Electrodynamic theory of fluorescence polarization of solutions: theory and application to the...

Electrodynamic theory of fluorescence polarization ofsolutions: theory and application to the determination

of protein–protein separation

Edward Collett and Beth Schaefer*Department of Physics, Georgian Court University, 900 Lakewood Avenue,

Lakewood, New Jersey 08701-2697, USA

*Corresponding author: [email protected]

Received 11 September 2008; revised 15 December 2008; accepted 20 January 2009;posted 6 February 2009 (Doc. ID 101422); published 5 March 2009

The phenomenon of the fluorescence polarization of solutions has found numerous applications in bio-physics, biochemistry, immunology, and diagnostic and clinical medicine. The current theory to explainthe phenomenon of fluorescence polarization in solutions was developed by F. Perrin in 1926. Perrinbased his theory on the belief that fluorescence polarization is a manifestation of rotational Brownianmotion. Fluorescence polarization, however, is an electromagnetic radiation phenomenon. Using Max-well’s equations, suitably modified to account for the quantum behavior of fluorescence, E. Collett devel-oped a theory of fluorescence polarization (the electrodynamic theory) based on a model of dipole–dipoleinteractions. The electrodynamic theory is used to investigate protein–protein assays to determine theminimum and maximum binding distances between the proteins for (1) an estrogen receptor DNA boundto a fluorescein labeled estrogen response element and (2) a green fluorescent protein chimera ofS-peptide (S65T-His6) bound to a free S-protein. © 2009 Optical Society of America

OCIS codes: 170.6280, 260.0260, 260.2510, 260.5430, 300.6280.

1. Introduction

The phenomenon of fluorescence (in solutions) wasfirst extensively studied by G. G. Stokes, who wasled to enunciate his empirical law for fluorescence,namely, the wavelength of the fluorescence wasalways greater than the wavelength of the incidentradiation [1]. During the course of his investigationsStokes also reported that the fluorescence was unpo-larized when the solution was irradiated with eitherunpolarized or linearly polarized light. As a resultthe belief was held for many years that fluorescencewas unpolarized. This belief continued untilF. Weigert discovered that in viscous solutions thefluorescence became partially polarized [2]. Shortlyafter Weigert’s discovery, F. Perrin developed atheory to explain fluorescence polarization and pro-posed that the phenomenon was a manifestation of

rotational Brownian motion [3]. Using a model of arotating fluorescent molecule, Perrin derived anequation for the degree of polarization, P, for theemitted radiation (fluorescence),

1P−13¼

�1P0

−13

��1þ

�RTηV

�τ�; ð1Þ

where R is the gas constant, T is the absolute tem-perature, V is the volume of the fluorescence mole-cule, η is the viscosity of the solvent, τ is the meanlifetime of the fluorescence, and P0 is the “limitingdegree of polarization.” Perrin’s theory for fluores-cence polarization remained practically dormant inthe scientific literature until G. Weber began re-search on fluorescence in proteins and used Perrin’stheory and equation (above) to explain his results [4].Since that time, the work of Perrin and Weber hascompletely permeated the fields of biochemistryand biophysics. This has led to the application of

0003-6935/09/081553-12$15.00/0© 2009 Optical Society of America

10 March 2009 / Vol. 48, No. 8 / APPLIED OPTICS 1553

the Perrin–Weber theory to investigations such asmembrane/fluidity/rigidity, microviscosity in liquidsand polymers, receptor–ligand binding, fluorescenceimmunoassays, antigen–antigen binding, DNA hy-bridization and detection, DNA–protein binding,and protein–protein binding. A very readable ac-count and description of Perrin’s theory along withWeber’s work and the numerous applications of fluor-escence polarization can be found in Ref. [5].Fluorescence is an electromagnetic radiation phe-

nomenon and can be described by Maxwell’s equa-tions suitably modified to account for the quantumbehavior. A theory to describe fluorescence polariza-tion based on Maxwell’s equations and a model of di-pole–dipole interactions along with the application ofMaxwell–Boltzmann statistics were developedearlier by E. Collett. This formulation of fluorescencepolarization is called the electrodynamic theory ofthe fluorescence polarization of solutions [6].One of the major applications of fluorescence polar-

ization in biochemistry is to fluorescence polarizationassays of protein–protein interactions and theirbinding affinity. There is, however, nothing in Per-rin’s theory that describes the observed bindingbetween proteins and, in fact, any link between mo-lecular Brownian rotation and binding is completelyabsent. The appearance of fluorescence polarizationand, in particular, binding reactions between fluores-cent biological molecules as well as proteins can bereadily explained and described by the electrody-namic theory. In this paper we show that the electro-dynamic theory, along with chemical equilibriumequations, leads to very simple equations to deter-mine the separation between proteins in solution.The electrodynamic theory also predicts that the de-gree of polarization, expressed as a binding curve,follows a sigmoidal S-curve in complete agreementwith the S-curves that are observed in numerous pro-tein–protein assays (see Fig. 4). We believe that thisis the first time that the appearance of the S-curve influorescence polarization assays has been explained.From this behavior we conclude that fluorescencepolarization can be explained by dipole–dipole(protein–protein) interactions and therefore is amanifestation of van der Waals forces (permanentdipole–permanent dipole forces).The objective of the present paper is twofold. The

first is to describe the electrodynamic theory and torelate it to the chemical equilibrium equations for as-says. The second objective is to apply the resultingequations to determine the binding distance betweenproteins by analyzing the fluorescence polarizationassay curves for (1) the binding of an estrogen recep-tor DNA bound to a fluorescein labeled estrogen re-sponse element and (2) the binding of a GFP chimeraof S-peptide (S65T-His6) bound to a free S-protein.

2. Outline of the Electrodynamic Theory

In this section the electrodynamic theory of the fluor-escence polarization of solutions is briefly outlined. Amore detailed analysis of the electrodynamic theory

is presented in Appendix A of this paper. Maxwell’sequations show that the radiated electric field is de-scribed by the inhomogeneous wave equation [7],

∇2Eðr; tÞ − μεc2

∂2Eðr; tÞ∂t2

¼�4πμc2

�∂jðr; tÞ∂t

; ð2Þ

where ∇2 is the Laplacian operator, Eðr; tÞ is the vec-tor electric field, μ and ε are the permeability andpermittivity of the medium, c is the speed of light,and jðr; tÞ is the current density. The time derivativeof jðr; tÞ, ∂jðr; tÞ=∂t, on the right-hand side of Eq. (2),shows that the source of the electric field (and itspolarized components) arises from the accelerationof electrical charges. The solution of Eq. (2) for theradiation field components Eðr; tÞ is described inAppendix A. Using these results and a model ofdipole–dipole interaction, along with the assumptionof thermal equilibrium so that the molecular orienta-tions are described by a Maxwell–Boltzmann distri-bution, leads to the following Stokes vector for thefluorescence radiation:

S ¼S0

S1

S2

S3

0BB@

1CCA

¼ 215

�ωnm4

c4R2

�ðα3 − α1Þ2I0

1þ 1342

μ2aμ2bk2T2r6ab

− 114

μ2aμ2bk2T2r6

ab

00

0BBBB@

1CCCCA: ð3Þ

In Eq. (3), I0 is the intensity of the incident light (ex-citation) that is linearly horizontally polarized, ωnmis the frequency of the emitted fluorescence (n andmrefer to the energy levels of the fluorescent molecule),c is the speed of light in a vacuum, R is the distancebetween the dipole pair and the observer, α3 and α1are the polarizabilities of the fluorescent molecule, μaand μb are the dipole moments of the fluorescentand host molecule, respectively, k is Boltzmann’sconstant, T is the temperature, and rab is the dipoleseparation. The form of Eq. (3) is especially interest-ing. The first and second Stokes parameters, S0 andS1, are equal to the sum and difference of the hori-zontal and vertical intensities, IH and IV , respec-tively. We see then the electrodynamic theoryshows that if a fluorescent molecule is irradiatedwith incident linearly horizontal (or vertical) polar-ized light, it is necessary to measure only the firsttwo Stokes parameters in a fluorescence polarizationassay.

Equation (3) shows immediately that as the di-pole separation increases to infinity, rab → ∞, theequation reduces to

1554 APPLIED OPTICS / Vol. 48, No. 8 / 10 March 2009

S ¼S0

S1

S2

S3

0BB@

1CCA ¼ 2

15

�ωnm4

c4R2

�ðα3 − α1Þ2I0

1000

0BB@

1CCA: ð4Þ

Equation (4) is the Stokes vector for unpolarizedlight. This result explains Stokes’ observation thatfor a weak concentration of fluorescent quinine sul-phate dissolved in an aqueous solution, the fluores-cence is unpolarized [1]. We note in Eq. (3) thatjS0j > jS1j, so, in general, the fluorescence is partiallypolarized. In Appendix A, the degree of polarization,P, for the fluorescence represented by the Stokes vec-tor, Eq. (3), is shown to be

P ¼��S1

S0

��¼114

μ2aμ2bk2T2r6

ab

1þ 1342

μ2aμ2bk2T2r6

ab

; ð5Þ

where j…j is the absolute magnitude sign.Equation (5) shows that the magnitude of P dependsupon the interacting dipoles, μa and μb, and the di-pole separation rab (which appears as an inversesixth power relation). Thus the electrodynamic the-ory shows that fluorescence polarization can arisefrom molecular dipoles that are bound together. InAppendix A the limit (maximum value) of the degreeof polarization is P0 ¼ Pmax ¼ 3=13 ≈ 0:231; inPerrin’s theory it is 1=3. Equation (5) can be solvedfor rab in terms of P to determine the separationbetween two dipoles,

rab ¼�μ2aμ2bð3 − 13PÞ

42Pk2T2

�16

: ð6Þ

Equation (6) is a major result of the electrodynamictheory and shows that fluorescence polarization canbe used to determine the distance between dipoles(or ligands and receptors (binding sites)), that is, pro-teins. While we use the electrodynamic theory to de-scribe protein–protein separation later in this paper,we first consider an application of Eq. (6), namely, thedetermination of the dipole–dipole separation offluorescent perylene dissolved in ethanol. Thedipole moments for perylene μa and ethanol μb are,respectively [8],

μa ¼ 2:10 × 10−18 esu · cm;

μb ¼ 2:89 × 10−18 esu · cm:ð7Þ

Using values of T ¼ 293K, k ¼ 1:38 × 10−16 ergK−1

and for, say, a degree of polarization ofP ¼ 0:117ð¼ 11:7% or 117mPÞ, the dipole separa-tion is found from Eq. (6) to be

rab ¼�μ2aμ2bð3 − 13PÞ

42Pk2T2

�16 ¼ 4:35 × 10−8 cm ¼ 4:35Å:

ð8Þ

Both perylene and ethanol are very small moleculesand have small dipole moments, so this value for thedipole separation appears to be reasonable. It is alsoof interest to plot Eq. (5) in terms of the degree ofpolarization P for increasing values of the dipoleseparation, rab. This is shown in Fig. 1(a).

For proteins the dipole moments are much largerand, typically, range from 250D (Debye) to 900D. InFig. 1(b) a plot of Eq. (5) is made for two proteins withdipole moments of 350D and 800D with the abscissaplotted in terms of dipole separation (angstroms). Inthemeasurement of protein titrations it is customaryto express the abscissa logarithmically because of thewide range of concentrations. Consequently, Fig. 1(c)is a plot of Eq. (5) with the abscissa plotted logarith-mically. This final plot shows the familiar S-curveobtained in protein titrations (see Fig. 4).

3. Relation of the Degree of Polarization toFluorescence Polarization Assays

One of the major applications of fluorescence polar-ization is to analyze macromolecular assays in whichone of the reactants is labeled with a fluorophore.The reactants form a fluorescent complex from whichthe equilibrium constant Kd can then be determined.The electrodynamic theory describes the interactionbetween a single fluorescent molecule and a hostmolecule. This configuration is, of course, highlyidealized. In practice, for a solution this “single in-teraction” is replaced with an enormous numberof molecules. Thus it is necessary to relate the

Fig. 1. Plots using Eq. (5) of the degree of polarization (in mP)versus the dipole separation (in angstroms). Figure 1(a) is a plotfor fluorescent perylene dissolved in ethanol. The respective dipolemoments are 2:10 × 10�18 esu cm and 2:89 × 10�18 esu cm; the tem-perature T is 293K. Figures 1(b) and 1(c) describe two proteinswith dipole moments of 350D and 800D with the abscissa plottedin terms of decreasing dipole separation (corresponding to increas-ing concentration). Figure 1(c) is plotted logarithmically. In parti-cular, the final plot shows the familiar S-curve obtained in proteintitrations (see Fig. 4).


nonchemical electrodynamic variables (parameters),which are the dipole (protein) moments, μa, μb, andthe separation between the proteins, rab, to the che-mical parameters of the solution, which are concen-trations and equilibrium constants. In assays, theresults are described in terms of the binding con-stant, the concentrations of the reactants, and theequilibrium constant. The relationship between theelectrodynamic parameters and the assay para-meters is developed in the following paragraphs.For a fluorescence polarization assay, the reaction

between free ligands, ½LF �, and free receptors, ½RF�,can be described by the equilibrium equation,

LF þ RF↔LF · RF ¼ B; ð9Þ

where the bound ligand to the free receptor is repre-sented by “B”, and “↔” describes the equilibrium be-havior. In Eq. (9) the dot, “·”, indicates the bindingbetween the free ligand and receptor. The formula-tion for the reaction given by Eq. (9) and the relationof B in terms of a binding constant f B was first con-sidered by A. Clark and is discussed in Appendix B[9]. The law of mass action leads to the followingequation, Eq. (10), that relates the binding ratiof B, defined as the ratio of the bound/total numberof molecules (½B�=½RT �) to the concentration of the freeligands ½LF� and the equilibrium constant Kd (theconcentration is represented by ½…�),

½B�½RT �

¼ f B ¼ ½LF�½Kd þ LF�

; 0 ≤ f B ≤ 1: ð10Þ

The binding parameter f B can also be expressed interms of the degree of polarization, P. In practice,the degree of polarization P for an assay is foundto be between Pmax and Pmin, the maximum andmini-mum degrees of polarization. The appearance of Pmaxand Pmin can be explained as due to the movementand separation of the individual dipoles in solutionthat are restricted by the surrounding dipoles. Thusit is expected that the limiting degree of polarizationcan never be attained.The maximum degree of polarization, Pmax, is as-

sociated with the bound molecules and the minimumdegree of polarization, Pmin, is associated with thefree molecules, so P ¼ Pmaxf B þ Pminf F. The boundmolecules can also be expressed in terms of the freemolecules by the equation f F ¼ 1 − f B. Using thesetwo relations then leads to the following relationfor f B, namely,

f B ¼ P − Pmin

Pmax − Pmin: ð11Þ

Equation (11) is now equated to Eq. (10), so wehave

f B ¼ P − Pmin

Pmax − Pmin¼ ½LF�

½Kd þ LF�: ð12Þ

The range of the degree of polarization, P, isexpressed by

Pmin ≤ P ≤ Pmax: ð13Þ

For an ideal protein–protein reaction, Pmin ¼ 0 andPmax ¼ 3=13, so Eq. (13) becomes

0 ≤ P ≤ 3=13: ð14Þ

Similarly, using Eqs. (11) and (14) leads to the fol-lowing range relation for the binding constant f B:

0 ≤ f B ≤ 1: ð15Þ

Equation (12) can be solved for P and is found to be(the concentration symbol ½…� is suppressed)

P ¼ PminKd þ PmaxLF

Kd þ LF: ð16Þ

Finally, Eq. (16) can be substituted into Eq. (6) andyields

rab ¼�μ2aμ2bð3 − 13XÞ

42k2T2X

�16

; ð17Þ

where X is given by the right-hand side of Eq. (16).Three forms of the binding parameter f B can be de-

fined. The first relates the measured polarizationdata obtained from the assay data and is writtenas fMB:

fMB ¼ P − Pmin

Pmax − Pmin: ð18Þ

The second form of the binding parameter is thetheoretical value written as f TB and is used todescribe an ideal solution so that Pmin ¼ 0 andPmax ¼ 3=13:

f TB ¼ 133

114

μ2aμ2bk2T2r6

ab

1þ 1342

μ2aμ2bk2T2r6

ab

: ð19Þ

The third and final form of the binding value is fora nonideal solution in which Pmin and Pmax are notequal to zero and is simply written as f B:

f B ¼ P − Pmin

Pmax − Pmin¼

0@ 1

14

μ2aμ2b

k2T2r6ab

1þ1342

μ2aμ2b

k2T2r6ab

1A − Pmin

Pmax − Pmin; ð20Þ

where Eq. (5) has been used. Equation (20) can be


solved for rab in terms of the binding value f B to yield

rabðf BÞ ¼� μ2aμ2b42k2T2

�16�3− 13Pmin − 13f BðPmax −PminÞ

Pmin þ f BðPmax −PminÞ�1

6

:

ð21Þ

Equation (21) can be used to determine the mini-mum, maximum, and midrange values of the proteinseparation rabðf BÞ for f B ¼ 1, f B ¼ 1=2, and f B ¼ 0.Using these conditions, Eq. (21) yields

rabðf B ¼ 1Þ ¼ rabðminÞ ¼� μ2aμ2b42k2T2

�16�3 − 13Pmax

Pmax

�16

;

ð22Þ

rab

�f B ¼ 1

2

�¼ rabðmidÞ

¼� μ2aμ2b42k2T2

�16

2643 − 13

�PmaxþPmin

2

�

PmaxþPmin2

375

16

;

ð23Þ

rabðf B ¼ 0Þ ¼ rabðmaxÞ ¼� μ2aμ2b42k2T2

�16�3 − 13Pmin

Pmin

�16

:

ð24Þ

For an ideal solution, Pmax ¼ 3=13 and Pmin ¼ 0, soEq. (21) reduces to

rabðf TBÞ ¼�13μ2aμ2b42k2T2

�16

�1 − f TB

fTB

�16

: ð25Þ

Of particular interest is the determination of theprotein separation where the theoretical bindingparameter has the value f TB ¼ 1=2. For this value,Eq. (25) reduces to

rab

�f TB ¼ 1

2

�¼

�13μ2aμ2b42k2T2

�16

: ð26Þ

Equation (26) can be used to determine a typicalprotein–protein separation. For proteins, the dipolemoments are extremely large, of the order of 300Dto 800D (D ¼ Debye unit ¼ 10−18 esu · cm). Assum-ing, for example, specific dipole moments of μa ¼500D and μb ¼ 350D, and a temperature ofT ¼ 293K, rab for f TB ¼ 1=2 in Eq. (26) is found to be

rab

�f TB ¼ 1

2

�¼

�13μ2aμ2b42k2T2

�16 ¼ 134:0Å: ð27Þ

The minimum separation between two proteinscan also be determined directly from the degree of

polarization P, Eq. (6). For an ideal conditionP ¼ Pmax ¼ 3=13, which according to Eq. (6) leadsto rab ¼ 0, the smallest possible separation. In prac-tice, this cannot be reached so an arbitrary conditionof 0:99Pmaxð¼ 0:228Þ is chosen. For this conditionEq. (6) reduces to

rabð0:99PmaxÞ ¼ 0:382

�μ2aμ2bk2T2

�16

: ð28Þ

Substituting the above values μa, etc., into Eq. (28)yields rab ¼ ð0:99PmaxÞ ¼ 62:3Å.

We now consider the application of the previousequations to the determination of rab for two assays.The source of the data for the first assay, an estrogenreceptor DNA bound to a fluorescein labeled estrogenresponse element, is from the Beacon ApplicationsGuide [10], and for the second assay the source isthe reference for a GFP chimera of S-peptide(S65T-His6) bound to a free S-protein [11].

4. Determination of Protein Separations

A. Binding of Estrogen Receptor DNA Binding Domain toa Fluorescein Labeled Estrogen Response Element

The first application of the electrodynamic theory isto the binding of human recombinant estrogen recep-tor (hER) to a fluorescein-labeled estrogen responseelement. Fluorescence polarization has been used tostudy the equilibrium binding between hER and adoubled stranded, fluorescein-labeled oligonu-cleotide containing an estrogen response element(ERE-F) ([10], pg. 4-3). The fluorescein-labeledDNA was incubated with increasing amounts ofthe binding protein, and the polarization of theDNA-protein complex was measured after each pro-tein addition, allowing for the construction of anequilibrium binding curve. At low protein concentra-tions, most of the labeled DNA remained free andshowed a low degree of polarization. As more DNAbinding domain was added, more of the DNA wasbound in the complex, and the degree of polarizationwas observed to increase.

The reaction can be described by the followingequilibrium equation:

Labeled DNA ðunboundÞþDNA Binding Protein↔Labeled DNA ðboundÞ· DNA Binding Protein

In this equation “↔” represents the forward andreverse equilibrium reaction and “·” indicates thatthe molecules are bound to each other. Specifically,the molecules that were used were ER DNA (recom-binant, from E. Coli) and ERE50ds-F, fluorescein-labeled 50 base-pair oligonucleotide containing anestrogen responsive element made up of two singlestrands, ERE1-F and ERE2-F. In most fluorescencepolarization assays the assay is not described interms of the degree of polarization parameter,


P, but in terms of another parameter, namely, the an-isotropy, A. The two are related by P ¼ ð3AÞ=ð2þ AÞ.In this paper the data are described only in termsof P.The measured data were plotted in terms of the de-

gree of polarizationmillipols (mP) and the DNA bind-ing concentration in nanomolar (nM) and is shown inFig. 2. The binding plot, fMB, using the dataextracted from the curve shown on pg. 4-3 inRef. [10] and Eq. (18) is shown in Fig. 3.The data given in Table 1 using Eq. (17) and the

right-hand side of Eq. (16) lead to

rabðLFÞ ¼

8>>><>>>:μ2aμ2b

�3−13ð1:46×10−9þ0:156LFÞ

23:0×10−9þLF

�

42

�ð1:46×10−9þ0:156LFÞ

23:0×10−9þLF

�k2T2

9>>>=>>>;

16

; ð29Þ

where LF is the concentration of the free ligands. Inorder to evaluate rabðLFÞ, Eq. (29), it is necessary toknow the dipole moments of the labeled DNA and theDNA binding protein, μa and μb. Accepted identifica-tion codes (PDB-ID) for proteins are tabulated in theProtein Data Base (PDB) [12]. The correspondingcalculated dipole moments can then be found onthe Protein Dipole Moments Server [13]. Based onsimilar molecules listed in the PDB we took thevalues of the dipole moments to be 264D and926D, respectively. With these values for μa andμb, Eq. (29) is evaluated at the concentrationsof LF ¼ 0:1 nM; 1:0nM;…; 1000nm, as shown inTable 2.The minimum, midrange, and maximum binding

distances can be determined from Eqs. (22)–(24).We find

rabðf B ¼ 1Þ ¼ rabðminÞ ¼� μ2aμ2b42k2T2

�16�3 − 13Pmax

Pmax

�16

¼ 132:4� 0:4Å;

ð30Þ

rab

�f B ¼ 1

2

�¼ rabðmidÞ

¼� μ2aμ2b42k2T2

�16

�3 − 13

�PmaxþPmin

2

�

PmaxþPmin2

�16

¼ 152:2� 0:5Å; ð31Þ

rabðf B ¼ 0Þ ¼ rabðmaxÞ ¼� μ2aμ2b42k2T2

�16�3 − 13Pmin

Pmin

�16

¼ 176:1� 0:6Å:

ð32Þ

Fig. 2. Plot of the assay data for the binding of estrogen receptorDNA binding domain to ERE50ds-F. The numerical values of thedata points were obtained from the figure shown on pg. 4-3 of theBeacon Applications Guide [10].

Fig. 3. Binding curve derived from Fig. 2 and using Eq. (18) isshown. The limits on f B are seen to be 0.0 and 1.0, as expected.We note that at fMB ¼ 1=2, the data point corresponds to the dis-sociation constant Kd ¼ 23:0 × 10−8 M listed in Table 1.

Table 1. Binding Parameters of Human RecombinantEstrogen Receptor (hER) to a Fluorescein-Labeled

Estrogen Response Elementa

BindingParameter

Value� std:err(mP)

Degree ofPolarization

P (all free) 63:5� 1:1 0:0635� 0:0011P (all bound) 156:4� 1:0 0:1564� 0:0010logKd 1:362� 0:019 -Kd(nM)(range of 1 SE) 23:0ð22:0 − 24:1Þ -aThe minimum and maximum degrees of polarization, Pmin and

Pmax, are shown in the third column. The table is reproduced frompg. 4-3 of the Beacon Applications Guide [10].


Finally, using the “99% of the degree of polariza-tion” condition, Eq. (28), and the above values forthe protein dipole moments, the limiting separationis found to be

rabð0:99PmaxÞ ¼ 0:382

�μ2aμ2bk2T2

�16 ¼ 69:6Å: ð33Þ

B. Binding of a GFP Chimera of S-Peptide (S15-GFP(S65T)-His6) with a Free S-Protein

In this application we consider a very interestingfluorescence polarization assay that has been carriedout using a variant (S65T) of green fluorescent pro-tein (GFP) bound to a free S-protein [11]. Specifically,a GFP chimera of S-peptide (S15-GFP (S65T)-His6)interacted with a free S-protein first in an aqueoussolution and then in a saline solution of 0:01MNaCl.By carrying out an assay of the GFP chimera withthe free protein, the equilibrium curves were deter-mined and found to be sigmoidal. The present appli-cation differs from the previous application becausethe data for the measured degree of polarization arenot given. On the other hand, the binding plot of thebound fraction f B is given. This still allows the pro-tein separations to be determined. The binding plotgiven in Ref. [11] is shown in Fig. 4.The protein–protein separation rab for an ideal so-

lution (Pmax ¼ 3=13 and Pmin ¼ 0 can be expressed interms of the concentration LF and the dissociationKdby substituting Eq. (5) into Eq. (12), which leads to

rabðLF;KdÞ ¼�13μ2aμ2bKd

42LFk2T2

�16

: ð34Þ

We note that at f TB ¼ 1=2 [see Eq. (25)] we have thecondition LF ¼ Kd, and Eq. (34) reduces to Eq. (26).In order to determine rab, the dipole moments of

the GFP chimera and the S-protein (peptide) mustbe known. The dipole moments of the GFP chimeraand the S-peptide are chosen from similar moleculesin the PDB to be 375D and 256D, respectively. Then,substituting the dipole values into Eq. (34) yields(LF ¼ Kd)

rabðLF ;KdÞ ¼�13μ2aμ2bKd

42LFk2T2

�16 ¼ 109:7Å: ð35Þ

With the dipole values given above and consider-ing, for example, a concentration of, say, LF ¼ 1:00 ×10−7 M along with the values for Kd for an aqueousand saline solution, the protein separation rab inthe respective solutions are found from Eq. (34) to be

KdðaqueousÞ ¼ 1:1 × 10−8 M → rab ¼ 75:9Å; ð36Þ

KdðsalineÞ ¼ 4:2 × 10−8 M → rab ¼ 95:0Å: ð37ÞThe addition of the salt to the aqueous solution in-

creases the separation between the GFP chimera andthe S-peptide. These values can be shown to corre-spond to the binding values f B ¼ 0:901 for theaqueous solution and f B ¼ 0:704 for the saline solu-tion. The results shown in Eqs. (36) and (37) supportthe conclusion expressed by the authors in the citedreference that the addition of NaCl reduced thebinding affinity, that is, increased the protein–protein separation [11]. In Table 2 the dipole separa-tions rab are calculated using Eq. (34) for differentconcentrations and dissociation constants(KdðaqueousÞ ¼ 1:1 × 10−8 M and KdðsalineÞ ¼4:2 × 10−8 M).

5. Conclusion

In this paper the electrodynamic theory of the fluor-escence polarization of solutions has been describedand applied to the determination of distances be-tween proteins. The theory shows that the appear-ance of fluorescence polarization can be explainedas a manifestation of dipole–dipole binding. Incontrast, the current explanation for the appearanceof fluorescence polarization is based on the mechan-ism of rotational Brownian motion of a fluorophore.

Table 2. Values of the Protein Separation forDifferent Concentrations Using Eq. (29)a

LF , nM rab, Angstroms

0.1 175.81.0 173.710.0 160.7100.0 139.91000.0 133.5

aThe observation is made that as the concentration LF increases,the protein separation decreases, as expected. The additional ob-servation is made, however, that the separation between the pro-teins does not vary greatly between the maximum and minimumconcentrations of LF.

Fig. 4. Plots of f B for the GFP chimera; f B corresponds to fMB.The S-curve described by the open circles (∘) corresponds tothe aqueous solution and has a dissociation constant of1:1 × 10�8 M. The S-curve described by the solid circles (•) corre-sponds to the saline solution and has a dissociation constant of4:2 × 10−8 M (courtesy of Protein Science).


The electrodynamic theory shows that as the separa-tion between the fluorescent molecule and the hostmolecule goes from infinity to zero the degree of po-larization, P, goes respectively, from 0 to a maximum(limiting) value of 3=13 (in Perrin’s theory the valueis 1=3). The primary result of the electromagnetictheory is the equation for the dipole–pole separationthat is expressed in terms of the dipole moments, thedegree of polarization, and the ambient tempera-ture, Eq. (6).A major application of fluorescence is to analyze

macromolecular assays in which one of the reac-tants is labeled with a fluorophore. In this paperthe equations relating the electrodynamic para-meters to the assay (chemical) parameters are de-veloped and lead to equations to determine theminimum, mid-point and maximum separations be-tween proteins (dipoles) and binding sites. Further-more, when the degree of polarization versus thedipole separation is plotted the electrodynamictheory yields sigmoidal S-curves that also appearin assay titrations. Using data extracted from theS-curves for different assays and applying the equa-tions developed in this paper the protein separa-tions for these two protein interactions weredetermined. In particular, in the assay for greenfluorescent protein (GFP) the calculated differencein separations for aqueous and saline solutions wereshown to support the reported experimental obser-vations and conclusions.We also note the electrodynamic theory is based on

the same dipole model used in fluorescence reso-nance exchange theory (FRET) [14]. Thus both the-ories use the same mechanism of dipole–dipoleinteraction. However, in terms of the measurementof dipole (protein) separation, the application ofFRET appears to be limited to separations fromaround 10Å to 100Å, whereas in the electrodynamictheory there appears to be no limitation on the rangeof measured protein–protein separations.In the references a number of relatively recent

publications on fluorescence polarization and FRETare listed [15–20]. In particular, a very useful surveypaper on fluorescence polarization beginning withthe work of F. Perrin and G. Weber and its develop-ment and application to proteins is described inRef. [21].As we have shown in this paper, the protein se-

paration can be obtained directly from the degreeof polarization of fluorescence. It is our hope thatthe electrodynamic theory will encourage its applica-tion to the determination of protein–protein separa-tions with existing and future experimental data,and this will lead to increased understanding ofthe interactions between proteins as well as the in-teraction of drugs with proteins.

Appendix A. Outline of the Electromagnetic Theory ofthe Fluorescence Polarization of Solutions

In this appendix the electrodynamic theory is out-lined [6]. Maxwell’s equations can be shown to lead

to the following inhomogeneous wave equation forthe radiated electromagnetic field [7],

∇2Eðr; tÞ − μεc2

∂2Eðr; tÞ∂t2

¼�4πμc2

�∂jðr; tÞ∂t

; ð38Þ

where∇2 is the Laplacian operator, Eðr; tÞ is the elec-tric field vector, μ and ε are the permeability and per-mittivity of the medium, c is the speed of light, andjðr; tÞ is the current density. The time derivative ofjðr; tÞ on the right-hand side of Eq. (38) clearly showsthat the source of the electric field (and its polarizedcomponents) arises only from the acceleration ofcharges. The nonrelativistic solution of Eq. (38) isfound to be

E ¼ ec

�n × ðn × _βÞ

R

�; ð39Þ

where n ¼ R=R is a unit vector directed from the po-sition of the charge to the observation pointand _β ¼ _v=c.

Fluorescence is observed by allowing an incidenttransverse plane wave to interact with a fluorescentmolecule whereupon the molecule emits a sphericalwave. In a spherical coordinate system the compo-nents of Eq. (39) are found to be

Eθ ¼�

e

c2R

�½€xðtÞ cos θ − €zðtÞ sin θ�;

Eϕ ¼�

e

c2R

�½€yðtÞ�;

ð40Þ

whereEθ and Eϕ are the transverse field coordinates,€xðtÞ, €yðtÞ, and €zðtÞ are the acceleration components ofthe electric charge in Cartesian coordinates, and θ isthe angle of observation measured from the z axis.The angle ϕ is normally present in the radiationequations for Eθ and Eϕ, but here symmetry is as-sumed around the z axis, so ϕ is chosen to be equalto zero, and therefore the radiation equations reduceto the ones shown in Eq. (40). The acceleratingcharges can be written in terms of dipole momentsp ¼ er, where r ¼ xiþ yjþ k. Furthermore, the as-sumption is made that the dipoles can be representedas linear oscillators. In order to describe thequantum behavior of fluorescence it is necessary totransform the classical linear oscillators to theirquantum-mechanical analogs, namely,

xðtÞ → xnm ¼ xnmð0Þeiωnmt;

yðtÞ → ynm ¼ ynmð0Þeiωnmt;

zðtÞ → znm ¼ znmð0Þeiωnmt;

ð41Þ


where ωnm ¼ ðEn − EmÞ=ħ (Stokes’ law of fluores-cence). Substituting Eq. (41) into Eq. (40) then leadsto

Eθ ¼ −

�ωnm2

c2R

�ðpxnm cos θ − pznm sin θÞ;

Eϕ ¼ −

�ωnm2

c2R

�ðpynmÞ;

ð42Þ

where pinmði ¼ x; y; zÞ are the dipole moment compo-nents. Fluorescence is almost always observedperpendicular to the direction of the incident (excita-tion) beam so the observation angle θ is set toθ ¼ 90°. Furthermore, since we are only concernedwith the polarization and not with dipole transitions,we write px, etc., rather than pinm , etc. Thus Eq. (42)reduces to

Eθ ¼�ωnm

2

c2R

�ðpzÞ; Eϕ ¼ −

�ωnm2

c2R

�ðpyÞ: ð43Þ

The incident electric field E induces the dipole pz;yto oscillate, so we have

p0 ¼ ��α · E0; ð44Þ

where ��α is the diagonal molecular polarizibility ten-sor; primes are used to show that Eq. (44) refers tothe molecule’s coordinate system. Equation (44)can be written out as

px0py0px0

0@

1A ¼

α1 0 00 α2 00 0 α3

0@

1A Ez0

Ey0EZ0

0@

1A: ð45Þ

The molecule’s coordinate system, x0, y0, z0, mustnow be related to the observer’s coordinate system,x, y, z. This can be done using the formalism of anEuler angle rotation matrix so we have

px

py

px

0@

1A ¼ A−1

α1 0 00 α2 00 0 α3

0@

1AA

Ez

Ey

EZ

0@

1A; ð46Þ

where (we note that here θ refers to the angle be-tween the coordinate systems),

A ¼� cosψ cosϕ − cos θ sinϕ sinψ cosψ sinϕþ cos θ cosϕ sinψ sinψ sin θ− sinψ cosϕ − cos θ sinϕ cosψ − sinψ sinϕþ cos θ cosϕ cosψ cosψ sin θ

sin θ sinϕ − sin θ cosϕ cos θ

�: ð47Þ

The incident light is linearly horizontally polarizedlight and is represented by Exðz; tÞ, Eyðz; tÞ. Further-more, in order to simplify the calculations theassumption is made that α2 ¼ α1. SubstitutingEq. (47) into Eq. (46) leads Eq. (43) to become

Eθ ¼�ωnm

2

c2R

�ðα3 − α1Þðsin2 θ sinϕ cosϕÞEx;

Eϕ ¼ −

�ωnm2

c2R

�ðα3 − α1Þðsin θ cos θ sinϕÞEx:

ð48Þ

The intensity and the polarization of the fluores-cence can be described in terms of the Stokes polar-ization parameters. The four Stokes parameters forthe incident (exciting) plane wave are defined inCartesian coordinates to be [7],

S0 ¼ ExEx� þ EyEy

�; S1 ¼ ExEx� − EyEy

�;

S2 ¼ ExEy� þ EyEx

�; S3 ¼ iðExEy� − EyEx

�Þ;ð49Þ

where “�” indicates the complex conjugate andi ¼ ffiffiffiffiffiffi

−1p

. Similarly, the Stokes parameters for thefluorescent beam in a spherical coordinate systemare

S0 ¼ EθEθ� þ EϕEϕ

�; S1 ¼ EθEθ� − EϕEϕ

�;

S2 ¼ EθEϕ� þ EϕEθ

�; S3 ¼ iðEθEϕ� − EϕEθ

�Þ;ð50Þ

The following terms must now be calculated inorder to determine the Stokes parameters for thefluorescence:

EθEθ� ¼

�ωnm4

c4R2

�ðα3 − α1Þ2hsin4 θ sin2 ϕ cos2ϕiExEx

�;

EϕEϕ� ¼

�ωnm4

c4R2

�ðα3 − α1Þ2hsin2 θ cos2 θ sin2ϕiExEx

�;

EθEϕ� ¼

�ωnm4

c4R2

�

× ðα3 − α1Þ2hsin3 θ cos θ sin2ϕ cosϕiExEx�;

EϕEθ� ¼

�ωnm4

c4R2

�ðα3 − α1Þ2

× hsin3 θ cos θ sin2 ϕ cosϕiExEx�: ð51Þ


In Eq. (51) the orientation angles are placed withinangle brackets, h…i to indicate that an ensembleaverage of the molecular orientations must be taken.In order to calculate the ensemble averages, two as-sumptions are now made. The first is that the mole-cules are in thermal equilibrium, so the molecularorientations follow a Maxwell–Boltzmann distribu-tion. Thus the ensemble averages in Eq. (51) mustbe evaluated according to

h…i ¼Ra

Rbh…i exp

�−

Uða;bÞkT

�dτadτb

Ra

Rb exp

�−

Uða;bÞkT

�dτadτb

: ð52Þ

In Eq. (52) the integrations are carried out for theensemble averages of the molecules “a” and “b”,Uða; bÞ is the potential energy between the moleculesa and b, k is Boltzmann’s constant, and T is the ab-solute temperature. The second assumption is thatthe fluorescent molecules and the solvent moleculesinteract via a dipole–dipole interaction. The interac-tion potential is expressed as

Uðθa;ϕa;∶θb;ϕbÞ ¼μaμbrab3

½2 cos θa cos θb

− sin θa sin θb cosðϕa − ϕbÞ�;ð53Þ

where μa and μb are the permanent dipole momentsof the molecules a and b, respectively, and rab is thedistance (separation) between molecule a and mole-cule b. This interaction is shown in Fig. 5.The substitution of Eq. (53) into Eq. (52) does not

lead to integrals that can be evaluated in a closedform. However, to a very good approximation, the ex-ponential in Eq. (52) can be expanded to second ordersince Uða; bÞ=kT ≈ 1 and is written as

exp�−Uða; bÞkT

�≈ 1 −

Uða; bÞkT

þ 12Uða; bÞ2k2T2 : ð54Þ

Replacing the exponential in Eq. (52) with Eq. (54)and carrying out the integrations, the Stokes param-eters for the fluorescence are found from Eqs. (50)and (51) to be

S0 ¼�ωnm

4

c4R2

�ðα3 − α1Þ2I0

�215

þ 13315

μ2aμ2bk2T2r6ab

�;

S1 ¼�ωnm

4

c4R2

�ðα3 − α1Þ2I0

�−1105

μ2aμ2bk2T2r6ab

�; S2 ¼ 0;

S3 ¼ 0;

ð55Þ

where I0 ¼ ExEx� is the intensity of the incident ra-

diation. In Eq. (55), S0 is the total intensity of theemitted fluorescence and S1, S2, and S3 describethe polarization state of the fluorescence. The Stokesparameters in Eq. (55) can be written as a columnmatrix (Stokes vector),

S ¼S0

S1

S2

S3

0BB@

1CCA

¼ 215

�ωnm4

c4R2

�ðα3 − α1Þ2I0

1þ 1342

μ2aμ2bk2T2r6

ab

− 114

μ2aμ2bk2T2r6

ab

00

0BBBB@

1CCCCA: ð56Þ

Written in this form, the Stokes vector includesStokes’ law of fluorescence (the spectral shift ωnm),the fluorescence intensity as expressed by S0, andthe polarization behavior of the fluorescence ex-pressed by S1, S2, and S3.

The general expression for the degree of polariza-tion, P, can be written in terms of the four Stokesparameters and is given by

P ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS21 þ S2

2 þ S23

qS0

; 0 ≤ P ≤ 1: ð57Þ

Unpolarized light corresponds to P ¼ 0, completelypolarized light to P ¼ 1, and partially polarized lightto 0 < P < 1. The Stokes parameters also satisfy thefollowing relation:

S20 ≥ S2

1 þ S22 þ S2

3: ð58Þ

In Eq. (58) the equality sign “¼” describes comple-tely polarized light, and the greater than sign “>” de-scribes either unpolarized or partially polarizedlight. The degree of polarization for the fluorescence

Fig. 5. Dipole–dipole interaction between a fluorescent molecule,a, and a host molecule, b, separated by a distance rab showing theorientation polar angles θa and θb of the respective molecules. Theazimuthal angles ϕa and ϕb are not shown.


is seen from Eqs. (56) and (57) to be (S2 ¼ S3 ¼ 0),

P ¼��S1

S0

��¼114

μ2aμ2bk2T2r6ab

1þ 1342

μ2aμ2bk2T2r6

ab

: ð59Þ

Equations (56) and (59) are the starting points inthis paper, Eqs. (3) and (5).

Appendix B. Clark’s Theory of Ligand–ReceptorInteraction

In this paper we consider only dipole–dipole interac-tion between two different molecular species, andfluorescence polarization measurements, in practice,are based on a theory of ligand–receptor interactiondue to Clark [9]. The free ligands, LF, react with freereceptors, RF, to form bound molecules LF · RF ¼ B(the dot,“·” indicates that LF is bound to RF). The re-action between LF and RF is then expressed by theequation

LF þ RF↔LF · RF ¼ B: ð60ÞThe ratio of the bound/free molecules is given by

the law of mass action, so Eq. (60) is then expressedas an equilibrium equation in terms of concentra-tions as

½LF�½RF �½B� ¼ Kd; ð61Þ

whereKd is the dissociation constant. The total num-ber of free receptors RF can be expressed in terms ofthe total number of receptors, RT , and the bound re-ceptors, B, by Eq. (62),

RF ¼ RT − B: ð62ÞSubstituting Eq. (62) into Eq. (61) then yields

½LF�½RT − B�½B� ¼ Kd: ð63Þ

Equation (63) is now solved for the concentration ofthe bound molecules, B,

½B� ¼ ½RT �½LF�½Kd þ LF�

; ð64Þ

which can be rewritten as

½B�½RT �

¼ f B ¼ ½LF �½Kd þ LF�

: ð65Þ

In Eq. (65), f B is the fractional bound ratio shownin Eq. (12). The equation shows that as the concen-

tration of free ligands LF increases, the concentrationof the bound molecules also increases. In otherwords, as ½LF� increases the ratio ½B�=½RT � → 1, thereis complete binding of all the LF molecules to the freereceptor molecules RF. From Eq. (65), when½B�=½RT � ¼ 1=2 (corresponding to 50% of the receptorsites being occupied) the equilibrium (dissociation)constant is equal to the concentration of free ligands,that is, Kd ¼ LF. Equation (65) is plotted, in Fig. 6,for ½LF� from 0 to 10 and for Kd ¼ 1. In practice,the ligand concentration ranges over a much greaterrange than shown in Fig. 6. Most binding experi-ments range over four orders of concentrations,usually two orders above and two orders below Kd.Because of this relatively wide range in concentra-tion, it is useful to plot the abscissa shown in Fig. 6

Fig. 6. Plot of the binding curve given by Eq. (65). When f B ¼ 1=2,we have LF ¼ 1 and Kd ¼ 1.

Fig. 7. Change from a linear to a logarithmic abscissa not onlyallows the plotting over a large range of the free ligand concentra-tion but also gives rise to a sigmoidal or S-curve. The logarithmicplot is called a Klotz plot. Klotz plots make the determination ofthe dissociation constant Kd far easier to read than that obtainedwith the linear abscissa values shown in Fig. 6.


logarithmically. In Fig. 7, Eq. (65) is plotted with con-centrations from 10�2 to 102, and the dissociationconstant is taken to be Kd ¼ 1. Finally, in Fig. 7the range of the curve is seen from 10�2 to 102 to cor-respond to the following values for f BðLFÞ,

f Bð10−2Þ ¼ 0:99%; f Bð102Þ ¼ 99:1%: ð66ÞEquation (66) shows that between these two values

of concentration, the S-curve covers 98% of the data.

References1. G. G. Stokes, “On the change of refrangibility of light,” Proc.

Royal Soc. London vi, 195–208 (1852).2. F. Weigert, “Über polarisiertes fluoreszenzlicht,” Verh. Dtsch.

Phys. Ges. 23, 100–102 (1920).3. F. Perrin, “Polarisation de la lumière de fluorescence. Vie moy-

enne desmolécules dans l’etat excité,” J. Phys. Radium 7, 390–401 (1926).

4. G. Weber, “Polarization of the fluorescence of macromolecules:1. Theory and experimental method,”Biochemistry J. 51, 145–155 (1952).

5. J. R. Lackowicz, Principles of Fluorescence Spectroscopy (Ple-num, 1983).

6. E. Collett, “An electrodynamic theory for the emission offluorescence polarization by solutions,” Opt. Commun. 64,516–522 (1987).

7. E. Collett, Polarized Light: Fundamentals and Applications(Marcel Dekker, 1993).

8. A. L. McClellan, Table of Experimental Dipole Moments (W. H.Freeman, 1963).

9. A. J. Clark, “The Mode of Action of Drugs on Cells” (EdwardArnold, 1933).

10. “Beacon Applications Guide,” PanVera Corporation, Madison,Wisconsin, USA.

11. S.-H. Park and R. T. Raines, “Green fluorescent protein as asignal for protein–protein interactions,” Protein Sci. 6,2344–2349 (1997).

12. “RCSB Protein Data Bank,” www.rcsb.org.13. “Server and database for dipole moments of proteins,” http://

bip.weizmann.ac.il/dipol.14. T. Förster, “Intermolecular energy migration and fluores-

cence,” Ann. Phys. 2, 55–75 (1948).15. J. Lippincott-Schwartz and G. H. Patterson, “Development

and use of fluorescent protein markers in living cells,” Science300, 87–91 (2003).

16. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser,S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescentproteins at nanometer resolution,” Science 313, 1642–1645 (2006).

17. H. Lorenz, D. W. Hally, and J. Lippincott-Schwartz, “Fluores-cence protease protection of GFP chimeras to reveal proteintopology and subcellular localization,” Nature Meth. 3, 205–210 (2006).

18. G. Chirico, M. Collini, K. Tóth, N. Brun, and J. Langowski,“Rotational dynamics of curved DNA fragments studied byfluorescence polarization anisotropy,” Eur. Biophys. J. 29,597–606 (2001).

19. S. Zorilla, G. Rivas, A. U. Acuna, and M. P. Lillo, “Proteinself-association in crowded protein solutions: a time-resolvedfluorescence polarization study,” Protein Sci. 13, 2960–2969 (2004).

20. D. W. Piston and M. A. Rizzo, “FRET by fluorescencepolarization microscopy,” Methods Cell Biol. 85, 415–430 (2008).

21. T. L. Mann and U. J. Krull, “Fluorescence polarizationspectroscopy in protein analysis,” Analyst (Amsterdam)128, 313–317 (2003).


Date post:	02-Oct-2016
Category:	Documents
Upload:	beth
View:	217 times
Download:	1 times

Electrodynamic theory of fluorescence polarization of solutions: theory and application to the...

Documents