Molecular Fluorescence

146 5 Fluorescence polarization. Emission anisotropy

. Brownian di usion equation for a spherical particle

where Dr is the rotation di usion coe cient. In spherical coordinates, Eq. (5.36) can be rewritten as

Taking into account the initial condition cos2 oð0Þ ¼ 1, the solution of this equation is

Dr can be determined by time-resolved fluorescence polarization measurements, either by pulse fluorometry from the recorded decays of the polarized components Ik and I?, or by phase fluorometry from the variations in the phase shift between Ik and I? as a function of frequency (see Chapter 6). If the excited-state lifetime is unique and determined separately, steady-state anisotropy measurements allow us to determine Dr from the following equation, which results from Eqs (5.10) and (5.41):

This relationship is called Perrin’s equation, because it was established for the first time by Francis Perrin; it was written at that time with polarization ratios

148 5 Fluorescence polarization. Emission anisotropy

Box 5.3 Emission anisotropy of totally asymmetric rotors and ellipsoids

A totally asymmetric rotor has three di erent rotational di usion coe cients D1, D2 and D3, around the three principal di usion axes of the molecule (which are di erent from the laboratory axes x, y and z). In cases where the absorption and emission transition moments are not directed along one of the principal axes, the decay of rðtÞ is a sum of five exponential terms. There was originally some controversy about the expression of rðtÞa–d). The correct solution was derived by Belford et al.d)

6 X3

rðtÞ ¼ 5 i¼1 Ci expð t=tiÞ þ ½ðF þ GÞ=4& expð 6D 2DÞt

þ ½ðF GÞ=4& expð 6D þ 2DÞt

where D, the mean rotational di usion coe cient, D, Ci and F are given by

D ¼ ðD1 þ D2 þ D3Þ=3

D ¼ ðD12 þ D22 þ D32 D1D2 D1D3 D2D3Þ1=2

Ci ¼ ajakejek ði; j; k ¼ 123; 231 or 321Þ

where a1, a2 and a3 are the cosines of the angles formed by the absorption transition moments with the three principal axes, and e1, e2 and e3 are the corresponding direction cosines of the emission transition moments.

ti ¼ 1=ð3D þ 3DiÞ

X3

F ¼ ai2ei2 1=3

i¼1

X3

GD ¼ Diðai2ei2 þ aj2ek2 þ ak2ej2Þ D i 0j 0k i¼1

For a particle with spherical symmetry ðD1 ¼ D2 ¼ D3 ¼ DÞ, the above expression reduces to Eq. (5.41).

In the particular case of prolate and oblate ellipsoids, the number of exponentials is reduced to three because two of the three axes are equivalent. The rotation di usion coe cients around the axis of symmetry and the equatorial axis are denoted D1 and D2, respectively. The emission anisotropy can then be written as

rðtÞ ¼ a1 exp½ 6D2t& þ a2 exp½ ðD1 þ 5D2Þt& þ a3 exp½ ð4D1 þ 2D2Þt&

5.6 Effect of rotational Brownian motion 149

where the pre-exponential factors depend on the orientation of the transition moments with respect to the ellipsoid axes as follows:

a1 ¼ 0:1ð3 cos2 y1 1Þð3 cos2 y2 1Þ

a2 ¼ 0:3 sin 2y1 sin 2y2 cos f

a3 ¼ 0:3 sin2 y1 sin2 y2ðcos2 f sin2 fÞ

where y1 and y2 are the angles formed by the absorption and emission transitions moments, respectively, with the axis of symmetry of the ellipsoid, and f is the angle formed by the projections of the two moments in the plane perpendicular to the axis of symmetry.

When the absorption and emission transition moments are collinear (y1 ¼ y2 ¼ y and f ¼ 0), the pre-exponential factors become

a1 ¼ 0:1ð3 cos2 y 1Þ2

a2 ¼ 0:3 sin2 2y

a3 ¼ 0:3 sin4 y

In the case of an ellipsoid of revolution for which the absorption or emission transition moment is parallel to the axis of symmetry (y1 or y2 ¼ 0), the anisotropy decay is a single exponential

rðtÞ ¼ 0:4 expð 6D2tÞ

A more complex expression is obtained when the absorption and emission transition moments lie in a plane perpendicular to the axis of symmetry

rðtÞ ¼ 0:1 expð 6D2tÞ þ 0:3ð2 cos2 a 1Þ exp½ ð4D1 þ 2D2Þt&

where a is the angle between the absorption and emission transition moments. In the particular case where this angle is 45 , this expression reduces to the first term.

In principle, the shape parameters of asymmetric rotors can be estimated from time-resolved anisotropy decay measurements, but in practice it is di cult to obtain accurate anisotropy decay curves over much more than one decade, which is often insu cient to determine more than two rotational correlation times.

5.7 Applications 151

An approximate expression of the anisotropy decay is

where tc is the e ective relaxation time of rðtÞ, i.e. the time in which the initially photoselected distribution of orientations approaches the stationary distribution. This time is related to Dw and yc by:

where x0 ¼ cos yc.

In practice, the parameters r0, ry and tc are obtained from the best fit of IkðtÞ and I?ðtÞ given by Eqs (5.7) and (5.8), in which rðtÞ has the form of Eq. (5.49). Then, yc is evaluated from ry=r0, and Eq. (5.50) allows us to calculate the wobbling di usion constant Dw.

If data analysis with a single exponential decay is not satisfactory, a double exponential can be used, but such a decay must be considered as a purely mathematical model.

It is interesting to note that tcð1 ry=r0Þ is exactly the area A under ½rðtÞ ry&=r0. Therefore, even if the anisotropy decay is not a single exponential, Dw can be determined by means of Eq. (5.50) in which tcð1 ry=r0Þ is replaced by the measured area A. An example of application of the wobble-in-cone model to the study of vesicles and membranes is given in Chapter 8 (Box 8.3). More general theories have also been developed (see Box 5.4).

5.7

Applications

The various fields concerned with the applications of fluorescence polarization are listed in Table 5.1.

The fluorescence polarization technique is a very powerful tool for studying the fluidity and orientational order of organized assemblies (see Chapter 8): aqueous micelles, reverse micelles and microemulsions, lipid bilayers, synthetic non-ionic vesicles, liquid crystals. This technique is also very useful for probing the segmental mobility of polymers and antibody molecules. Information on the orientation of chains in solid polymers can also be obtained.

Fluorescence polarization is also well suited to equilibrium binding studies when the free and bound species involved in the equilibrium have di erent rotational rates (Scheme 5.1). Most molecular interactions can be analyzed by this method. It should be emphasized that, in contrast to other methods using tracers, fluorescence polarization provides a direct measurement of the ratio of bound and free tracer without prior physical separation of these species. Moreover, measure-

152 5 Fluorescence polarization. Emission anisotropy

Box 5.4 General model for hindered rotationsa–f )

The wobble-in-cone model has been generalized by introducing three autocorrelation functions G0ðtÞ, G1ðtÞ and G2ðtÞ into the expression for rðtÞ

rðtÞ ¼ r0½G0ðtÞ þ 2G1ðtÞ þ 2G2ðtÞ&

The values of these autocorrelation functions at times t ¼ 0 and t ¼ y are related to the two order parameters hP2i and hP4i, which are orientational averages of the secondand fourth-rank Legendre polynomial P2ðcos bÞ and P4ðcos bÞ, respectively, relative to the orientation b of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as

hP2i ¼ h3 cos2 b 1i=2

hP4i ¼ h35 cos4 b 30 cos2 b þ 3i=8

and the autocorrelation functions are given by

G0ðyÞ ¼ hP2i2

G1ðyÞ ¼ G2ðyÞ ¼ 0

It is assumed that the probe molecules undergo Brownian rotational motions with an angle-dependent ordering potential UðbÞ

UðbÞ ¼ kT½l2P2ðcos bÞ þ l4P4ðcos bÞ&

For a rod-like probe with its absorption transition moment direction coinciding with the long molecular axis, the rotational motion in this potential well is described by the di usion coe cient D?. The decay of the autocorrelation functions is then shown to be an infinite sum of exponential terms:

Xy

GkðtÞ ¼ bkm expð akmD?tÞ k ¼ 0; 1; 2

m¼0

5.7 Applications 153

The coe cients akm and bkm are complex functions of the parameters l2 and l4 that describe the ordering potential. In many practical situations, GkðtÞ is essentially mono-exponential:

GkðtÞ ¼ ½Gkð0Þ GkðyÞ& expð ak1D?tÞ þ GkðyÞ

The di usion constant D? with the underlying ‘microviscosity’, and the two order parameters hP2i, hP4i reflecting the degree of orientational constraint have been successfully determined from the fluorescence anisotropy decay in vesicles and liquid crystals.

ments are carried out in real time, thus giving information on the kinetics of association and dissociation reactions.

Immunoassays based on fluorescence polarization have become very popular because, in contrast to radioimmunoassays, they require no steps to separate free and bound tracer. In a fluoroimmunoassay, the fluorescently labeled antigen is

Tab. 5.1. Fields of application of fluorescence polarization