Lifetimes continuous distribution

To answer the question as to whether the fluorescence decay consists of a few distinct exponentials or should be interpreted in terms of a continuous distribution, it is advantageous to use an approach without a priori assumption of the shape of the distribution. In particular, the maximum entropy method (MEM) is capable of handling both continuous and discrete lifetime distributions in a single analysis of data obtained from pulse fluorometry or phase-modulation fluorometry (Brochon, 1994) (see Box 6.1). 

One can expect that the analysis of continuous distributions of electronic excited-state lifetimes will not only provide a higher level of description of fluorescence decay kinetics in proteins but also will allow the physical mechanisms determining the interactions of fluorophores with their environment in protein molecules to be elucidated. Two physical causes for such distributions of lifetimes may be considered ... 

That said, the established methods including those not mentioned above have their limits. PATFIT deals with discrete lifetimes. CONTIN [64, 65] and MELT provide continuous distribution of lifetimes. The merits and demerits of MELT and CONTIN were debated extensively [66-68]. Kansy developed an algorithm that can fit both, discrete lifetimes and log-normal distributions of lifetimes [69]. These challenges become increasingly severe as the range of lifetimes included in the data increases. To date, no method has addressed this issue satisfactorily. A number of approaches have been taken to overcome or circumvent the problems. However, their detailed discussion exceeds the scope of this chapter and will be presented elsewhere... 

The introduction of these light-induced states affects not only the carrier lifetimes but also the photoconductive response time Tq. Because of the continuous distribution of gap states in a-Si H, there are many more states that can trap photogenerated carriers close to the mobility edges than those that act as recombination centers. The response time depends on the density... 

Although the measurement of a single lifetime is trivial, severe problems appear if the system shows several lifetimes. In the worst case, the system does not even have discrete time constants but a continuous distribution of lifetimes. The difficulties do not emerge as the lack of a numerical solution but vice versa, as a continuum of solutions fitting well to the experimental results. This is due to the notoriously ill-posed character of the equations used for estimating the time constants from the experimental data. An obvious way to overcome these difficulties is to increase the precision of the measurements. Another, actually more successful. 

Although in many homogeneous systems fluorophores have distinct and discrete decay constants for their fluorescence, in heterogeneous systems the luminescent molecules have different environments and consequently different energy levels and also pathways for the energy dissipation. Moreover, in the RET processes the distance between the donor and acceptor is not constant but may vary slightly. Then it can be expected that the lifetimes are not sharply defined but they are actually continuously distributed. Mathematically this means that instead of the sum in (40), we have to use a Riemann-Stieltjes integral ... 

In special cases, instead of the above sum, the lifetime spectrum is approximated by a continuous distribution of exponentials ... 

The most characteristic example of continuous distribution of states is supplied by polymers. In an amorphous polymer, the size of the structural free volume (the hole between chains) varies randomly and so does the electron density in these free volumes. If Ps atoms are trapped by the holes, their lifetimes reflect the size distribution of free volumes (Tao 1974 Eldrup et al. 1981). 

The analysis involved deconvolution by iterative reconvolution, background subtraction, and optional correction for shift of the instrument response function. Statistical tests included chi-square, the Durbin-Watson test, the covariance matrix, a runs test, and the autocorrelation function [6]. An alternative form of data analysis involves distributions of lifetimes rather than a series of exponentials. Differentiation of systems obeying a decay law made up of three discrete components from systems where there exists a continuous distribution of lifetimes, or a distribution plus one or more discrete components, is a nontrivial analytical problem. Methods involving the minimization of the chi-square parameter are commonly used, but recently the maximum entropy method (MEM) has gained popularity [7]. Inherent in the MEM method is the theoretical lack of bias and the potential for recovering the coefficients of an exponential series with fixed lifetimes which are free of correlation effects and artificial oscillations. Recent work has compared the MEM with a new version of the exponential series method (ESM) which allows use of the same size probe function as the MEM and found that the two methods gave comparable results [8]. 

In some circumstances, it can be anticipated that continuous lifetime distributions would best account for the observed phenomena. Examples can be found in biological systems such as proteins, micellar systems and vesicles or membranes. If an a priori choice of the shape of the distribution (i.e. Gaussian, sum of two Gaussians, Lorentzian, sum of two Lorentzians, etc.) is made, a satisfactory fit of the experimental data will only indicate that the assumed distribution is compatible with the experimental data, but it will not demonstrate that this distribution is the only possible one, and that a sum of a few distinct exponentials should be rejected. 

Nonexponential luminescence decays are not well understood. However, regardless of the lack of understanding, it is a tradition to fit complex decays to sums of exponential functions either discrete or continuous (lifetime distributions). An important limitation of this approach is introduced by the nonorthogonal nature of the exponential function. The practice of fitting nonexponential luminescence decays to... 

J. R. Alcala, E. Gratton, and F. Prendergast, Interpretation of fluorescence decays in proteins using continuous lifetime distributions, Biophys. J. 51, 925-936 (1987). 

In the present work, we have found that a continuous Lorentzian lifetime distribution (two floating parameters) described by Eqn. 5, best modelled the experimental data ... 

Figure 4.5. Left side. Temperature dependence of the bimodal lifetime distribution parameters of Sulfolobus solfataricus (3-glycosidase. Long-lifetime component (squares) short lifetime component (circles). Right upper side W3/4H dependence on temperature thermal denaturation of Sulfolobus solfataricus p-glycosidase at pD 7.4 (continuous line) and pD 10.0 (dashed line). The lines were obtained by monitoring the amide l width calculated at 3A of the amide height (W3/4H) as a function of the temperature. Right bottom side dependence on temperature for Sulfolobus solfataricus p-glycosidase. (Likhtenshtein et al., 2000). Reproduced with permission.

In practice the lifetime distributions are usually obtained using a computer program such as the MELT [21] or CONTIN [22, 23] programs. The reliablity of these programs for measurring the o-PS lifetime distribution in polymers was shown by Cao et al [24]. A detailed description of these methods of data analysis is presented in Chapter 4. The advantage of the continuous lifetime analysis is that one can obtain free volume hole distributions rather that the average values obtained in the finite analysis. 

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