Multiexponential fitting parameters

Multiexponential Fitting Parameters for the Dipolar Time Correlation Functions of Water Molecules around the Three a-Helices of the Protein. Corresponding Parameters for Bulk Water is also Listed for Comparison... 

Lifetime heterogeneity can be analyzed by fitting the fluorescence decays with appropriate model function (e.g., multiexponential, stretched exponential, and power-like models) [39], This, however, always requires the use of additional fitting parameters and a significantly higher number of photons should be collected to obtain meaningful results. For instance, two lifetime decays with time constants of 2 ns, 4 ns and a fractional contribution of the fast component of 10%, requires about 400,000 photons to be resolved at 5% confidence [33],... 

To test this simple model, we fitted the experimental data by means of Eq. 2 (for convenience, instead of the raw data, we used the best multiexponential fits), n, the number of donor molecules in the first solvent shell, k/t, and kj, could be estimated. The other parameters, Pa, Pb, Pc (the probability of a donor molecule of being Da, Db or Dc respectively) and ksr, the intrinsic ET rate constant, were obtained from the fit. 

The stretched exponential function, A = Ao exp(—tfxf, has been applied to the fluorescence of unstained tissue [82-84], In particular, researchers at Paul French s group at Imperial college [82], show that the use of the stretched exponential, the parameters of mean, and the heterogeneity parameter (the inverse of the degree of stretch, ft) gives better tissue contrast and better fit than the mono- or multiexponential models. 

The preciseness of the primary parameters can be estimated from the final fit of the multiexponential function to the data, but they are of doubtful validity if the model is severely nonlinear (35). The preciseness of the secondary parameters (in this case variability) are likely to be even less reliable. Consequently, the results of statistical tests carried out with preciseness estimated from the hnal ht could easily be misleading—thus the need to assess the reliability of model estimates. A possible way of reducing bias in parameter estimates and of calculating realistic variances for them is to subject the data to the jackknife technique (36, 37). The technique requires little by way of assumption or analysis. A naive Student t approximation for the standardized jackknife estimator (34) or the bootstrap (31,38,39) (see Chapter 15 of this text) can be used. 

A typical result is shown in Fig. 5.67. It shows autofluorescence lifetime images of stratum comeum (upper row, 5 pm deep), and stratum spinosum (lower row, 50 mm deep). The multiexponential decay was approximated by a doubleexponential model, and the decay parameters determined by a Levenberg-Marquardt fit. The colour represents the fast lifetime component, Ti, the slow lifetime component, i2, the ratio of lifetime components, Zi /12, and the ratio of the amplitudes of the components, fli / fl2- The brightness of the pixels represents the intensity. 

Why is it difficult to resolve multie qx)nential decays Comparison of 7 (r) and /jCO indicates that the lifetimes and amplitudes are diffnent in each decay law. In fact, this is the problem. For a multiexponential decay, one can vary the lifetime to compensate feu the amplitude, or vice versa, and obtain similar intensity decays with diff ent values of a,- and X,. In mathematical terms, the values of a, and x, are said to be correlated. Thej oblem of cmrelated paramet has been described within the framework of general least squares fitting. " The unfortunate result is that the ability to determine the precise values of a/ and X,- is greatly hindered by parameter correlation. There is no way to avoid this problem, except by careful experimentation and conservative interpretation of data. 

In the vast majority of instances, the pyrene monomer fluorescence decays acquired with aqueous solutions of Py-WSPs, particularly when the pyrene labels are randomly incorporated into a WSP, are always multiexponential in nature. This experimental observation is a consequence of the distribution of distances between an excited pyrene and a ground-state pyrene or pyrene aggregate formed in water that leads to a distribution of rate constants for excimer formation by diffusion [30-32, 34]. Unfortunately, the analysis of multiexponential decays is notoriously difficult to handle and unless the decay times resulting from the photophysical processes are well resolved (i.e., separated by a factor of at least 2) as is the case for pyrene end-labeled short alkyl oligomers, the parameters retrieved from a triexponential fit should be considered with utmost caution [32]. Indeed early reports [38, 39] on the analysis of the fluorescence decays of Py-WSP based on the DMD model [40] or one of its variants introduced originally by Zachariasse to deal with the multiexponential decays of pyrene end-labeled oligomers yielded sets of parameters whose validity has been questioned [41]. 

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