A model for multiexponential tryptophan fluorescence intensity decay in proteins.

Tryptophan fluorescence intensity decay in proteins is modeled by multiexponential functions characterized by lifetimes and preexponential factors. Commonly, multiple conformations of the protein are invoked to explain the recovery of two or more lifetimes from the experimental data. However, in many proteins the structure seems to preclude the possibility of multiple conformers sufficiently different from one another to justify such an inference. We present here another plausible multiexponential model based on the assumption that an energetically excited donor surrounded by N acceptor molecules decays by specific radiative and radiationless relaxation processes, and by transferring its energy to acceptors present in or close to the protein matrix. If interactions between the acceptors themselves and back energy transfer are neglected, we show that the intensity decay function contain 2N exponential components characterized by the unperturbed donor lifetime, by energy transfer rates and a probability of occurrence for the corresponding process. We applied this model to the fluorescence decay of holo- and apoazurin, ribonuclease T1, and the reduced single tryptophan mutant (W28F) of thioredoxin. Use of a multiexponential model for the analysis of the fluorescence intensity decay can therefore be justified, without invoking multiple protein conformations.     

Orientational exchange approach to fluorescence anisotropy decay.

Fluorescence depolarization is a powerful technique in resolving dynamics of molecular systems. Data obtained in fluorescence depolarization experiments are highly complex. Mathematical models for analyzing data from depolarization due to rotational motion have been largely based on the rotational diffusion equation. These results have been verified by Monte Carlo simulations. It has been implicitly stated that a 90 degrees jump model between predefined orientations such as presented by G. Weber (1971. J. Chem. Phys. 55:2399-2411) should, for the specific case of fluorescence depolarization, give the same answer as the diffusion equation. Since the highly symmetric cases considered by G. Weber gave the same result as the diffusion equation, it has been desirable to use this method in cases where depolarization arises from both discrete processes and rotational diffusion. We have derived, in a compartmental formalism, the general result for excitation and emission dipoles not necessarily coincident with any of the principal rotational axes of the fluorophore from this exchange model, and have found it to be different from that of the diffusion equation approach. We have also verified this difference with a Monte Carlo simulation of our exchange model. This derivation allows us to define the limits of validity of the 90 degrees exchanges to model rotational diffusion. Also, for systems where movements may be jumps between a few preferred orientations, the actual physical mechanism of depolarization may not be accurately represented by continuous diffusion. The compartmental formalism developed here can be used to easily combine rotational motions with discrete position jumps or other level kinetics.(ABSTRACT TRUNCATED AT 250 WORDS)     

Confirmation that multiexponential fluorescence decay behavior of holoazurin originates from conformational heterogeneity

Homologous azurins from Pseudomonas fluorescens (ATCC 13525) and Pseudomonas aeruginosa (ATCC 10145) were examined by a number of electrophoretic techniques, and their copper to protein stoichiometry was determined by atomic absorption and amino acid analysis. Provided that the spectral ratio (A620/A280 or A625/A280) was 0.53 and there was no evidence of a Soret band in the absorption spectrum, then these criteria can be used to judge the homogeneity of the azurin sample. If the spectral ratio was less than 0.50, evidence suggested a nonreconstitutable, non-trypsin-digestible apoazurin was present. The fluorescence decay of these homogeneous holoazurins included three components, not two as previously reported [Szabo, A. G., et al. (1983) Biophys. J. 41, 233-244]. Whereas the decay times were nearly the same for the azurins from the different sources, the fractional fluorescence of each component varied with the azurin measured. The fluorescence of the corresponding apoazurins, prepared by a refined procedure, obeyed monoexponential decay kinetics. The temperature and pH effects on the fluorescence behavior of these homologous azurins are presented with the pH study suggesting an influence by a group which titrates between pH 5 and pH 7. When taken together these results confirm that the multiexponential decay behavior originates from conformational heterogeneity and not from contamination by an apo form.     

Analytical approach to the recovery of short fluorescence lifetimes from fluorescence decay curves.

Considerable effort in instrument development has made possible detection of picosecond fluorescence lifetimes by time-correlated single-photon counting. In particular, efforts have been made to narrow markedly the instrument response function (IRF). Less attention has been paid to analytical methods, especially to problem of discretization of the convolution integral, on which the detection and quantification of short lifetimes critically depends. We show that better discretization methods can yield acceptable results for short lifetimes even with an IRF several times wider than necessary for the standard discretization based on linear approximation (LA). A general approach to discretization, also suitable for nonexponential models, is developed. The zero-time shift is explicitly included. Using simulations, we compared LA, quadratic, and cubic approximations. The latter two proved much better for detection of short lifetimes and, in that respect, they do not differ except when the zero-time shift exceeds two channels, when one can benefit from using the cubic approximation. We showed that for LA in some cases narrowing the IRF beyond FWHM = 150 ps is actually counterproductive. This is not so for quadratic and cubic approximations, which we recommend for general use.     

On the wobbling-in-cone analysis of fluorescence anisotropy decay.

Interpretation of fluorescence anisotropy decay for the case of restricted rotational diffusion often requires a model. To investigate the extent of model dependence, two models are compared: a strict cone model, in which a fluorescent probe wobbles uniformly within a cone, and a Gaussian model, where the stationary distribution of the probe orientation is of a Gaussian type. For the same experimental anisotropy decay, analysis by the Gaussian model predicts a smaller value for the rate of wobbling motion than the strict cone analysis, but the difference is 35% at most; the cone angle obtained by the strict cone analysis agrees closely with the effective width of the Gaussian distribution. The results suggest that, when only two parameters (the rate and the angular range) are extracted from an experiment, the choice of a model is not crucial as long as the model contains the essential feature, e.g., the more-or-less conical restriction, of the motion under study. Model-independent analyses are also discussed.     

Anisotropy decay of fluorescence as an experimental approach to protein dynamics

This minireview makes an initial assessment of the progress made using anisotropy decay measurements for investigating the conformational changes and molecular dynamics in soluble systems. A critical analysis of available data is presented. The anisotropy decays of the tryptophan fluorescence of staphylococcal nuclease, adrenocorticotropin, melittin and of labeled transfer RNA were studied for investigating the functional conformational changes of these systems. The emissions of variously labeled immunoglobulins have been used to elucidate the conformations of these proteins before and after the binding of specific antibodies. Labeled myosin and its fragments have given information on the functional motions of the protein domains. The anisotropy decays of labeled and natural hemoglobin systems have been utilized for exploring the allosteric behavior of these molecules. The data suggest a wide applicability of this technique to the study of protein dynamics and conformational changes of macromolecules.     

A theory of fluorescence polarization decay in membranes.

Decay of fluorescence polarization after an impulsive excitation is correlated with wobbling motion of fluorescent molecules in membranes. The motion is characterized by two parameters, a "wobbling diffusion constant" and a "degree of orientational constraint" both of which can be determined directly from experimentally obtained decay. Detailed discussion, including theoretically calculated time-courses of polarization decay, is given for several types of molecules embedded in lipid bilayers; these types cover a large part of fluorescent probes available at present. The theory is useful for the analysis of fluorescence polarization decay in any system where the orientation of fluorophore is restricted by the surrounding structure.     

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.     

Constrained analysis of fluorescence anisotropy decay:application to experimental protein dynamics

Hydrodynamic properties as well as structural dynamics of proteins can be investigated by the well-established experimental method of fluorescence anisotropy decay. Successful use of this method depends on determination of the correct kinetic model, the extent of cross-correlation between parameters in the fitting function, and differences between the timescales of the depolarizing motions and the fluorophore's fluorescence lifetime. We have tested the utility of an independently measured steady-state anisotropy value as a constraint during data analysis to reduce parameter cross correlation and to increase the timescales over which anisotropy decay parameters can be recovered accurately for two calcium-binding proteins. Mutant rat F102W parvalbumin was used as a model system because its single tryptophan residue exhibits monoexponential fluorescence intensity and anisotropy decay kinetics. Cod parvalbumin, a protein with a single tryptophan residue that exhibits multiexponential fluorescence decay kinetics, was also examined as a more complex model. Anisotropy decays were measured for both proteins as a function of solution viscosity to vary hydrodynamic parameters. The use of the steady-state anisotropy as a constraint significantly improved the precision and accuracy of recovered parameters for both proteins, particularly for viscosities at which the protein's rotational correlation time was much longer than the fluorescence lifetime. Thus, basic hydrodynamic properties of larger biomolecules can now be determined with more precision and accuracy by fluorescence anisotropy decay.     

Extension of the performance of Laplace deconvolution in the analysis of fluorescence decay curves.

The original Laplace deconvolution of luminescence data, obtained with pulsed systems, is reviewed. The system of equations from which the luminescence parameters can be determined is generalized for the case that describes the relaxation by a sum of exponentials. Artifacts such as scatter and time-shift can be taken into account. A modification of the original method that eliminates the iterative procedure in the estimation of the cut-off correction is suggested. This modified Laplace method is no longer restricted to the cases where the cut-off error is rather small and the exciting flash has a low tail. The possibility of the combination of several discrete experiments in a single Laplace deconvolution, without introducing new parameters or normalization factors, is shown. The merits of this combination method are demonstrated on a time-resolved depolarization experiment.     

The phasor approach to fluorescence lifetime imaging analysis

Changing the data representation from the classical time delay histogram to the phasor representation provides a global view of the fluorescence decay at each pixel of an image. In the phasor representation we can easily recognize the presence of different molecular species in a pixel or the occurrence of fluorescence resonance energy transfer. The analysis of the fluorescence lifetime imaging microscopy (FLIM) data in the phasor space is done observing clustering of pixels values in specific regions of the phasor plot rather than by fitting the fluorescence decay using exponentials. The analysis is instantaneous since is not based on calculations or nonlinear fitting. The phasor approach has the potential to simplify the way data are analyzed in FLIM, paving the way for the analysis of large data sets and, in general, making the FLIM technique accessible to the nonexpert in spectroscopy and data analysis.     

Nanosecond decay studies of a fluorescence probe bound to apomyoglobin.

Excited state interactions of N-(p-tolyl)-2-aminonaphthalene-6-sulfonate (2, 6 p-TNS) bound to apomyoglobin were studied by nanosecond time-resolved emission spectroscopy. A dynamic interaction of the excited dye molecule with its binding site, associated with a significant change in the emission energy with time, was observed. The decay kinetics were found to be complex and consistent with the kinetic model for solvent relaxation as proposed by Bakhshiev et al. (Opt. Spectrosc. 21:307. 1966). The behavior of 2, 6 p-TNS bound to apomyoglobin was found to be qualitatively similar to that of the dye dissolved in a viscous solvent such as glycerol or adsorbed to egg lecithin vesicles. The detailed information obtained by following the changes in emission spectra of fluorescent probes on the nanosecond time scale leads to a better understanding of their interactions with biological systems.     

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     

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.     

Time-resolved fluorescence studies of flavodoxin. Fluorescence decay and fluorescence anisotropy decay of tryptophan in Desulfovibrio flavodoxins

The time-resolved fluorescence characteristics of tryptophan in flavodoxin isolated from the sulfate-reducing bacteria Desulfovibrio vulgaris and Desulfovibrio gigas have been examined. By comparing the results of protein preparations of normal and FMN-depleted flavodoxin, radiationless energy transfer from tryptophan to FMN has been demonstrated. Since the crystal structure of the D. vulgaris flavodoxin is known, transfer rate constants from the two excited states ¹ L _a and ¹ L _b can be calculated for both tryptophan residues (Trp 60 and Trp 140). Residue Trp 60, which is very close to the flavin, transfers energy very rapidly to FMN, whereas the rate of energy transfer from the remote Trp 140 to FMN is much smaller. Both tryptophan residues have the indole rings oriented in such a way that transfer will preferentially take place from the ¹ L _a excited state. The fluorescence decay of all protein preparations turned out to be complex, the parameter values being dependent on the emission wavelength. Several decay curves were analyzed globally using a model in which tryptophan is involved in some nanosecond relaxation process. A relaxation time of about 2 ns was found for both D. gigas apo- and holoflavodoxin. The fluorescence anisotropy decay of both Desulfovibrio FMN-depleted flavodoxins is exponential, whereas that of the two holoproteins is clearly non-exponential. The anisotropy decay was analyzed using the same model as that applied for fluorescence decay. The tryptophan residues turned out to be immobilized in the protein. A time constant of a few nanoseconds results from energy transfer from tryptophan to flavin, at least for D. gigas flavodoxin. The single tryptophan residue in D. gigas flavodoxin occupies a position in the polypeptide chain remote from the flavin prosthetic group. Because of the close resemblance of steady-state and time-resolved fluorescence properties of tryptophan in both flavodoxins, the center to center distance between tryptophan and FMN in D. gigas flavodoxin is probably very similar to the distance between Trp 140 and FMN in D. vulgaris flavodoxin (i.e. 20 Å). Offprint requests to: A.J.W.G. Visser     

A second generation global analysis program for the recovery of complex inhomogeneous fluorescence decay kinetics

A flourescence spectroscopy global data analysis environment is described. Within this analysis environment multidimensional fluoroscence decay data (time and frequency domains) can be analyzed in terms of a wide variety of photophysical models. A generalized compartmental analysis structure is utilized, where one can specify the functions used to link the various compartments together. All fitting parameters may be characterized by either discrete or distributed values. Applications of these new analysis programs to the examination of phase transitions in lipid/membrane systems are described.     

Anisotropy decay associated fluorescence spectra and analysis of rotational heterogeneity. 1. Theory and applications

Individual fluorescence spectra for species in a heterogeneous system can be determined by using differences between the rotational correlation times of those components. Each spectrum derived is associated with a particular fluorescence anisotropy decay function; hence, they are anisotropy decay associated spectra (ADAS). We have previously shown [Knutson, J. R., Walbridge, D. G., & Brand, L. (1982) Biochemistry 21, 4671-4679] that a system containing different decay functions for total intensity can be resolved into constituent decay-associated spectra. ADAS extends the technique into the realm of fluorescence polarization, making use of the often disparate Brownian rotations found in heterogeneous biochemical systems. In this paper, we present the basic theory for ADAS in various heterogeneous systems and then present an example of ADAS resolving a binary mixture of macromolecules into "fast-rotor" (smaller or more mobile) and "slow-rotor" (larger or less mobile) components. They correctly superimpose spectra taken for the unmixed components. In the companion paper [Davenport, L., Knutson, J. R., & Brand, L. (1986) Biochemistry (following paper in this issue)], a specific application to a problem of importance of lipid biochemistry--e.g., the origin of the membrane probe order parameter in lipid bilayers--is presented, demonstrating the role rotational heterogeneity may play in biochemical fluorescence.     

A theoretical approach to the analysis of axonal transport.

A theoretical model of intra-axonal transport is proposed that presupposes a carrier system moving down the axon in a distal direction. Protein and particle transport is achieved by their reversible association with the distally moving carriers. Mathematical equations representing the concentrations of moving carriers and proteins and/or particles within the axon at any position and time are proposed. Analysis of the equations demonstrates that a traveling wave solution for the particle concentration (an experimental fact) is possible provided the chemical interaction between particles and carriers exhibits positive cooperativity. The phase velocity of the wave solution is interpreted as the observed velocity of the intra-axonal transport, known to be independent of position of observation. In addition, the theory predicts a spectrum of transport velocities for different proteins, in agreement with observations. The velocity of a given protein is dependent on its affinity to the carrier.