Exponential components

Fig. 6. A typical correlation function obtained for IFABP in 20 mM phosphate buffer at pH 7.3 at room temperature. The experiment was performed using a ConfoCor 2 LSM combination instrument (Carl Zeiss-Evotec, Jena, Germany) and the correlation function data (G(t)) were fitted to the form G(r) = G(0)/(1 + x/rD), where Td is the characteristic diffusion time. An additional exponential component improves the fit. 

The non-mono-exponentiality of the transverse relaxation in muscle and meat has most often been solved by decomposing the relaxation decay into two or three exponential components. In general, this has resulted in the detection of a major relaxation component characterised by a time constant around 35-50 ms, which corresponds to approximately 80-95% of the relaxation, and a slower relaxing component characterised by a time constant around 100-250 ms, which represents approximately 5-15% of the relaxation. In addition, a fast relaxing component with a time constant between 0 and 10 ms, which corresponds to about 5% of the relaxation, has been observed.9,11 The presence of three relaxation components in the... 

Proteins having one chromophore per molecule are the simplest and most convenient in studies of fluorescence decay kinetics as well as in other spectroscopic studies of proteins. These were historically the first proteins for which the tryptophan fluorescence decay was analyzed. It was natural to expect that, for these proteins at least, the decay curves would be singleexponential. However, a more complex time dependence of the emission was observed. To describe the experimental data for almost all of the proteins studied, it was necessary to use a set of two or more exponents.(2) The decay is single-exponential only in the case of apoazurin.(41) Several authors(41,42) explained the biexponentiality of the decay by the existence of two protein conformers in equilibrium. Such an explanation is difficult to accept without additional analysis, since there are many other mechanisms leading to nonexponential decay and in view of the fact that deconvolution into exponential components is no more than a formal procedure for treatment of nonexponential curves. 

The fluorescence decay is multiexponential.(199 200) This is unequivocal evidence that the wyebutine base can be bound in at least two different conformations with different solvent shielding. Wells and Lakowicz(200) resolved two exponential components. They measured the normalized amplitudes and lifetimes for the wyebutine fluorescence at two different concentrations of added Mg2+ S° = 0.50, t, = 1.7 ns, = 0.50, and t2 = 5.9 ns with no added Mg2+ present, and S°i = 0.16, t, =0.6 ns, S2 = 0.84, and r2 = 6.0ns with 10 nM Mg2+. Since the 6ns component is the longest lifetime present, it must represent the conformation that shields the wyebutine to the greatest extent and is generally believed to involve a 3 stack of bases 34-38.w 199-2011 The fraction of the tRNA in this conformation increases when Mg2 + is added to the solution. This structure is also observed in crystal structures which include Mg2+.(202 204) In the other conformation(s), the wyebutine is more exposed to the solvent. A 5 stack, which does not include bases 37 and 38, is one possibility. The wyebutine base would be more exposed to the solvent and have a shorter fluorescence lifetime as a result. However, both NMR data(205 206) and chemical modification studies(207) are inconsistent with a 5 stack. For the moment, this matter is unresolved. 

The results from fitting the anisotropy decay support the above conclusions. Wells and Lakowicz(200) resolved two exponential components in the anisotropy decay. They obtained ro = 0.11, t r = 0.3 ns, rj = 0.15, and t = 18.5 ns for the sample with no added Mg2+, and ro = 0.05, t R = 0.4 ns, r J = 0.17, and t"R = 17.4 ns for a sample with 10 mM Mg2+. Here r 0 and r o are the amplitudes of the fast and slow components. The longer rotational relaxation time corresponds to overall tumbling of the tRNA, although its amplitude is reduced by much more rapid local motions. The shorter relaxation time corresponds directly to a rapid local motion. Upon addition of Mg2+, the relative amplitude of die rapid local motion decreases, while that of the overall tumbling increases. This implies that the wyebutine base is held in a more rigid or constrained state, such as a 3 stack, in the presence of Mg2+. In that state, the amplitude of local angular motion is substantially diminished in comparison with that in the alternate state that prevails in the absence of Mg2+. As noted before, the exact nature of these conformation(s) is unresolved. 

Takeda et al. [104] found that the steady-state irradiation of A-octadecyl 6,8-dinitro-BlPS resulted in the formation of the closed form. This is in agreement with the femtosecond studies. However, the fade reaction showed three components when monitored at 500 nm in 1,2-dichloroethane. The first component was manifest as a bleach that occurred within the instrument response time of 25 nsec. This was followed by two longer components with 240-nsec and 3.4 psec lifetimes. The spiropyran ring-closed form evolved with two exponential components with lifetimes of 620 nsec and 4 p-sec. The Af-octadecyl group may affect the outcome compared to the A-methylated form. 

The introduction of reversible photoassociation at higher fluor (or quencher) concentrations invalidates the analysis of fluorescence decay curves of either emitting species in terms of a single decay constant since these species do not relax independently accordingly the decay curves each represent the sum of two exponential components and the corresponding decay constants Ax and A2 are related to the photoassociation rate constants kDM and kMD and to the lifetimes rp and t of both emitting species. The treatment given below follows that of Birks, Dyson, and Munro.1... 

Figure 15.2 2H NMR relaxation function obtained in a PDMS model network in the relaxed state, showing the two components the fast relaxing component (the curve with alternatively long and short dashes) is pseudo-solid and belongs to elastic chains (which have restricted motions), the slowly relaxing, exponential component (dashed line), is liquid-like and belongs to dangling chains. The total signal (black points) is a superposition of both contributions, which gives the fraction of dangling chains...

We note that the distribution is a simple exponential superimposed on the constant background concentration of the solute, J0/v (see Figure 6.2). The effective thickness of the exponential component is seen to be identical in form (but with DT replacing D) to that found for field-flow fractionation, Eq. 6.20 f = DTI u. ... 

Thus the annihilation lifetime spectrum has two exponential components the unperturbed decay from the m= 1 states and the quenched decay from the m= 0 state (at 4 kG the lifetime is about 30 nsec). Measurement of these decay rates, Ay and Ay, at gas pressures ranging from 200 — 1400 torr allows one, after extrapolation to zero gas density, to solve (3) for As- Measurements were made at three different magnetic fields and the average result is As = 7.994 0.011 nsec-1, in agreement with the A2 calculation at the 1400 ppm level. 

The presence of hard and soft domains in segmented polyurethanes also has been confirmed by experimental results using pulsed NMR and low-frequency dielectric measurements. Assink (55) recently has shown that the nuclear-magnetic, free-induction decay of these thermoplastic elastomers consists of a fast Gaussian component attributable to the glassy hard domains and a slow exponential component associated with the rubbery domains. Furthermore, the NMR technique also can be used to determine the relative amounts of material in each domain. 

Lastly, we consider the diffusive contribution to the signal. Since this portion of the signal arises from molecular reorientation, it should be completely depolarized unless these diffusive reorientational dynamics also have a significant DID component. The orientational decay will be made up of exponential components, the number of which depends on the molecular symmetry and the relationship between the principal axes of the diffusion and polarizability tensors of the molecules (8). If these tensors share no axes, the orientational decay will be composed of a sum of five exponentials. If the tensors share one axis, the decay will be composed of three exponentials. If the tensors share all three axes, the decay will be composed of two exponentials. If the molecule is further a symmetric top, then reorientation about the axis of symmetry cannot be observed, and the decay will be composed of a single exponential. In principle, considerably more information is available when the principal axes of the diffusion and polarizability tensors are not shared however, in practice it is virtually impossible to find a unique fit to the sum of five exponentials, some of which may have very small amplitudes. In the remainder of this chapter we will therefore concentrate on symmetric-top liquids. 

Figure 2 A series of FID resolution procedures into several component decays. The longest relaxation decay, showing a simple exponential function at the longer timescale, was subtracted from the observed FID (A). The other exponential component at the middle timescale was also fitted as amorphous relaxation (B).

Figure 3.4 illustrates two lifetime spectra collected by methods similar to those outlined above, (a) exhibits the non-exponential shoulder region associated with the annihilation of non-thermalised positrons. After thermalisation (essentially at time zero for condensed matter) the spectra are sums of exponential components associated with each decay mode, and a background component B, A] = 2 A, exp(-Ajt,) + B. For long lifetime components (> Ins) each X can be extracted by non-linear least squares fitting. For short X values characteristic of condensed matter, however, a... 

Dyads (1 - 4) were synthesized by the method described previously. Fluorescence spectra were measured in degassed acetonitrile solutions at 2S °C with a Hitachi 8S0 spectrofluorometer. The excitation wavelength was the Soret maximum. Fluorescence lifetimes were measured at 25 °C using a Horiba NAES-500 ns-fluorometer interfaced to an NEC PC-9801 RX personal computer. The excitation light below 420 nm was cut off with a glass filter. The fluorescence was detected by a single-photon counting system and analyzed as the sum of two exponential components after deconvolution of the instrument response function. NMR spectra were recorded on a JEOL JNM GX-270 NMR spectrometer. 

If the annealing process is of this nature, and there is a finite number of types of recombination loci, then in principle the annealing curve can be resolved into that many exponential components. It is clear that by this procedure any arbitrary monotonic experimental curve can be fitted provided a suflSciently large number of exponential terms, Eq. (5), are taken however, a necessary condition is that extrapolation of the linear portions of the curves thus resolved for different temperatures must give the same intercept values (at f = 0), since the quantities rioi, n 2,. .. are not affected by the temperature of annealing. 

The traditional and most commonly used approach to the positron lifetime spectrum consists in its decomposition into a sum of several exponential components ... 

The metabolism and tissue distribution of cesium-137 were studied in rats injected intraperitoneally and sacrificed 1-300 days postinjection. In a chronic study, rats were administered cesium-137 in their drinking water daily. In the acute study, with the exception of the brain, muscle, and total animal, all tissues showed retention curves resolvable into three exponential components with half-lives of 1.5-2, 5-8, and 15-17 days. Retention in muscles was resolvable into a two-exponential function with half-lives of 8 and 16 days. In the chronic study, the highest equilibrium cesium-137 concentrations, 10% of the average daily intake per gram, occurred in the muscle. The authors concluded that the muscle should be considered the formal critical organ for cesium-137. 

It has been proposed that the monomer emission decay is predominantly exponential with an additional minor late exponential component attributed to excimer dissociation (10,11). The contribution of the late component is less than 1% of the population of... 

The rate constants were derived from multi-exponential fits to open and closed time probability density functions (25). The number of open and closed states represents the number of exponential components required to fit each of the probability density functions, e.g. fitting the open and closed time distributions for the control data (lO M L-glutamate alone) required 3 and 4 components respectively. Numbers in brackets are percentages of channel populations occupying given open or closed states. 

While HAN shows an ESIPT reaction between two groups both located on the molecular frame of the dye and does not need solvation for the occurrence of internal proton motion, 1- and 2-naphthol, which are among the most studied aromatic systems, show an excited-state intermolecular proton transfer to the medium (water, alcohols) [88-96]. The produced anionic structure emits at the blue side of the normal form. The effects of CD on the intermolecular proton-transfer reaction from 1-naphthol (1-NP) [97, 98] to water and from water to 1-aminopyrene (1-AP) [98] have been studied by emission spectroscopy. For 1-NP in pure water, the decay of the 360 nm emission band (that of the neutral reactive species leading to the anionic one emitting at the blue side, 460 nm) was fitted with a 36 ps exponential component [98]. In the presence of (S-CD, the decay at 370 nm needed two exponential functions with time constants of 700 ps (75%) and 1600 ps (25%) [98]. The average time constant for deprotonation of 1-NP in the presence of (S-CD (1 1... 

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