Multi-exponential decay

The Laplace inversion (LI) is the key mathematical tool of the DDIF experiment. The ability to convert the measured multi-exponential decay into a distribution of decay times is crucial to the DDIF pore size distribution application. However, unlike other mathematical operations, the Laplace inversion is an ill-conditioned problem in that its solution is not unique, and is fairly sensitive to the noise in the input data. In this light, significant research effort has been devoted to optimizing the transform and understanding its boundaries [17, 53, 54],... 

Kremers, G. J., Van Munster, E. B., Goedhart, J. and Gadella, T. W. (2008). Quantitative lifetime unmixing of multi-exponentially decaying fluorophores using single-frequency FLIM. Biophys. J. 95, 378-89. 

The analysis of the histograms of photon arrival times is equivalent in both cases and relies on fitting appropriate model functions to the measured decay. The selection of the fitting model depends on the investigated system and on practical considerations such as noise. For instance, when a cyan fluorescent protein (CFP) is used, a multi-exponential decay is expected furthermore, when CFP is used in FRET experiments more components should be considered for molecules exhibiting FRET. Several thousands of photons per pixel would be required to separate just two unknown fluorescent... 

General relations for single exponential and multi-exponential decays For a single exponential decay, the b-pulse response is... 

For a multi-exponential decay with n components, the (5-pulse response is... 

In practice, initial guesses of the fitting parameters (e.g. pre-exponential factors and decay times in the case of a multi-exponential decay) are used to calculate the decay curve the latter is reconvoluted with the instrument response for comparison with the experimental curve. Then, a minimization algorithm (e.g. Marquardt method) is employed to search the parameters giving the best fit. At each step of the iteration procedure, the calculated decay is reconvoluted with the instrument response. Several softwares are commercially available. 

Data analysis in phase fluorometry requires knowledge of the sine and cosine of the Fourier transforms of the b-pulse response. This of course is not a problem for the most common case of multi-exponential decays (see above), but in some cases the Fourier transforms may not have analytical expressions, and numerical calculations of the relevant integrals are then necessary. 

The maximum entropy method has been successfully applied to pulse fluorometry and phase-modulation fluorometry3- . Let us first consider pulse fluorometry. For a multi-exponential decay with n components whose fractional amplitudes are a , the d-pulse response is... 

This can easily be extended to multi-exponential decays of I(t) and r(t). 

In phase fluorometry, the phase (and modulation) data are recorded at a given wavelength and analyzed in terms of a multi-exponential decay (without a priori assumption of the shape of the decay). The fitting parameters are then used to calculate the fluorescence intensities at various times, 2 > 3 > The procedure is repeated for each observation wavelength X, X2, A3,... It is then easy to reconstruct the spectra at various times. 

Transverse relaxation of musculature is relatively fast compared with many other tissues. Measurements in our volunteers resulted in T2 values of approximately 40 ms, when mono-exponential fits were applied on signal intensities from images recorded with variable TE. More sophisticated approaches for relaxometry revealed a multi-exponential decay of musculature with several T2 values." Normal muscle tissue usually shows lower signal intensity than fat or free water as shown in Fig. 5c. Fatty structures inside the musculature, but also water in the intermuscular septa (Fig. 5f) appear with bright signal in T2-weighted images. 

The phasor method associates the decay dynamics with a vector in a so-called phasor space. In particular, purely exponential decay corresponds to a phasor with its end point on a semicircle of radius 1/2 and centered at (1/2, 0). Tuning of the decay time from zero to infinity results in a counterclockwise displacement of the end point from (1,0) to (0,0) along the semicircle. Multi-exponential decay is equivalent to a point inside the semicircle, but its dependence on the weight-averaged decay... 

E.g. tryptophane residues of proteins excite at 290-295 mn but they emit photons somewhere between 310 and 350 mn. The missing energy is deposited in the tryptophane molecular enviromuent in the form of vibrational states. While the excitation process is complete in pico-seconds, the relaxation back to the initial state may take nano-seconds. While this period may appear very short, it is actually an extremely relevant time scale for proteins. Due to the inherent thermal energy, proteins move in their (aqueous) solution, they display both translational and rotational diffusion, and for both of these the characteristic time scale is nano-seconds for normal proteins. Thus we may excite the protein at time 0 and recollect some photons some nano seconds later. With the invention of lasers, as well as of very fast detectors, it is completely feasible to follow the protein relax back to its ground state with sub-nano second resolution. The relaxation process may be a simple exponential decay, although tryptophane of reasons we will not dwell on here display a multi-exponential decay. 

Figure 2a shows a few fluorescence upconversion transients as measured for 1 dissolved in n-heptane, under magic angle conditions. The transients show multi-exponential decay behavior... 

The initial steep change in the extinction is ascribed to the temperature equilibration of the sample. Following this, a multi-exponential decay in the extinction is observed, indicating the complex relaxation kinetics of the homogeneous reaction between the various cationic SE s. Finally, the long-term single exponential relaxation... 

In this application of the BWR theory, Hudson and Lewis assume that the dominant line-broadening mechanism is provided by the modulation of a second rank tensor interaction (i.e., ZFS) higher rank tensor contributions are assumed to be negligible. R is a 7 X 7 matrix for the S = 7/2 system, with matrix elements written in terms of the spectral densities J (co, rv) (see reference [65] for details). The intensity of the i-th transition also can be calculated from the eigenvectors of R. In general, there are four transitions with non-zero intensity at any frequency, raising the prospect of a multi-exponential decay of the transverse magnetization. There is not a one-to-one correspondence between the... 

This is simply a multi exponential decay with the ampHtude of the k-th mode given by ak Tk (1 - e tplTk). In the short and in the long exposure limit, this reduces to... 

Once the scaling relation of Eq. (39) is known, the molar mass distribution can, at least in principle, be obtained from a Laplace inversion of the multi-exponential decay function as defined in Eq. (40). At this point, the differences between PCS and TDFRS stem mainly from the different statistical weights and from the uniform noise level in heterodyne TDFRS, which does not suffer from the diverging baseline noise of homodyne PCS caused by the square root in Eq. (38). 

Figure 5. Sketch of the model suggested for the multi-exponential decay observed for the Ru(bpy)32+ luminescence probe molecule decay in the interlamelar space. To a square lattice for quencher ions is superimposed a (1/2, 1/2) translated square lattice for the probe molecules. (Reprinted with permission from ref. 29. Copyright 1984 Royal Society of Chemistiy.)...

Among the best well-known examples of photostability after UV radiation, the ultrafast nonradiative decay observed in DNA/RNA nucleobases, has attracted most of the attention both from experimental and theoretical viewpoints [30], Since the quenched DNA fluorescence in nucleobase monomers at the room temperature was first reported [31] new advances have improved our knowledge on the dynamics of photoexcited DNA. Femtosecond pump-probe experiments in molecular beams have detected multi-exponential decay channels in the femtosecond (fs) and picosecond (ps) timescales for the isolated nucleobases [30, 32-34], The lack of strong solvent effects and similar ultrafast decays obtained for nucleosides and nucleotides suggest that ultrashort lifetimes of nucleobases are intrinsic molecular properties, intimately... 

A number of experiments measuring the decay of the P state have revealed complicating factors in addition to the multi-exponential decay of the state described above. In measurements monitoring the decay of P using short (SOnlOOfs) laser pulses conducted by Vos and co-workers, oscillations were observed superimposed on the resulting kinetic traces (Vos et al., 1994a,b,c 1993,1991). These oscillations have been attributed to coherent nuclear motion associated with the P state (Vos and Martin, 1999 ... 

Time-resolved emission spectra exhibit multi-exponential decay at 340 and 437 nm corresponding to lifetimes in the range 0.1-10 ns [107]. The longer lifetimes agree with those reported for other ZnS powders [108] and for colloids [26, 94]. It is recalled that multi-exponential decay may originate not only in different emitting states but also in the presence of various particle sizes [109-111]. 

Causes of dual (or multi) exponential decay include... 

It is clear from the evidence presented above that a number of factors may influence emission from proteins, these being multi-exponential decays of single tryptophan residues, heterogeneity of environment, presence of other emitting species and nanosecond fluctuations of the macromolecule structure. Only when the contributions of these phenomena have been assessed can any definite conclusions be made about the influence of structure on the photophysics of biopolymers. 

Triplet states of xanthone depend upon the composition of the solvent the presence of water enhances the yield of 2 un state Dual phosphorescence from 2-(2 -hydroxyphenyl) benzoxazole is due to keto-enol tautomerism and the kinetics multi-exponential decay is due to differences of environment . Triplet state properties and triplet state-oxygen interactions of the biologically interesting linear and angular furocoumarins are useful in view of possible clinical application . 

The emission-lifetime measurements in ns-time region were also carried out for the two emission bands. Multi-exponential decay behavior was observed for both the emission bands. Fast decay component at >.=480 nm less than the order of ns was attributed to the recombination of electrons and holes. Slow decay component at >,=480 nm in the order of a few ns was attributed to thermal detrapping of the electron from the surface states to the conduction band since such thermal activation could enhance the lifetime at the band-edge emission. The emission lifetime at >.=480 nm increased as excess Cshallow trap sites. 

In particular, the application of multi-exponential decay kinetics anticipated from models that assume distinct photophysical species within polymer chains may be inappropriate in some cases. The possibility of non-exponential fluorescence decay behaviour arising from energy migration and trapping (11) should also be considered. Additional studies of the mobilities of fluorescent probes incorporated in PMA using time-resolved fluorescence anisotropy measurements provide further support for a "connected cluster" model to describe the conformation of this polyelectrolyte in aqueous solution at low pH. 

The experimental results revealed that typically there is no saturated part in the transient of the response even for comparatively long (about 10-15 minutes) exposure to the headspace air. In general, a sophisticated curving shape of the transients displays both the rise and the fall of the response signals. It is clearly seen that the response to the headspace does not correspond to the pure multi-exponential decay. Moreover, most of the transients can be split into a linear combination of two curves, each of which is described by a pure multi-exponential decay. 

A complex shape of the curve displays the complete transient of the response to the volatile compounds of the explosive materials. The possibility to apply the method of the multi-exponential decay is highly complicated. Therefore, an analysis of the graphical images of the sensor responses to the headspace of the explosives has been omitted from this study. 

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