The fluorescence lifetime value of tryptophan residues varies by more than a factor of 100 in different proteins and is determined by several factors, which include solvent exposure and interactions with other elements of the protein matrix. Because of the variety of different elements that can alter the lifetime value and the sensitivity to the particular environment of the tryptophan residue, it is likely that non-unique lifetime values result in protein systems. The emission decay of most proteins can be satisfactorily described only using several exponential components. Here it is proposed that continuous lifetime distributions can better represent the observed decay. An approach based on protein dynamics is presented, which provides fluorescence lifetime distribution functions for single tryptophan residue proteins. First, lifetime distributions for proteins interconverting between two conformations, each characterized by a different lifetime value, are derived. The evolution of the lifetime values as a function of the interconversion rate is studied. In this case lifetime distributions can be obtained from a distribution of rates of interconversion between the two conformations. Second, the existence of a continuum of energy substates within a given conformation was considered. The occupation of a particular energy substate at a given temperature is proportional to the Boltzmann factor. The density of energy states of the potential well depends upon the width of the well, which determines the degree of freedom the residue can move in the conformational space. Lifetime distributions can be obtained by association of each energy substate with a different lifetime value and assuming that the average conformation can change as the energy of the substate is increased. Finally, lifetime distributions for proteins interconverting between two conformations, each characterized by a quasi-continuum of energy substates, are presented. The origin of negative components of the lifetime distribution is also discussed. In the companion paper that will follow (Alcala, J. R., E. Gratton, and F. J.Prendergast, 1987, Biophys. J., in press) lifetime distributions obtained here are used to fit experimental data.

The decay of the tryptophanyl emission in proteins is often complex due to the sensitivity of the tryptophan excited state to its surroundings. The traditional analysis of the decay curve using exponential components is based on the identification of each component with a particular protein conformation. An alternative approach assumes that proteins can exhibit a large number of conformations and that, at room temperature, the interconversion rate between conformations can be of the same order of magnitude as the excited-state decay rate. Following this assumption, the analysis of the protein emission was performed using continuous distributions of lifetime values. The number of average protein conformations, the range of mobility around each conformation, and the rate of interconversion between conformations determines the characteristics of the lifetime distribution. The fluorescence decay from some single tryptophan proteins was measured using multifrequency phase fluorometry and analyzed using a sum of exponentials, unimodal and bimodal probability-density functions, and the analytical form for lifetime distribution obtained for a model in which the tryptophan residue can move in a single potential well. For ribonuclease T1 and neurotoxin variant 3, the sum of two exponentials and bimodal probability-density functions gave comparable results, whereas for phospholipase A2, the description of the decay required three exponentials or bimodal probability-density functions. Also the temperature dependence of the fluorescence decay was investigated. It was found that the lifetime distribution was broader and shifted toward longer lifetime values at lower temperature. The analysis of the decay of tryptophan in buffer and of some tryptophan derivatives gave single-exponential decays. The single-potential well lifetime distribution, which has only three adjustable parameters, gave good fits for all cases investigated, but in the case of phopholipase A2, the temperature dependence of the parameters that describe the single-potential well distribution indicated the inadequacy of this model at lower temperature, suggesting that multiple potential wells can describe better the decay for this protein.

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.

We describe the construction and operation of a cross-correlation phase and modulation fluorometer which uses the harmonic content of a high repetition rate mode-locked laser as the excitation source.A mode-locked argon ion laser is used to synchronously pump a dye laser. The pulse train output from the dye laser is amplitude modulated by an acousto-optic modulator and then frequency doubled

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