Single exponential phase

Hassler et al. [40, 41] observed that the stretched exponential behavior was related to the active phase of the enzyme (busy phase) while the less active phase (lazy phase) exhibited a single exponential phase (Fig. 4.20). The behavior which was originally observed by Edman et al. 1999 [39] at a much lower sensitivity had been observed again, however, with a much better intensity and time resolution. As can be seen from the trace, the turnover frequency is more than 4-fold higher in the busy periods. It is tempting to relate this to the memory effect decribed in the next chapter (Fig. 4.21). 

For single exponential fluorescence decay, as is expected for a sample containing just one fluorophore, either the phase shift or the demodulation can be used to calculate the fluorescence lifetime t. When the excitation light is modulated at an angular frequency (o = 2itv, the phase angle f, by which the emission modulation is shifted from the excitation modulation, is related to the fluorescence lifetime by ... 

If the signal decay is a single-exponential curve, equations 16 and 17 result in values for X that are in agreement with each other. Dissimilar values indicate multiexponential decay, which usually means that the sample contains more than one fluorophore. Multiexponential decay can be resolved by using a phase fluorometer with phase sensitive detection. A time-independent, direct-current signal is produced that is proportional to the cosine of the difference between the phase angle of the detector ( D) and the phase angle of the fluorescence ( ) ... 

With some further assumptions, it is possible to use single frequency FLIM data to fit a two-component model, and calculate the relative concentration of each species, in each pixel [16], To simplify the analysis, we will assume that in each pixel of the sample we have a mixture of two components with single exponential decay kinetics. We assume that the unknown fluorescence lifetimes, iq and r2, are invariant in the sample. In each pixel, the relative concentrations of species may be different and are unknown. We first seek to estimate the two spatially invariant lifetimes, iq and t2. We make a transformation of the estimated phase-shifts and demodulations as follows ... 

The principles of pulse and phase-modulation fluorometries are illustrated in Figures 6.5 and 6.6. The d-pulse response I(t) of the fluorescent sample is, in the simplest case, a single exponential whose time constant is the excited-state lifetime, but more often it is a sum of discrete exponentials, or a more complicated function sometimes the system is characterized by a distribution of decay times. For any excitation function E(t), the response R(t) of the sample is the convolution product of this function by the d-pulse response ... 

An efficient way of overcoming this difficulty is to use a reference fluorophore (instead of a scattering solution) (i) whose fluorescence decay is a single exponential, (ii) which is excitable at the same wavelength as the sample, and (iii) which emits fluorescence at the observation wavelength of the sample. In pulse fluorometry, the deconvolution of the fluorescence response can be carried out against that of the reference fluorophore. In phase-modulation fluorometry, the phase shift and the relative modulation can be measured directly against the reference fluorophore. 

Fig. 6.13. Data obtained by the phase-modulation technique with a Fluorolog tau-3 instrument (Jobin Yvon-Spex) operating with a xenon lamp and a Pockel s cell. Note that because the fluorescence decay is a single exponential, a single appropriate modulation frequency suffices for the lifetime determination. The broad set of frequencies permits control of the proper tuning of the...

The time of data collection depends on the complexity of the (5-pulse response. For a single exponential decay phase fluorometry is more rapid. For complex 5-pulse responses, the time of data collection is about the same for the two techniques in pulse fluorometry, a large number of photon events is necessary, and in phase fluorometry, a large number of frequencies has to be selected. It should be emphasized that the short acquisition time for phase shift and modulation ratio measurements at a given frequency is a distinct advantage in several situations, especially for lifetime-imaging spectroscopy. 

In the case of a single exponential decay, the lifetime can be rapidly calculated by either the phase shift modulation ratio M by means of Eqs (6.25) and (6.26) established in Chapter 6 (Section 6.2.3) ... 

Figure 6.10 Frequency-dependent phase shifts and demodulations (bottom panel) for IR-I44 (open symbols) and DOTCI (solid symbols), together with the best single exponential lit to each dataset. The residuals for the 1R-144 data set are depicted in the upper panel. Reproduced from Ref. 25 with permission.

The apparent lifetimes calculated by these expressions are the true lifetimes only if the fluorophore obeys single exponential decay kinetics. In the case of a single exponential decay, the apparent lifetimes as determined from the two equations should be the same. If the apparent phase and modulation lifetimes are not equal, more than one decay process is indicated. 

The mobility of tyrosine in Leu3 enkephalin was examined by Lakowicz and Maliwal/17 ) who used oxygen quenching to measure lifetime-resolved steady-state anisotropies of a series of tyrosine-containing peptides. They measured a phase lifetime of 1.4 ns (30-MHz modulation frequency) without quenching, and they obtained apparent rotational correlation times of 0.18 ns and 0.33 ns, for Tyr1 and the peptide. Their data analysis assumed a simple model in which the decays of the anisotropy due to the overall motion of the peptide and the independent motion of the aromatic residue are single exponentials and these motions are independent of each other. 

Johnson and Borisy first showed that the lag phase in the plot of turbidity (i.e., polymer weight concentration) versus time accounted for only 5—10% of the entire amplitude obtained upon completion of the polymerization process. By fitting the elongation phase to a single exponential process, these investigators arrived at the correct conclusion that microtubule number concentration becomes relatively stable within the first minutes... 

All carbon resonances of pure PMMA, PS(OH), and their blends showed single-exponential decays in both Ti and Tjp. The Tj and Tjp values for the main resonance lines are shown in Figs. 13 and 14, respectively. For each of PMMA and PS(OH) containing different amounts of hydroxyl, the protons attached to different carbons have similar relaxation times, which indicates that the spin diffusion equalizes the relaxation rates of all protons. In addition, the Tj and Tip values for PS(OH) show a gradual decrease and increase, respectively, as the hydroxyl content in PS(OH) is increased. The difference in either Tj or Tj between PMMA and PS(OH) is substantial, encouraging the gathering of information about the phase structure of their blends by NMR relaxation time analysis. 

The curve fitting programs cope better with fewer variables in the equations. Try to reduce the number of variables. For example, suppose you have to fit a multiphasic curve to three exponentials that are moderately separated in time. There are seven unknowns three rate constants three amplitudes and an endpoint. If the slowest phase is sufficiently separated from the second, first fit the tail of the slowest phase to a single exponential. Then fit the whole curve to a triple exponential equation in which the rate constant and the amplitude that were derived for the third phase are used as constants. Use a time window that focuses on the first two phases and not the whole time course. Similarly, if the first phase is much faster than the second and third, fit the tail of the process to two exponentials. Then fit the fast time region to a triple exponential in which the last two phases have fixed rate constants and amplitudes. 

We have implemented the discrimination in the frequency domain. As is known in multifrequency phase fluorometry,17 the time-delayed fluorescence acquires a phase shift >p and a reduction in amplitude Mp upon increasing the modulation frequency m = 2irf of the sinusoidally modulated excitation. For a simple single exponential decay, this phase shift

[Pg.385]

Gas phase experiments utilized the same laser system but employed a different sample cell, a 1.5 cm long stainless steel cell with CaF2 windows. A turbomolecular pump was used to evacuate the cell down to the vapor pressure of the W(CO)6. The cell was heated slightly to 326 K to increase the vapor pressure and produce an optical density of 1.0. The gas phase decays are not single exponentials, but rather tri-exponentials. This triexponential character will be discussed in the results section. 

The decay of the CO stretch is a single exponential when W(CO)6 has substantial interactions with a solvent. A single exponential (aside from orientational relaxation in liquids) is observed even when very fast pulses are used in the experiments (81). In the gas phase, the transition frequency of the CO stretch evolves over a range of frequencies because of its time-dependent interaction with the low-frequency modes. When a buffer gas or solvent is added, collisions cause the coherent evolution of the slow modes to be interrupted frequently, possibly averaging away the perturbation responsible for the observed fast time dependence. Thus, the fastest and slowest components of the tri-exponential decay are inherently low-pressure, gas phase phenomena. 

In the gas phase, the asymmetric CO stretch lifetime is 1.28 0.1 ns. The solvent can provide an alternative relaxation pathway that requires single phonon excitation (or phonon annihilation) (102) at 150 cm-1. Some support for this picture is provided by the results shown in Fig. 8. When Ar is the solvent at 3 mol/L, a single exponential decay is observed with a lifetime that is the same as the zero density lifetime, within experimental error. While Ar is effective at relaxing the low-frequency modes of W(CO)6, as discussed in conjunction with Fig. 8, it has no affect on the asymmetric CO stretch lifetime. The DOS of Ar cuts off at "-60 cm-1 (108). If the role of the solvent is to open a relaxation pathway involving intermolecular interactions that require the deposition of 150 cm-1 into the solvent, then in Ar the process would require the excitation of three phonons. A three-phonon process would be much less probable than single phonon processes that may occur in the polyatomic solvents. In this picture, the differences in the actual lifetimes measured in ethane, fluoroform, and CO2 (see Fig. 3) are attributed to differences in the phonon DOS at 150 cm-1 or to the magnitude of the coupling matrix elements. 

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