Fluorescence polarization

If fluorophores are excited with linearly polarized light , they - with a few exceptions - also emit linearly polarized light. This phenomenon is called fluorescence polarization .

If the fluorophores are movable and not fixed in space, the fluorescence polarization is influenced by the rotation of the movable fluorophores, i.e. by the rotational diffusion constant : The lifetime of the excited state, i.e. the time between absorption of a photon and emission of a photon, the so-called Fluorescence lifetime is very short - it is in the nanosecond range - but the average speed of rotation of the mobile fluorophores is usually large enough that it influences the measured fluorescence polarization.

Determination of the fluorescence polarization

Basic metrological structure

Schematic diagram for determining and . ${\displaystyle I_{\parallel }}$ ${\displaystyle I_{\perp }}$

The excitation light is a polarizer linearly polarized and is then incident on the sample. The emission light is analyzed with a second polarizer - the analyzer. For this purpose, the intensity of the emission light is measured in two positions of the analyzer relative to the position of the polarizer:

1. If the polarizer and analyzer are parallel to each other, the fluorescence intensity is measured parallel to the plane of the excitation light. This intensity - the parallel radiation - is referred to as . ${\displaystyle I_{\parallel }}$
2. If the polarizer and analyzer are perpendicular to one another, the fluorescence intensity is measured perpendicular to the plane of the excitation light. This intensity - the perpendicular radiation - is referred to as . ${\displaystyle I_{\perp }}$

Polarization, anisotropy, total intensity

The difference D between and is used as a measure of the degree of polarization of the emission light. ${\displaystyle I_{\parallel }}$ ${\displaystyle I_{\perp }}$

If D is zero, the rotational speed of the fluorophores examined is so fast that the orientations of the fluorophores are stochastically distributed within the fluorescence lifetime of the excited fluorophore: completely unpolarized emission light is then measured.

If D is equal to one, the rotational speed of the fluorophores examined is so slow that the orientation of the fluorophores does not change within the fluorescence lifetime of the excited fluorophore: the polarization of the excitation light is retained in the emission light. To do this, however, the emission light must be emitted from the fluorophore at the same angle as the excitation light. This is mostly not the case, that is, there is an intrinsic rotation of the emitted light to the absorbed light by the fluorophore, even if it is not rotating.

The difference D is always normalized with a factor. Here, two different values have been established, which differ in their standardization: the polarization P and the anisotropy A .

The polarization P is defined as:

${\displaystyle P={\frac {I_{\parallel }-G\cdot I_{\perp }}{I_{\parallel }+G\cdot I_{\perp }}}}$.

The weighting factor G is a device factor to be determined separately. The measured values ​​for and deviate from the ideal values, since the sensitivity ratio of the detector system for parallel and perpendicular radiation can be different. In the ideal case, G = 1. ${\displaystyle I_{\parallel }}$ ${\displaystyle I_{\perp }}$

The anisotropy A is defined as:

${\displaystyle A={\frac {I_{\parallel }-G\cdot I_{\perp }}{S}}={\frac {I_{\parallel }-G\cdot I_{\perp }}{I_{\parallel }+2\cdot G\cdot I_{\perp }}}}$.

The total intensity S is defined as:

${\displaystyle S=I_{\parallel }+2\cdot G\cdot I_{\perp }}$.

The device factor G is:

${\displaystyle G={\frac {i_{\perp }}{i_{\parallel }}}}$.

The G factor is determined using a fluorescent sample before the actual measurement. The two intensities and are determined exactly the opposite of the intensities and : ${\displaystyle i_{\perp }}$ ${\displaystyle i_{\parallel }}$ ${\displaystyle I_{\perp }}$ ${\displaystyle I_{\parallel }}$

1. If it is determined that the polarizer is at 90 ° and the analyzer is at 0 °, it is determined when the polarizer is at 0 ° and the analyzer is at 90 °. ${\displaystyle I_{\perp }}$ ${\displaystyle i_{\perp }}$
2. If it is determined that the polarizer is at 90 ° and the analyzer is at 90 °, it is determined when the polarizer is at 0 ° and the analyzer is at 0 °. ${\displaystyle I_{\parallel }}$ ${\displaystyle i_{\parallel }}$

Relationship between polarization and anisotropy

The following relationships exist between polarization and anisotropy:

${\displaystyle P={\frac {3\,A}{2+A}}}$

${\displaystyle A={\frac {2\,P}{3-P}}}$

The polarization and the anisotropy can therefore be transformed directly into one another.

Heterogeneous fluorophore populations

If different fluorophore populations are present, a mixed polarization or a mixed anisotropy is measured. ${\displaystyle {\overline {P}}}$ ${\displaystyle {\overline {A}}}$

According to Gregorio Weber, the following relationship can be written for mixed polarization : ${\displaystyle {\overline {P}}}$

${\displaystyle {\frac {1}{\overline {P}}}-{\frac {1}{3}}={\frac {1}{\sum _{i=1}^{n}{\frac {f_{i}}{{\frac {1}{P_{i}}}-{\frac {1}{3}}}}}}}$

Here, the polarization of the i th fluorophore population and the proportion of the i th fluorophore population of the total intensity S : ${\displaystyle P_{i}}$ ${\displaystyle f_{i}}$

${\displaystyle f_{i}={\frac {S_{i}}{\sum _{j=1}^{m}S_{j}}}={\frac {S_{i}}{S}}}$.

Because of the relationship between polarization and anisotropy, Weber's formula can be formed for mixed anisotropy : ${\displaystyle {\overline {A}}}$

${\displaystyle {\overline {A}}=\sum _{i=1}^{n}f_{i}\,A_{i}}$

A special case of a heterogeneous fluorophore population is when a background intensity is to be subtracted from the actual measurement signal:

${\displaystyle P={\frac {(I_{\parallel }-I_{\parallel ,\,\mathrm {Hintergrund} })-G\,(I_{\perp }-I_{\perp ,\,\mathrm {Hintergrund} })}{(I_{\parallel }-I_{\parallel ,\,\mathrm {Hintergrund} })+G\,(I_{\perp }-I_{\perp ,\,\mathrm {Hintergrund} })}}}$

${\displaystyle A={\frac {(I_{\parallel }-I_{\parallel ,\,\mathrm {Hintergrund} })-G\,(I_{\perp }-I_{\perp ,\,\mathrm {Hintergrund} })}{(I_{\parallel }-I_{\parallel ,\,\mathrm {Hintergrund} })+2\,G\,(I_{\perp }-I_{\perp ,\,\mathrm {Hintergrund} })}}}$

The intensities of the parallel and perpendicular radiation of the background must be determined in a separate measurement.

Dependence of the fluorescence polarization on the mobility of the fluorophore

The dependence of the fluorescence polarization on the mobility of the fluorophore in stationary fluorescence measurements was derived by Francis Perrin in 1926 from the theory of Brownian molecular motion. The Perrin equation named after him describes the relationship between the measured polarization, the fluorescence lifetime and the rotation relaxation time . The Perrin equation is: ${\displaystyle \tau }$ ${\displaystyle \rho }$

${\displaystyle \left({\frac {1}{P}}-{\frac {1}{3}}\right)=\left({\frac {1}{P_{0}}}-{\frac {1}{3}}\right)\,\left(1+{\frac {3\,\tau }{\rho }}\right)}$

Here is the intrinsic polarization of the immobile fluorophore. ${\displaystyle P_{0}}$

Because of the relationship between the polarization P and the anisotropy A - and the relationship between the rotational relaxation time and the rotational correlation time - the Perrin equation can be rewritten as: ${\displaystyle \rho =3\,\theta }$ ${\displaystyle \rho }$ ${\displaystyle \theta }$

${\displaystyle {\frac {1}{A}}={\frac {1}{A_{0}}}\,\left(1+{\frac {\tau }{\theta }}\right)}$

It is the intrinsic anisotropy of the fluorophore immovable, analogous to . ${\displaystyle A_{0}}$ ${\displaystyle P_{0}}$

The following representation of the Perrin equation is usually preferred because it is more compact than the original formulation of the equation:

${\displaystyle {\frac {A_{0}}{A}}=1+{\frac {\tau }{\theta }}}$

The fluorescence lifetime is a fixed substance size for every fluorophore, provided that no dynamic quenching processes occur. If , that is, if the fluorophores practically no longer rotate (for example in a highly viscous solution), then the quotient tends towards one, i.e. A becomes equal to the intrinsic anisotropy A ₀ . If on the other hand , i.e. the rotation of the fluorophores is infinitely fast, then the anisotropy A also tends towards 0. Because we can only assume values ​​greater than or equal to zero and because of the linear relationship of the Perrin equation, it follows that the anisotropy A only values ​​between zero and the intrinsic anisotropy A ₀ . Similarly, the polarization P can also only assume values ​​between zero and the intrinsic polarization P ₀ . ${\displaystyle \tau }$ ${\displaystyle \theta \rightarrow \infty }$ ${\displaystyle A/A_{0}}$ ${\displaystyle \theta \rightarrow 0}$ ${\displaystyle \tau }$ ${\displaystyle \theta }$

For a spherical molecule in aqueous solution, the following relationship can be established for the rotational correlation time: ${\displaystyle \theta }$

${\displaystyle \theta ={\frac {\eta \,V}{R\,T}}}$

Where is the viscosity of the solvent, T is the temperature, R is the gas constant and V is the molecular volume of the fluorophore. The general relationships follow from the Perrin equation under these conditions: ${\displaystyle \eta }$

• The anisotropy A increases as the volume of the fluorophore increases.
• The anisotropy A increases as the viscosity of the solvent increases.
• The anisotropy A decreases as the temperature increases.
• The anisotropy A decreases as the fluorescence lifetime increases. ${\displaystyle \tau }$

literature

• Joseph R. Lakowicz : Principles in Fluorescence Spectroscopy. 3rd edition, Springer, 2006, ISBN 0-387-31278-1 .
• Michel Daune: Molecular Biophysics. Vieweg, 1997, ISBN 3-528-06689-X .

credentials

1. ^ Gregorio Weber: Polarization of the Fluorescence of Macromolecules. 1. Theory and Experimental Method , Biochemical Journal , 51 , 145-155, (1952).
2. ^ Francis Perrin: Polarization de la Lumiére de Fluorescence. Vie Moyenne des Molécules dans L'Etat Exité , Journal de Physique, 7 , no. 12, 390-401, (1926).