Stern-Volmer equation

Stern-Volmer plot : the term was plotted against the concentration of the quencher . The increase in the linear course (red solid and dotted line) of the Stern-Volmer equation corresponds to the Stern-Volmer constant . If measurements are made of , coated with a known concentration of the quencher in the Stern-Volmer plot (in the diagram drawn points) and a linear regression carried out, the Stern-Volmer constant can be determined experimentally. ${\displaystyle (F_{0}/F)-1}$ ${\displaystyle \mathrm {[Q]} }$ ${\displaystyle K_{SV}}$ ${\displaystyle F_{0}}$ ${\displaystyle F}$

In physical chemistry, the Stern-Volmer equation describes the dependence of the quantum yield or the intensity of the fluorescence of a fluorescent dye on the concentration of substances that quench the fluorescence (so-called quenchers ). Under certain circumstances, the Stern-Volmer equation can also be used to describe the dependence of the lifetime of the excited state of a fluorescent dye on the concentration of the quencher.

The equation comes from a collaboration between the physical chemists Otto Stern and Max Volmer at the physicochemical institute of the Berlin University under Walther Nernst . The Stern-Volmer equation was described for the first time by Stern and Volmer in 1919 in the article “About the decay time of fluorescence” , which appeared in the scientific journal Physikalische Zeitschrift .

The equation in its classic form is:

${\displaystyle {\frac {F_{0}}{F}}=1+K_{SV}\cdot [\mathrm {Q} ]}$

It is the fluorescence intensity of the fluorescent dye (fluorophore) in the absence of the quencher, the fluorescence intensity thereof in the presence of the quencher, the concentration of the quencher and the Stern-Volmer constant. ${\displaystyle F_{0}}$ ${\displaystyle F}$ ${\displaystyle \mathrm {[Q]} }$ ${\displaystyle K_{SV}}$

The following notation of the Stern-Volmer equation is often preferred:

${\displaystyle {\frac {F_{0}}{F}}-1=K_{SV}\cdot \mathrm {[Q]} }$

If the term is plotted against the concentration , a simple linear relationship results. The slope of the straight line is then the Stern-Volmer constant . ${\displaystyle (F_{0}/F)-1}$ ${\displaystyle \mathrm {[Q]} }$ ${\displaystyle K_{SV}}$

An important prerequisite for the validity of the Stern-Volmer equation is that all molecules of the fluorophore can be reached equally by the quencher: the same Stern-Volmer constant must apply to all molecules of the fluorophore . If some of the molecules of the fluorophore are more or less accessible for the quencher - and thus their fluorescence is easier or more difficult to erase - then the Stern-Volmer equation in the above form cannot be used. It then has to be modified. ${\displaystyle K_{SV}}$

Another important requirement of the Stern-Volmer equation is that the quencher must only quench the fluorescence in one way. If the quencher quenches the fluorescence in different ways, the Stern-Volmer equation in the above form cannot be used. It then has to be modified .

The Stern-Volmer constant for dynamic fluorescence quenching

Stern-Volmer plot for dynamic fluorescence quenching. The term (shown in red) and the term (shown in blue) were plotted against the concentration of the quencher. With dynamic fluorescence quenching , the two courses are exactly the same. ${\displaystyle (F_{0}/F)}$ ${\displaystyle (\tau _{0}/\tau )}$ ${\displaystyle \mathrm {[Q]} }$

During dynamic fluorescence quenching - also known as dynamic quenching or dynamic quenching - the quencher collides with the fluorophore. If the fluorophore is in its excited state during the collision, the fluorophore returns from the excited state to its ground state without emitting photons. This is why dynamic fluorescence quenching is also known as collision quenching. The dynamic fluorescence quenching is based on the fact that the energy of the excited fluorophore is given off without radiation.

The Stern-Volmer constant for dynamic fluorescence quenching is:

${\displaystyle K_{SV}=K_{d}=\tau _{0}\,k_{q}}$

This is the bimolecular quenching constant and the lifetime of the excited state of the fluorophore in the absence of the quencher. ${\displaystyle k_{q}}$ ${\displaystyle \tau _{0}}$

The bimolecular quenching constant can be calculated directly from the lifetime of the undisturbed fluorophore ( = 0) and the lifetime of the fluorophore for the quencher concentration : ${\displaystyle k_{q}}$ ${\displaystyle \tau _{0}}$ ${\displaystyle \mathrm {[Q]} }$ ${\displaystyle \tau }$ ${\displaystyle \mathrm {[Q]} }$

${\displaystyle k_{q}={\frac {1}{\mathrm {[Q]} }}\cdot \left({\frac {1}{\tau }}-{\frac {1}{\tau _{0}}}\right)}$

The bimolecular quenching constant is also represented as: ${\displaystyle k_{q}}$

${\displaystyle k_{q}=\gamma \cdot k_{0}}$

It is the bimolecular rate coefficient for collisions: this coefficient indicates how likely the Fluorophormolekül and the quencher molecule collide. The bimolecular rate coefficient can be calculated using the Smoluchowski equation . The parameter is the quenching efficiency and indicates the probability with which the quencher will quench the excited fluorophore in the event of a collision. The quenching efficiency can assume values ​​between zero and one: If it is equal to zero, the excited fluorophore is never quenched by the quencher, regardless of how often it hits the fluorophore. The quencher is then no quencher at all in the sense of dynamic fluorescence quenching. If the quenching efficiency is equal to one, the quencher quenches the excited fluorophore with every collision that occurs. ${\displaystyle k_{0}}$ ${\displaystyle k_{0}}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma }$

The decrease in the lifetime of the excited state in the presence of the quencher is characteristic of dynamic fluorescence quenching . The longer the excited state, the more likely a collision between the quencher and the excited fluorophore and thus also its extinction. The following ratio therefore applies to dynamic fluorescence quenching: ${\displaystyle \tau }$

${\displaystyle {\frac {F_{0}}{F}}={\frac {\tau _{0}}{\tau }}=1+K_{d}\cdot \mathrm {[Q]} }$

For the same concentration of the quencher, with dynamic fluorescence quenching as the temperature rises, the value for the is in principle greater, i.e. H. the quencher extinguishes more at a higher temperature than at a lower temperature: the diffusion speed of the quencher - and thus also the bimolecular rate coefficient - increases with increasing temperature, which means that the number of collisions with the fluorophore and thus the number of extinguishing processes also increases. This is an important distinguishing feature between dynamic and static fluorescence quenching, since it is the other way around with static fluorescence quenching. ${\displaystyle K_{d}}$ ${\displaystyle k_{0}}$

The previous one applied to a homogeneous fluorophore population . If there is more than one fluorophore population - i.e. if several fluophores are present or if the members of a fluorophore species are in different chemical environments and their fluorescence behavior is therefore significantly different. H. measurable, distinguish - then the Stern-Volmer equation for dynamic fluorescence quenching for a heterogeneous fluorophore population reads :

${\displaystyle {\frac {F}{F_{0}}}=\sum _{i=1}^{n}{\frac {f_{i}}{1+K_{d,\,i}\,\mathrm {[Q]} }}}$

It is the Stern-Volmer constant for the dynamic quenching of the ith Fluorophorpopulation and the percentage of the ith Fluorophorpopulation of the total intensity of fluorescence: ${\displaystyle K_{d,\,i}}$ ${\displaystyle f_{i}}$

${\displaystyle f_{i}={\frac {F_{i}}{\sum _{i=1}^{n}F_{i}}}}$

Here is the fluorescence intensity of the ith fluorophore population. ${\displaystyle F_{i}}$

Derivation

The fluorophore F passes through the absorption of a photon - symbolized by the expression - into the excited state F ^* : ${\displaystyle h\cdot \nu _{\mathrm {Absorption} }}$

${\displaystyle \mathrm {F} +h\cdot \nu _{\text{Absorption}}\ {\xrightarrow {k_{a}}}\ \mathrm {F^{*}} }$

It is the velocity constant of absorption, with which the fluorophore changes from the ground state to the excited state. The reaction constant expresses how many educts are converted into products per second . ${\displaystyle k_{a}}$

From this excited state the fluorophore can return to the ground state in various ways:

${\displaystyle \mathrm {(A)} \quad \mathrm {F^{*}} \ {\xrightarrow {k_{f}}}\ \mathrm {F} +h\cdot \nu _{\text{Emission}}}$

${\displaystyle \mathrm {(B)} \quad \mathrm {F^{*}} \ {\xrightarrow {k_{w}}}\ \mathrm {F} +{\text{Wärme}}}$

${\displaystyle \mathrm {(C)} \quad \mathrm {F^{*}+Q} \ {\xrightarrow {k_{q}}}\ \mathrm {F+Q^{'}} }$

In (A) the fluorophore changes to the ground state through the emission of a photon (fluorescence), in (B) the energy of the excited state turns into heat without radiation through other processes and in (C) the excited state becomes radiationless through the quencher transferred to the basic state. The quencher absorbs the energy of the excited state of the fluorophore, symbolized by Q '. ${\displaystyle \mathrm {Q} }$

The three reaction pathways (A), (B) and (C) have the respective rate constants , and . ${\displaystyle k_{f}}$ ${\displaystyle k_{w}}$ ${\displaystyle k_{q}}$

The fluorescence quantum yield is defined as:

${\displaystyle \phi ={\frac {I_{e}}{I_{a}}}}$

Here is the number of photons emitted by fluorescence per unit of time and is the number of photons absorbed by the fluorophore per unit of time. ${\displaystyle I_{e}}$ ${\displaystyle I_{a}}$

The number of emitted photons per unit of time is determined by (A) from the concentration of excited fluorophore and  : ${\displaystyle \mathrm {[F^{*}]} }$ ${\displaystyle k_{f}}$

${\displaystyle I_{e}=k_{f}\cdot \mathrm {[F^{*}]} }$

( see also: first order reaction )

The number of absorbed photons can be determined, analogously to , from the concentration of the fluorophore F and the rate constant . Since the excited fluorophores F * return to the ground state through the above-mentioned processes (A), (B) and (C), the two processes absorption and loss of the excitation energy through the three processes (A), (B) and ( C) equal in equilibrium : ${\displaystyle I_{e}}$ ${\displaystyle \mathrm {[F]} }$ ${\displaystyle k_{a}}$

${\displaystyle I_{a}=k_{a}\cdot \mathrm {[F]} =k_{f}\cdot \mathrm {[F^{*}]} +k_{w}\cdot \mathrm {[F^{*}]} +k_{q}\cdot \mathrm {[Q]\cdot [F^{*}]} }$

Reactions (A) and (B) are first-order reactions, while reaction (C) is a second-order reaction .

By replacing and the quantum yield of fluorescence after shortening by : ${\displaystyle I_{e}}$ ${\displaystyle I_{a}}$ ${\displaystyle \mathrm {[F^{*}]} }$

${\displaystyle \phi ={\frac {k_{f}}{k_{f}+k_{w}+k_{q}\cdot \mathrm {[Q]} }}}$

In the absence of the quencher - d. H. - the quantum yield is the same: ${\displaystyle \mathrm {[Q]} =0}$

${\displaystyle \phi _{0}={\frac {k_{f}}{k_{f}+k_{w}}}}$

The ratio is the same: ${\displaystyle (\phi _{0}/\phi )}$

${\displaystyle {\frac {\phi _{0}}{\phi }}={\frac {k_{f}+k_{w}+k_{q}\cdot \mathrm {[Q]} }{k_{f}+k_{w}}}=1+{\frac {k_{q}\cdot \mathrm {[Q]} }{k_{f}+k_{w}}}}$

The fluorescence lifetime of the excited state in the absence of the quencher is the inverse sum of the two reaction rates , and . First order reaction constants have the unit [1 / s] (read: per second), second order reaction constants have the unit [1 / (mol s)] (read: per second and mol ): ${\displaystyle k_{f}}$ ${\displaystyle k_{w}}$ ${\displaystyle k_{q}\cdot \mathrm {[Q]} }$

${\displaystyle \tau ={\frac {1}{k_{f}+k_{w}+k_{q}\cdot \mathrm {[Q]} }}}$

In the absence of the quencher - d. H. - the fluorescence lifetime is the same: ${\displaystyle \mathrm {[Q]} =0}$

${\displaystyle \tau _{0}={\frac {1}{k_{f}+k_{w}}}}$

If the fluorescence lifetime is used in the ratio, the following equation results after conversion: ${\displaystyle (\phi _{0}/\phi )}$ ${\displaystyle \tau _{0}}$

${\displaystyle {\frac {\phi _{0}}{\phi }}=1+\tau _{0}\,k_{q}\cdot \mathrm {[Q]} }$

If, on the other hand, and in the ratio , we get: ${\displaystyle \tau }$ ${\displaystyle \tau _{0}}$ ${\displaystyle (\phi _{0}/\phi )}$

${\displaystyle {\frac {\phi _{0}}{\phi }}={\frac {\tau _{0}}{\tau }}}$

Because of the direct proportionality of the quantum yield of the fluorescence to the intensity of the fluorescence it follows: ${\displaystyle \phi }$ ${\displaystyle F}$

${\displaystyle {\frac {\phi _{0}}{\phi }}={\frac {F_{0}}{F}}=1+\tau _{0}\,k_{q}\cdot \mathrm {[Q]} }$

and:

${\displaystyle {\frac {F_{0}}{F}}={\frac {\tau _{0}}{\tau }}}$

The Stern-Volmer constant for static fluorescence quenching

Stern-Volmer plot for static fluorescence quenching. The application was carried out as in the picture for dynamic fluorescence quenching. The static fluorescence quenching can be distinguished from the dynamic fluorescence quenching based on the ratio of the fluorescence lifetimes.

In the case of static fluorescence quenching - also known as static quenching or static quenching - a complex is formed from the fluorophore and the quencher that does not itself fluoresce. This can be described using a chemical equation: ${\displaystyle \mathrm {F} }$ ${\displaystyle \mathrm {Q} }$ ${\displaystyle \mathrm {FQ} }$

${\displaystyle \mathrm {F\;+\;Q\ \leftrightharpoons \ FQ} }$

The chemical equilibrium between the fluorophore, the quencher and the complex of fluorophore and quencher is formed according to the law of mass action and is equal to the Stern-Volmer constant : ${\displaystyle K_{s}}$ ${\displaystyle K_{SV}}$

${\displaystyle K_{SV}=K_{s}=\mathrm {\frac {[FQ]}{[F][Q]}} }$

This is the concentration of the complex of fluorophore and quencher, the concentration of the unbound fluorophore and the concentration of the unbound quencher. ${\displaystyle \mathrm {[FQ]} }$ ${\displaystyle \mathrm {[F]} }$ ${\displaystyle \mathrm {[Q]} }$

In the case of static fluorescence quenching, the ratio of and , in contrast to dynamic fluorescence quenching, is equal to one: ${\displaystyle \tau _{0}}$ ${\displaystyle \tau }$

${\displaystyle {\frac {\tau _{0}}{\tau }}=1}$

This is because static fluorescence quenching only reduces the number of stimulable fluorophores, while dynamic fluorescence quenching reduces the lifetime of the excited state. The ratio of the lifetimes therefore remains constant with static fluorescence quenching. This fact is an important differentiator for both types of fluorescence quenching.

For the same concentration of the quencher, with increasing temperature, the value for the decreases in principle with static fluorescence quenching . I.e. The quencher does not quench as much at higher temperatures as at lower temperatures: At higher temperatures, the quencher binds to the fluorophore more poorly than at lower temperatures, which is why the number of quenching processes decreases with increasing temperature. This is an important feature in order to distinguish static fluorescence quenching from dynamic, since dynamic fluorescence quenching works the other way around. ${\displaystyle K_{s}}$

Derivation

The association constant is: ${\displaystyle K_{s}}$

${\displaystyle K_{s}=\mathrm {\frac {[FQ]}{[F][Q]}} }$

The total concentration of the fluorophore is made up of the concentration of the unbound fluorophore and the concentration of the fluorophore bound with the quencher : ${\displaystyle \mathrm {[F]_{0}} }$ ${\displaystyle \mathrm {[F]} }$ ${\displaystyle \mathrm {[FQ]} }$

${\displaystyle \mathrm {[F]_{0}=[F]+[FQ]} \,}$

If this equation is rearranged and then inserted into the equation of the association constant , the result is: ${\displaystyle \mathrm {[FQ]} }$ ${\displaystyle K_{s}}$

${\displaystyle K_{s}=\mathrm {{\frac {[F]_{0}}{[F][Q]}}-{\frac {1}{[Q]}}} }$

This equation is now rearranged according to: ${\displaystyle \mathrm {([F]_{0}/[F])} }$

${\displaystyle \mathrm {\frac {[F]_{0}}{[F]}} =1+K_{s}\cdot \mathrm {[Q]} }$

Because of the direct proportionality of the fluorescence intensity of the fluorophore to its concentration, it follows: ${\displaystyle F}$ ${\displaystyle \mathrm {[F]} }$

${\displaystyle \mathrm {\frac {[F]_{0}}{[F]}} ={\frac {F_{0}}{F}}=1+K_{s}\cdot \mathrm {[Q]} }$

Stern-Volmer equation with simultaneous dynamic and static fluorescence quenching

Stern-Volmer plot with simultaneous dynamic and static fluorescence quenching. The application was carried out as in the picture for dynamic fluorescence quenching.

If dynamic and static fluorescence quenching occur simultaneously, the Stern-Volmer equation in its above form cannot be used. Here the Stern-Volmer equation of combined deletion must be used:

${\displaystyle {\frac {F_{0}}{F}}=(1+K_{d}\cdot \mathrm {[Q]} )(1+K_{s}\cdot \mathrm {[Q]} )}$

It is the Stern-Volmer constant for the dynamic and the Stern-Volmer constant Static quenching. The plot of the Stern-Volmer equation is no longer linear for combined deletion. A non-linear behavior of the Stern-Volmer plot therefore indicates the combined deletion. ${\displaystyle K_{d}}$ ${\displaystyle K_{s}}$

Because of the relationship between dynamic fluorescence quenching and fluorescence lifetime , the Stern-Volmer equation of combined quenching can also be written as: ${\displaystyle \tau }$

${\displaystyle {\frac {F_{0}}{F}}={\frac {\tau _{0}}{\tau }}\cdot (1+K_{s}\cdot \mathrm {[Q]} )}$

The value of the ratio of the lifetimes for the combined quenching must lie between the value for the dynamic and static fluorescence quenching: ${\displaystyle (\tau _{0}/\tau )}$

${\displaystyle {\frac {F_{0}}{F}}>{\frac {\tau _{0}}{\tau }}>1}$

This behavior is therefore also an indication of the combined fluorescence quenching.

If the ratio of the lifetimes is known, the equilibrium constant of the static fluorescence quenching can be determined for the combined quenching using the Stern-Volmer equation. ${\displaystyle (\tau _{0}/\tau )}$ ${\displaystyle K_{s}}$

By rearranging the Stern-Volmer equation for the combined deletion, a linearized form of the equation is obtained:

${\displaystyle \left({\frac {F_{0}}{F}}-1\right)\cdot {\frac {1}{\mathrm {[Q]} }}=\left(K_{d}+K_{s}\right)+K_{d}\cdot K_{s}\cdot \mathrm {[Q]} }$

If the left term of the equation is plotted against the concentration , a simple linear relationship is again obtained in the Stern-Volmer plot. The values ​​for and can then be determined from the rise and the intercept . ${\displaystyle \mathrm {[Q]} }$ ${\displaystyle (K_{d}\cdot K_{s})}$ ${\displaystyle (K_{d}+K_{s})}$ ${\displaystyle K_{d}}$ ${\displaystyle K_{s}}$

Applications of the Stern-Volmer equation

In macromolecules such as B. proteins , fluorophores, such as. B. the amino acid tryptophan can be reached differently for different quenchers. This accessibility of the fluorophore depends, among other things, on the charge and the size of the quencher.

Thus, a tryptophan residue in proteins for the loaded quencher iodide I ^- only be achieved if the tryptophan residue ranges at the surface of the protein in the aqueous medium. Iodide is difficult to penetrate into hydrophobic areas. The acrylamide quencher can only reach a tryptophan residue if it is on the surface and not in a pocket that is too small: Due to its size, acrylamide cannot penetrate into every "corner" of the protein. The quencher O ₂ (bimolecular oxygen), on the other hand, can also delete tryptophans that are hidden deep in the protein because it is small enough and uncharged.

This knowledge can be used to determine the relative location of fluorophores, such as tryptophan, in proteins. For this purpose, the Stern-Volmer plots for different quenchers can be compared or the protein can be examined in the folded and unfolded state so that previously inaccessible fluorophores become accessible to the quencher as the protein unfolds.

See also

• Kinetics (chemistry) , fluorescence

literature

• Joseph R. Lakowicz : Principles of Fluorescence Spectroscopy , 3rd edition, Springer, 2006, ISBN 978-0-387-31278-1 .
• Kirsten Lotte: 3D fluorescence spectroscopy with tryptophan and tryptophan analogues: from solvent influences to protein conformations. Bielefeld University Faculty of Chemistry (Dissertation), 2004, URN (NBN) urn: nbn: de: hbz: 361-5533 .

Web links

• Derivations and further information

credentials

1. ↑ Otto Stern, Max Volmer: About the decay time of fluorescence . Physikalische Zeitschrift, 20 , 183-188, (1919)
2. ^ A. Young Moon, Douglas C. Poland, Harold A. Scheraga: Thermodynamic Data from Fluorescence Spectra. I. The System Phenol-Acetate , The Journal of Physical Chemistry, 69 , 2960-2966, (1965)