Molecular electronic transitions are characterized by absorption and emission dipoles (and their associated transition moments) fixed with respect to the molecular frame. When an assembly of randomly oriented molecules is illuminated with polarized light of suitable frequency the probability of absorption is proportional to cos2y, where y represents the angle between the absorption transition moment and the exciting electric vector. This anisotropy in the photoabsorption process causes preferential excitation (photoselection) of those molecules oriented forming angles y close to zero. The orientational anisotropy is a maximum at the instant of excitation (t = o) and will decrease as a function of time due, among other things, to the Brownian motion experienced by the excited molecules. Since these random motions depend on the shape and size of the molecules and on the viscosity and temperature of the solvent, the decrease of the orientational anisotropy will be related to these parameters. Therefore, any method able to detect the initial orientation of the photoselected population and monitor its temporal evolution will give important information about the dynamic properties of the molecule. When the photoselected molecules are fluorescent the disorientation caused by the Brownian motion may be determined by recording the polarized fluorescence components In and I± which represent, respectively, the intensities of emission parallel and perpendicular to the exciting vector. Then, the emission anisotropy, r, is defined by:
The maximum value of r (the intrinsic anisotropy r0) depends on the relative orientation of the excitation and emission transition moments of the molecule and corresponds to the emission anisotropy observed in the absence of other depolarizing process taking place during the lifetime of the excited state (t). In most cases, decreasing this initial value depends only on the ratio between t and the speed of the rotational diffusion by an expression originally formulated by Perrin (1926) and later applied and extended by Weber (1953) and Jablonski (i960):
where 9 is the rotational correlation time, which is related to the rotational diffusion coefficient, D (9 = 1/6D), and, therefore, to the molecular dimensions and to the viscosity and temperature of the solvent. Many fluorescent molecules in water solution have correlation times much shorter than the lifetime (9 « t). Then, the measured anisotropy (r) is zero (the random orientation of the excited molecules is recovered before fluorescence emission). However, for large molecules such as proteins, the rotational motions occur in the same time-scale as the fluorescence lifetime and, consequently, the fluorescence light that they emit will be partially polarized (r ^ o).
Equation 1.12 was deduced assuming spherical rotors in isotropic media and from experiments under continuous illumination, that is, under steady-state conditions. Its use in the determination of any one of the parameters involved in the equation requires the explicit knowledge of all the others parameters and it is limited to fluorophores in isotropic solvents having specific symmetry, with a single fluorescence lifetime. Steady-state anisotropy can also be used to monitor lipid-peptide interactions, as long as the peptides contain an extrinsic or intrinsic fluorophore. The interaction of the peptide with the membrane slows down and partially prevents the rotation of the macromolecule-bounded fluorophore, causing an increasing of the anisotropy value. In addition, steady-state anisotropy measurements have been extensively used to detect alterations induced by peptide and proteins in membrane systems through incorporation of extrinsic probes into the lipid vesicles (e.g. Ahn et al. 2002; Zhao et al. 2001). The anisotropy values obtained under these conditions contain structural and dynamic information about the peptide and the lipid membrane which cannot be separated from a simple steady-state experiment (Jahnig 1979). This and much more information is available if one measures directly the anisotropy decay, that is, the values of r(t) after pulse excitation. In the case of spherical fluorophores dissolved in isotropic solvents and in the absence of other depolarization processes r(t) is a single exponential:
For more complex structures (non-spherical fluorophores, fluorophores located in an anisotropic environment such as a lipid membrane, macromolecule-bound fluorophores having internal flexibility, etc.), r(t) can be described as a multi-exponential decay plus a constant term (Dale 1977; Heyn 1989):
where r^ is the residual anisotropy obtained for t ^ ^ when the rotational motion of the fluorophore is restricted and, consequently, depolarization is not complete (which is the usual case for fluorophores inserted in lipid membranes). The anisotropy decay parameters (^f, Pf, and rJ) contain information about the shape, size, flexibility and location of the fluorophore or of the bound macromol-ecule, and/or about the rigidity of the molecular environment where the fluorophore resides (e.g. a lipid membrane). To extract this information it is necessary to propose a rotational model for the fluorophore and compare the exponential terms corresponding to this model with those obtained from the fit of the experimental anisotropy decay to Eq. 1.14. Some of these rotation models and their associated anisotropy decay laws have been extensively revised by Lakowicz (1999).
Although Brownian motion is the main source of fluorescence depolarization, additional processes such as energy transfer, fluorescence reabsorption or light scattering can indirectly modify the orientation of the emission dipoles affecting the anisotropy value. The extent to which the anisotropy is decreased due to these mechanisms can be minimized if one works with dilute samples and suitable excitation and emission wavelengths.
Together with FRET, quenching is the most common methodology in the area of biological applications of fluorescence. In the case that fluorescence quenching is a dynamic process, i.e. diffusion dependent (static mechanisms will be addressed later), this implies the resolution of Fick's diffusion equation. In homogeneous solution the resulting complex formalisms, such as the so-called "radiation boundary condition" can be experimentally verified.
However, in biological systems, specifically the problem of both fluorophore and quencher diffusing in a membrane, "transient effects" due to diffusion cannot be studied in detail due to the following: (1) diffusion is no longer isotropic; (2) the decay of most samples in the absence of a quencher is not a single exponential but is intrinsically complex, which among other reasons can be due to populations in different microenvironments in the membrane, such as different transversal locations (in addition, for the more relevant fluorescent amino acids, tryptophan and tyrosine, this complexity is also due, among other reasons, to the emission of different ground-state rotamers - see Sect. 188.8.131.52); (3) in most experimental situations there is strong scattering due to the vesicles, and in this way the very beginning of the decay is biased due to this artefact. Although this can be taken into account in the time-resolved analysis, it is always a critical problem.
In this way the well-known Stern-Volmer equation is in most situations the starting point, and from time-resolved data a linear relationship (Eq. 1.15) should be obtained:
where t0 and t are the fluorescence lifetimes in the absence and presence of the quencher, kq is the bimolecular rate constant, and [Q] is the effective quencher concentration in the membrane which is obtained from knowledge of its partition coefficient (see Sect. 1.4.1) . For this rate constant, and related to the "transient effects" described above, the approximation of Umberger and Lamer (1945) can be considered, kq = 4nNk(2Rc)(2D)[l + 2Rcl(2x0D)m] (1.16)
where Rc is the collisional radius (taken as the sum of the fluorophore and quencher radius), and in this way the diffusion coefficient D can be obtained. In the case that the quenching is not diffusion controlled, the above definition for kq is multiplied by an efficiency factor y. This approximation takes into account the fact that only a fraction of the collisions is effective in the excited state deactivation.
As mentioned above, only exceptionally is the intrinsic decay of the fluorophore not complex, and is usually empirically described by a sum of exponentials (Eq. 1.2). Generally, the quenching of each component cannot be detailed, and the use of mean or average lifetimes, which is not uncommon to find described in the wrong way in the literature, is the usual approach.
In Eq. 1.15 above, the lifetime quantum yields (see Eq. 1.3) (integration of the area under the decay) should be used on the left-hand side of the equation, while the mean lifetime should be used on the right-hand side,
I a, if which in fact is the one that describes the mean lifetime that the molecule spends in the excited state, so is the relevant one to consider in the diffusion process. The mean lifetime can be even more refined, such as described by Sillen and Engelborghs (1998).
Regarding the static components that can be present in the steady state, the first step is to consider a sphere of action, which accounts for the statistical pairs existing at the moment of excitation. So, from the time-resolved linear Stern-Volmer plot (Eq. 1.15), and the fitting of Eq. 1.18 to the steady-state data I0II, the volume of the sphere V is obtained,
In the case that the recovered sphere radius is greater than the sum of the col-lisional radii of fluorophore and quencher (~ 10 A for typical molecular pairs), a model considering the formation of a molecular complex should be invoked. The general formalism for the case where the equilibrium constant Ka is not very high, or the quencher concentration is low, is given in Castanho and Prieto (1998).
A specific application of quenching in membranes is related to the determination of the transversal position of molecules, i.e. the in-depth location in a membrane. The most well-known approach of this type of differential quenching is the so-called "parallax method", which will be briefly addressed in Sect. 1.4.5.
Much less used to obtain structural information in membranes is the self-quenching mechanism. However, this is a very common interaction in membrane studies, since the compartmentalization effect of this heterogeneous medium leads to high effective concentrations of molecules in a membrane.
Self-quenching can happen in cases that a protein or a peptide (1) was deriva-tized with a fluorophore that undergoes self-quenching (BODIPY is an example, Fernandes et al. 2003), or (2) it can also happen in the case of a non-derivatized peptide, since tryptophan can be quenched in an intermolecular way, by another peptide segment, which has in its structure a residue that is a suitable quencher of tryptophan, such as a cysteine or a lysine (e.g. de Almeida et al. 2004).
When studying self-quenching, the Stern-Volmer relationship (Eq. 1.15) for dynamic quenching still holds, where now the quencher concentration [Q], is replaced by the fluorophore concentration [F], and the value of t0 is the limiting value for a diluted solution, the situation where there is no self-quenching. However, the same formalism is not valid for the steady state, since the total intensity increases due to the increase in fluorophore, but there is a concomitant decrease due to the self-quenching interaction. This leads to the following hyperbolic-type relationship, where 7f is the fluorescence intensity and C is a constant, and in the case that the effective concentration is high enough, the static contribution can be taken in the framework of the sphere of action model.
The above formalisms assumed that diffusion in the membrane happens in a three-dimensional medium. The solution for a bidimensional system (Razi-Naqvi 1974), when applied to a membrane, is relevant in the case that the fluorescence lifetime was on the order of hundreds of nanoseconds, which is not the case for most probes or fluorescent amino acids.
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