence decay data in which a single exponential decay is used in place of the usual excitation profile is described. For analysis by the method of moments, the resulting decay lifetimes are identical to those in the multiexponential decay, while the pre-exponential factors are a simple function of the true values and the parameters of the single exponential decay. This approach, which we call the F/F deconvolution method, is capable of eliminating the errors in decay analyses which arise from the wavelength dependence of the instrument response function.     

The fluorescence anisotropy decay due to energy transfers occuring in the ethidium bromide-DNA complex. Determination of the deformation angle of the DNA helix

It has been shown in a preceding work that the fluorescence anisotropy decay of ethidium bromide-DNA complex is accelerated by energy migration between dyes bound to the same DNA molecule. In the present work, this result is confirmed. A quantitative analysis has been performed in the following way. The spectroscopic term of the transfer rate constant has been accurately reevaluated by quantum yield and spectral measurements. One assumes that the dye intercalates between two adjacent base pairs and that its distribution is random along the DNA molecule. One introduces the deformation angle δ of the DNA helix induced by the ethidium bromide intercalation. For several values of δ, the energy migration contribution to the anisotropy decay is computed by a Monte Carlo method. In multiplying these computed functions by the measured brownian anisotropy, one obtains the anisotropy decay curve. Comparison with the experimental data leads to the conclusion that the ethidium bromide molecule unwinds the DNA helix by an angle δ = ?16°. This result is m agreement with the work of other authors. We think that the method used here may provide accurate information on the spatial distribution of an array of chromophores bound to a rigid structure.     

Maximum likelihood method for the analysis of time-resolved fluorescence decay curves

The usefulness of fluorescence techniques for the study of macromolecular structure and dynamics depends on the accuracy and sensitivity of the methods used for data analysis. Many methods for data analysis have been proposed and used, but little attention has been paid to the maximum likelihood method, generally known as the most powerful statistical method for parameter estimation. In this paper we study the properties and behavior of maximum likelihood estimates by using simulated fluorescence intensity decay data. We show that the maximum likelihood method provides generally more accurate estimates of lifetimes and fractions than does the standard least-squares approach especially when the lifetime ratios between individual components are small. Three novelties to the field of fluorescence decay analysis are also introduced and studied in this paper: a) discretization of the convolution integral based on the generalized integral mean value theorem: b) the likelihood ratio test as a tool to determine the number of exponential decay components in a given decay profile; and c) separability and detectability indices which provide measures on how accurately, a particular decay component can be detected. Based on the experience gained from this and from our previous study of the Padé-Laplace method, we make some recommendations on how the complex problem of deconvolution and parameter estimation of multiexponential functions might be approached in an experimental setting. Offprint requests to: F. G. Prendergast     

The use of moment index displacement in analyzing fluorescence time-decay data

Moment index displacement automatically corrects a number of significant nonrandom instrumental errors in fluorescence time-decay measurements. Three-component data, obtained by measuring the fluorescence decay of three different species mixed in the same solution, were used as a test sample. It was shown, as predicted by theory, that moment index displacement corrects three nonrandom instrumental errors: (1) the presence of scatter in the data; (2) time origin shifts between lamp and fluorescence data; and (3) lamp drift, or time-dependent changes in the shape of the excitation curve. The data clearly show that the use of the method of moments with moment index displacement to analyze fluorescence decay data is not a curve-fitting procedure. This procedure will accurately obtain decay parameters for multiple-exponential decays from certain badly distorted data, yielding a calculated curve very different from the actual data.     

A nomogram for deconvolution of single exponential fluorescence decays.

An extremely rapid technique for deconvolving single exponential luminescence decay data is described that involves essentially no mathematical manipulation of the experimental data. The method permits "real time" measurement of deconvolved luminescence lifetimes with conventional pulsed, lifetime-fluorometers and phosphorimeters. The method assumes that the true luminescence decay of the chromophore is accurately represented by a single exponential decay function.     

Analysis of fluorescence decay curves by means of the Laplace transformation.

A computational procedure is described for the analysis of fluorescence decay data convolved with a lamp flash of finite width. The computer program calculates the ratio of the Laplace transforms of the decay and the lamp flash for different values of s to give the transforms of the impulse response for each value of s. These are set equal to the analytical Laplace transforms of the decay law involved. Solution of the nonlinear simultaneous equations yields the desired decay parameters. The method can be modified to analyze data that contains a component due to scattered light and can also provide essential information regarding transit time changes of the photomultiplier with changes in emission wavelength. The method was tested by the analysis of real and simulated data. The accuracy of the analysis depends on the degree of correlation among the parameters.     

Estimating Orangutan Densities Using the Standing Crop and Marked Nest Count Methods: Lessons Learned for Conservation

Reliable estimates of great ape abundance are needed to assess distribution, monitor population status, evaluate conservation tactics, and identify priority populations for conservation. Rather than using direct counts, surveyors often count ape nests. The standing crop nest count (SCNC) method converts the standing stock of nests into animal densities using a set of parameters, including nest decay rate. Nest decay rates vary greatly over space and time, and it takes months to calculate a site-specific value. The marked nest count (MNC) method circumvents this issue and only counts new nests produced during a defined period. We compared orangutan densities calculated by the two methods using data from studies in Sumatra and Kalimantan, Indonesia. We show how animal densities calculated using nest counts should be cautiously interpreted when used to make decisions about management or budget allocation. Even with site-specific decay rates, short studies using the SCNC method may not accurately reflect the current population unless conducted at a scale sufficient to include wide-ranging orangutan movement. Density estimates from short studies using the MNC method were affected by small sample sizes and by orangutan movement. To produce reliable results, the MNC method may require a similar amount of effort as the SCNC method. We suggest a reduced reliance on the traditional line transect surveys in favor of feasible alternative methods when absolute abundance numbers are not necessary or when site-specific nest decay rates are not known. Given funding constraints, aerial surveys, reconnaissance walks, and interview techniques may be more cost-effective means of accomplishing some survey goals.     

Comparison of Linear,Nonlinear and Semiparametric Mixed‐effects Models for Estimating HIV Dynamic Parameters

The potency of antiretroviral agents in AIDS clinical trials can be assessed on the basis of an early viral response such as viral decay rate or change in viral load (number of copies of HIV RNA) of the plasma. Linear, parametric nonlinear, and semiparametric nonlinear mixed‐effects models have been proposed to estimate viral decay rates in viral dynamic models. However, before applying these models to clinical data, a critical question that remains to be addressed is whether these models produce coherent estimates of viral decay rates, and if not, which model is appropriate and should be used in practice. In this paper, we applied these models to data from an AIDS clinical trial of potent antiviral treatments and found significant incongruity in the estimated rates of reduction in viral load. Simulation studies indicated that reliable estimates of viral decay rate were obtained by using the parametric and semiparametric nonlinear mixed‐effects models. Our analysis also indicated that the decay rates estimated by using linear mixed‐effects models should be interpreted differently from those estimated by using nonlinear mixed‐effects models. The semiparametric nonlinear mixed‐effects model is preferred to other models because arbitrary data truncation is not needed. Based on real data analysis and simulation studies, we provide guidelines for estimating viral decay rates from clinical data. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A Four-Compartment Model for Ca2+ Dynamics: An Interpretation of Ca2+ Decay after Repetitive Firing of Intact Nerve Terminals

In the presynaptic nerve terminals of the bullfrog sympathetic ganglia, repetitive nerve firing evokes [Ca²⁺] transients that decay monotonically. An algorithm based on an eigenfunction expansion method was used for fitting these [Ca²⁺] decay records. The data were fitted by a linear combination of two to four exponential functions. A mathematical model with three intraterminal membrane-bound compartments was developed to describe the observed Ca²⁺ decay. The model predicts that the number of exponential functions, n, contained in the decay data corresponds to n – 1 intraterminal Ca²⁺ stores that release Ca²⁺ during the decay. Moreover, when a store stops releasing or starts to release Ca²⁺, the decay data should be fitted by functions that contain one less exponential component for the former and one more for the latter than do the fitting functions for control data. Because of the current lack of a parameter by which quantitative comparisons can be made between two decay processes when at least one of them contained more than one exponential components, we defined a parameter, the overall rate (OR) of decay, as the trace of the coefficient matrix of the differential equation systems of our model. We used the mathematical properties of the model and of the OR to interpret effects of ryanodine and of a mitochondria uncoupler on Ca²⁺ decay. The results of the analysis were consistent with the ryanodine-sensitive store, mitochondria, and another, yet unidentified store release Ca²⁺ into the cytosol of the presynaptic nerve terminals during Ca²⁺ decay. Our model also predicts that mitochondrial Ca²⁺ buffering accounted for more than 86% of all the flux rates across various membranes combined and that there are type 3 and type 1 and/or type 2 ryanodine receptors in these terminals.     

Closed-time distribution of ionic channels. Analytical solution to a one-dimensional defect-diffusion model.

A one-dimensional version of the model recently proposed by L?uger (1988) to explain the closed-time distribution of ionic channels in cell membranes is solved analytically. While the probability density f(t) for closed-time lengths may show a well-defined exponential behavior at short times, a power-law decay is predicted at long times. The influence of an additional random distribution of defects in the current-conducting protein is investigated and found to be dominating at long times. Explicit expressions that may be used for fitting experimental data are given for the closed-time distribution. Some of the available data are discussed and shown to be in good agreement with the predictions of the model.     

Time-resolved room temperature protein phosphorescence: nonexponential decay from single emitting tryptophans.

The single room temperature phosphorescent (RTP) residue of horse liver alcohol dehydrogenase (LADH). Trp-314, and of alkaline phosphatase (AP), Trp-109, show nonexponential phosphorescence decays when the data are collected to a high degree of precision. Using the maximum entropy method (MEM) for the analysis of these decays, it is shown that AP phosphorescence decay is dominated by a single Gaussian distribution, whereas for LADH the data reveal two amplitude packets. The lifetime-normalized width of the MEM distribution for both proteins is larger than that obtained for model monoexponential chromophores (e.g., terbium in water and pyrene in cyclohexane). Experiments show that the nonexponential decay is fundamental; i.e., an intrinsic property of the pure protein. Because phosphorescence reports on the state of the emitting chromophore, such nonexponential behavior could be caused by the presence of excited state reactions. However, it is also well known that the phosphorescence lifetime of a tryptophan residue is strongly dependent on the local flexibility around the indole moiety. Hence, the nonexponential phosphorescence decay may also be caused by the presence of at least two states of different local rigidity (in the vicinity of the phosphorescing tryptophan) corresponding to different ground state conformers. The observation that in the chemically homogeneous LADH sample the phosphorescence decay kinetics depends on the excitation wavelength further supports this latter interpretation. This dependence is caused by the wavelength-selective excitation of Trp-314 in a subensemble of LADH molecules with differing hydrophobic and rigid environments. With this interpretation, the data show that interconversion of these states occurs on a time scale long compared with the phosphorescence decay (0.1-1.0 s). Further experiments reveal that with increasing temperature the distributed phosphorescence decay rates for both AP and LADH broaden, thus indicating that either 1) the number of conformational states populated at higher temperature increases or 2) the temperature differentially affects individual conformer states. The nature of the observed heterogeneous triplet state kinetics and their relationship to aspects of protein dynamics are discussed.