Fluorescence Perrin equation

The Perrin equation 48 describes the relationship between the rotational correlational time, 0, and the fluorescence lifetime, r, for steady-state measurements ... 

If the aromatic group is bound tightly within the protein molecule, then one may obtain information on the rotational diffusion of the whole molecule from fluorescence polarization studies. Such investigations, which were started by Weber,(68) were widely popular in the 1960s and 1970s. Correlation times D of macromolecule rotations were determined according to the Perrin equation ... 

The rx term is the anisotropy at times long compared to the fluorescence lifetime, whereas in Eq. (5.9) 2 will be long. If there is no rM, then Eq. (5.8) reduces to the familiar Perrin equation for an isotropic rotator. Earlier, some confusion existed in this field since it was not recognized that an rro term was required for the case of membrane lipid bilayers. For the most part, time-resolved anisotropy measurements have a short rotational correlation time and an term. However, it has been recognized that a more adequate description may be to use two rotational correlation times, where the second may be quite long but not infinite as the rm implies/35 36 ... 

W. van der Meer, R. B. van Hoeven, and W. J. van Blitterswijk, Steady-state fluorescence polarization data in membranes. Resolution into physical parameters by an extended Perrin equation for restricted rotation of fluorophores, Biochim. Biophys. Acta 854, 38-44 (1986). 

FPA results obtained at different salt conditions may not be directly comparable because the fluorescence properties of 6-MI, including the lifetime (t), are salt dependent. The salt dependence of the FPA of a helix in a complex construct should thereby be normalized relative to the FPA of a short control duplex of the same sequence of the targeted helix to account for salt effects on the local environment of the 6-MI fluorophore. The normalization ratio, rnoml, can be calculated as the ratio between the apparent rotational correlation time, 9, of the constructs and the control duplex only, rnomi = construct/ control- is related to the rate of anisotropy decay, with larger 9 associated with higher anisotropy. If the basic Perrin equation for a sphere (Eq. (14.3)) is used to simplify calculation, then... 

These values can then be used in conjunction with the Perrin equation [19] to estimate the rate of rotation of a spherical fluorescent molecule, Rs, as follows ... 

Fluorescence polarization cannot attain the +1 theoretical limits for maximum beam polarization owing to the nature of the absorption and emission processes, which usually correspond to electric dipole transitions. Although the excitation with linearly polarized radiation favours certain transition dipole orientations (hence certain fluorophore orientations, and the so-called photoselection process occurs), a fairly broad angular distribution is still obtained, the same happening afterwards with the angular distribution of the radiation of an electric dipole. The result being that, in the absence of fluorophore rotation and other depolarization processes, the polarization obeys the Lev shin-Perrin equation,... 

Researches on theoretical topics have not been reported very extensively. A few papers are mentioned here and some others at appropriate points later in the article. Weber has re-examined the famous Perrin equation for quantifying the rotational depolarization of fluorescence. The arguments presented in the paper are applied to the temperature dependence of the local motions of tyrosine and tryptophan residues observed in proteins. 

The relations between polarization, fluorescence lifetime, and rotational diffusion were explored by F. Perrin, no doubt inspired by the work of his father, J. Perrin, on the translational diffusion of macroscopic particles. The Perrin equation may be written... 

EFFECTS OF ROTATIONAL DIFFUSION ON FLUORESCENCE ANISOTROPIES THE PERRIN EQUATION... 

Fluorescence polarization immunoassays (FPIs) are widely used to measure the amounts of drugs (D) and small molecides in chnical samples (Chapter 19). FPIs are based on the changes in polarizadon (or anisotropy) which occur when a labeled < g analog (F-D) binds to an antibody specific filT that drag O igure 20.20). The anisotropy of the lafaded drag can be estiinated fiom the Perrin equation. 

Figure 35 (a) Principle of a fluorescence polarization immunoassay and (b) the Perrin equation defining the expected anisotropy (r),... 

If the only significant process for depolarization is rotational relaxation, then the fluorescence anisotropy, for a single molecule or fluorophore, may be given by a form of the Perrin equation ... 

If this approach is followed, the general form of the Perrin equation can be obtained, which relates the steady-state fluorescence anisoflopy (rj) of a spherical fluorophore rotating in an isotropic medium to its rotational correlation time () ... 

The relationships derived with the Perrin equation apply only to spherical fluorophores, which generally does not include fluorescent dyes [9]. If a fluorescent molecule has a planar structure, serious deviations from the Perrin equation can occur. It is therefore essential to determine which molecular motions result in the observed depolarizations and how the latter relate to the structure of the fluorescent probe. The rate of rotation of the fluorophore is an average of the inplane (Vip) and out-of-plane (Vop) rates of rotation [9]. The in-plane rotation is about an axis normal to the ring plane, whereas the out-of-plane rotation is about an axis contained in the ring plane at right angles to the absorption oscillator. There are three cases of interest in relation to the Perrin equation ... 

If the medium is highly ordered or anisotropic, the Perrin equation does not hold and microviscosity can be used only qualitatively. In this case, a fluorophore is restricted to certain motions that will occur to different extents in different dimensions. An example of such a phenomenon would be a fluorescent molecule that would align itself along the fatty acid chains of phospholipids in the phospholipid bilayer of a liposome. For fluorophores in an anisotropic medium, the time-resolved fluorescence anisotropy does not fall to zero (see Fig. 4, curve a), that is, the depolarizing rotations of the fluorophore will not attain an isotropic distribution at infinite time. The anisotropy will decay to an infinite-time anisotropy (r ) rather than to zero and will have the form... 

In order to apply the Perrin equation [Eq. (16)] the value for the fluorescence lifetime of the excited state at different temperatures has to be determined. In the absence of a fluorometer with time-resolved measurement capabilities, this is performed by measuring the total fluorescence intensity [Eq. (3)] as a function of temperature. This was performed only for paraffin oil, because the other samples showed high interference. The fluorescence intensity reaches a plateau at 0-10°C. The value for the lifetime of DPH in the absence of dynamic quenching processes (at low temperatures) is 11.4 ns (Xo) and is proportional to the plateau... 

The temperature dependence of the DPH fluorescence lifetime in paraffin oil is shown in Figure 9. The lifetime values derived from this experiment were used in the derivation of Perrin equations for all systems [Eq. (16)]. Perrin plots for canola oil, canola margarine fat, and heavy white paraffin oil are shown in Figure 10. These curves can be used for the determination of viscosities at different temperatures by measuring steady-state anisotropies. The correlations obtained are presented in Table 1. 

Fluorescence polarization is frequently expressed in terms of anisotropy (r) due to the comparatively simple equations describing rotational depolarization. The dependence of fluorescence anisotropy on molecular rotation can be described quantitatively with the well-known Perrin equation. 

When a direct determination of time-resolved fluorescence anisotropy is not possible, for example because of photo-bleaching, Perrin equation (in a system with a single lifetime and rotational correlation time)... 

F. Perrin Theory of fluorescence polarization (sphere). Perriris equation Indirect determination of lifetimes in solution. Comparison with radiative lifetimes... 

The changes have been used to provide information about the enviromnent of the fluorescent probe and to follow changes in conformation of the macromolecule. In other work the study of the fluorescence polarization properties of the attached probe under steady state illumination and the application of Perrin s equation enable calcu-latnn of the rotary Brownian motion of the polymer. This technique has been extended by Jablonski and Wahl to the use of time-resolved fluorescence polarization measurements to calculate rotational relaxation times of molecules These experiments are discussed fiilly in the fdlowing section of this review. 

The Combined Stern-Volmer and Perrin Model A model has been proposed by Morishima et al. [97] which takes account of Manning s theory [98] of polyelectrolytes and introduces a modification into the Stem-Volmer equation to describe sphere-of-action (Perrin) quenching this has been termed combined Stern-Volmer and Perrin Analysis and has been adopted [95,96] in an effort to describe quenching of fluorescence from labeled PMAA by T1+ ions, for example. 

Assuming a spherical shape for the fluorescent molecule, the degree of change in the rotational Brownian motion is given by Eq. (3.25), where v is the volume of the spherical molecule, r)0 is the solvent viscosity, r is the fluorescence lifetime of the chromophore, and T is the temperature. The values of r0 and r/v can be obtained from a plot of Mr versus T/rj0. Thus, if the fluorescence lifetime of the chromophore is known, it is possible to determine the hydrodynamic volume of the rotating molecule and its rotational diffusion constant D,. This data treatment is known as the Perrin-Weber approximation,25 after the two scientists who first derived the equations in the case of protein chromophores. 

HPhe polarization of fluorescence technique employing dye-macromolecule x conjugates is a sensitive hydrodynamic method for studying the structure and interactions of proteins 19, 20, 41, 54, 65) and synthetic polypeptides 26, 30, 31, 49)- The relationship describing the dependence of polarization of fluorescence upon the Brownian rotational diffusion of the macromolecule was developed by Perrin 50) and extended by Weber 65) in the form of the equations (for excitation with natural light) ... 

A similar treatment in terms of a Perrin-type formulation for static quenching was applied to the fullerene-donor systems without invoking contributions due to the excitation of ground state fullerene-donor complexes [88], In room-temperature toluene, upward curvatures were observed in quenchings of Cgo and C70 fluorescence intensities by DMA and DEA (Fig. 30). The upward curvatures were also treated by including static fluorescence quenching in terms of the equation as follows [88,94]. 

The degree of fluorescence polarization, P, of TNS aqueous solution with amylose was measured with changing solvent viscosity at constant temperature (25 C). Figure 3 shows Perrin plot constructed from the data obtained. Since the plot is linear, the mean rotational relaxation time,

, can be calculated using the following equation (6, 8)... 

As described by the Perrin s equation (13), the molecular rotational correlation time (0) can be obtained by the values of steady-state anisotropy (r) and fluorescence lifetime (r). An inspection of Fig. 4 and Table 2 shows that in buffered solution the steady state anisotropy and fluorescence lifetime of free TPPS are 0.01 and 9.7 ns, respectively. In the presence of high CD concentration, the increase of the steady-state anisotropy and of the fluorescence lifetime indicates that the rotational correlation time 9 is increasing. These findings would suggest the entanglement of TPPS in the CD aggregates. 

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