FRET - Förster Resonance Energy Transfer || Optimizing the Orientation Factor Kappa-Squared for...

4Optimizing the Orientation Factor Kappa-Squared for MoreAccurate FRET MeasurementsB. Wieb van der Meer, Daniel M. van der Meer, and Steven S. Vogel

4.1Two-Thirds or Not Two-Thirds?

Two-thirds or not two-thirds? This is the question. Kappa-squared can vary between0 and 4, but when the orientations of donor and acceptor dipoles randomize withinthe lifetime of the excited state, its value is 2/3. Many authors of FRETpapers adoptthis assumption. However, there is strong evidence that kappa-squared is definitelynot equal to 2/3 in many cases. For example, in Ref. [1], a series of DNA conjugatesin which a donor (stilbene dicarboxamide) and an acceptor (perylene dicarboxamide)are covalently attached to opposite sites of an A:T base pair duplex domain consistingof 4–12 base pairs yield a FRET efficiency that is strongly nonlinear with varyingdistance. For 7–9 base pairs the efficiency drops to almost zero consistent with anear-zero value of kappa-squared; whereas for 5 and 10 base pairs the efficiencyreaches a maximum consistent with a kappa-squared value of 1 [1]. In anotherexample [2], a Cy3 donor and a Cy5 acceptor are attached to the 50-termini of duplexDNA via a 3-carbon linker to the 50-phosphate so that they are predominantly stackedonto the ends of the helix in the manner of an additional base pair [2]. A cartoonillustrating the first two examples is shown in Figure 4.1a; the third example isshown in Figure 4.1b. The transition dipoles are essentially perpendicular to thehelical axis, and the periodicity is on the order of 5 base pairs. As a result, kappa-squared changes dramatically with the donor–acceptor distance, approaching zero at13 and 18 base pairs. The graph of FRET efficiency versus donor–acceptor distancelooks like the graph of the height of a bouncing ball versus time (dashed curve inFigure 4.18). In reality, there is somemotional averaging so that for none of the basepair choices does the efficiency dip to zero, but there are clear maxima and minimain the efficiency versus distance curve at predictable donor–acceptor distances (wewill come back to this trend with Figure 4.18 in Section 4.9). The error in thedistance by assuming kappa-squared¼ 2/3 is about 25% at 13 base pairs [2]. In athird example [3], a Cy3 donor and a Cy5 acceptor are rigidly attached to DNA in sucha way that the dipoles are essentially parallel to the axis of the DNAmolecule. In twocases, a configuration of collinear donor and acceptor dipoles was engineered: onewith Cy3 and Cy5 on the same B-DNA strand and separated by three helical turns

FRET – Förster Resonance Energy Transfer: From Theory to Applications, First Edition.Edited by Igor Medintz and Niko Hildebrandt.� 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

j63

(sample 1 in Ref. [3]) and the other with Cy3 and Cy5 on opposite strands andseparated by 2.5 turns (sample 2 in Ref. [3]). In both cases, the two oscillatingdipoles are expected to be collinear for which configuration kappa-squaredreaches its maximum value of 4. The experimentally obtained kappa-squaredvalues (from the known distance, the measured efficiency, and the known F€orsterdistance R0 ¼ 6:34ðk2Þ1=6 nm) was 3.2 for sample 1 and 3.5 for sample 2,indicating the near-parallel alignment of the dipoles with the line connectingdonor and acceptor [3]. These three examples of having a kappa-squared differentfrom 2/3 are for traditional donors and acceptors attached to DNA. Furthermore,the 2/3 assumption also fails in FRET experiments using fluorescent proteins asdonors and acceptors, which undergo, usually restricted, rotation independent ofeach other that is slow relative to the lifetime of the excited state in the presenceof acceptor.

Figure 4.1 (a) A cartoon for the FRET situationin Refs [1,2]. The double-headed arrowsrepresent transition moments for donor oracceptor. The angle between them depends onthe donor–acceptor distance relative to thehelical pitch, reaching near-zero (k2 � 1) whenthe distance equals an even integer times a

quarter pitch and 90� when the distance equalsan odd integer times a quarter pitch (k2 � 0).(b) A cartoon for the FRET situation in Ref. [3](samples 1 and 2) where the transitionmoments are essentially aligned with thedonor–acceptor separation vectorcorresponding to k2 � 4.

64j 4 Optimizing the Orientation Factor Kappa-Squared for More Accurate FRET Measurements

4.2Relevant Questions

Kappa-squared needs attention, but this orientation factor problem does not need tobe disorienting! A few relevant questions must be asked. Do the orientations ofdonor and acceptor change during the time within which transfer may take place,that is, effectively during the lifetime of the excited state in the presence of theacceptor(s)? And, if they do, how? Do they change rapidly during this time? Does thedynamic averaging regime apply or the static averaging regime or neither? Is thefluorescence polarized? To what extent? Can kappa-squared be measured? Candepolarization factors be measured? Are simulations available or is structuralinformation obtainable that may exclude certain orientations? First, we need toknow how to visualize kappa-squared.

4.3How to Visualize Kappa-Squared?

Kappa-squared for a given donor–acceptor pair depends on the direction of theemission transition moment of the donor, the absorption transition moment of theacceptor, and the line connecting the centers of the donor and the acceptor. We canintroduce unit vectors: d along the emission transitionmoment of the donor, a alongthe absorption transition moment of the acceptor, and r pointing from the center ofthe donor to the center of the acceptor. These three unit vectors are shown inFigure 4.2.To visualize better the implications for kappa-squared of this three-dimensional

geometry, the following exercise is recommended. Hold your two index fingers infront of your face and simulate the donor dipole with your left index finger and that

Figure 4.2 The unit vectors d, a, and r; d isalong the emission transition moment of thedonor, a is along the absorption transitionmoment of the acceptor, and r points from the

center of the donor to the center of theacceptor. The d and a vectors are displayed atarbitrary orientations within the spheresrepresented at their locations.

4.3 How to Visualize Kappa-Squared? j65

of the acceptor with the other index finger. You can then rotate one hand in threedimensions around its wrist and estimate (see below) how kappa-squared changesfor the given orientation of the other finger and for any of the orientations of thefinger of the one hand, and then rotate the other hand while maintaining theabsolute orientations of the fingers constant and evaluate the orientation factoragain. Rotating the hands around each other demonstrates that significant changesin kappa-squared also result for transfer in different directions, that is, for differentorientations of the separation vector r. The angle between d and a is qT, that betweend and r is qD, and that between a and r is qA. There are three common ways ofexpressing kappa-squared (k2) in angles:

k2 ¼ ðcos qT � 3 cos qD cos qAÞ2; ð4:1Þk2 ¼ ðsin qD sin qA cos w� 2 cos qD cos qAÞ2; ð4:2Þk2 ¼ ð1þ cos2qDÞcos2v; ð4:3Þ

where w is the angle between the projections of d and a on a plane perpendicular to randv is the angle between the electric dipole field due to the donor at the location ofthe acceptor and a. The electric field is along 3rcos qD � d, and the unit vector alongthis direction is eD ¼ ð3r cos qD � dÞ= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 3 cos2qDp

. The angles appearing inEquations 4.1–4.3 are illustrated in Figure 4.3. The dependence of kappa-squaredon cos qD and cos v expressed in Equation 4.3 is illustrated in Figure 4.4.Figure 4.3 illustrates in particular the different planes formed by the vectors: the

DR plane through d and r, the AR plane through a and r, the DAplane through d anda, and the EDA plane through eD and a. Note that eD lies always in the DR plane andthat, whenever a is perpendicular to this plane, a is also perpendicular to d, r, and eD,so that kappa-squared is zero. Equation 4.3 gives insight into the distribution ofkappa-squared values. The highest value possible, 4, can only be realized if r and d

Figure 4.3 In this illustration of the angles in Equations 4.1–4.3, uT ¼ 97:18�, uD ¼ uA ¼ 60�,w ¼ 120�, and v ¼ 48:59�, yielding k2 ¼ 0:7625. The unit vectors d, r, and eD are in the DRplane, a and r in the AR plane, and, eD and a are in the EDA plane.


are parallel or antiparallel yielding cos2qD ¼ 1 and if, at the same time, a and eD areparallel or antiparallel, ensuring that cos2v is also equal to 1. On the other hand,whenever a and eD are perpendicular to each other, kappa-squared equals zero.Therefore, if we consider all possible orientations, there is a very high probabilitythat kappa-squared is low and amuch smaller probability that it is high. Accordingly,the isotropic average is 2/3: if all orientations are equally probable in threedimensions, the average of cos2v is 1/3 and that of cos2qD is also equal to 1/3,so Equation 4.3 predicts the isotropically averaged value of kappa-squared as 2/3.Expressions of kappa-squared in terms of unit vectors and dot products are alsorelevant:

k2 ¼ ða � d � 3ða � rÞðr � dÞÞ2 ¼ ða � eDÞ2ð1þ 3ðr � dÞ2Þ2: ð4:4ÞThe right-hand side is the vector form of Equation 4.3 and the expression in themiddle is the vector form of Equation 4.1.From these forms, it is clear that kappa-squared does not change if we

1) flip the donor transition moment, d ! �d,2) flip the acceptor transition moment, a ! �a,3) allow the donor and acceptor to trade places, r ! �r, and4) interchange the donor and acceptor transition moments, a $ d.

The transition moments can be visualized as rod-like molecular antennas or evenindex fingers. It is instructive to choose different orientations and evaluate thecorresponding kappa-squared values as in the examples shown in Figure 4.5.

Figure 4.4 Kappa-squared versus the absolute values of cos uD and cos v.

4.3 How to Visualize Kappa-Squared? j67

4.4Kappa-Squared Can Be Measured in At Least One Case

Dale has shown that for the special case depicted in Figure 4.6, kappa-squared can bemeasured using time-resolved fluorescence depolarization [4].The donor and acceptor are assumed not to move with respect to the macro-

molecule, but the whole system can rotate around its axes, exhibiting rotationaldiffusion around the symmetry axis at D== and around any axis perpendicular tothat at D?, where D== and D? are rotational diffusion constant. If both donor andacceptor fluoresce, three different time-resolved anisotropies can be measured:rD,rA, and rT. rD is the donor fluorescence anisotropy (donor is excited and donorfluorescence is measured), rA is the acceptor fluorescence anisotropy (acceptor isexcited and acceptor fluorescence is measured), and rT is the transfer anisotropy(donor is excited and sensitized acceptor fluorescence is observed). These aniso-tropies vary with t, the time after an ultrashort flash, and are given by thefollowing:

rD ¼ rDðtÞ ¼ b1De�ð2D?þ4D==Þt þ b2De

�ð5D?þD==Þt þ b3De�6D?t: ð4:5aÞ

b1D ¼ 310

sin4 qDb2D ¼ 310

sin22 qD; b3D ¼ 410

32cos2 qD � 1

2

� �2

: ð4:5bÞ

Figure 4.5 Examples of donor and acceptororientations with corresponding k2 values. Thedonor dipole is along the bar in the center ofeach circle with various examples of acceptordipoles, also depicted as bars, along thecircumference. For the first circle (a), the donorand acceptor transition moments are parallel toeach other in the same plane and k2 variesbetween 4 and 0, depending on the location ofthe acceptor on the circle. These values arelabeled next to the acceptor. Note that evenwhen the donor and acceptor transitionmoments are parallel, k2 can still be zero. For

the circle (b), with donor and acceptor dipolesagain lying in the same plane, the acceptordipole is oriented along the electric field of theexcited donor at the location of the acceptorwith k2 values between 1 and 4. For the circles(c) and (d), the orientation factor is 0 for eachexample as the acceptor dipole is perpendicularto the donor electric field. For the circle (c), thedonor and the acceptor are oriented in thesame (DR) plane, but for the circle (d), theacceptor is perpendicular to the DR plane. Ineach circle, the electric field lines of the exciteddonor are shown.


rA ¼ rAðtÞ ¼ b1Ae�ð2D?þ4D==Þt þ b2Ae

�ð5D?þD==Þt þ b3Ae�6D?t: ð4:5cÞ

b1A ¼ 310

sin4qA; b2A ¼ 310

sin2 2qA; b3A ¼ 410

32cos2qA � 1

2

� �2

: ð4:5dÞ

rT ¼ rTðtÞ ¼ b1Te�ð2D?þ4D==Þt þ b2Te

�ð5D?þD==Þt þ b3Te�6D?t: ð4:5eÞ

b1T ¼ 310

sin2 qD sin2 qAcos 2w

¼ 310

2ðcos qT � cos qDcos qAÞ2 � sin2qD � sin2qAh i

: ð4:5f ðiÞÞ

b2T ¼ 310

sin 2qD sin 2qA cos w ¼ 65cos qDcos qA cos qT � cos qD cos qA½ �:

ð4:5f ðiiÞÞ

b3D ¼ 410

32cos2 qD � 1

2

� �32cos2 qA � 1

2

� �: ð4:5f ðiiiÞÞ

Global analysis allows one to obtain the rotational diffusion constants and cos qD(from rD), cos qA (from rA), and cos qT (from rT), so that kappa-squared can becalculated using Equation 4.1 [4]. The conclusion for now is that kappa-squared can

Figure 4.6 A cartoon of a macromolecule witha donor (D, emission transition dipole alongthe unit vector d) and an acceptor (A,absorption transition dipole along the unitvector a) rigidly attached to the symmetry axisof the macromolecule, which undergoesrotational diffusion around the symmetry axis at

D== and around any axis perpendicular to that atD?. D== and D? are rotational diffusionconstants. The angles appearing in theanisotropy decays (uD, uA, uT, and w) are asshown in Figure 4.4 and defined nearEquation 4.1. This cartoon is similar to the onepresented by Dale [4].

4.4 Kappa-Squared Can Be Measured in At Least One Case j69

be measured in this specific case. However, this approach can probably be extendedto more and perhaps all cases. This particular case is unique in that the orientationsof the transition moments are fixed in the frame of the macromolecule. In mostcases, some motion occurs. The range and frequencies of such motions may differ.The concept of averaging regimes is useful for understanding the implications ofmotion for FRET.

4.5Averaging Regimes

The initial steps of a FRETexperiment involve the absorption of a photon by a donorfluorophore. Absorption of a photon is rapid, typically occurring within a femto-second (10�15 s), and results in the elevation of a ground-state electron into a myriadof potential electronic and vibrational excited states. Over the next few hundredfemtoseconds, this array of potential excited-state electronic and rotational–vibra-tional energy sublevels are consolidated into the Boltzmann rotational–vibrationallevel manifold of the lowest-energy singlet excited state, as a result of vibrationalenergy loss due to subsequent kinetic interactions between the excited fluorophoreand surrounding molecules. Fluorophores, in general, spend from picoseconds totens of nanoseconds in this relatively long-lived lowest singlet excited state beforeeventually transitioning back to a ground-state sublevel. With their return to a groundstate, excess excited-state energy will be either emitted as a photon (donor fluores-cence), transferred to a nearby acceptor (FRET), or it will be utilized by some othernonradiative mechanisms. To understand the factors that can influence the proba-bility of energy transfer by FRET, one must understand the types of events that canoccur while a fluorophore is in its excited state. In relation to kappa squared, themain factors that must be considered is to what extent donor and acceptorfluorophores can move relative to each other while in the excited state – specifically,how fluorophore motion may influence the position of an acceptor relative to theorientation of the donor emission dipole, and how itmay influence the orientation ofthe acceptor absorption dipole relative to the orientation of the donors excited-stateelectric field.When every donor and every acceptor can take up its entire range of orientations

during the lifetime of the excited donor state in the presence of acceptors, the systemis said to be in the dynamic averaging regime, and the dynamic averaging conditionapplies. In this regime, kappa-squared can be replaced by an appropriate averagevalue, and the average FRET efficiency is given by

Eh idynamic ¼3=2ð Þ k2h i�R6

0

3=2ð Þ k2h i�R60 þ r6DA

; ð4:6Þ

where the brackets denote an average, �R0 is the F€orster distance when k2 ¼ 2=3, andrDA is the donor–acceptor distance. The isotropic condition applies when allorientations are equally probable. The dynamic isotropic average of kappa-squaredequals 2/3 in the one-, two-, and three-dimensional cases [5]. As discussed above, the


isotropic average is not always valid. The dynamic averaging regime is discussedfurther in Sections 4.6–4.9.When the rates of rotation are small compared to the rate of donor decay in the

presence of acceptor, the system is in the static averaging regime, and the staticaveraging condition applies. In this regime, kappa-squared cannot be replaced by auniversal average value, and the average FRET efficiency is given by

Eh istatic ¼3=2ð Þk2�R6

0

3=2ð Þk2�R60 þ r6DA

* +; ð4:7Þ

where the symbols have the same meaning as in Equation 4.6. Note, however, thedifference between calculating an average static efficiency and obtaining an averagedynamic efficiency. Also note that at large distances when r6DA � ð3=2Þðk2�R6

0Þ, thedifferences between Equations 4.7 and 4.6 vanish. Effective values for kappa-squared in the static and dynamic averaging regimes have been derived by Dalefor random spatial distributions of separations of free donors and acceptors insolutions of three, two, and one dimension with, as appropriate, random three- andtwo-dimensional orientational distributions or, for the one-dimension spatial, one-dimension orientational case, the inline configuration [4,6]:

Orientationaldistribution

Spatialsolutiondistribution

Dynamicaverage ofkappa-squared

Static averageof kappa-squared

3D 3D 2/3ffiffiffiffiffik2

pD E2ffi 0:69012 ffi 0:4762


3p

i3 ffi 0:73973 ffi 0:4048D


6p

i6 ffi 0:83056 ffi 0:3281D


3p

i3 ffi 0:94622 ffi 0:8471D


6p

i6 ffi 0:94566 ffi 0:7151D

1D 1D 4ffiffiffiffiffik2

6p

i6 ¼ 4D

In the inline configuration (1D, 1D), a single value for kappa-squared applies, so thatin this case the dynamic and static values are identical.The first report on FRET in 2D free donor, free acceptor solutions was published

by Tweet et al. [7]. Loura et al. [8] confirmed 1D solution FRET in an experimentalsystem, with the theory given in detail in Ref. [9].It is possible that the average rate of transfer is on the same order of magnitude as

a dominant rate of rotation for the donor or acceptor. In this case, the system isneither in the dynamic regime nor in the static regime. This case is discussed inSection 4.12.

4.5 Averaging Regimes j71

4.6Dynamic Averaging Regime

Dale et al. [10] and Haas et al. [11] have shown that in the dynamic averaging regime,fluorescence depolarization data allow one to remove some of the uncertainty in theFRET distance resulting from kappa-squared. Dale et al. emphasize depolarizationbecause of rapid restricted rotations [10], whereas Haas et al. [11] mainly considerexcitation into overlapping transitions as the reason for low polarization values.However, in the dynamic averaging regime, depolarization due to degeneracy or overlapof transitions is essentially indistinguishable from depolarization resulting fromreorientations. Indeed, it has been shown that the equation for the average kappa-squaredderivedbyDale et al., Equation4.10, is the sameas theonederivedbyHaas et al.,if cylindrical symmetry in the transitions is assumed [12]. In the dynamic averagingregime, the donor emissionmoment, which is along the unit vector d, fluctuates rapidlyaround d

X(unit vector, called donor axis) and the acceptor absorptionmoment, which is

along the unit vector a, fluctuates rapidly around aX (unit vector, called acceptor axis).These fluctuations may represent rapid restricted rotations or coupling between over-lapping transitions. As a result, the kappa-squared value fluctuates around an averagevalue. And, this average value, which may be used instead of the kappa-squared valueappearing in the FRETefficiency, depends on two parameters d and a (defined below)and on three variables W, HD, and HA (defined below). The parameter d is the axialdepolarization factor for the donor emission moment:

d ¼ dXD� � ¼ 3

2cos2 yD � 1

2

� �¼ðp0

32cos2yD � 1

2

� �sin yDFD yDð ÞdyD; ð4:8Þ

where yD is the fluctuating angle between d and dX, and FD yDð Þ is a distribution

function. Similarly, the axial depolarization factor for the acceptor absorptionmoment is

a ¼ dXA� � ¼ 3

2cos2yA � 1

2

� �¼ðp0

32cos2yA � 1

2

� �sin yAFA yAð ÞdyA; ð4:9Þ

where yA is the fluctuating angle between a and ax, and FA yAð Þ is the distributionfunction. Note that it is assumed here that the distributions are cylindricallysymmetrical. The parameters d and a are second rank orientational order parameterswith values between�0.5 and 1 : 1when the transitionmoment is completely alignedwith its axis, 0 when the angle between the transition moment and its axis is equal tothe magic angle at all times or when the transition moment is completely random,and �0.5 when the angle between the transition moment and its axis is 90�, that is,when the transitionmoment is degenerate in a plane perpendicular to the axis.HD isthe angle between the donor axis and the line connecting the centers of donor andacceptor,HA is the angle between this connection line and the acceptor axis, andW isthe angle between the projections of the donor and acceptor axes on a planeperpendicular to the connection line. The average value of the orientation factor


in the dynamic averaging regime is [10]:

k2� � ¼ 2

3� 13

d þ að Þ þ d 1� að Þcos2HD þ a 1� dð Þcos2HA þ adk21;1; ð4:10aÞ

with k21;1 ¼ sin HD sin HA cos W� 2 cos HD cos HAð Þ2: ð4:10bÞThedepolarization factors d anda can be obtained fromfluorescence depolarization

measurements [10]. The “time-zero” value of the donor fluorescence anisotropy valueis proportional to d2 and that of the acceptor is proportional to a2. Here “time-zero”must be understood in the context of the dynamic averaging regime, and the so-calledB€urkli–Cherry [13] plot illustrates the concept of “zero-time anisotropy.”Figure 4.7 may suggest that time-resolved fluorescence anisotropy measurements

are necessary in order to obtain depolarization factors. However, Corry et al. haveshown that steady-state confocal microscopy also enables one to measure suchfactors and that kappa-squared can even be obtained if some knowledge of therelative geometry is assumed [14].Because of the proportionality between measured fluorescence anisotropy values

and the square of d or a, the experimentally obtained depolarization factors can beeither positive or negative if d2 and a2 are between 0 and 0.25. These signambiguities may be resolved if independent structural or spectroscopic informationis available.

Figure 4.7 A log–log plot of fluorescenceanisotropy versus time after a flash excitation.This is also called a B€urkli–Cherry plot [13]. Itshows a stepwise decrease of the anisotropywith time and nicely illustrates that “zero-timeanisotropy” in the context of FRET refers to theanisotropy value reached after the completionof rotations with frequencies higher than the

average transfer rate. Note that the lifetime ofthe excited state (of donor or acceptor) seemsnot to matter in this graph, but it is relevant inpractice because of noise: after a few lifetimes,the time-resolved anisotropy becomes verynoisy and is completely unreliable for timeslarger than about five lifetimes.

4.6 Dynamic Averaging Regime j73

Dale et al. found the maxima and minima of h k2 i in Equation 4.10 numerically andpresented a contour plot allowing to read the highest and lowest h k2 i value for eachcombination of depolarization factors [10]. It is also possible to find these maximaand minima analytically by setting the derivatives of hk 2 i with respect to HD , HA , andW equal to zero, solving the set of three resulting equations [15]. As shown at http://www.FRETresearch.org, there are six candidates for maxima and minima:

k2A ¼23 þ 23 a þ 2

3 d þ 2ad : ð4: 11aÞ

k2P ¼23 � 13 a � 1

3 d : ð4: 11bÞ

k2H ¼23 � 13 a � 1

3 d þ ad : ð4 :11c Þ

k2M ¼23 þ 16 a þ 1

6 d � ad þ 1

2 j a � d j : ð4: 11dÞ

k2L ¼23 þ 16 a þ 1

6 d � ad � 1

2 j a � d j ð4:11e Þ

k2T ¼19 ð1 � aÞð1 � d Þ þ 4

9

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 � aÞð1 � d Þð1 þ 2aÞð1 þ 2d Þ

p: ð4: 11f Þ

The distributions of transition moments can be visualized as ellipsoids with thesymmetry axis equal to 1 þ 2d for the donor and 1 þ 2a for the acceptor and with anyaxis perpendicular to the symmetry axis equal to 1 � d for the donor and 1 � a forthe acceptor. As a result, in the extreme situation where the depolarization factorequals 1, the distribution behaves like a needle-like “molecular antenna, ” and in theother extreme where it is �0.5, the transition dipole distribution resembles a disk-like “antenna.” Figure 4.8 illustrates the meanings of the six candidates using suchellipsoids and verbal descriptions.Careful comparison (http://www.FRETresearch.org) of the magnitudes of one

candidate relative to those of the others in all points of the plane formed byparameter values �ð1=2Þ d 1 and �ð1=2Þ a 1 leads to the conclusionthat there are nine different regions where the maxima and minima can becalculated using the expressions for the six candidates in Equations 4.11a–4.11f.These regions have borders expressed as d ¼ 0, a ¼ 0, C ¼ 0, E ¼ 0, F ¼ 0, orG ¼ 0, where C, E, F, and G are defined as follows:

C ¼ aþ d � 12: ð4:12aÞ

E ¼ 3aþ 3d þ 5ad þ 1: ð4:12bÞF ¼ 2d � 3aþ 2ad � 1: ð4:12cÞG ¼ 2a� 3d þ 2ad þ 1: ð4:12dÞ

The regions are shown in Figure 4.9.


http://www.fretresearch.org



The meaning of the symbols and the properties of the different regions arespecified in Table 4.1.Note that if both depolarization factors, for the donor and the acceptor, are

positive, the minimum hk2i is k2P and the maximum is k2A, as pointed out by Dale

et al. [10]. The reader may wonder why we split up this region in a centralAP

1

zone and three sections ofAP

2 around it. The reason is that the six candidates are

not only possiblemaxima andminima but are also potential answers to the question:

Figure 4.8 Description of the candidates for maxima and minima of the average kappa-squaredin the dynamic regime, specified by Equation 4.11. They are also candidates for the most probablekappa-squared in this averaging regime.

4.6 Dynamic Averaging Regime j75

What is themost probable kappa-squared value? Figure 4.9 also serves as the startingpoint in our approach to this question.

4.7What Is the Most Probable Value for Kappa-Squared in the Dynamic Regime?

What is the “most probable” kappa-squared value? This is an ambiguous question! Ifwe are trying to find the most probable value per se of kappa-squared, that is,independent of and in isolation from any other FRET parameter, we will get oneanswer; but if we want the most probable k2 corresponding to the most probableseparation derived for a given efficiency, a completely different answer emerges. It iswell established that the probability density of kappa-squared per se for a pair oflinear donor and acceptor transition moments (a ¼ d ¼ 1 in Equation 4.10a)exhibits an infinitely high peak at k2 ¼ 0 (see Equation 4.21 and Figure 4.19)(also refer to Refs [10,12,16]). Nevertheless, if we consider any nonzero efficiency,however small, whether or not obtained in an actual experimental situation, derivingeither from a transfer efficiency or a transfer rate, the most probable kappa-squared

Figure 4.9 Regions in the ða; dÞ plane showing in each a column with the kappa-squaredmaximum indicated by the top letter and the minimum by the bottom letter. Table 4.1 givesdetails.


value cannot be equal to zero, because k2 ¼ 0 means that the efficiency also equalszero. Interestingly, the explanation of this paradox is based on the link betweenkappa-squared and the “relative distance,” which is defined by

r “Relative distance” Actual FRET distanceDistance assuming k2 ¼ 2=3

¼ 32k2

� �1=6

: ð4:13Þ

In formal mathematical terms, we need three probability functions for this explan-ation: the range probability, PR k2min ! k2

� �, the probability density for kappa-

squared, p k2ð Þ, and the probability density for the relative distance, Q rð Þ:

PR k2min ! k2� � ¼ probability that kappa-squared has a value between k2min and k2:

ð4:14Þp k2� �

dk2 ¼ probability that kappa-squared has a value between k2 and k2 þ dk2:

ð4:15ÞQ rð Þdr ¼ probability that the relative distance has a value between r and rþ dr:

ð4:16Þ

Table 4.1 Maxima and minima in the dynamic averaging regime.

Name ofregion

Maximumhk2i in thatregion

Minimumhk2i in thatregion

Definition of region Are allcandidatesvalid there?

AP

1 k2A k2P fa > 0; d > 0;F �G� C > 0g Yes

AP

2 k2A k2P fa � 0; d � 0;F �G� C 0g All but not k2T

MH

3 k2M k2H fa� d 0;C � 0g All but not k2T

MT

4 k2M k2T fa� d < 0;E � C < 0;F � G < 0g Yes

ML

5 k2M k2L fa� d < 0; F �G � 0g All but not k2T

HA

6 k2H k2A fa 0; d 0;E � 0g All but not k2T

MA

7 k2M k2A fa� d < 0;E 0g All but not k2T

HL

8 k2H k2L fa 0; d 0;F �G 0g All but not k2T

HT

9 k2H k2T fa 0; d 0;F �G� E < 0g Yes

4.7 What Is the Most Probable Value for Kappa-Squared in the Dynamic Regime? j77

Both p k2ð Þ and Q rð Þ are proportional to the derivative of PR k2min ! k2� �

. And,because of the link between the relative distance and kappa-squared as expressed inEquation 4.13, it follows that

Q rð Þ ¼ 4r5 � p k2� �

: ð4:17ÞTherefore, the mathematical explanation of the above-mentioned apparent paradoxis that the derivative of the p k2ð Þ will in general not be zero when the derivative ofQ rð Þ equals zero because of the factor 4r5 in Equation 4.17. It is difficult to visualizethis apparent paradox for the general case of any choice for a and d. However, for thecase that both of these depolarization factors are equal to 1, PR k2min ! k2

� � ¼PR 0 ! k2ð Þ is the area under the curve that is obtained when cutting the three-dimensional plot in Figure 4.3 at a certain kappa-squared level and projecting the cutin the cos qDj j � cos vj j plane, as shown in Figure 4.10.In this special case, constant-k2 curves can be calculated fromEquation4.3. The area

of the square formed by all possible values of jcos qDj and jcos vj between 0 and 1represents the total probability of 1 that k2 has any value between0 and4, 0 for cos v ¼0 and 4 for jcos vj ¼ jcos qDj ¼ 1. We can divide up the square in, say, a hundred

strips by drawing the 101 curves jcos vj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2=ð1þ cos 2qDÞ

pinside the square

choosing k2 equal to 0� 4/100, 1� 4/100, . . . , 99� 4/100, and 100� 4/100. Thisway the area of each strip represents the probability that kappa-squared has a value

Figure 4.10 Lines of constant kappa-squared inthe cos uDj j � cos vj j are shown for the case inwhich both depolarization factors are equal to 1.The area below the curve labeled 1/3 is equal to

the probability that kappa-squared is between 0and 1/3, the area between this curve and the 2/3curve represents the probability that kappa-squared is between 1/3 and 3/2, and so on.


be tween the k 2 for the lowe r bou nd ary a nd that f or the upp er b oundary. Such ad i vi s i o n h a s b e e n i n i t i a t e d i n Fig u r e 4 . 1 1 a . T h e v e r y fi rst s trip between the 0 a nd0.04 curves has by far the large st area and succ essive st rips r apidly de cre ase inare a, indicating that th e most probable value for the orientat ion factor is 0.Howe ve r, nothing i s s aid a bout the distanc e or th e ef fi ciency. Over th at ve ry fi rs tstrip, the relative distance is 0 at the lower boundary, but 0.626 at the higherbou nd ary, whereas the maximum relative distanc e is 1.35. As a r esu lt, the veryfi rst s trip in k 2 represe n ts 46% of al l dist ance ch oices.It seems more appropriate, therefore, to translate the combination of a measured

efficiency and an independently obtained F€orster distance to a distance with anorientational uncertainty specified by Equation 4.10. In terms of the example of

Figure 4.7, this means we should divide up the square by drawing 101 curves jcos v j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið 2=3 Þr 6ð Þ= ð1 þ cos 2 qD Þp

inside the square by choosing r equal to 0 � 61=6 =100,

1 � 61=6 = 100, . . . , 99 � 61=6 =100, and 100 � 61=6 = 100. This way the area of each striprepresents the probability that the relative distance has a value between the r for thelower boundary and that for the upper boundary. This is indicated in Figure 4.11b,where careful analysis shows that the strip straddling the left upper corner of the

square has the biggest area, corresponding to r ¼ 3=2ð Þ1=6 ¼ 1:07 and to k2 ¼ 1. Ingeneral, the location of the maximum of Q will differ dramatically from that of themaximum of p, as in the example of Figure 4.11. Therefore, themost probable kappa-squared per se will differ from the most probable kappa-squared at a given efficiency.An algorithm to find the most probable kappa-squared in the second case (at a

given efficiency) is brie fly as follows (http://www.FRETresearch.org):

1) Choose the a; d pair that best describes the depolarization properties of the actualsystem. (See Equations 4.12c and 4.12d, and the explanations near these. Note thataxial symmetry is assumed. If axial symmetry cannot be justified, see Ref. [11].)

Figure 4.11 Pair of diagrams illustrating thatthe question “What is the most probable kappa-squared value?” is ambiguous. This examplerefers to the dynamic regime and a ¼ d ¼ 1.(a) The diagram is for unknown efficiency with

k2 ¼ 0 as the most probable value. (b) Thediagram corresponds to having a knownefficiency and thus a link between the orientationfactor and relative distance resulting in k2 ¼ 1being the most probable kappa-squared value.



2) Because�ð1=2Þ a 1 and�ð1=2Þ d 1, this pairmust lie within one of theregions shown in Figure 4.6 and Table 4.1. Use this figure or table to decide which

regionAP

1;

AP

2; . . . ; or

HT

9

� �applies. Choose B, the number of bins,

and vary the bin number i from 1 to B, obtaining bins with relative distance valuesbetween rmin þ ði� 1Þðrmax � rmin Þ=B and rmin þ iðrmax � rmin Þ=B, with

rmin ¼ ð3=2Þk2min

� �1=6and rmax ¼ ð3=2Þk2max

� �1=6.

3) For n ¼ 1 to B, setQðnÞ to 0. ThisQðnÞ is the nth component of a B-dimensionalarray, which will in the end become a histogram approximating the frequencydistribution of the relative distance, QðrÞ.

4) Choose N, and in so doing pick N3 points, varying j, ‘, and m from 1 to N,calculating cos HD ¼ ðj � 1Þ=ðN � 1Þ, cos HA ¼ ð‘� 1Þ=ðN � 1Þ, and W ¼pðm � 1Þ=ðN � 1Þ Substitute these into Equation 4.7 to calculate hk2i valuesand from there relative distance values r ¼ ð3=2Þhk2ið Þ1=6. Compare each r withthe lower and upper boundary of each bin. Place each r in the appropriate bin byadding 1 to each QðnÞ whenever r > rmin þ ðn� 1Þðrmax � rmin Þ=B and r rmin þ nðrmax � rmin Þ=B with 1 n B.

5) NormalizeQ by dividing each component by N3. As a result, the sum of allQðnÞvalues will become 1, signifying that the probability thatQ has any value equals 1.(For n ¼ 1 to B, QðnÞ ¼ QðnÞ=N3.)

We have examined graphs of QðrÞ obtained with this algorithm for a largenumber of points in the plane formed by the depolarization factors a and d, varyingthese between �0.5 and 1. Results for the most probable kappa-squared are shownin Figure 4.12.The definition of the most probable kappa-squared in Figure 4.8 is that value

corresponding to the highest peak in QðrÞ:

most probable k2 ¼ 23r6peak: ð4:18Þ

Whenever one or both of the depolarization factors is negative, the mostprobable kappa-squared is k2P. In the region where both depolarization factorsare positive, there is a rather large central region where it is k2L, surrounded byfour regions with k2H as the best value and two regions where k2M is the mostprobable value. The uncertainty in the distance as a result of variations in theorientation factor has two aspects: the most probable kappa-squared may deviatefrom 2/3, that is, the location of the peak may differ from r ¼ 1, and, the peakmay be fairly broad, that is, the 67% confidence interval (CI) may have consider-able width (the 67% confidence interval is the range of r-values near the peakwhere the total QðrÞ adds up to 67%). It is appropriate to call the first aspect a“peak location error” (PLE) and the second a “broad distribution error” (BDE). Wenote that r ¼ 1 is the relative distance value that corresponds to k2 ¼ 2=3 and,thus, define the PLE as

PLE ¼ 1� rPEAKð Þ � 100%: ð4:19Þ


PLE> 0 means that k2 ¼ 2=3 overestimates the most probable distance, andPLE< 0 signifies that this assumption underestimates the distance at the peak.Figure 4.13 shows examples of distributions and PLE values.Our definition of the “broad distribution error” is

BDE ¼ 1=2 67%CIð Þ ¼ rUL � rLLð Þ � 50%; ð4:20Þwhere rLL is the lower and rUL is the upper limit of the 67% CI for QðrÞ. In somecases, the peak is near the center of the confidence interval, but relative distancedistributions can also be highly asymmetric with the peak at the upper or lower limitof this interval. Figure 4.14 shows examples.Figure 4.15 shows lines of equal “peak location error” in the ða; dÞ plane.Near a ¼ d ¼ 1 and a ¼ d ¼ �1=2, the PLE is negative, but in the majority of

points, the PLE is positive with the most probable distance smaller than the one atk2 ¼ 2=3. A very high positive PLE of about 30% occurs near a ¼ d ¼ 0:96, close tothe red line. On the red line,Q has two equally high peaks. The red line is the borderbetween two regions where rpeak is calculated differently. As a result, PLE changesdiscontinuously at this border. The most dramatic change is at a ¼ d ¼ 0:96 wherethe PLE is�6%, corresponding to k2H, at the side where the factors are slightly higherthan 0.96, and þ30%, corresponding to k2L, at the side where the depolarization

Figure 4.12 Map indicating the most probablekappa-squared in the dynamic regime, where thismost probable value is defined as the kappa-squared for which the relative distance is the mostlikely (see text). When one or both of thedepolarization factors for the donor or acceptorare negative, k2P is the best. The region where bothdepolarization factors are positive consists of fourregions labeled H, where k2H is the most probable,two regions labeled M where k2M is the best, and

one labeled L, where k2L is the most probablekappa-squared value. The border between the Hand L regions near the top right corner is welldescribed by aþ d � 0:985ad ¼ 1:012. Thecurved border between H and M on top and L onthe bottom, starting at d ¼ 0; a ¼ 0:79 andending near d ¼ 0:81; a ¼ 1 follows the trenda ¼ 0:79þ 0:504d4:28; and the one with H andM on the right and L on the left is described byd ¼ 0:79þ 0:504a4:28.


factors are slightly smaller than 0.96. The green lines are also borders betweenregions where the peak is calculated differently, but with a continuous change inPLE. A large discontinuous change in PLE also implies a fairly broad distancedistribution and, therefore, a relatively large BDE. Results for BDE are shown inFigure 4.16.This diagram shows data for the 67%CI, obtained with our program for findingQ

(available on theWeb site) at any choice for the depolarization factors a and d. To runthis program, one must choose an a and a d, a value for B (the number of bins, thatis, the number of bars in the histogram approximation for the Q -function), and avalue for N (a measure for how many times the relative distance is evaluated; thenumber of points is N3). After locating the peak (allowing one to confirm the resultsof Figure 4.12), the CI is obtained by moving away from the peak in both directionswhile adding the Q-values of the bars in the histogram until 0.67 has been reached.Near the axes, a ¼ 0 or d ¼ 0, the peak is extremely asymmetric with the relativedistance at the peak, rPEAK, coinciding with the lower limit of the CI, rLL, at positive aor d, and matching the upper limit rUL at negative a or d, as shown in Figure 4.13.Away from these axes, say for a > 0:1; d > 0:1 or a > 0:1; d < �0:1, ora < �0:1; d > 0:1, or a < �0:1; d < �0:1, the peak is more symmetric and rPEAKis close to the center of the CI. A completely different problem arises near the redborders shown in Figure 4.15. At the red borders, theQ-function has two peaks thatare exactly of equal height. For example, at a ¼ d ¼ 0:96, the Q-function has a peakcorresponding to hk2i ¼ k2H ¼ 0:948ðrPEAK ¼ 1:06Þ and an equally high peak forhk2i ¼ k2L ¼ 0:065ðrPEAK ¼ 0:68Þ. In such cases, the center of the CI should be

Figure 4.13 Examples of probability density ofthe relative distance illustrating the definition ofthe systematic error. The graph on the left is fora ¼ d ¼ 0:73, where the main peakcorresponds to k2L with a relative distancesmaller than 1 so that PLE is positive (this Qhas a secondary maximum corresponding to

k2H). The distribution in the center is for a ¼1; d ¼ 0:5 or d ¼ 1; a ¼ 0:5, showing one peakmatching k2H ¼ k2M ¼ 2=3, yielding r ¼ 1 andPLE¼ 0. The graph on the right is for a ¼ d ¼ 1,with a peak at r ¼ 1:07 corresponding to k2H ¼1 and a negative PLE.


chosen at the average of the two rPEAK values, and the CI should be built up fromthere. Examples of graphs for the frequency distributions Q versus the relativedistance r are shown in Figure 4.17.

4.8Optimistic, Conservative, and Practical Approaches

For assessing the kappa-squared-induced error in the FRET distance, there is an“optimistic” approach that assuming kappa-squared equals 0.67 introduces little orno error, and there is a “conservative” method based on depolarization factorsresulting in a minimum and maximum kappa-squared (with corresponding mini-mum and maximum distances) without the ability to pinpoint the most probablekappa-squared in this range. The optimistic method is that of Haas et al. [11] andSteinberg et al. [17], and the conservative approach is that of Dale et al. [10]. Thisclassification is an oversimplification, of course, as both the first group and thesecond group of authors have provided a detailed and versatile discussion of errors

Figure 4.14 Examples of probability density forthe relative distance illustrating the definition ofthe random error. In each, the 67% confidenceinterval (CI) is shaded dark gray and runs fromrLL, the lower limit of the CI, to rUL, the upperlimit of the CI. The three graphs have the samescale in r and Q. The one in the center isrelatively broad and low. The other two arenarrow and high, actually extremely high, as

both go to infinity at one point on the interval.The graph on the left is for a ¼ �0:5; d ¼ 0 ord ¼ �0:5; a ¼ 0 with its peak at rUL and a BDEof about 1%. The distribution in the center is fora ¼ d ¼ 1 with its peak near the average of rLLand rUL and a BDE of about 24%. The graph onthe right is for a ¼ 1; d ¼ 0 or d ¼ 1; a ¼ 0 witha peak at rLL, and a BDE of about 7%.

4.8 Optimistic, Conservative, and Practical Approaches j83

resulting from the orientation factor. Nevertheless, neither group has pointed outthat there are at least two different aspects associated with the kappa-squared-induced error: the PLE, introduced in Equation 4.19, and the BDE, introduced inEquation 4.20. For lack of a better name, we would like to call the procedureintroduced in the previous section the “practical” approach. Table 4.3 compares the“optimistic,” “conservative,” and “practical” approaches for a range of cases.Note that the PLE in itself is not a problem because when the depolarization

factors are known, this error can be accurately predicted using Figure 4.12 and itsdefinition (Equation 4.19). However, the discontinuous jump in the systematic errornear ða; dÞ ¼ ð0:96; 0:96Þ may cause serious problems, as a value of 0.96 isexperimentally almost undistinguishable from 1 and thus a slight uncertainty inthe depolarization factors near this value may cause the PLE to shift from �6to þ30%. For such high values of a and d, the BDE is also high (see Figure 4.15).Comparing the confidence interval for ða; dÞ ¼ ð1; 1Þ with that for (0.81,0.95) inFigure 4.15 illustrates a problem related to the discontinuity in the PLE: a relativelyminor variation in the depolarization factors may cause this interval to shift fromone that is centered around r ¼ 1 to one that is centered around 0.85. In Ref. [12], itwas assumed that the case a ¼ d ¼ 1 was the worst-case scenario. This is a logical

Figure 4.15 Lines of constant “peak locationerror” are shown with the value of PLE givennext to the lines in percent. At the red curves,the relative distance frequency distribution hastwo equally high peaks. These curves areborders between regions where the mostprobable kappa-squared is calculated differently,as indicated in Figure 4.12. At one side of a red

curve, one of the peaks is highest and on theother side, the other peak is highest. As a result,the PLE changes discontinuously when a redline is crossed. The blue lines are also bordersbetween regions where the most probablekappa-squared is calculated differently, but witha continuous change in the value of the peakand of PLE.


assumption as a ¼ d ¼ 0 is the best-case scenario and the kappa-squared-inducederror gets worse and worse when onemoves away from a ¼ d ¼ 0. After Ref. [6] waspublished, it became possible to generate plots of the frequency distribution ofdistances and kappa-squared with a few keystrokes on a computer. So, now we mustset the record straight: a ¼ d ¼ 1 is not the worst-case scenario as far as theorientation-induced error is concerned in the dynamic regime; it appears thataþ d � 0:985ad ¼ 1:012, the red line in Figure 4.15 where the PLE jumps from30 to 6%, is the worst-case scenario in this regime.

4.9Comparison with Experimental Results

It is imperative to be keenly aware of the assumptions underlying any method onewants to apply. For example, in Refs [1,2], the depolarization factors are not given,

Figure 4.16 Regions with lower and higher“broad distribution errors” (BDE), defined inEquation 4.13. A high BDE corresponds to abroad QðrÞ, and low BDE values indicategraphs for QðrÞ with a narrow peak. On thegreen lines, BDE equals 5%. The long green lineconnects the points (�0.5, 0.31), (�0.4, 0.45),(�0.3, 0.6), (�0.2, 0.66), (�0.1, 0.71), (0, 0.78),(0.1, 0.72), (0.2, 0.66), (0.3, 0.58), (0.4, 0.58),(0.54, 0.54), (0.58, 0.4), (0.58, 0.3), (0.66, 0.2),(0.72, 0.1), (0.78, 0), (0.71,�0.1), (0.66, �0.2),(0.6, �0.3), (0.45, �0.4), and (0.31,�0.5). Theshort green line passes through (�0.5, �0.25),(�0.33, �0.33), and (�0.25, �0.5). In the

region between the green lines, BDE is smallerthan 5% reaching 0% at a ¼ d ¼ 0. At (�0.5,�0.5), BDE¼ 8%. In between the long greenline and the red lines, BDE varies between 5%(on green) and 10% (on red). At (1, �0.5) and(�0.5, 1), BDE is about 12% and on the shortred curves near these points, RE equals 10%. At(1, 1), BDE¼ 24%, and BDE decreases withdecreasing a and/or d reaching BDE¼ 10% onthe red line connecting (0.45, 1), (0.52, 0.94),(0.66, 0.9), (0.65, 0.83), (0.7, 0.77), (0.74, 0.74),(0.77, 0.7), (0.83, 0.65), (0.9, 0.66), (0.94, 0.52),and (1, 0.45).

4.9 Comparison with Experimental Results j85

Figure 4.17 Ex a mp le s o f fr e qu enc yd is tr ib u ti o ns f or th e r el at iv e di st anc e wi th th e67 % co nf i de nc e i nt e r va l ( C I) in d i ca te d in e a chas an area bet we en red li ne s. A ll gr aphs haveth e sa me v er t ic al sc ale a nd th e sa meh or iz on t al sc al e. Th e wi dt h o f ea ch bo x r e fe r sto a r e la ti ve di st a nc e of 1. 4 , a nd t he h ei gh t o feach box is 9. 5 in Q-u ni t s. Fo r mo st ch oi ce sof a and d , a d is tr i bu t io n wi th o ne do mi nan t

pea k i s f ou nd ; bu t fo r pa ra m et er c ho ic e s ne a rth e re d li nes i n Fi g ur e 4. 15 , mo re t ha n o neequally pronounced peaks may occur, as isshown in the right bottom corner for ða; dÞ ¼ð0:81; 0:95Þ or (0.95, 0.81). Data for theseplots are shown in Table 4.2. The readers willbe able to generate their own graphs for anychoice of a and d by visiting the Web siteht tp :/ /w ww. FRE Tr e se a r ch .o r g.


http://www.FRETresearch.org

but they should be positive. Therefore, the “practical approach ” would suggest thatthe best kappa-squared value should be k2H , k

2L , or k2M. However, in the practical

approach, it is assumed that nothing is known about cos HD , cos H A , or W, thevariables appearing in Equation 4.10. In the spirit of information theory, it isassumed in this approach that all values of these “hidden variables” are equallyprobable when no information about them is available. Nevertheless, for the systemsin Refs [1,2], information about the relative orientation of donor and acceptor isavailable: the transition dipoles are essentially perpendicular to the axis of a helix andthe angle between the dipoles should depend on the pitch of the helix. This isactually an example of the case where the transfer depolarization is known, withinlimits, as introduced and analyzed by Dale et al. [10], and in which case equations forkappa-squared have been derived [13]. The geometry of the donor– acceptor pair inRef. [2] suggests that the best kappa-squared value should be a hybrid between k 2Hand k 2P , k

2 ¼ k 2HP ¼ ð2=3 Þ � ð1 =3Þ a � ð1= 3Þd þ ad cos 2 HT (Equation 38 i n R ef.[15]), and that t he angle HT b etwee n the p ref erre d direc tions of th eir trans itionmoments s hould be equ al to 180� rDA =pit ch. Sub stituting this value for HT, andk 2 ¼ k2HP , into the expression for the FRET effi ciency allows us to mo d el th eef fi c iency versus distance tre n d. Such a n att empt to model the ef fi ciency expect edfor the system in Ref . [2] is s hown in Figure 4.18.

Table 4.2 Data for Figure 4.17.

a or d d or a k2minimum k2maximum rminimum rmaximum rLL rPEAK rUL QðrPEAK Þa)

12 0 1

2 1 0.953 1.070 0.953 0.953 1.013 1

�12 �1

214

54 0.894 1.110 0.958 1.066 1.108 5.613

�12 0 1

356 0.891 1.038 0.986 1.038 1.038 1

�12

12

16

1712 0.794 1.134 0.930 0.994 1.056 6.602

1 �12 0 2 0 1.201 0.829 0.949 1.069 1b)

1 0 13

43 0.891 1.122 0.891 0.891 1.026 1

1 12

16

83 0.794 1.260 0.887 0.994 1.101 4.295

12

12

13

116 0.891 1.184 0.929 0.976 1.020 10.077

1 1 0 4 0 1.348 0.809 1.051 1.294 2.091

0.81 0.95 0.08 3.379 0.702 1.311 0.702 0.848 1.006 2.691

N ¼ 300 and B ¼ 100, but when a or d ¼ 0, data have been calculated analytically (see http://www.FRETresearch. org).a) The frequency distribution of kappa-squared is proportional to that for the relative distance [12]

according to the relation pðk2Þ ¼ ð1=4ÞQðrÞ=r5, with r ¼ 3k2=2ð Þ1=6.b) Mathematically, one can show that the Q-value at the peak is 1, but numerical values depend on B

and N.




The kappa-squared in samples 1 and 2 of Ref. [3] is also an example of thecase where the transfer depolarization is known. This kappa-squared should beessentially equal to k2A (Equation 4.11a) with values for the depolarization factorsclose to unity.The history of the kappa-squared fluorescence depolarization relationship is

interesting and relevant here. In the late 1970s, when time-resolved fluorescencedepolarization was virtually nonexistent, both Dale et al. [10] and Haas et al. [11]realized that information from fluorescence depolarization can be useful forunraveling distance effects from orientation effects in FRET. At that time, bothgroups had steady-state fluorescence depolarization data in mind. Fairly recently,Dale revisited the kappa-squared fluorescence depolarization relationship [4] andcame to the conclusion that this unraveling can be much more effective thanpreviously thought when the full time dependence of fluorescence anisotropy istaken into account. In terms of Figure 4.7, it is fair to say that a glimpse at a shortinterval on one of the plateaus of anisotropy versus time allows one to put limits onkappa-squared, but a full view of the this curve has the potential to pin down theorientation factor completely.

Figure 4.18 The FRET efficiency versusdistance expected for the system of Ref. [2].Assuming k2 ¼ k2HP ¼ ð2=3Þ � ð1=3Þa�ð1=3Þd þ adcos 2QT and QT ¼ 180�rDA=pitch,with g ¼ rDA=pitch and h ¼ �R0=pitch (�R0¼F€orster distance when k2 ¼ 2=3) yields a FRETefficiency of the form: E ¼ 1� ð1=½ 2Þa�

ð1=2Þd þ ð3=2Þadcos 2180�g�h6=g6 þ 1� ð1=2Þa� ð1=2Þd þ ð3=2Þadcos 2½

180�g�h6g. For this graph we chose h ¼ 2, anda ¼ d ¼ 0:8 (solid line) and a ¼ d ¼ 1 (dashedline). The trend described by the solid line issimilar to that of the experimental data inRef. [2].


4.10Smart Simulations Are Superior

Any available information that allows one to exclude certain donor–acceptor orienta-tions will help to narrow down the range of possible kappa-squared values. Simula-tions can be a powerful tool in this exclusion process. A case in point is themoleculardynamics simulations performed by Lillo et al. [21]. Following the “conservativeapproach,” these authors found for donors and acceptors at specific sites in a PGK(phosphoglycerate kinase) a fairly large rangeof possible kappa-squared values and thecorresponding donor–acceptor distances, but they noticed that some of the values forHD,HA, andW appearing inEquation4.10were inconsistentwith the crystal structureof PGK and the excluded volume of the probes at the known sites in PGK. Theyperformedmolecular dynamic simulations of kappa-squared utilizing Equation 4.10,measureddepolarization factors, thecrystal structureofPGK, and theknown locationsof the donors and acceptors, and found the most probable values ofHD,HA, andW,resulting in an improved kappa-squared value and more precise donor–acceptordistances [21]. In the same spirit, Borst et al. built structuralmodels of the FRET-basedcalcium sensor YC3.60 and noticed that minor structural changes induced by slightlyrotating the fluorescent protein around a flexible linker while keeping the sameaverage distance between the donor and the acceptor gave rise to any value of kappa-squared between 0 and 3, but a fivefold change in orientation factor (from 0.5 to 2.5)brings only about a 1.3-fold increase in critical distance indicating that the FRETprocess in YC3.60 ismainly distance dependent [22]. Gustiananda et al. [23] presentedFRETresults froman intrinsic tryptophan donor to a dansyl acceptor attached to theN-terminus in model peptides containing the second deca-repeat of the prion proteinrepeat system from marsupal possum. They used simulations for finding the bestkappa-squared in this system and extended theirmolecular dynamics simulations outto 22ns to help ensure adequate sampling of the dansyl and tryptophan ring rotations.They found good agreement of the simulated kappa-squared value with 2/3, exceptat the lowest temperatures [23]. Deplazes et al. performed molecular dynamicssimulations of FRET from AlexaFluor 488 donors to AlexaFluor 568 acceptors [24].In their system, the isotropic dynamic condition was met, meaning that all possibleorientations of the transition moments of donor and acceptor and of the lineconnecting their centers are equally probable and sampled within a time shortcompared to the inverse transfer rate. The frequency distribution (Figure 4.19) ofkappa-squared from the simulation data showed excellent agreement with thetheoretical distribution [12]:

pðk2Þ ¼1

2ffiffiffiffiffiffiffi3k2

p ln ð2þffiffiffi3

pÞ; 0 k2 1;

1

2ffiffiffiffiffiffiffi3k2

p ln2þ ffiffiffi

3pffiffiffiffiffi

k2p

þffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 1

p� �

; 1 k2 4:

8>><>>: ð4:21Þ

Their results show that even in their simple situation, simulations lasting longer than200 ns would be required to accurately sample the fluorophore separations andkappa-squared if only a single donor–acceptor pair had been included. Many aspects


of FRETwere simulated in this study, including frequency distributions of relevantangles, donor–acceptor distance, and FRETefficiency. As expected, very low correla-tionwas foundbetweendonor–acceptor distance and orientation factor [24]. VanBeeket al. did find such a correlation in a molecular dynamics simulation of a coumarindonor and an eosin acceptor, both attached to HEWL (hen egg-white lysozyme) [25].In the dynamic regime, it is implicitly assumed that kappa-squared is independent ofthe donor–acceptor distance. (In the static regime, an indirect correlation betweendistance and kappa-squared is expected, as discussed near Equation 4.31). Thecorrelation between orientation and distance in the molecular dynamics study ofVanbeek et al. is quite strong and involves both the sign and the magnitude of kappa(k ¼ cos qT � 3cos qAcos qD, the square of which is given in Equation 4.1, where theangles are also defined). This correlation is illustrated in Figure 4.20, which is amodification of Figure 6 of Ref. [25], graciously made available for this chapter by Dr.Krueger. An additional advantage of molecular dynamics simulations is that noassumptions about timescales need to be made, whereas in the interpretations ofFRETexperiments, the results do depend onwhether the system in is the dynamic orstatic regime.Note that the FRET efficiency also shows a relationship with kappa and the donor–acceptor distance in this illustration. The kappa-squared concept is based on theideal dipole approximation that is known to fail when molecules get “too close” toeach other. Mu~noz-Losa et al. performedmolecular dynamics simulations to find outhow “too close” should be defined [26]. They showed that the ideal dipole approxi-mation performs well down to about a 2 nm separation between donor and acceptorfor the most common fluorescent probes, provided the molecules sample anisotropic set of relative orientations. If the probe motions are restricted, however,

Figure 4.19 The probability density p k2ð Þ versus k2, as described by Equation 4.21, in thedynamic regime for the case a ¼ d ¼ 1.

4.10 Smart Simulations Are Superior j91

this approximation performs poorly even beyond 5 nm. In the case of such restrictedmotion, FRET practitioners should worry not only about kappa-squared but alsoabout the failure of the ideal dipole approximation [26]. In a more recent paper fromthe same laboratory, an improved construction of experimental observables frommolecular dynamics sampling has been proposed [27]. Hoefling et al. have intro-duced a similar analysis [28].

4.11Static Kappa-Squared

Webegin our consideration of the impact ofmolecularmotion during the excited stateon FRET by considering how the rate of FRET is influenced by the separation betweendonor and acceptor (rDA) as well as by the orientation of donor and acceptor dipolesrelative to each other (k2). The rate of energy transfer by FRET, kT, is dependent on theinverse sixth power of the separation between donor and acceptor [29,30]:

kT ¼ 1t0D

R0

rDA

� �6

; ð4:22Þ

Figure 4.20 Modification of Figure 6 of Ref. [25]: a scatter plot of the donor–acceptor distanceagainst kappa showing the correlation between this distance, kappa, and the FRET efficiency. Thecolor code on the right is for the efficiency. This graph has been prepared by Dr. Brent Krueger.


where t0D is the fluorescence lifetime of the donor molecule and R0 is the F€orsterdistance, the separation at which 50% of the donor excitation events result in energytransfer to the acceptor. Furthermore, the R0 value used for any specific donor–acceptor FRETpair always assumes that the dipole–dipole coupling orientation factor(k2) will have a value of 2/3, but in reality it can have any value from0 to 4 in biologicalexperiments, and can be expressed as [12,17]

k2 ¼ 1þ 3x2� �

z2: ð4:23Þ

This is Equation 4.3 with the abbreviations z ¼ cos vj j and z ¼ cos qDj j, where qD isthe angle between the donor emission dipole orientation and the donor–acceptorseparation vector and v is the angle between the donor electric field vector at theacceptor location and the acceptor absorption dipole orientation. For a typical donorfluorophore with a fluorescence lifetime of 3 ns, more than 99% of excited fluo-rophores have returned to their ground state within approximately 15 ns, that is,within about five lifetimes. It is therefore reasonable to consider (i) if the separationrDA can change during this period, (ii) if the position of the acceptor relative to thedonor emission dipole orientation can change during the excited state and therefore(except for change only along the separation vector) the value of qD, and, pari passu,that of the field vector at the acceptor location, (iii) if the orientation of the donoremission dipole changes, and thus again the value ofqD changes, andfinally (iv) if theangle between the donor electric field vector at the acceptor location and the acceptorabsorption dipole orientation (v) changes within this 15 ns period. Changes in rDAand/or in qD can be caused by significant lateral motion of the acceptor fluorophorerelative to the position of the donor fluorophore. Thus, our first consideration shouldbe how far afluorophore canmove by diffusion in 15 ns?Diffusion is a function of themass of the molecule, its hydrodynamic shape, and the temperature, as well as theviscosity of the milieu. Assuming a temperature of 20 �C and an essentially aqueouslocal environment, a small fluorophore may have a diffusion coefficient between100–1000 mm2/s, while a larger fluorophore like GFPwill have a diffusion coefficientof 70mm2/s. Under these conditions, one might expect that a free fluorophore coulddiffuse a distance between 1.4–5.5 nm during a 15 ns excited state. Clearly, suchmotion could influence the effective value of both rDA and qD in a FRETexperiment.In practice, however, most donor fluorophores will return to the ground state in amuch shorter time span, with a median value (50% of excited states lost) ofln 2ð Þ � t0D , in this instance �2 ns, effectively limiting the distance that mostmolecules (about 80%) can diffuse by up to 0.5–2.0 nm away from their originallocation. Furthermore, when one considers that fluorophores used in biologicalFRET experiments are typically coupled to much larger molecules such as proteincomplexes or nucleic acids with much smaller diffusion coefficients in aqueoussolution, and even smaller in cell cytoplasm inwhich the local viscosity for these largemolecules ismuchhigher than that ofwater, it is typically assumed that lateralmotionduring the excited state will be so limited that it will not be responsible for anyalterations in the rDA or qD values for a specific pair of molecules tagged with donorand acceptor fluorophores.

4.11 Static Kappa-Squared j93

In addition to lateral diffusion, another type that must be considered is that due tomolecular rotation. Specifically, we will consider if donor and acceptor fluorophorescan rotate during the excited state, and if so, the impact of this rotational motion onthe values of qD and v, and thus on FRET. Molecular rotation is typically parame-terized by a rotational correlation time ðtrotÞ, the average time that it takes amoleculeto rotate 1 rad around a specific axis. For spherical molecules, the rotationalcorrelation time will be the same around all three axes. Nonspherical moleculescan have different diffusion coefficients for each principal axis of rotation. These areidentical for spherical molecules, leading to a monoexponential decay of theemission anisotropy of a fluorescent probe rigidly associated with this structure,with a rotational correlation time proportional to the inverse of the diffusioncoefficient. For ellipsoids of revolution, there are two different ones, one for theaxis of symmetry and another for the two principal axes perpendicular to this,leading, in general, to a triple exponential decay of the anisotropy with threedifferent correlation times: one associated with rotation of the unique axis beingproportional to the inverse of the diffusion coefficient of the axis of symmetry, andtwo associated with both this rotation and that of the equivalent perpendicularprincipal axes and differing in the contributing weights of their summed diffusioncoefficients defining the inverse of the correlation times. Rotational correlationtimes of fluorescent molecules can be measured experimentally by monitoring thedecay of fluorescence anisotropy as a function of time after a transient excitationpulse [31]. In the absence of homo-FRET, this decay is primarily caused bymolecularrotation. By fitting the anisotropy decay to a triexponential model, the rotationalcorrelation time or times can be estimated. In the case of ellipsoids of revolution, thegeneral form of the time-dependent decay of the fluorescence anisotropy, r(t), isgiven by (compare Equation 4.5)

r tð Þ ¼ r0 �Xi¼3

i¼1

ai � e�t=troti ; ð4:24Þ

where r0 is the limiting anisotropy, the initial anisotropy at the instant of photo-excitation prior to any rotational depolarization, ai is the amplitude of the ith decaycomponent, and troti is the rotational correlation time of the ith decay component.The contribution of all three components to the decay depends on the orientations ofthe absorption and emission transition dipoles in the molecular frame: each onealone or any combination of pairs may appear and, in addition, the anisotropy maydecay monotonically from either positive or negative values or start at zero or apositive or negative value, then increase or decrease before changing directiontoward zero at long enough times, or even cross zero before turning over andmoving toward zero [32]. In practice, differences in rotational correlation times forthe three axes for most fluorophores are hard to experimentally distinguish, and,more typically, the monoexponential anisotropy decay will statistically adequately fitthe anisotropy decay data, that is, when rotational diffusion coefficients are similarenough, the extent of similarity required depending on the level down to which the


anisotropy is accurately recovered [33]:

r tð Þ ¼ r0 � e�t=trot : ð4:25ÞHere the value of trot is a function of the solution viscosity (g), temperature (T), andthe molar volume of the rotating molecule (V) [33]:

trot ¼ gVRT

; ð4:26Þ

where R is the gas constant. For example, small fluorophores, such as fluorescein(332.31 g/mol), will have a rotational correlation time of �140 ps in water at roomtemperature, while a large 28 000Da fluorophore such as Venus (a yellow GFPderivative) has a rotational correlation time of �15 ns under the same conditions[31], presumably because the volume of Venus is approximately 100 times greaterthan the fluorescein. When a population or randomly oriented fluorophores(isotropic) are photoselected using a linearly polarized light source, the highestanisotropy value theoretically possible (fundamental anisotropy) is 0.4 with one-photon excitation and 0.57 with two-photon excitation [31]. In practice, other factorscan reduce the value of the initial anisotropy value at time¼ 0. Thus, the limitinganisotropy measured in a time-resolved anisotropy measurement is usually smallerthan the fundamental anisotropy expected from theory. With time, measuredanisotropy values for fluorophores in solution that are free to rotate in any directionwill decrease as a single exponential with an asymptote at 0. This value indicates thepoint where all remaining molecules in the excited state are randomly oriented. Thespeed of this orientational randomization is parameterized by the rotational correla-tion time. For a system decaying as a single exponential, this occurs at �5X, therotational correlation time. Thus, for a small molecule like fluorescein, near-complete orientational randomization can occur within 700 ps, well within theexcited-state lifetime of fluorescein (4.1 ns). In contrast, for Venus under the sameconditions, this would require 75 ns, much longer than its lifetime of 3 ns. Asmentioned above, most of the excited donor fluorophores in a FRETexperiment willreturn to the ground state in a much shorter time span, with a median value ofln 2ð Þ � t0D , (for Venus, 2 ns). With a rotational correlation time of 15 ns, free Venusis only expected to rotate �11� in 2 ns. Furthermore, Venus will rotate even slowerwhen attached to another protein, or if situated in the more viscous cytoplasm foundin cells. Thus, Venus is not expected to rotate much during its excited state. Incontrast, a small fluorophore like fluorescein may be able to rotate during its excitedstate. Thus, when considering the value of k2 to use in interpreting a FRETexperiment, it is important to note that the values qD and v may be average valuesover many possible angles when small fluorophores are used as FRET donors andacceptors, while the values for qD and v may be static for any particular donor–acceptor pair composed of fluorescent protein donors and acceptors. At this point, itis worth noting and yet again emphasizing that the 2/3 value for k2, so ubiquitouslyused in FRETexperiments, is based on two assumptions: (i) That, bar a fortuitouslyoccurring set of static relative orientations leading to this value, qD and v haverandom values (i.e., they come from isotropic distributions). (ii) That the values of


qD and v are changing rapidly relative to the fluorescence lifetime (dynamic). Fromthe above calculations, it should be clear that these assumptions (isotropic dynamic)might be valid for some FRETexperiments using small fluorophores like fluoresceinthat can rotate rapidly, provided that they are attached via flexible-enough chainslinking them to the protein backbone and/or donor and acceptor to different small-enough segmentally flexible units in the protein, but are not generally expected to bevalid for FRET experiments using fluorescent proteins as donors and acceptorsbecause being bound into the alpha-helical backbone of the protein, they hardlyrotate at all during their fluorescent lifetimes (static, with the possible exception thatthey may be bound to separate subunits of the protein, one or both of which mayexhibit rapid segmental flexibility).What k2 value should be used in a FRETexperiment if one assumes that the values

of qD and v are randomly selected from isotropic populations, but the donor andacceptors are in the static regime, that is, they are hardly rotating during the excited-state lifetime of the donor? Steinberg et al. [17] have shown that in the static regime,k2h i for an isotropic population varies with separation in a sigmoid fashion, startingessentially at zero at very low distance, eventually leveling off at a value of 2/3 at verylarge distances [12]. Dale has derived an approximation for an effective-kappa-squared value for use in the static regime: k2h ief f � 2/3 1� Eh ið Þ [34]. Recently, MonteCarlo simulations were used to address this same issue [35]. This study confirmedSteinberg’s finding that no single value of k2h i can be used to predict the energytransfer behavior of a static population (and is in good agreement with the Daleeffective kappa-squared approximation [34]), rather it was found that a k2h i valuemust be calculated from the random values of qD andv on a FRETpair by FRETpairbasis for each pair in the population.What emerged from this study is that even for apopulation that has a homogeneous separation that strongly favors energy transferby FRET (rDA<R0), because the most probable value of k2h i for an isotropicpopulation is zero [12], a large fraction of FRET pairs in a population will only transfera negligible fraction of their excitation energy by FRET, and the population behavior willbe heterogeneous with some FRET pairs having very efficient transfer and somehaving none (k2¼ 0) or essentially none (k2 near 0). With respect to FLIMmeasurements of donor lifetimes from an isotropic static population of donorsand acceptors, a simple single-exponential decay is expected only if the rDA value ismuch larger than the R0 value (approximately no FRET). In this case, the simplelifetime decay would be the same as the decay of donor alone. If the rDA value is shortenough to support a significant amount of FRET, a multiexponential decay is expectedeven when only a single fixed rDA value is present in the population. In this instance,the average FRET efficiency calculated from the multiexponential decay may besmaller than that obtained through steady-state intensity measurements for tworeasons. First, low kappa-squared values are more common than high values, andsecond, because a fraction of donor–acceptor pairs with relatively large kappa-squared values will exhibit such efficient transfer (approaching unity) that the donorlifetimes will be beyond the resolution of the measurement and so not appear in thedecay curve.


Is there experimental evidence for static FRET behavior in experiments withfluorescent protein donors and acceptors? Specifically, for FRET in the staticisotropic regime, we expect to observe (i) a complexmultiexponential donor lifetime,even for a homogeneous population of FRET pairs and (ii) a large fraction of FRETpairs in the population should fail to transfer energy by FRET because of theprevalence of low k2 values expected in an isotropic population and the absence ofappreciable rotational motion during the excited state, even when separationbetween donors and acceptors are short. In Figure 4.21a, a three different constructsare depicted, each engineered to express in cells a Cerulean [36] FRETdonor (a blueGFP derivative) covalently attached to a Venus [37] acceptor (a yellowGFP derivative)via a 5-, 17-, or 32-amino acid linker. These constructs are called C5V, C17V, andC32V, respectively [38,39]. As a negative FRET control, a single point mutation wasintroduced into Venus at Y67 (from Y to C) to form “Amber,” a protein that isthought to have the same structure as Venus, but cannot form the Venus fluo-rophore and does not act as a dark absorber in the region of Cerulean fluorescence[40]. This Amber mutation was then used to create three more constructs; C5A,C17A, and C32A. While the Cerulean decays of C5A, C17A, and C32A areindistinguishable (Figure 4.21b), the decays of C5V, C17V, and C32V were allfaster than the Cerulean–Amber constructs, with C5V having the fastest decay,and C32V having the slowest. Using these Cerulean decays in the presence andabsence of acceptor (Venus), C5V with its short 5-amino acid linker had thehighest average FRET efficiency [(43 2)%], the FRET efficiency of C17V wasintermediate [(38 3)%], and C32V, with the longest linker separating the donorfrom the acceptor, had the lowest FRET efficiency [(31 2)%] [38]. Note that C5V,C17V, and C32V all have complex decays that are clearly not single exponential,even though every expressed molecule in the population should have one Ceru-lean donor covalently attached to one Venus acceptor. These complex multi-exponential fluorescence lifetime decays for donor covalently attached to acceptorssuggest that the underlying distribution of FRET efficiencies in these populationsis heterogeneous. While this complex decay behavior is consistent with the firstprediction of FRET in the static isotropic regime, somewhat awkward is theobservation that the lifetimes of the three corresponding Cerulean–Amber con-structs did not decay as a purely single exponential as the F€orster theory predictsfor donor-only constructs. This might arise from more complicated photophysicsfor fluorescent protein fluorophores, perhaps indicating multiple excited states forthese fluorophores. While such complicated donor-alone decay behavior is prob-lematic, it is quite typical for decays of isolated fluorescent proteins, not to sayubiquitous, and has been observed in experiments measuring FRET betweenspectral variants of many different fluorescent proteins [41]. Regardless, to test thesecond prediction of FRET in the static isotropic regime, an analysis method isneeded that can account for the complex decay behavior of the donor-alone. Tolook for a fraction of molecules in a population that does not undergo energytransfer, the data plotted in Figure 4.21b were transformed and replotted as thetime-resolved FRET efficiency (TRE) (Figure 4.21c) This transformation involves


Figure 4.21 (a) Cartoons depicting the FRET-positive protein constructs C5V, C17V, andC32V and their FRET-negative analogues C5A,C17A, and C32A, where C stands for Cerulean(donor), V for Venus (acceptor), and A forAmber (VenusY67C), a nonabsorbing Venuswith a single-point mutation that preventschromophore formation. The number betweenC and V and C and A denotes the number ofamino acids in the linker connecting them. (b)Donor fluorescence intensity, IDA, versus timeafter donor excitation in the presence of energytransfer to Venus for C5V, C17V, and C32V, andintensity, ID, versus time in the absence ofenergy transfer for C5A, C17A, and C32A.

(c) Experimental TRE versus time for C5V, C17V,and C32V compared to C5A, C17A, and C32A.(d) Theoretical TRE versus time based onEquation 4.28 in which IDA(t) is calculatedassuming that the excited donor ischaracterized by a monoexponential decay withtime constant equal to the appropriate averageof the measured lifetimes for a population ofdonor–acceptor pairs over which kappa-squared is randomly distributed in the staticlimit, with choices for the relative distance (R0/rDA with R0 defined for kappa-squared¼ 2/3)that yield a strong resemblance to theexperimental curves in panel (c).


calculating the time-dependent change in FRET efficiency normalized to thefluorescence lifetime decay of the donor:

TRE tð Þ ¼ ID tð Þ � IDA tð ÞID tð Þ ; ð4:27Þ

where ID(t) is the fluorescence decay of the donor-alone and IDA(t) is the fluorescencedecay of the donor in the presence of acceptor. Note that ID(t) does not have to bemonoexponential, it could just as well have a more complex decay resulting from thesum of multiple excited states. Similarly, IDA(t) can also be a complex decay resultingfrom multiple decay components, but including a component, or components, repre-senting energy transfer by FRET from the donor (ormultiple donor excited states) to anacceptor (or multiple acceptor ground states). If every donor or donor excited state thatregisters in the IDA(t) determinationundergoesFRET, theTREcurveswill start at a valueof 0 at time0 and eventually asymptote at a TREvalue of 1. In contrast, if somedonorsordonor excited states never transfer energy by FRET, as predicted for energy transfer inthe static isotropic regime, theTREcurvewill still start at a valueof 0at time0, but appearto asymptote to a TREvalue that is less than 1. This difference represents the fraction ofmolecules in the population that do not transfer energy by FRETand/or that transferswith very high efficiency and is not detected. In Figure 4.21c, we can see that the TREcurves for the decay data presented in panel (b) for C5V (and C5A), C17V (and C17A),and C32V (also C32A) all seem to asymptote to a value that is between 0.71 and 0.73,indicating that for these constructs approximately 27–29% of the donors either do nottransfer energy and/or very efficiently transfer it by FRET (or any other additionalmechanism). This type of behavior is consistent with the predictions of FRET in thestatic isotropic regime, but other sources of population heterogeneity [35] may alsoparticipate in producing a TRE curve asymptote of less than 1.Themain advantage of TRE analysis over directly examining fluorescence lifetime

decay curves is that TRE analysis facilitates discriminating between populationFRET behavior in the dynamic and static regimes. If all donor–acceptor pairs in thesample behave similarly and are expected to have the same overall efficiency, theTRE curve will be 1 minus a single exponential. In contrast, if a distribution ofefficiency values is present in the system, a sharp deviation of this trend will be seen.It is expected that a single-exponential TRE curve could be a signature for thedynamic regime, whereas the static regimemay be characterized by a more complexTRE curve appearing to asymptote to a value less than 1. In the static isotropicregime, theory predicts that the TRE curve should follow the following trend:

TRE ¼ 1�ð10

dxð10

dze�z2 1þ3x2ð Þy ¼ 1�ffiffiffip

p2

ð10

dxerf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3x2ð Þyp� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3x2ð Þyp ; ð4:28Þ

where x and z are introduced in Equation 4.16, erf denotes the error function, and yis given by

y ¼ 32

R0

rDA

� �6 tt0D

; ð4:29Þ


where t is the time, t0D is the average donor lifetime in the absence of transfer,R0 is theF€orster distancewhen k2 ¼ 2=3. The rDAvalues estimatedby TRE analysis assuming astatic isotropic regime for C5V, C17V, andC32V (5.0, 5.3, and 5.5nm, respectively) arelower than the rDA values estimated from the average efficiency and fluorescencelifetime decay analysis assuming a dynamic isotropic regime (5.7, 5.9, and 6.2nm,respectively). This is expected because a large fraction of the FRETpairs in an isotropicstatic regime population will have k2 values close to zero. It is clear that theexperimental TRE data (Figure 4.21c) are not in perfect agreementwith the theoreticalTRE results based on Equation 4.28 (compare Figure 4.21c and d). While the basis ofthese small discrepancies is not known, we speculate that fluctuations in the separa-tion between donors and acceptors, or deviations from a purely isotropic distributionofqD andv angles, which are not taken into account inEquation4.28,may explain thisdiscrepancy. Regardless, it is quite remarkable that with only one adjustable parame-ter, 3R6

0

� �= 2t0D r

6DA

� �, the agreement between theory and experiment is as good as it is,

clearly indicating, we believe, that the static regime character of kappa-squared is themajor reason forwhy the time-resolved efficiency for C5V,C17V, andC32Vdeviates sodramatically from a single exponential rising from 0 to 1.If it is known that a population of FRET pairs are in the static isotropic regime,

with a few assumptions it is also possible to estimate the donor–acceptorseparation from experimentally measured k2h i values using as our starting pointan estimate of the average kappa-squared in the static regime introduced bySteinberg et al. [17]:

Eh i ¼ 3=2ð Þ k2h i�R60

3=2ð Þ k2h i�R60 þ r6DA

; ð4:30Þ

The angle brackets in this equation denote an average, �R0 ¼ R0 is the F€orsterdistance when k2 ¼ 2=3, and rDA is the donor-acceptor distance. Steinberg et al.have shown, in a graph, that k2h i varies with distance in a sigmoid fashion in thestatic regime, starting essentially at zero at very low distance, then rising slowly untilabout rDA ¼ 2=5R0, where k2h i starts to increase more strongly with increasingdistance until about rDA ¼ 7=5R0, where k2h i begins to level off reaching 2/3 at verylarge distances [12]. Between rDA ¼ 2=5R0 and rDA ¼ 7=5R0, k2h i varies linearlywith distance and is approximately equal to 2=3 rDA=R0 � ð2=5Þð Þ [12]. For example,the distances between the Cerulean and Venus fluorophores in C5V, C17V, andC32V most likely fall in this range between 0:4R0 and 1.4R0 (2.2–7.7 nm). Substi-tuting k2h i ¼ 2=3 u� 2=3ð Þ (with u rDA=R0) into (4.23) yields Equation 4.31 for u:

u6 ¼ 1� Eh iEh i u� 2

5

� �: ð4:31Þ

Solving this equation numerically using the measured average efficiencies andR0 ¼ 5:4 nm, the estimated rDA values are found to be 5.1, 5.4, and 5.8 nm for C5V,C17V, and C32V, respectively, in excellent agreement with distance estimatesderived from TRE analysis (5.0, 5.3, and 5.5 nm respectively), and with rDA valuesobtained after substituting Dale’s approximation, k2h i � 2=3ð Þ 1� Eh ið Þ [34], into


equation 4.30 (5.2, 5.4, and 5.8 nm respectively). Interestingly, this substitution

leads to rDA � �R0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Eh ið Þ2= Eh i6

q.

With regard to FRET in the static regime, it is important to realize that it ispossible to be in the static regime even when the FRET donor and acceptor used inan experiment are small fluorophores like fluorescein. Clearly, experimental factorssuch as high viscosity or short rigid linkers can restrain the motion of a smallfluorophore. Similarly, if fluorescent protein donors and acceptors are attached tointeracting proteins via a short rigid linker, the values of qD and v, and thus k2, maybe fixed and identical for every FRETpair in the population. If this is the case, FLIM–

FRET analysis will reveal a simple exponential decay that is faster than the decay ofthe donor alone, and TRE curves will asymptote from 0 to a value of 1. In this case,we are still in the static regime, but since the kappa-squared value is unique, theisotropic assumption is obviously no longer valid.

4.12Beyond Regimes

It is possible that the average rate of transfer is on the same order of magnitude as adominant rate of rotation for the donor or acceptor. In this case, the system is neitherin the dynamic regime nor in the static regime. Molecular dynamics simulations areextremely useful in this intermediate regime [19–23]. Analysis is still possible bybuilding mathematical models based on the idea that a system of donors andacceptors undergoing translational and/or rotational motion during the transfertime (inverse of the average transfer rate) can be described as a collection of stateswith transitions between them [42]. These states can be visualized as snapshots: at acertain moment, a donor is excited and has a particular orientation, while theacceptor has another orientation. This donor–acceptor pair is then in a D�A state. Alittle later the donor or acceptor changes its orientation, that is, a rotational transitionto another D�A state has occurred. FRETcorresponds to a transition to a DA� state. Asystematic description of such time developments implies selecting a representativeset of orientation states, evaluating kappa-squared values, and identifying transferrates and rates of rotation. This approach leads to a matrix equation for which theeigenvectors and eigenvaluesmust be found, so that intensities and anisotropies canbe calculated [42]. The following example illustrates this method. A donor andacceptor are at a fixed distance rDA from each other. The acceptor’s absorptionmoment has an isotropic degeneracy. The donor’s emission moment is linear andcan only have two orientation states: parallel to the “connection” line (line connect-ing the centers of donor and acceptor) or perpendicular to it. The rate of rotation ofthis moment is 1=tR. The FRETrate is ð3=2Þk2t�1

0D r�6, where t0D is the fluorescence

lifetime of the donor in the absence of FRET and r is the relative distance (rDAdivided by the F€orster distance if kappa-squared would be equal to 2/3). In thisexample, k2 equals either 4=3 or 1=3: 4=3 when the donor is in the “parallel” statewith its moment parallel to the connection line and 1=3 when the donor is in the

4.12 Beyond Regimes j101

“perpendicular” state with its moment perpendicular to this line. IDA , the fluores-cence intensity of the donor in the presence of acceptor after excitation with a veryshort pulse of light, is proportional to y== þ y ? . Here, y == is the fraction of the donorswith its moment parallel to the connection line and y? is the fraction with thismoment perpendicular to the connection line. For ID, the fluorescence intensity ofthe donor in the absence of FRET, y== ¼ y ? ¼ 1=2 at all times, but for I DA, y == ¼y? ¼ 1=2 only at time zero when the system is excited by the flash, whereas at latertimes y== > y ? until they both decay to zero at times much larger than t 0D . The rateequation for this example is

ddt

y==y?

� �¼ t �1

0D þ t �1R 1 þ ð8=3Þx½ � t�1

R

t �1R t �1

0D þ t� 1R 1 þ ð2=3Þx½ �

� �y==y?

� �;

ð4: 32Þ

where the differentiation is with respect to the time t and x ¼ ð3= 4Þ r �6 t �10D tR . The

time-resolved ef ficiency TRE (defined in Equation 4.27) can be calculated from thesolution of (4.32) in terms of the two eigenvectors with the initial condition y== ¼y? ¼ 1=2 and for this example reads (see FRETresearch.org for details)

TRE ¼ 1� 12

1� 1ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p !

e�ðt=tRÞ 1þð5=3Þxþffiffiffiffiffiffiffiffi1þx2

p� �

�12

1þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p !

e�ðt=tRÞ 1þð5=3Þx�ffiffiffiffiffiffiffiffi1þx2

p� �: ð4:33Þ

The special cases for this example are as follows:

No FRET with x ¼ 0 and TRE ¼ 0.The static regime with x ! 1, tR ! 1, while x=tR remains at ð3=4Þr�6t�1

0D ,yielding TRE ¼ TRESTATIC ¼ 1� ð1=2Þe�2r�6t�1

0Dt � ð1=2Þe�ð1=2Þr�6t�1

0Dt.

The dynamic regime with x ! 0, tR ! 0, while x=tR remains at ð3=4Þr�6t�10D ,

yielding TRE ¼ TREDYNAMIC ¼ 1� e�ð5=4Þr�6t�10D

t.

4.13Conclusions

In FRET situations where the transition moments of donor and acceptor areisotropically degenerate or reorient rapidly and completely within a time comparableto the lifetime of the excited donor state in the presence of acceptor, one can becertain that kappa-squared equals 2/3. Often this simplification is not warranted.However, we have indicated which methods can be utilized to diagnose the potentialproblems caused by the orientation factor, which alternative value can be used if theexperimental conditions allow one to find an average or actual kappa-squared value,and what can be done in cases where an average value is poorly defined. The concept



of time-resolved FRET efficiency (Equation 4.27) can be useful, especially whencombined with the idea of relaxation of the probability density p k2ð Þ: In the dynamicregime, p k2ð Þ relaxes from a relatively broad distribution to a narrow peak within atime much shorter than the lifetime of the excited state of the donor, whereas in thestatic regime this relaxation takes much longer than this lifetime.

Acknowledgments

We wish to thank Dr. Bob Dale for many helpful suggestions and stimulatingdiscussions. Bob also made us aware of important papers we had overlooked. Drs.Manual Prieto and David Lilley mentioned relevant papers as well. We thank themfor that. Dr. Klaus Suhling gave us useful ideas to improve the explanation ofrelevant issues. Dr. David Piston suggested to add electric field lines to Figure 4.5.We wish to acknowledge Dr. Paul Blank for stimulating discussions on strategiesfor fitting TRE decays to characterize separation distance in the static isotropicregime and Dr. Brent Krueger for designing Figure 4.20 especially for us,allowing us to use it in this chapter, and for stimulating discussions aboutkappa-squared and molecular dynamics simulations. We are indebted toDr. Phil Womble for writing a program, allowing us, with help from SandeepKothapalli, to obtain some preliminary data for the preparation of Figure 4.12. Wethank Sarah Witten Rogers for valuable help in calculating frequency distribu-tions for the relative distance. S.S.V. was supported by the intramural program ofthe National Institute of Health, National Institute on Alcohol Abuse andAlcoholism, Bethesda, MD.

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