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Kappa-squared: from nuisance to new sense
Reviews in Molecular Biotechnology 82 Ž2002. 181᎐196
Kappa-squared: from nuisance to new sense B.W. van der Meer U Department of Physics and Astronomy, Vanderbilt Uni¨ ersity, Nash¨ ille, TN 37235, USA
Abstract The orientation factor, which is commonly called kappa-squared, is often considered to be a nuisance because it represents a significant uncertainty in the distance obtained with the FRET technique. It is shown that this uncertainty is rather small in almost all cases of practical interest if one takes the width of a 67% confidence interval ŽCI. for the distance distribution as a measure of uncertainty. Kappa-squared has the potential to open up new information on orientations and rotations from time-resolved studies of donor and acceptor anisotropies. One can make sense of such data by designing matrix models for the transitions between states describing various orientations and positions of donors and acceptors in the system. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Orientation factor; Fluorescence depolarization; Distance distribution; Averaging regimes; Uncertainty in the distance; Most probable distance
1. Acronym The subject of this issue is known by a large number of names and acronyms most of which are listed in Table 1. This list is not complete, since every combination of deletions of the first seven letters of FSSRREEET will generate a valid acronym with a corresponding name Žvan der Meer et al., 1994.. The most common acronyms are FRET, RET and FET. A strong argument in favor of FRET is that, if one reads it U
Corresponding author. On leave from Western Kentucky University, Bowling Green, KY 42101, USA. Fax: q1-270745-2014. E-mail address: [email protected] ŽB.W. van der Meer..
as Fluorescence Resonance Energy Transfer, each of the four letters has significance and is essential Ždos Remedios and Moens, 1999.. One could say that RET is not an adequate name because you cannot do this technique without Fluorescence, and FET is not good because Resonance plays a key role. A strong argument in favor of RET is that it is not misleading in the sense that it does not suggest that the transferred energy is in the form of fluorescence. Fluorescence serves to detect transfer but is not a condition for transfer. Therefore, FETs Fluorescence Energy Transfer and FRET s Fluorescence Resonance Energy Transfer are somewhat dubious. However, a new way of reading FRET as ‘Fluorescence with Resonance Energy Transfer’ is both descriptive Žreveals all the essential elements. and not mislead-
1389-0352r02r$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 9 - 0 3 5 2 Ž 0 1 . 0 0 0 3 7 - X
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B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
Table 1 Names and acronyms for fluorescence with resonance energy transfer Name
Acronym
Fluorescence with resonance energy transfer Forster resonance energy transfer ¨ Fluorescence resonance energy transfer Resonance energy transfer Radiationless energy transfer Fluorescence energy transfer Forster energy transfer ¨ Energy transfer Excitation energy transfer Electronic excitation energy transfer Singlet᎐singlet energy transfer Singlet energy transfer Non-radiative energy transfer Forster excitation energy transfer ¨ Long-range resonance transfer Inductive resonance energy transfer Non-radiative excitation energy transfer Forster singlet᎐singlet radiationless resonance ¨ electronic excitation energy transfer
FRET FRET FRET RET RET FET FET ET EET EEET SSET SET NET FEET LRT IRET NEET FSSRREEET
ing Žcorrectly indicates the role played by fluorescence ..
2. Kappa-squared for one pair The orientation factor, 2 , for one donor᎐acceptor pair depends on three orientations, shown in Fig. 1: the emission transition dipole moment of the donor Ž D-direction.; the absorption transition dipole moment of the acceptor Ž A-direction.; and the line connecting the centers of the donor fluorophore and the acceptor chromophore Ž R-direction. ŽForster, 1948, ¨ 1993 ŽEnglish translation of the 1948 paper.; Dale et al., 1979; van der Meer et al., 1994, 1999.. The angle between D and A is T , that between D and R is D , and the angle between A and R is A Žsee Fig. 2.. The planes formed by D and A, by D and R, and by A and R are the DA-plane, DR-plane and AR-plane, respectively. The angle between the DR-plane and the AR-plane is . The oscillating electric field associated with D Žor one could say, caused by the donor dipole. is along the line ED , which is sometimes parallel to D, sometimes parallel to R, but in general can be
constructed as follows Žsee Fig. 3.. Project a unit vector along D onto the R direction. Measure this projection and mark off three times this distance along the R direction. Add to this vector Žhaving magnitude 3cos D and direction along R . a unit vector in the negative D-direction. This vector sum is along the ED direction. The angle between ED and A is . We can evaluate kappa-squared in three different ways in terms of the various angles: 2 s Ž cosT y 3cos D cos A .
2
2 s Ž sin D sin A cos y 2cos D cos A . 2 s cos 2 Ž 1 q 3cos 2 D . .
Ž1. 2
Ž2. Ž3.
The highest value for kappa-squared is four, which occurs if both D and A are parallel Žor antiparallel . to R. The lowest value is 0, which refers to a situation where A is perpendicular to ED . The case 2 s 4 can occur in a few ways w D, R and A must be Žanti. parallelx, but 2 s 0 can be realized in an infinite number of ways wwhenever A and ED are perpendicular to each
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
ˆ Aˆ and Rˆ are along the D-, A- and Fig. 1. The unit vectors D, ˆ is along the emission transition R-directions, respectively. D moment of the donor, Aˆ is along the absorption transition moment of the acceptor, and Rˆ is along the line from the center of the donor fluorophore to that of the acceptor ˆ Aˆ and Rˆ can lie in one plane, but are in chromophore. D, general not co-planar.
otherx. Eq. Ž3. for kappa-squared depends on only two variables, and D . This form of kappasquared refers to the electric field of the donor. Kappa-squared is proportional to the square of this field, which yields the factor 1 q 3cos 2 D , and is also proportional to cos 2 , where is the angle between this field and the acceptor transition moment. This means that if we pick a point for the location of the acceptor with respect to a donor place in the origin, then aligning the acceptor orientation A with the donor’s electric field
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line passing through that point will yield the highest kappa-squared value that can be achieved in that point: 1 q 3cos 2 D . The cos 2 proportionality implies that kappa-squared is zero whenever A is along a line perpendicular to an electric field line. Eq. Ž3. also allows for a geometrical construction of 2 Žvan der Meer, 1999.. An alternative procedure illustrating the relation between the relevant orientations and the magnitude of kappa-squared is shown in Fig. 3. Since the average of the cos 2 of a polar angle is 13 , and both and D are independent polar angles, it follows from Eq. Ž3., that the average value of 2 is 23 , if D, A and R are random. It is clear from Eqs. Ž1. ᎐ Ž3. and the definitions of the angles used in these equations that kappasquared does not change its value if we perform the following operations: 1. 2. 3. 4.
flip the donor transition moment; flip the acceptor transition moment; let the donor and acceptor trade places; and interchange the donor and acceptor transition moments.
3. Averaging regimes Averaging conditions or averaging regimes usually refer to rotational motion; when every donor and every acceptor can take up its entire range of
ˆ and R, ˆ A is between Aˆ and R, ˆ T is Fig. 2. The polar angles D , A , T and , and the azimuthal angle : D is between D ˆ and A, ˆ is between Aˆ and the electric field associated with the donor emission moment, and is between the between D 2 ˆ and Aˆ on a plane perpendicular to R. ˆ EˆD s Ž3 Rcos ˆ ˆ. projections of D D y D r 1 q 3cos D is a unit vector along the electric field ˆ and R, ˆ the AR-plane through Aˆ associated with the donor emission moment Žsee Fig. 3.. The DR-plane is the plane through D ˆ the DA-plane through Dˆ and A, ˆ and the ED A-plane through EˆD and A. ˆ and R,
'
184
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
where the brackets denote averages. Averaging regimes and averaging procedures not only are related to the orientation factor, but also depend on whether the distance is fixed or not Žvan der Meer et al., 1994.. A general approach to deal with the complexities of moving donors and acceptors is discussed below in the section called ‘beyond regimes’.
4. The average kappa-squared in the dynamic regime
Fig. 3. The relation between the ED -direction and kappaˆ on Rˆ is 3cos D . The vector sum squared. The projection of D ˆ ˆ of 3 Rcos D and yD is along the ED -direction. The projection of Aˆ on the ED -direction is cos. The area of the larger square is 1 q 3cos 2 D , that of the smaller square is cos 2 . The product of these two areas equals kappa-squared.
orientations during the transfer time Žthe inverse of the average transfer rate., the system is said to be in the dynamic averaging regime, and the dynamic averaging condition applies. In that regime, the average rate of rotations k R , is much larger than the average rate of transfer, k T : k R 4 k T . In this case, the orientation factor can be replaced by its appropriate average value ŽDale et al., 1979; van der Meer et al., 1994, 1999.. On the other hand, when the rates of rotation are slow compared to the rate of transfer, that is if k T 4 k R , the system is in the static averaging regime. The average transfer efficiencies in these regimes are Žvan der Meer et al., 1994.:
if k R 4 k T , average Es
3 2 6 ² : R 0 2 3 2 6 ² : R 0 q R 6 2
Ž4.
In the dynamic averaging regime the D-direction fluctuates rapidly around an orientation called the donor axis Ž D X -direction., and similarly, the A-direction fluctuates around the acceptor axis Ž A X -direction.. If these fluctuations take place in a time-scale that is short compared to the transfer time Žthe inverse of the average rate of transfer ., kappa-squared can be replaced by ² 2 :, the dynamic average of the orientation factor ŽDale et al., 1979; van der Meer, 1999.. This average depends on two parameters, d and a Ždefined below., and three variables, ⌽, ⌰D and ⌰A Ždefined below.. The parameter d is the axial depolarization factor for the donor transition moment,
¦ 32 cos y 12 ; 3 1 s H ž cos y / sin F Ž . d 2 2
ds ² d DX : s
if k T 4 k R , average Es
−
3 2 6 R0 q R6 2
< Ž5.
D
2
D
0
D
D
D
D
Ž6.
where D is the fluctuating angle between the donor transition moment and the donor axis, and FD Ž D . is a distribution function. Similarly, the axial depolarization factor for the acceptor transition moment is
¦ d : s ¦ 32 cos y 12 ; 3 1 s H ž cos y / sin F Ž . d 2 2 X A
as 3 2 6 R0 2
2
0
2
A
2
A
A
A
A
A
Ž7.
where A is the fluctuating angle between the acceptor transition moment and the acceptor axis,
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
and FAŽ A . is a distribution function. Note that it is assumed here that the distributions of the transition moments around the axis are cylindrically symmetric. The parameter dŽ a. is a second rank orientational order parameter with values between y0.5 and 1: ds 1 Ž as 1. when the transition moment is completely aligned with its axis; ds 0 Ž as 0. when the angle between the transition moment and its axis is equal to the magic angle at all times or when the transition moment is completely random; and ds y1r2Ž a s y1r2. when the angle between D and D X Ž A and A X . is 90⬚ at all times, that is, when the transition moment is degenerate in a plane perpendicular to the axis. The factors d and a can be obtained from fluorescence depolarization measurements ŽDale et al., 1979; van der Meer et al., 1994, 1999.. If one takes the z-axis along the R-direction, then the polar angle ⌰D is the angle between the donor axis and the z-axis, ⌰A is that between the acceptor axis and the z-axis. ⌰ is the azimuthal angle between the projections of the donor and acceptor axis on the xy-plane. The dynamical average of kappa-squared is given by ŽDale et al., 1979.: ² 2 : s
2 1 1 y dy aq d Ž 1 y a. cos 2 ⌰D 3 3 3 q aŽ 1 y d . cos 2 ⌰A q ad 2
= Ž sin⌰D sin⌰A cos⌽ y 2cos⌰D cos⌰A . . Ž8. Note that sin⌰D sin⌰A cos⌽ y 2cos⌰D cos⌰A s cos⌰ T y 3cos⌰D cos⌰A , where ⌰T is the angle between the donor axis and acceptor axis. Haas et al. Ž1978. and Steinberg et al. Ž1983. have a different approach to dealing with the orientation factor. They make use of the fact that many chromophores show mixed polarization in their spectral behavior. In other words, they assume that the chromophores do not move, but that the donor emission and acceptor absorption across the relevant spectral range of overlap are not characterized by single transition dipoles but by combinations of two or more incoherent dipole moments. In their formulation V1 , V2 and V3 are
185
the oscillating dipoles which characterize the donor fluorescence. They are mutually perpendicular and V12 q V22 q V33. Similarly, W1 , W2 and W3 are the oscillating dipoles characterizing the acceptor absorbance, which are also mutually perpendicular and have W12 q W22 q W33 . In this approach the average kappa-squared is: 3
² 2 : s
3
Ý Ý Vi 2 Wj 2 Žcos⌰i j y 3cos⌰D i cos⌰A j .
2
is1 jy1
Ž9. where ⌰i j is the angle between donor dipole i and acceptor dipole j, ⌰D i is the angle between donor dipole i and the line connecting the centers of donor and acceptor, ⌰A j , is the angle between acceptor dipole j and the line connecting the centers. This expression allows for complete asymmetry in the transition moments. However, this asymmetry is difficult to measure, and if we ignore it, that is, if we assume V12 s V22 s 13 Ž1 y d ., V32 s 31 Ž1 y 2 d ., W12 s W22 s 31 Ž1 y a. and W32 s 31 Ž1 y 2 a., the expression in Eq. Ž9. is completely equivalent to the one in Eq. Ž8.. This equivalence is shown in the Appendix A. In the dynamic averaging regime the fluorescence depolarization experiment cannot distinguish between degeneracy of the transition dipoles and fast rotational motion. Only one energy transfer rate constant is observable in this regime, and that is the average rate constant ² k T :, given by: ² kT : s ² k 2 : k D
6 3 R rR 2Ž 0 .
Ž 10.
where k D is the rate of donor decay in the absence of acceptor, R 0 is the Forster distance ¨ for 2 s 2r3, and R is the donor᎐acceptor distance, which is assumed to be unique and essentially constant during times of the order of 1rk D . In the dynamic averaging regime the donor intensity at time t after excitation with a flash of light is proportional to: eyŽ ² k T :qk D .t .
Ž 11.
Minima and maxima of the average kappasquared correspond with the lower and upper
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
186
limits of the distance. These extremities can be found by setting the derivatives of kappa-squared in Eq. Ž8. with respect to ⌽, ⌰D and ⌰A to zero, and solving the resulting equations. There are six possible candidates for maxima and minima. These are:
In any point in the Ž d,a.-plane, one of these
2A s
2 2 2 q aq dq 2 ad 3 3 3
Ž 12.1.
2M s
2 1 1 1 q aq dy adq < ay d < 3 6 6 2
Ž 12.2.
extremities is the maximum and another is the minimum, but it depends on the values of d and a which one of the six extremities is the maximum and which is the minimum. Accordingly, the Ž d,a.-plane is divided into different regions defined below in Table 2. This table refers to the case where the transfer depolarization is not known. The case that this depolarization is known has been discussed by Dale et al. Ž1979. and van der Meer Ž1999..
2L s
2 1 1 1 q aq dy ady < ay d < 3 6 6 2
Ž 12.3.
5. The average kappa-squared in the static regime
2H s
2 1 1 y ay dq ad 3 3 3
Ž 12.4.
2T s
1 4 Ž 1 y a.Ž 1 y d . q 9 9
In the static averaging regime, the average kappa-squared depends on the distance ratio RrR 0 , that is the donor᎐acceptor distance divided by the Forster distance for 2 s 2r3. Stein¨ Ž . berg et al. 1983 found this ² 2 :-value. This average orientation factor is the one that would be used in Forster’s equation for the average ¨ efficiency of FRET between donor᎐acceptor pairs for which rotational diffusion is frozen. In their analysis, they assumed that each pair is isolated from all other pairs and that the electronic transition of each chromophore is a single dipole, which is randomly oriented in space. Their result is a plot of ² 2 : vs. RrR 0 ŽFig. 1 in Steinberg et al., 1983., which essentially is the following trend:
2P s
='Ž 1 y a.Ž 1 y d .Ž 1 q 2 a.Ž 1 q 2 d .
Ž 12.5.
2 1 1 y ay d. 3 3 3
Ž 12.6.
For A 2 : the axis of both the donor and the acceptor are Aligned with the separation vector. For M 2 : the axis of the Most prolate distribution Ždonor or acceptor. is aligned with the separation vector; the other Žacceptor or donor. is perpendicular to the separation vector. In other words, if d) a, the donor axis is aligned with the separation vector and the acceptor axis is perpendicular to it, and vice versa if a) d. For L 2 : the axis of the Least prolate distribution Ždonor or acceptor. is aligned with the separation vector; the other Žacceptor or donor. is perpendicular to the separation vector. For H 2 : the donor and acceptor are in a ‘H-configuration’, that is their axis are parallel to each other, but perpendicular to the separation vector. For T 2 : the axis of both donor and acceptor are Tilted at some angle from the separation vector, but are co-planar with this vector. For p : the separation vector is Perpendicular to the axis of the donor, which is perpendicular to that of the acceptor, which is also perpendicular to the separation vector. 2
¡0 2
~ 3 ž R y 25 R
² 2 : f
¢
2 3
2 R 5 0 2 7 R F . 5 0 R F 5 R0 7 R F R0 5 Ž 13. RF
0
/ rR
0
If the dipoles are not random, but distributed in a cylindrically symmetric way around axis following distribution functions FD Ž D . and FAŽ A . Žintroduced above., the intensity emitted by the donor in the static averaging regime at time t,
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
187
Table 2 Regions with different kappa-squared maxima and minima in the plane of all possible depolarization factors for donors and acceptors Region
Maximum
Minimum
0 F dF 1, 0 F aF 1 y 12 F dF 0, 12 y dF aF 1 1 1 2 F dF 1, 2 y dF aF 0 1 q 3d 1 y 2 F dF y 13 , y F aF 12 y d 3 q 5d 1 q 3d y 13 F dF 0, F aF 12 y d 2q2 d 1 q 3d 1y2 d 0 F dF 12 , y F aF 3 q 5d 2 dy 3 1 q 3d 1 1 2 F dF 1, y 3 q 5d F aF 2 y d 1 q 3d y 13 F dF 0, 0 F aF 2q2 d 1y2 d 0 F dF 12 , F aF 0 2 dy 3 1 q 3d y 12 F dF y 13 , 0 F aF 3 q 5d 1 q 3d 0 F dF 1, y 12 F aF y 3 q 5d 1 q 3d 1 1 y 2 F dF y 3 , F aF 0 2q2 d 1y2 d 1 1 y 2 F dF 0, y 2 F aF 2 dy 3 1y2 d 1 q 3d 1 1 y 2 F dF y 3 , F aF 2 dy 3 2q2 d 1y2 d 1 q 3d y 13 F dF 0, F aF y 2 dy 3 3 q 5d
A 2 M 2 M 2
P2 H 2 H 2
M 2
T 2
M 2
T 2
M 2
T 2
M 2
T 2
M 2
L 2
M 2
L 2
M 2
A 2
M 2
A 2
H 2
L 2
H 2
L 2
H 2
T 2
H 2
T 2
after excitation with a flash of light, is proportional to
yŽ k D qk T .t
H0 H0 e
FD Ž D .
=FA Ž A . sin D sin A d D d A .
Ž 14.
6. Assessing the orientational error The main goal of FRET is to find the distance or distribution of distances between donor and acceptor. In the dynamic averaging regime it is possible to deduce the uncertainty in the distance resulting from assuming an average value for 2 . There are three probability functions that are directly related to 2 ; the range probability, PR Ž 2MIN ª 2 ., the probability distribution, pŽ 2 ., and the relative distance distribution, QŽ ..
PR Ž 2MIN ª 2 . is the probability of finding a value for the average kappa-squared in the range from its lowest value to 2 , pŽ 2 .d 2 is the probability of encountering a value for the average kappasquared in the interval 2 to 2 q d 2 , and QŽ .d is the probability of finding a value for the relative distance between and q d. The relative distance is the ratio of the actual donor᎐acceptor distance over the distance obtained from FRET data assuming 2 s 23 . These and related distances are defined in Table 3. The function Q was introduced by Haas et al. Ž1978. and discussed by Steinberg et al. Ž1983., but is slightly modified here; instead of considering Q as a function of R⬘rR, we take Q as a function of RrR⬘. The advantage of the latter choice is that RrR⬘ has a limited range, varying between 0 and 1.348 in the case of perfect order, whereas R⬘rR varies over the much wider range from 0.742 to infinity. There is a one-to-one relation between a
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
188 Table 3 Distances Symbol
Definition
R R0 R0 R⬘ MI N MA X m p l o h i
Distance between a donor᎐acceptor pair Forster distance Ždepends on kappa-squared. ¨ Forster distance for 2 s 23 ¨ Apparent distance, obtained by assuming 2 s 23 Relative distance, RrR⬘ Relative distance corresponding with lowest 2 Relative distance corresponding with highest 2 Relative distance at which Q has its maximum Lower limit of the 67% CI for Upper limit of the 67% CI for
particular kappa-squared value and the donor᎐acceptor distance via the following equations: 2 s
2 6 3
or:
s
3 2 2
ž /
1r6
.
Ž 15.
This relation has a dramatic effect on the interdependence between pŽ 2 . and QŽ ., as illustrated in Fig. 4. The peak in pŽ 2 . is always at or near the minimum kappa-squared value, but as a result of Eq. Ž15., the maximum of QŽ . is shifted dramatically towards distances corresponding with higher kappa-squared values in many cases. However, the characteristic features of QŽ . vary widely as the depolarization factors change between 0 and 1 as illustrated in Fig. 5. QŽ . has a very sharp peak near s 1 if both donor and acceptor depolarization factors are small. If one of the depolarization factors is close to zero and the other is larger than that, QŽ . has a peak at the lowest value for the relative distance, slightly below s 1. On the other hand, if both factors are approximately equal and bigger than zero, the most probable relative distance is slightly above s 1. In determining the FRET distance and its uncertainty due to kappa-squared, it is not only relevant to find the minimum and maximum value of the distance, but is also important to find the most probable distance and the 67% CI. These distances and intervals are studied for the following cases in detail below: as ds 1; as d< 1
Fig. 4. The effect of 2 s 23 6 on the relation between pŽ K 2 . and QŽ ..
and the case where the two depolarization factors equal zero. as ds 1 is the worst case scenario where the orientational uncertainty is the highest. PR Ž 2MIN ª 2 . in this case is given by:
¡
2 ln Ž 2 q 3 . 3
( ' ~( ( ' ¢ ž' ' / 2 y 1 q 3
PR Ž 0 ª 2 . s
ln
2 3
2q 3
q y1 2
0 F 2 F 1
2
. 1 F 2 F 4 Ž 16.1.
For as ds 1, pŽ 2 ., the derivative of PR with respect to 2 , is:
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
189
The graph of QŽ . for as ds 1 is shown in Fig. 6. The most probable relative distance, m p s 1.070, where QŽ . of Eq. Ž16.3. has its maximum. The 67% CI for the relative distance is from lo s 0.813 wcorresponding to PR s 2r3 lnŽ2 q '3 . s 13 x to h i s MAX s 1.348. The most probable is slightly below 1.081, the middle of this interval. In the case where as d< 1, PR Ž 2MIN ª 2 . is given by:
'
PR Ž 0 ª 2 . s
¡ Fig. 5. The characteristics of the relative distance distribution, QŽ ., for various values of the depolarization factors. Details of QŽ . for as ds 1 , for as ds 0.25, and for as 1, ds 0 Ž as 0, ds 1. are shown in Figs. 6᎐8, respectively.
¢
pŽ2 . s
¡
1
'3
~2
¢2'3
2
1
2
ln Ž 2 q '3 . ln
ž'
2 q '3
2 q ' 2 y 1
¡ '2 lnŽ2 q '3 . 2
Q Ž . s '2 ln 2
¢ž 32 /
(
ž
2 1 y a 3 3
/
2 4 q a 3 3 Ž 17.1.
0 F 2 F 1
/
.
where is:
Ž 16.2.
)
1F F4 2
Since pŽ 2 .d 2 and QŽ .d both denote the fraction of the population of donor᎐acceptor pairs with the orientation factor between 2 and 2 q d 2 , we have QŽ .d s pŽ 2 .d 2 or QŽ . s pŽ 2 .d 2rd s pŽ 2 s 23 6 .45.Therefore, QŽ . in this case is given by:
~
~
2 2 2 y 3 q 3a 4 a 2 2 y aF 2 F 3 3 y1 Ž . Ž q y tan 1q2 . 4 2 1 y aF 2 F 3 3
0FF
3 2
(
0
2 1 y a 3 3 s a
(
2 a 6 y 1 y . 3a 2 Ž 17.2.
ž
pŽ2 . s
.
y1 Ž
.
2 2 2 1 y aF 2 F q a 3 3 3 3 . 2 1 2 4 2 y aF F q a 3 3 3 3 Ž 17.3.
QŽ . in this case is given by:
1r6
F F 6 1r6 Ž 16.3.
/
For as d< 1, pŽ 2 . is:
¡ ~ 4a ¢4a y 1a tan
1r6
ž /
2 q '3 2 6 2 6 q y1 3 3
s
2 y
QŽ . s
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
190
1
pŽ2 . s
(ž
2 a 2 y
2 1 q a 3 3
/
2 1 2 2 y aF 2 F y a. 3 3 3 3
Ž 18.2.
QŽ . in this case has the following form: QŽ . s
2 5'3
(
2 a 6 y 1 q
Fig. 6. Q vs. for as ds 1. The striped area corresponds with the 67% CI for the relative distance. MI N s 0, l o s 0.813, m p s 1.070, the center of the 67% CI is at s 1.081, and h i s MAX s 1.348.
¡
~
¢ ž
1 1 aF 2 F 1 q a 6 12 . 1 1 1q aF 2 F 1 q a 12 3 Ž 17.4.
1y
5
4a 5 1 1 y tany1 Ž . a a
/
The graph of QŽ . for as ds 0.25, which is essentially the upper limit of this interval, is shown in Fig. 7. The most probable relative distance 1 equals m p s 1 q a, is that where QŽ . of 12 Eq. Ž17.4. has its maximum. The 67% CI for the 1 relative distance is from l o s 1 y 16 y a 3 2 2 corresponding to PR s w rŽ4 a.xŽ 2 y 3 q 3 a. s 1r34 to h i s MAX s 1 q 13 a. The most probable 1 1 is slightly below l o s 1 y y a, the 12 6 middle of this interval. In the case ds 0 Žor as 0. PR Ž 2MIN ª 2 . is:
ž
ž
PR Ž 0 ª
2.
s
)
2 y
ž
ž 1 y 12 a /
1 a 2
/
1r6
F F Ž1 q a.
Ž 18.3. 1r6
.
The graph of QŽ . for as 1 Žor ds 1., which corresponds with the worst error in this case, is shown in Fig. 8. QŽ . has its peak at m p s MIN s Ž1 y 12 a.1r6 Žs 0.891 for as 1.. The 67% CI for the relative distance is from m p s MIN s Ž1 y 1 .1r6 to h i s Ž1 y 12 a.1r6 Žs 1.026 for as 1. 2a Žfrom PR s Ž 2 y 23 q 13 a. ras 23 .. The maximum relative distance is MA X s Ž1 y a.1r6 Žs 1.122 for as 1. . Steinberg et al. Ž1983. proposed to take the width of the peak of QŽ . as a measure of the orientational error in the distance. Note, however, that the peak of QŽ . in this case is infinitely high. Therefore, the width of the peak at half-maximum height is zero here,
'
/
/
2 1 q a 3 3 a
2 1 2 2 y aF 2 F y a. 3 3 3 3 For ds 0 Žor as 0. Ž 2 . is given by:
Ž 18.1.
Fig. 7. Q vs. for as ds 0.25. The striped area corresponds with the 67% CI for the relative distance. MI N s 0.953, l o s 0.984, m p s 1.020, the center of the 67% CI is at s 1.027, and h i s MAX s 1.070.
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
191
Fig. 8. Q vs. for as 1, ds 0 Ž as 0, ds 1.. The striped area corresponds with the 67% CI for the relative distance. MI N s l o s m p s 0.891, h i s 1.026, the center of the 67% CI is at s 0.959, and MA X s 1.122.
and is in general, not a good measure for the error in the distance. Taken together, these case studies suggest the following trends: 1. The 67% CI for the relative distance is significantly smaller than the full range from the minimum to the maximum relative distance. Figs. 9 and 10 illustrate this point. 2. The center of the 67% CI is always close to the most probable distance with a difference of less than 7%. This center is slightly higher than the most probable distance Žsee Figs. 9 and 10.. 3. The center of the 67% CI is always close to the distance corresponding to 2 s 23 with a difference of less than 8%. If the donor and acceptor depolarization factors are roughly equal to each other, the most probable distance is slightly higher than the 2 s 23 distance ŽFig. 9.. If one of the two depolarization factors is close to zero, the most probable distance is slightly lower than the 2 s 23 distance ŽFig. 10.. 4. The width or the half-width of the 67% CI is a good measure for the uncertainty in the distance due to kappa-squared. This error is relatively small for most realistic depolarization values. However, this uncertainty exceeds
Fig. 9. MA X , MIN , h i , l o and m p versus one of the depolarization factors for the case that the two depolarization factors are equal to each other. Curves drawn as thicker lines represent values that are completely known, and curves drawn as thinner lines denote interpolated values.
20% if one or both of the depolarization factors approach their maximum values. This trend is shown in Figs. 11 and 12.
7. Beyond regimes What do you do when your system is in between the static and dynamic averaging regimes? You build models: mathematical models that result in matrix equations Žvan der Meer et al., 1993.. A system of donors and acceptors undergoing translational andror rotational motion during the transfer time Žinverse of average transfer rate. can be thought of as a collection of states with transitions between them. Think of these states as snapshots; imagine that you can observe a donor᎐acceptor pair. You might see at a certain moment a donor that is excited and has a certain
192
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
Fig. 12. The uncertainty in the distance due to kappa-squared versus one of the depolarization factors for the case that the other factor is zero. Here the error is taken as the full width of the CI, Ž h i y l o . = 100%.
Fig. 10. MA X , MIN , h i , l o and m p versus one of the depolarization factors for the case that the other factor is zero.
Fig. 11. The uncertainty in the distance due to kappa-squared versus one of the depolarization factors for the case that the two depolarization factors are equal to each other. Here, the 1 error is taken as half of the width of the CI, Ž h i y l o . = 2 100%. The part of the curve drawn as a thicker line represents values that are completely known, and the thinner part of the curve denotes interpolated values.
orientation while the acceptor has another orientation Žthe pair is in a DUA state.. A nanosecond later the donor may still be more or less in the same situation, but the acceptor has quite a different orientation Ža rotational transition to another DUA state has taken place.. A few nanoseconds later the donor has lost its excitation energy and the acceptor is excited Žan energy transfer transition to a DAU state has occurred.. A systematic description of such time developments implies selecting a representative set of orientational states, evaluating kappa-squared values, identifying transfer rates and other rate constants. This approach leads to a matrix equation for which the eigenvectors and eigenvalues must be found, so that intensities and anisotropies can be calculated. To illustrate this procedure let us consider the following example. Suppose a macromolecule contains a donor that can have two distinct orientations, 1 and 2 with an angle ␥ between them, and an acceptor with a fixed orientation at a fixed distance from the donor. The macromolecule itself has a random orientation in the sample and undergoes rotations too slow to observe on the time scale of the donor fluorescence. In this case we have two DUA states corre-
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
Ž SQ y ⌬ k T . 2 cos 2 z1
sponding with the two donor orientations. Let the rate of rotational jumps between the two donor orientations be k R , and the rates of transfer be k T q ⌬ k T and k T y ⌬ k T for state 1 to the acceptor and for state 2 to the acceptor, respectively. If the rate of donor emission in the absence of acceptor is k D , and the probability that state 1 Ž2. is excited is x 1 Ž x 2 ., then the time-derivatives of x 1 and x 2 are given by:
␣ 12 s
ž
x1 . x2
ž /
Ž 19.
At time zero, when the system is excited with a flash of vertically polarized light, the values for x 1 and x 2 are x l s cos 2 z1 and x 2 s cos 2 z 2, where z1 Ž z 2 . is the angle between the vertical and a donor orientation 1 Ž2.. The solution of Eq. Ž19. with these boundary conditions is based on the eigenvalues and eigenvectors of the matrix in Eq. Ž19. and has the following form: x 1 s ␣ 11 ey t q ␣ 12 eyŽq.t
Ž 20.1.
x 2 s ␣ 21 ey t q ␣ 22 eyŽq.t
Ž 20.2.
where the eigenvalues and q are determined by: s k D q k R q k T y SQ ,
SQ s
'k
2 Ž . R q ⌬ kT
2
and s 2 SQ Ž 21. and the coefficients in Eqs. Ž20.1. and Ž20.2. are: ␣ 11 s
k R2 cos 2 z1 q k R Ž SQ y ⌬ k T . cos 2 z 2 k R2 q Ž SQ y ⌬ k T .
2
Ž 22.1.
k R2 q Ž SQ y ⌬ k T .
Ž 22.2.
2
2
␣ 21 s
ž /
kR yk D y k R y k T q ⌬ k T
yk R Ž SQ y ⌬ k T . cos 2 z 2
k R Ž SQ y ⌬ k T . cos 2 z1
d x1 s dt x 2 yk D y k R y k T y ⌬ k T kR
193
␣ 22 s
/
q Ž SQ y ⌬ k T . cos 2 z 2 k R2 q Ž SQ y ⌬ k T .
Ž 22.3.
2
yk R Ž SQ y ⌬ k T . cos 2 z1 q k R2 cos 2 z 2 k R2 q Ž SQ y ⌬ k T .
2
. Ž 22.4.
The time-resolved donor intensity observed through a vertical emission polarizer, IV , is the following linear combination of x 1 and x 2 : IV s c - ␣ 11cos 2 z1 q ␣ 21cos 2 z 2 ) ey t qc - ␣ 12 cos 2 z1 q ␣ 22 cos 2 z 2 ) eyŽq.t
Ž 23.
where c is a constant proportional to the concentration, molar extinction coefficient and quantum yield of the donor in the absence of acceptor. The brackets indicate an average over all polar angles z1 and z 2 under the constraint that the angle between donor orientation 1 and 2 is ␥ for all orientations of the protein. The time-resolved donor intensity observed through a horizontal emission polarizer, IH , is also a linear combination of x l and x 2 : IH s c - ␣ 11cos 2 x1 q ␣ 21cos 2 x 2 ) ey t
Ž 24. qc - ␣ 12 cos 2 x1 q ␣ 22 cos 2 x 2 ) eyŽq.t
where x l Ž x 2 . is the angle between donor orientation one Žtwo. and a line perpendicular to the vertical. Here, the brackets have a similar meaning as in Eq. Ž23. except that the averaging is not only over z1 and z 2 , but also over x l and x 2 , under the constraint that the horizontal and vertical direction are perpendicular to each other. Using the averages,
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194
1 , 5 ²cos 2 x1cos 2 z 2 : s ²cos 2 z1cos 2 x 2 : 3 2 s y sin 2 ␥ 15 15 and ²cos 2 z1cos 2 z 2 : s ²cos 2 z1cos 2 x1 :
The donor intensity and anisotropy in the static averaging regime follow from Eqs. Ž26. and Ž27. by taking the limit: k Rr⌬ k T ª 0 at which ª 1:
²cos 4 z1 : s ²cos 4 z 2 : s
Ž 25.
Static averaging regime: ID is proportional to eyŽ k Dqk Ty⌬ k T . Ž 1 q ey2 ⌬ k T t . t
s ²cos 2 z 2 cos 2 x 2 : 1 s , 15
Static averaging regime:
Ž 31.
rD s
2 . 5
Ž 32.
the donor intensity, ID s IV q 2 IH , and anisotropy, rD , are derived from Eqs. Ž23. and Ž24.:
This specific example illustrates the following general points:
ID is proportional to ey t Ž 1 q ey t .
1. The donor anisotropy in the presence and absence of acceptor indicates which averaging regime applies: if this anisotropy is virtually constant in time, then, most likely, the static averaging regime applies; if the donor anisotropy in the absence of acceptor is essentially identical to that in the presence of acceptor, the dynamic averaging regime applies. 2. Anisotropies give information about kappasquared. In the dynamic averaging regime this information allows one to assess the uncertainty in the FRET distance due to kappa-squared ŽHaas et al., 1978; Dale et al., 1979; Steinberg et al., 1983, see also above.. Outside of this averaging regime the combination of time-resolved donor Žand acceptor. intensities and anisotropies gives information about the distribution of kappa-squared values, and should allow for a more precise determination of the donor᎐acceptor distance. 3. Outside the dynamic and static averaging regimes time-resolved donor Žand acceptor. anisotropies give information about the orientational distributions and rotational motions of the donors and acceptors in an unique and interesting way. For example, such anisotropies should be capable of determining the angular size of rotational jumps occurring in such systems Žvan der Meer et al., 1993..
Ž 26.
2 3 3 1 y sin 2 ␥ q sin 2 ␥ 5 4 4
rD s
½
q
3 2 3 sin ␥ q 1 y sin 2 ␥ ey t 4 4
ž
/
5
r 1 q ey t 4 Ž 27.
with
s
ž
SQ y ⌬ k T y k R SQ y ⌬ k T q k R
2
/
.
Ž 28.
The donor intensity and anisotropy in the dynamic averaging regime follow from Eqs. Ž26. and Ž27. by taking the limit: ⌬ k T rk R ª 0 at which ª 0 yielding: Dynamic averaging regime: ID is proportional to eyŽ k Dqk T .
t
Ž 29.
Dynamic averaging regime: rD s
2 3 3 1 y sin 2 ␥ q sin 2 ␥ ey2 k R t . 5 4 4
ž
/
Ž 30.
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cos⌰i1cos⌰A1 q cos⌰i2 cos⌰A2
Appendix A Substituting V12 s V22 s 13 Žl y d ., V32 s 13 Žl q 2 d ., W12 s W22 s 13 Žl y a., and W32 s 13 Ž1 q 2 a., into Eq. Ž9. transforms this equation into: 2 s
1 Ž 1 q 2 d . Ž 1 q 2 a. Tp p q Ž 1 y a. Tp m 4 9 q
195
s cos⌰D i y cos⌰i3 cos⌰A
rule 6.
Evaluating the square in the definition of Tp m yields: 2
Tp m s 9cos 2 ⌰D
1 Ž 1 y d . Ž 1 q 2 a. Tm p q Ž 1 y a. Tm m 4 9
Ý cos 2 ⌰A j js1
Ž A1.
2
y 6cos⌰D
2
Ý cos⌰ 3 j cos⌰A j q Ý cos 2 ⌰ 3 j . js1
js1
Ž A4.
where we have utilized the abbreviations given in Eqs. ŽA2. and ŽA3.. Tp p ' Ž 3cos⌰D cos⌰A y cos⌰T .
2
2
Tp m '
Ý Ž3cos⌰D cos⌰A j y cos⌰ 3 j .
Ž A2.
2
Tp m s 9cos 2 ⌰D y 9cos 2 ⌰D cos 2 ⌰A
js1
y 6cos⌰D Ž cos⌰D y cos⌰D cos⌰A .
2
Tm p '
Ý Ž3cos⌰D i cos⌰A y cos⌰i3 .
2
q1 y cos 2 ⌰ T s 1 q 3cos 2 ⌰D y Tp p .
is1 2
Tm m '
Using rule 1 for the first term, rule 6 for the second term and rule 4 for the third term, simplifies Tp m as follows:
2
Ý Ý Ž3cos⌰D i cos⌰A j y cos⌰i j .
2
Ž A3.
Ž A5.
Similarly, we find:
is1 js1
Tm p s 1 q 3cos 2 ⌰A y Tp p . The following projection rules allow us to simplify Tp m , Tm p and Tp p : cos 2 ⌰A1 q cos 2 ⌰A2 s 1 y cos 2 ⌰A
Evaluating the square in the definition of Tm m yields:
rule 1 2
cos 2 ⌰D1 q cos 2 ⌰D 2 s 1 y cos 2 ⌰D
rule 2
2
Tm m s 9 Ý cos 2 ⌰D i is1
Ý cos 2 ⌰A j js1
2
cos ⌰ 1 j q cos ⌰ 2 j s 1 y cos ⌰ 3 j 2
2
2
rule 3
rule 4
2
y 6 Ý cos⌰D i is1 2
cos 2 ⌰i1 q cos 2 ⌰i2 s 1 y cos 2 ⌰i3
Ž A6.
qÝ
Ý cos⌰ 3 j cos⌰A j js1
2
Ý cos 2 ⌰i j .
Ž A7.
is1 js1
cos⌰ 1 j cos⌰D1 q cos⌰ 2 j cos⌰D 2 s cos⌰A j y cos⌰ 3 j cos⌰D
rule 5
Employing rules 1 and 2 for the first term, rule 6 for the second term and rule 4 for the third term reduces Eq. ŽA7. to:
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196
Tm m s 9 Ž 1 y cos 2 ⌰D .Ž 1 y cos 2 ⌰A .
References
2
y6 Ý cos⌰D i Ž cos⌰D i y cos⌰i3 cos⌰A . is1 2
q Ý Ž 1 y cos 2 ⌰i3 . s 9 Ž 1 y cos 2 ⌰D . is1 2
= Ž 1 y cos 2 ⌰A . y 6 Ý cos 2 ⌰D i
Ž A8.
is1 2
q6 Ý cos⌰D i cos⌰i3 cos⌰A is1 2
q Ý Ž 1 y cos 2 ⌰i3 . . is1
Using rule 2 in the 2nd term following the last equal sign in Eq. ŽA8., rule 5 in the next term and rule 3 in the last term, simplifies Tm m to: Tm m s 9 Ž 1 y cos 2 ⌰D .Ž 1 y cos 2 ⌰A . y 6 q 6cos 2 ⌰D q 6cos 2 ⌰A q 1 q cos 2 ⌰A s 4 y 3cos 2 ⌰D y 3cos 2 ⌰A q Tp p .
Ž A9.
Substituting Tp m s 1 q 3cos 2 ⌰D y Tp p , Tm p s 1 q 3cos 2 ⌰A y Tp p and Tp p s 4 y 3cos 2 ⌰D y 3cos 2 ⌰A q Tp p into Eq. ŽA1., and rearranging terms yields: 2 s 23 y 13 dy 13 aq dŽ1 y a.cos 2 ⌰D q aŽ1 y d .cos 2 ⌰A q adTp p , which is identical to Eq. Ž8..
Acknowledgements Support by NASA NCCW60 is gratefully acknowledged.
Dale, R.E., Eisinger, J., Blumberg, W.E., 1979. The orientational freedom of molecular probes: the orientation factor in intramolecular energy transfer. Biophys. J. 26, 161᎐194. ŽAppendix B was corrected in 1980, Biophys. J. 30, 365.. dos Remedios, C.G., Moens, P.D.J., 1999. Resonance energy transfer in proteins. In: Andrews, D.L., Demidov, A.A. ŽEds.., Resonance Energy Transfer. John Wiley and Sons Ltd, Chichester, pp. 1᎐55. Forster, T., 1948. Zwischenmolekulare Energiewanderung und ¨ Fluoreszenz. Ann. Phys. 2, 55᎐75. Forster, T., 1993. Intermolecular energy migration and fluo¨ rescence. In: Mielczarek, E.V., Greenbaum, E., Knox, R.S. ŽEds.., Biological Physics. American Institute of Physics, New York, pp. 183᎐221. Translation of Forster, T., 1948. ¨ Haas, E., Katchalski-Katzir, E., Steinberg, I.Z., 1978. Effect of the orientation of donor and acceptor on the probability of energy transfer involving electronic transitions of mixed polarization. Biochemistry 17, 5065᎐5070. Steinberg, I.Z., Haas, E., Katchalski-Katzir, E., 1983. Longrange non-radiative transfer of electronic excitation energy. In: Cundall, R.B., Dale, R.E. ŽEds.., Time-Resolved Fluorescence Spectroscopy in Biochemistry and Biology. Plenum Press, New York, pp. 411᎐450. van der Meer, B.W., Raymer, M.A., Wagoner, S.L., Hackney, R.L., Beechem, J.M., Gratton, E., 1993. Designing matrix models for fluorescence energy transfer between moving donors and acceptors. Biophys. J. 64, 1243᎐1263. van der Meer, B.W., Coker III, G., Chen, S.-Y.S., 1994. Resonance Energy Transfer: Theory and Data. VCH Publishers, Inc., New York. van der Meer, B.W., 1999. Orientational aspects in pair energy transfer. In: Andrews, D.L., Demidov, A.A. ŽEds.., Resonance Energy Transfer. John Wiley and Sons Ltd, Chichester, pp. 151᎐172.
Kappa-squared: from nuisance to new sense B.W. van der Meer U Department of Physics and Astronomy, Vanderbilt Uni¨ ersity, Nash¨ ille, TN 37235, USA
Abstract The orientation factor, which is commonly called kappa-squared, is often considered to be a nuisance because it represents a significant uncertainty in the distance obtained with the FRET technique. It is shown that this uncertainty is rather small in almost all cases of practical interest if one takes the width of a 67% confidence interval ŽCI. for the distance distribution as a measure of uncertainty. Kappa-squared has the potential to open up new information on orientations and rotations from time-resolved studies of donor and acceptor anisotropies. One can make sense of such data by designing matrix models for the transitions between states describing various orientations and positions of donors and acceptors in the system. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Orientation factor; Fluorescence depolarization; Distance distribution; Averaging regimes; Uncertainty in the distance; Most probable distance
1. Acronym The subject of this issue is known by a large number of names and acronyms most of which are listed in Table 1. This list is not complete, since every combination of deletions of the first seven letters of FSSRREEET will generate a valid acronym with a corresponding name Žvan der Meer et al., 1994.. The most common acronyms are FRET, RET and FET. A strong argument in favor of FRET is that, if one reads it U
Corresponding author. On leave from Western Kentucky University, Bowling Green, KY 42101, USA. Fax: q1-270745-2014. E-mail address: [email protected] ŽB.W. van der Meer..
as Fluorescence Resonance Energy Transfer, each of the four letters has significance and is essential Ždos Remedios and Moens, 1999.. One could say that RET is not an adequate name because you cannot do this technique without Fluorescence, and FET is not good because Resonance plays a key role. A strong argument in favor of RET is that it is not misleading in the sense that it does not suggest that the transferred energy is in the form of fluorescence. Fluorescence serves to detect transfer but is not a condition for transfer. Therefore, FETs Fluorescence Energy Transfer and FRET s Fluorescence Resonance Energy Transfer are somewhat dubious. However, a new way of reading FRET as ‘Fluorescence with Resonance Energy Transfer’ is both descriptive Žreveals all the essential elements. and not mislead-
1389-0352r02r$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 9 - 0 3 5 2 Ž 0 1 . 0 0 0 3 7 - X
182
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
Table 1 Names and acronyms for fluorescence with resonance energy transfer Name
Acronym
Fluorescence with resonance energy transfer Forster resonance energy transfer ¨ Fluorescence resonance energy transfer Resonance energy transfer Radiationless energy transfer Fluorescence energy transfer Forster energy transfer ¨ Energy transfer Excitation energy transfer Electronic excitation energy transfer Singlet᎐singlet energy transfer Singlet energy transfer Non-radiative energy transfer Forster excitation energy transfer ¨ Long-range resonance transfer Inductive resonance energy transfer Non-radiative excitation energy transfer Forster singlet᎐singlet radiationless resonance ¨ electronic excitation energy transfer
FRET FRET FRET RET RET FET FET ET EET EEET SSET SET NET FEET LRT IRET NEET FSSRREEET
ing Žcorrectly indicates the role played by fluorescence ..
2. Kappa-squared for one pair The orientation factor, 2 , for one donor᎐acceptor pair depends on three orientations, shown in Fig. 1: the emission transition dipole moment of the donor Ž D-direction.; the absorption transition dipole moment of the acceptor Ž A-direction.; and the line connecting the centers of the donor fluorophore and the acceptor chromophore Ž R-direction. ŽForster, 1948, ¨ 1993 ŽEnglish translation of the 1948 paper.; Dale et al., 1979; van der Meer et al., 1994, 1999.. The angle between D and A is T , that between D and R is D , and the angle between A and R is A Žsee Fig. 2.. The planes formed by D and A, by D and R, and by A and R are the DA-plane, DR-plane and AR-plane, respectively. The angle between the DR-plane and the AR-plane is . The oscillating electric field associated with D Žor one could say, caused by the donor dipole. is along the line ED , which is sometimes parallel to D, sometimes parallel to R, but in general can be
constructed as follows Žsee Fig. 3.. Project a unit vector along D onto the R direction. Measure this projection and mark off three times this distance along the R direction. Add to this vector Žhaving magnitude 3cos D and direction along R . a unit vector in the negative D-direction. This vector sum is along the ED direction. The angle between ED and A is . We can evaluate kappa-squared in three different ways in terms of the various angles: 2 s Ž cosT y 3cos D cos A .
2
2 s Ž sin D sin A cos y 2cos D cos A . 2 s cos 2 Ž 1 q 3cos 2 D . .
Ž1. 2
Ž2. Ž3.
The highest value for kappa-squared is four, which occurs if both D and A are parallel Žor antiparallel . to R. The lowest value is 0, which refers to a situation where A is perpendicular to ED . The case 2 s 4 can occur in a few ways w D, R and A must be Žanti. parallelx, but 2 s 0 can be realized in an infinite number of ways wwhenever A and ED are perpendicular to each
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
ˆ Aˆ and Rˆ are along the D-, A- and Fig. 1. The unit vectors D, ˆ is along the emission transition R-directions, respectively. D moment of the donor, Aˆ is along the absorption transition moment of the acceptor, and Rˆ is along the line from the center of the donor fluorophore to that of the acceptor ˆ Aˆ and Rˆ can lie in one plane, but are in chromophore. D, general not co-planar.
otherx. Eq. Ž3. for kappa-squared depends on only two variables, and D . This form of kappasquared refers to the electric field of the donor. Kappa-squared is proportional to the square of this field, which yields the factor 1 q 3cos 2 D , and is also proportional to cos 2 , where is the angle between this field and the acceptor transition moment. This means that if we pick a point for the location of the acceptor with respect to a donor place in the origin, then aligning the acceptor orientation A with the donor’s electric field
183
line passing through that point will yield the highest kappa-squared value that can be achieved in that point: 1 q 3cos 2 D . The cos 2 proportionality implies that kappa-squared is zero whenever A is along a line perpendicular to an electric field line. Eq. Ž3. also allows for a geometrical construction of 2 Žvan der Meer, 1999.. An alternative procedure illustrating the relation between the relevant orientations and the magnitude of kappa-squared is shown in Fig. 3. Since the average of the cos 2 of a polar angle is 13 , and both and D are independent polar angles, it follows from Eq. Ž3., that the average value of 2 is 23 , if D, A and R are random. It is clear from Eqs. Ž1. ᎐ Ž3. and the definitions of the angles used in these equations that kappasquared does not change its value if we perform the following operations: 1. 2. 3. 4.
flip the donor transition moment; flip the acceptor transition moment; let the donor and acceptor trade places; and interchange the donor and acceptor transition moments.
3. Averaging regimes Averaging conditions or averaging regimes usually refer to rotational motion; when every donor and every acceptor can take up its entire range of
ˆ and R, ˆ A is between Aˆ and R, ˆ T is Fig. 2. The polar angles D , A , T and , and the azimuthal angle : D is between D ˆ and A, ˆ is between Aˆ and the electric field associated with the donor emission moment, and is between the between D 2 ˆ and Aˆ on a plane perpendicular to R. ˆ EˆD s Ž3 Rcos ˆ ˆ. projections of D D y D r 1 q 3cos D is a unit vector along the electric field ˆ and R, ˆ the AR-plane through Aˆ associated with the donor emission moment Žsee Fig. 3.. The DR-plane is the plane through D ˆ the DA-plane through Dˆ and A, ˆ and the ED A-plane through EˆD and A. ˆ and R,
'
184
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
where the brackets denote averages. Averaging regimes and averaging procedures not only are related to the orientation factor, but also depend on whether the distance is fixed or not Žvan der Meer et al., 1994.. A general approach to deal with the complexities of moving donors and acceptors is discussed below in the section called ‘beyond regimes’.
4. The average kappa-squared in the dynamic regime
Fig. 3. The relation between the ED -direction and kappaˆ on Rˆ is 3cos D . The vector sum squared. The projection of D ˆ ˆ of 3 Rcos D and yD is along the ED -direction. The projection of Aˆ on the ED -direction is cos. The area of the larger square is 1 q 3cos 2 D , that of the smaller square is cos 2 . The product of these two areas equals kappa-squared.
orientations during the transfer time Žthe inverse of the average transfer rate., the system is said to be in the dynamic averaging regime, and the dynamic averaging condition applies. In that regime, the average rate of rotations k R , is much larger than the average rate of transfer, k T : k R 4 k T . In this case, the orientation factor can be replaced by its appropriate average value ŽDale et al., 1979; van der Meer et al., 1994, 1999.. On the other hand, when the rates of rotation are slow compared to the rate of transfer, that is if k T 4 k R , the system is in the static averaging regime. The average transfer efficiencies in these regimes are Žvan der Meer et al., 1994.:
if k R 4 k T , average Es
3 2 6 ² : R 0 2 3 2 6 ² : R 0 q R 6 2
Ž4.
In the dynamic averaging regime the D-direction fluctuates rapidly around an orientation called the donor axis Ž D X -direction., and similarly, the A-direction fluctuates around the acceptor axis Ž A X -direction.. If these fluctuations take place in a time-scale that is short compared to the transfer time Žthe inverse of the average rate of transfer ., kappa-squared can be replaced by ² 2 :, the dynamic average of the orientation factor ŽDale et al., 1979; van der Meer, 1999.. This average depends on two parameters, d and a Ždefined below., and three variables, ⌽, ⌰D and ⌰A Ždefined below.. The parameter d is the axial depolarization factor for the donor transition moment,
¦ 32 cos y 12 ; 3 1 s H ž cos y / sin F Ž . d 2 2
ds ² d DX : s
if k T 4 k R , average Es
−
3 2 6 R0 q R6 2
< Ž5.
D
2
D
0
D
D
D
D
Ž6.
where D is the fluctuating angle between the donor transition moment and the donor axis, and FD Ž D . is a distribution function. Similarly, the axial depolarization factor for the acceptor transition moment is
¦ d : s ¦ 32 cos y 12 ; 3 1 s H ž cos y / sin F Ž . d 2 2 X A
as 3 2 6 R0 2
2
0
2
A
2
A
A
A
A
A
Ž7.
where A is the fluctuating angle between the acceptor transition moment and the acceptor axis,
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
and FAŽ A . is a distribution function. Note that it is assumed here that the distributions of the transition moments around the axis are cylindrically symmetric. The parameter dŽ a. is a second rank orientational order parameter with values between y0.5 and 1: ds 1 Ž as 1. when the transition moment is completely aligned with its axis; ds 0 Ž as 0. when the angle between the transition moment and its axis is equal to the magic angle at all times or when the transition moment is completely random; and ds y1r2Ž a s y1r2. when the angle between D and D X Ž A and A X . is 90⬚ at all times, that is, when the transition moment is degenerate in a plane perpendicular to the axis. The factors d and a can be obtained from fluorescence depolarization measurements ŽDale et al., 1979; van der Meer et al., 1994, 1999.. If one takes the z-axis along the R-direction, then the polar angle ⌰D is the angle between the donor axis and the z-axis, ⌰A is that between the acceptor axis and the z-axis. ⌰ is the azimuthal angle between the projections of the donor and acceptor axis on the xy-plane. The dynamical average of kappa-squared is given by ŽDale et al., 1979.: ² 2 : s
2 1 1 y dy aq d Ž 1 y a. cos 2 ⌰D 3 3 3 q aŽ 1 y d . cos 2 ⌰A q ad 2
= Ž sin⌰D sin⌰A cos⌽ y 2cos⌰D cos⌰A . . Ž8. Note that sin⌰D sin⌰A cos⌽ y 2cos⌰D cos⌰A s cos⌰ T y 3cos⌰D cos⌰A , where ⌰T is the angle between the donor axis and acceptor axis. Haas et al. Ž1978. and Steinberg et al. Ž1983. have a different approach to dealing with the orientation factor. They make use of the fact that many chromophores show mixed polarization in their spectral behavior. In other words, they assume that the chromophores do not move, but that the donor emission and acceptor absorption across the relevant spectral range of overlap are not characterized by single transition dipoles but by combinations of two or more incoherent dipole moments. In their formulation V1 , V2 and V3 are
185
the oscillating dipoles which characterize the donor fluorescence. They are mutually perpendicular and V12 q V22 q V33. Similarly, W1 , W2 and W3 are the oscillating dipoles characterizing the acceptor absorbance, which are also mutually perpendicular and have W12 q W22 q W33 . In this approach the average kappa-squared is: 3
² 2 : s
3
Ý Ý Vi 2 Wj 2 Žcos⌰i j y 3cos⌰D i cos⌰A j .
2
is1 jy1
Ž9. where ⌰i j is the angle between donor dipole i and acceptor dipole j, ⌰D i is the angle between donor dipole i and the line connecting the centers of donor and acceptor, ⌰A j , is the angle between acceptor dipole j and the line connecting the centers. This expression allows for complete asymmetry in the transition moments. However, this asymmetry is difficult to measure, and if we ignore it, that is, if we assume V12 s V22 s 13 Ž1 y d ., V32 s 31 Ž1 y 2 d ., W12 s W22 s 31 Ž1 y a. and W32 s 31 Ž1 y 2 a., the expression in Eq. Ž9. is completely equivalent to the one in Eq. Ž8.. This equivalence is shown in the Appendix A. In the dynamic averaging regime the fluorescence depolarization experiment cannot distinguish between degeneracy of the transition dipoles and fast rotational motion. Only one energy transfer rate constant is observable in this regime, and that is the average rate constant ² k T :, given by: ² kT : s ² k 2 : k D
6 3 R rR 2Ž 0 .
Ž 10.
where k D is the rate of donor decay in the absence of acceptor, R 0 is the Forster distance ¨ for 2 s 2r3, and R is the donor᎐acceptor distance, which is assumed to be unique and essentially constant during times of the order of 1rk D . In the dynamic averaging regime the donor intensity at time t after excitation with a flash of light is proportional to: eyŽ ² k T :qk D .t .
Ž 11.
Minima and maxima of the average kappasquared correspond with the lower and upper
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186
limits of the distance. These extremities can be found by setting the derivatives of kappa-squared in Eq. Ž8. with respect to ⌽, ⌰D and ⌰A to zero, and solving the resulting equations. There are six possible candidates for maxima and minima. These are:
In any point in the Ž d,a.-plane, one of these
2A s
2 2 2 q aq dq 2 ad 3 3 3
Ž 12.1.
2M s
2 1 1 1 q aq dy adq < ay d < 3 6 6 2
Ž 12.2.
extremities is the maximum and another is the minimum, but it depends on the values of d and a which one of the six extremities is the maximum and which is the minimum. Accordingly, the Ž d,a.-plane is divided into different regions defined below in Table 2. This table refers to the case where the transfer depolarization is not known. The case that this depolarization is known has been discussed by Dale et al. Ž1979. and van der Meer Ž1999..
2L s
2 1 1 1 q aq dy ady < ay d < 3 6 6 2
Ž 12.3.
5. The average kappa-squared in the static regime
2H s
2 1 1 y ay dq ad 3 3 3
Ž 12.4.
2T s
1 4 Ž 1 y a.Ž 1 y d . q 9 9
In the static averaging regime, the average kappa-squared depends on the distance ratio RrR 0 , that is the donor᎐acceptor distance divided by the Forster distance for 2 s 2r3. Stein¨ Ž . berg et al. 1983 found this ² 2 :-value. This average orientation factor is the one that would be used in Forster’s equation for the average ¨ efficiency of FRET between donor᎐acceptor pairs for which rotational diffusion is frozen. In their analysis, they assumed that each pair is isolated from all other pairs and that the electronic transition of each chromophore is a single dipole, which is randomly oriented in space. Their result is a plot of ² 2 : vs. RrR 0 ŽFig. 1 in Steinberg et al., 1983., which essentially is the following trend:
2P s
='Ž 1 y a.Ž 1 y d .Ž 1 q 2 a.Ž 1 q 2 d .
Ž 12.5.
2 1 1 y ay d. 3 3 3
Ž 12.6.
For A 2 : the axis of both the donor and the acceptor are Aligned with the separation vector. For M 2 : the axis of the Most prolate distribution Ždonor or acceptor. is aligned with the separation vector; the other Žacceptor or donor. is perpendicular to the separation vector. In other words, if d) a, the donor axis is aligned with the separation vector and the acceptor axis is perpendicular to it, and vice versa if a) d. For L 2 : the axis of the Least prolate distribution Ždonor or acceptor. is aligned with the separation vector; the other Žacceptor or donor. is perpendicular to the separation vector. For H 2 : the donor and acceptor are in a ‘H-configuration’, that is their axis are parallel to each other, but perpendicular to the separation vector. For T 2 : the axis of both donor and acceptor are Tilted at some angle from the separation vector, but are co-planar with this vector. For p : the separation vector is Perpendicular to the axis of the donor, which is perpendicular to that of the acceptor, which is also perpendicular to the separation vector. 2
¡0 2
~ 3 ž R y 25 R
² 2 : f
¢
2 3
2 R 5 0 2 7 R F . 5 0 R F 5 R0 7 R F R0 5 Ž 13. RF
0
/ rR
0
If the dipoles are not random, but distributed in a cylindrically symmetric way around axis following distribution functions FD Ž D . and FAŽ A . Žintroduced above., the intensity emitted by the donor in the static averaging regime at time t,
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187
Table 2 Regions with different kappa-squared maxima and minima in the plane of all possible depolarization factors for donors and acceptors Region
Maximum
Minimum
0 F dF 1, 0 F aF 1 y 12 F dF 0, 12 y dF aF 1 1 1 2 F dF 1, 2 y dF aF 0 1 q 3d 1 y 2 F dF y 13 , y F aF 12 y d 3 q 5d 1 q 3d y 13 F dF 0, F aF 12 y d 2q2 d 1 q 3d 1y2 d 0 F dF 12 , y F aF 3 q 5d 2 dy 3 1 q 3d 1 1 2 F dF 1, y 3 q 5d F aF 2 y d 1 q 3d y 13 F dF 0, 0 F aF 2q2 d 1y2 d 0 F dF 12 , F aF 0 2 dy 3 1 q 3d y 12 F dF y 13 , 0 F aF 3 q 5d 1 q 3d 0 F dF 1, y 12 F aF y 3 q 5d 1 q 3d 1 1 y 2 F dF y 3 , F aF 0 2q2 d 1y2 d 1 1 y 2 F dF 0, y 2 F aF 2 dy 3 1y2 d 1 q 3d 1 1 y 2 F dF y 3 , F aF 2 dy 3 2q2 d 1y2 d 1 q 3d y 13 F dF 0, F aF y 2 dy 3 3 q 5d
A 2 M 2 M 2
P2 H 2 H 2
M 2
T 2
M 2
T 2
M 2
T 2
M 2
T 2
M 2
L 2
M 2
L 2
M 2
A 2
M 2
A 2
H 2
L 2
H 2
L 2
H 2
T 2
H 2
T 2
after excitation with a flash of light, is proportional to
yŽ k D qk T .t
H0 H0 e
FD Ž D .
=FA Ž A . sin D sin A d D d A .
Ž 14.
6. Assessing the orientational error The main goal of FRET is to find the distance or distribution of distances between donor and acceptor. In the dynamic averaging regime it is possible to deduce the uncertainty in the distance resulting from assuming an average value for 2 . There are three probability functions that are directly related to 2 ; the range probability, PR Ž 2MIN ª 2 ., the probability distribution, pŽ 2 ., and the relative distance distribution, QŽ ..
PR Ž 2MIN ª 2 . is the probability of finding a value for the average kappa-squared in the range from its lowest value to 2 , pŽ 2 .d 2 is the probability of encountering a value for the average kappasquared in the interval 2 to 2 q d 2 , and QŽ .d is the probability of finding a value for the relative distance between and q d. The relative distance is the ratio of the actual donor᎐acceptor distance over the distance obtained from FRET data assuming 2 s 23 . These and related distances are defined in Table 3. The function Q was introduced by Haas et al. Ž1978. and discussed by Steinberg et al. Ž1983., but is slightly modified here; instead of considering Q as a function of R⬘rR, we take Q as a function of RrR⬘. The advantage of the latter choice is that RrR⬘ has a limited range, varying between 0 and 1.348 in the case of perfect order, whereas R⬘rR varies over the much wider range from 0.742 to infinity. There is a one-to-one relation between a
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188 Table 3 Distances Symbol
Definition
R R0 R0 R⬘ MI N MA X m p l o h i
Distance between a donor᎐acceptor pair Forster distance Ždepends on kappa-squared. ¨ Forster distance for 2 s 23 ¨ Apparent distance, obtained by assuming 2 s 23 Relative distance, RrR⬘ Relative distance corresponding with lowest 2 Relative distance corresponding with highest 2 Relative distance at which Q has its maximum Lower limit of the 67% CI for Upper limit of the 67% CI for
particular kappa-squared value and the donor᎐acceptor distance via the following equations: 2 s
2 6 3
or:
s
3 2 2
ž /
1r6
.
Ž 15.
This relation has a dramatic effect on the interdependence between pŽ 2 . and QŽ ., as illustrated in Fig. 4. The peak in pŽ 2 . is always at or near the minimum kappa-squared value, but as a result of Eq. Ž15., the maximum of QŽ . is shifted dramatically towards distances corresponding with higher kappa-squared values in many cases. However, the characteristic features of QŽ . vary widely as the depolarization factors change between 0 and 1 as illustrated in Fig. 5. QŽ . has a very sharp peak near s 1 if both donor and acceptor depolarization factors are small. If one of the depolarization factors is close to zero and the other is larger than that, QŽ . has a peak at the lowest value for the relative distance, slightly below s 1. On the other hand, if both factors are approximately equal and bigger than zero, the most probable relative distance is slightly above s 1. In determining the FRET distance and its uncertainty due to kappa-squared, it is not only relevant to find the minimum and maximum value of the distance, but is also important to find the most probable distance and the 67% CI. These distances and intervals are studied for the following cases in detail below: as ds 1; as d< 1
Fig. 4. The effect of 2 s 23 6 on the relation between pŽ K 2 . and QŽ ..
and the case where the two depolarization factors equal zero. as ds 1 is the worst case scenario where the orientational uncertainty is the highest. PR Ž 2MIN ª 2 . in this case is given by:
¡
2 ln Ž 2 q 3 . 3
( ' ~( ( ' ¢ ž' ' / 2 y 1 q 3
PR Ž 0 ª 2 . s
ln
2 3
2q 3
q y1 2
0 F 2 F 1
2
. 1 F 2 F 4 Ž 16.1.
For as ds 1, pŽ 2 ., the derivative of PR with respect to 2 , is:
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189
The graph of QŽ . for as ds 1 is shown in Fig. 6. The most probable relative distance, m p s 1.070, where QŽ . of Eq. Ž16.3. has its maximum. The 67% CI for the relative distance is from lo s 0.813 wcorresponding to PR s 2r3 lnŽ2 q '3 . s 13 x to h i s MAX s 1.348. The most probable is slightly below 1.081, the middle of this interval. In the case where as d< 1, PR Ž 2MIN ª 2 . is given by:
'
PR Ž 0 ª 2 . s
¡ Fig. 5. The characteristics of the relative distance distribution, QŽ ., for various values of the depolarization factors. Details of QŽ . for as ds 1 , for as ds 0.25, and for as 1, ds 0 Ž as 0, ds 1. are shown in Figs. 6᎐8, respectively.
¢
pŽ2 . s
¡
1
'3
~2
¢2'3
2
1
2
ln Ž 2 q '3 . ln
ž'
2 q '3
2 q ' 2 y 1
¡ '2 lnŽ2 q '3 . 2
Q Ž . s '2 ln 2
¢ž 32 /
(
ž
2 1 y a 3 3
/
2 4 q a 3 3 Ž 17.1.
0 F 2 F 1
/
.
where is:
Ž 16.2.
)
1F F4 2
Since pŽ 2 .d 2 and QŽ .d both denote the fraction of the population of donor᎐acceptor pairs with the orientation factor between 2 and 2 q d 2 , we have QŽ .d s pŽ 2 .d 2 or QŽ . s pŽ 2 .d 2rd s pŽ 2 s 23 6 .45.Therefore, QŽ . in this case is given by:
~
~
2 2 2 y 3 q 3a 4 a 2 2 y aF 2 F 3 3 y1 Ž . Ž q y tan 1q2 . 4 2 1 y aF 2 F 3 3
0FF
3 2
(
0
2 1 y a 3 3 s a
(
2 a 6 y 1 y . 3a 2 Ž 17.2.
ž
pŽ2 . s
.
y1 Ž
.
2 2 2 1 y aF 2 F q a 3 3 3 3 . 2 1 2 4 2 y aF F q a 3 3 3 3 Ž 17.3.
QŽ . in this case is given by:
1r6
F F 6 1r6 Ž 16.3.
/
For as d< 1, pŽ 2 . is:
¡ ~ 4a ¢4a y 1a tan
1r6
ž /
2 q '3 2 6 2 6 q y1 3 3
s
2 y
QŽ . s
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190
1
pŽ2 . s
(ž
2 a 2 y
2 1 q a 3 3
/
2 1 2 2 y aF 2 F y a. 3 3 3 3
Ž 18.2.
QŽ . in this case has the following form: QŽ . s
2 5'3
(
2 a 6 y 1 q
Fig. 6. Q vs. for as ds 1. The striped area corresponds with the 67% CI for the relative distance. MI N s 0, l o s 0.813, m p s 1.070, the center of the 67% CI is at s 1.081, and h i s MAX s 1.348.
¡
~
¢ ž
1 1 aF 2 F 1 q a 6 12 . 1 1 1q aF 2 F 1 q a 12 3 Ž 17.4.
1y
5
4a 5 1 1 y tany1 Ž . a a
/
The graph of QŽ . for as ds 0.25, which is essentially the upper limit of this interval, is shown in Fig. 7. The most probable relative distance 1 equals m p s 1 q a, is that where QŽ . of 12 Eq. Ž17.4. has its maximum. The 67% CI for the 1 relative distance is from l o s 1 y 16 y a 3 2 2 corresponding to PR s w rŽ4 a.xŽ 2 y 3 q 3 a. s 1r34 to h i s MAX s 1 q 13 a. The most probable 1 1 is slightly below l o s 1 y y a, the 12 6 middle of this interval. In the case ds 0 Žor as 0. PR Ž 2MIN ª 2 . is:
ž
ž
PR Ž 0 ª
2.
s
)
2 y
ž
ž 1 y 12 a /
1 a 2
/
1r6
F F Ž1 q a.
Ž 18.3. 1r6
.
The graph of QŽ . for as 1 Žor ds 1., which corresponds with the worst error in this case, is shown in Fig. 8. QŽ . has its peak at m p s MIN s Ž1 y 12 a.1r6 Žs 0.891 for as 1.. The 67% CI for the relative distance is from m p s MIN s Ž1 y 1 .1r6 to h i s Ž1 y 12 a.1r6 Žs 1.026 for as 1. 2a Žfrom PR s Ž 2 y 23 q 13 a. ras 23 .. The maximum relative distance is MA X s Ž1 y a.1r6 Žs 1.122 for as 1. . Steinberg et al. Ž1983. proposed to take the width of the peak of QŽ . as a measure of the orientational error in the distance. Note, however, that the peak of QŽ . in this case is infinitely high. Therefore, the width of the peak at half-maximum height is zero here,
'
/
/
2 1 q a 3 3 a
2 1 2 2 y aF 2 F y a. 3 3 3 3 For ds 0 Žor as 0. Ž 2 . is given by:
Ž 18.1.
Fig. 7. Q vs. for as ds 0.25. The striped area corresponds with the 67% CI for the relative distance. MI N s 0.953, l o s 0.984, m p s 1.020, the center of the 67% CI is at s 1.027, and h i s MAX s 1.070.
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191
Fig. 8. Q vs. for as 1, ds 0 Ž as 0, ds 1.. The striped area corresponds with the 67% CI for the relative distance. MI N s l o s m p s 0.891, h i s 1.026, the center of the 67% CI is at s 0.959, and MA X s 1.122.
and is in general, not a good measure for the error in the distance. Taken together, these case studies suggest the following trends: 1. The 67% CI for the relative distance is significantly smaller than the full range from the minimum to the maximum relative distance. Figs. 9 and 10 illustrate this point. 2. The center of the 67% CI is always close to the most probable distance with a difference of less than 7%. This center is slightly higher than the most probable distance Žsee Figs. 9 and 10.. 3. The center of the 67% CI is always close to the distance corresponding to 2 s 23 with a difference of less than 8%. If the donor and acceptor depolarization factors are roughly equal to each other, the most probable distance is slightly higher than the 2 s 23 distance ŽFig. 9.. If one of the two depolarization factors is close to zero, the most probable distance is slightly lower than the 2 s 23 distance ŽFig. 10.. 4. The width or the half-width of the 67% CI is a good measure for the uncertainty in the distance due to kappa-squared. This error is relatively small for most realistic depolarization values. However, this uncertainty exceeds
Fig. 9. MA X , MIN , h i , l o and m p versus one of the depolarization factors for the case that the two depolarization factors are equal to each other. Curves drawn as thicker lines represent values that are completely known, and curves drawn as thinner lines denote interpolated values.
20% if one or both of the depolarization factors approach their maximum values. This trend is shown in Figs. 11 and 12.
7. Beyond regimes What do you do when your system is in between the static and dynamic averaging regimes? You build models: mathematical models that result in matrix equations Žvan der Meer et al., 1993.. A system of donors and acceptors undergoing translational andror rotational motion during the transfer time Žinverse of average transfer rate. can be thought of as a collection of states with transitions between them. Think of these states as snapshots; imagine that you can observe a donor᎐acceptor pair. You might see at a certain moment a donor that is excited and has a certain
192
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Fig. 12. The uncertainty in the distance due to kappa-squared versus one of the depolarization factors for the case that the other factor is zero. Here the error is taken as the full width of the CI, Ž h i y l o . = 100%.
Fig. 10. MA X , MIN , h i , l o and m p versus one of the depolarization factors for the case that the other factor is zero.
Fig. 11. The uncertainty in the distance due to kappa-squared versus one of the depolarization factors for the case that the two depolarization factors are equal to each other. Here, the 1 error is taken as half of the width of the CI, Ž h i y l o . = 2 100%. The part of the curve drawn as a thicker line represents values that are completely known, and the thinner part of the curve denotes interpolated values.
orientation while the acceptor has another orientation Žthe pair is in a DUA state.. A nanosecond later the donor may still be more or less in the same situation, but the acceptor has quite a different orientation Ža rotational transition to another DUA state has taken place.. A few nanoseconds later the donor has lost its excitation energy and the acceptor is excited Žan energy transfer transition to a DAU state has occurred.. A systematic description of such time developments implies selecting a representative set of orientational states, evaluating kappa-squared values, identifying transfer rates and other rate constants. This approach leads to a matrix equation for which the eigenvectors and eigenvalues must be found, so that intensities and anisotropies can be calculated. To illustrate this procedure let us consider the following example. Suppose a macromolecule contains a donor that can have two distinct orientations, 1 and 2 with an angle ␥ between them, and an acceptor with a fixed orientation at a fixed distance from the donor. The macromolecule itself has a random orientation in the sample and undergoes rotations too slow to observe on the time scale of the donor fluorescence. In this case we have two DUA states corre-
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Ž SQ y ⌬ k T . 2 cos 2 z1
sponding with the two donor orientations. Let the rate of rotational jumps between the two donor orientations be k R , and the rates of transfer be k T q ⌬ k T and k T y ⌬ k T for state 1 to the acceptor and for state 2 to the acceptor, respectively. If the rate of donor emission in the absence of acceptor is k D , and the probability that state 1 Ž2. is excited is x 1 Ž x 2 ., then the time-derivatives of x 1 and x 2 are given by:
␣ 12 s
ž
x1 . x2
ž /
Ž 19.
At time zero, when the system is excited with a flash of vertically polarized light, the values for x 1 and x 2 are x l s cos 2 z1 and x 2 s cos 2 z 2, where z1 Ž z 2 . is the angle between the vertical and a donor orientation 1 Ž2.. The solution of Eq. Ž19. with these boundary conditions is based on the eigenvalues and eigenvectors of the matrix in Eq. Ž19. and has the following form: x 1 s ␣ 11 ey t q ␣ 12 eyŽq.t
Ž 20.1.
x 2 s ␣ 21 ey t q ␣ 22 eyŽq.t
Ž 20.2.
where the eigenvalues and q are determined by: s k D q k R q k T y SQ ,
SQ s
'k
2 Ž . R q ⌬ kT
2
and s 2 SQ Ž 21. and the coefficients in Eqs. Ž20.1. and Ž20.2. are: ␣ 11 s
k R2 cos 2 z1 q k R Ž SQ y ⌬ k T . cos 2 z 2 k R2 q Ž SQ y ⌬ k T .
2
Ž 22.1.
k R2 q Ž SQ y ⌬ k T .
Ž 22.2.
2
2
␣ 21 s
ž /
kR yk D y k R y k T q ⌬ k T
yk R Ž SQ y ⌬ k T . cos 2 z 2
k R Ž SQ y ⌬ k T . cos 2 z1
d x1 s dt x 2 yk D y k R y k T y ⌬ k T kR
193
␣ 22 s
/
q Ž SQ y ⌬ k T . cos 2 z 2 k R2 q Ž SQ y ⌬ k T .
Ž 22.3.
2
yk R Ž SQ y ⌬ k T . cos 2 z1 q k R2 cos 2 z 2 k R2 q Ž SQ y ⌬ k T .
2
. Ž 22.4.
The time-resolved donor intensity observed through a vertical emission polarizer, IV , is the following linear combination of x 1 and x 2 : IV s c - ␣ 11cos 2 z1 q ␣ 21cos 2 z 2 ) ey t qc - ␣ 12 cos 2 z1 q ␣ 22 cos 2 z 2 ) eyŽq.t
Ž 23.
where c is a constant proportional to the concentration, molar extinction coefficient and quantum yield of the donor in the absence of acceptor. The brackets indicate an average over all polar angles z1 and z 2 under the constraint that the angle between donor orientation 1 and 2 is ␥ for all orientations of the protein. The time-resolved donor intensity observed through a horizontal emission polarizer, IH , is also a linear combination of x l and x 2 : IH s c - ␣ 11cos 2 x1 q ␣ 21cos 2 x 2 ) ey t
Ž 24. qc - ␣ 12 cos 2 x1 q ␣ 22 cos 2 x 2 ) eyŽq.t
where x l Ž x 2 . is the angle between donor orientation one Žtwo. and a line perpendicular to the vertical. Here, the brackets have a similar meaning as in Eq. Ž23. except that the averaging is not only over z1 and z 2 , but also over x l and x 2 , under the constraint that the horizontal and vertical direction are perpendicular to each other. Using the averages,
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194
1 , 5 ²cos 2 x1cos 2 z 2 : s ²cos 2 z1cos 2 x 2 : 3 2 s y sin 2 ␥ 15 15 and ²cos 2 z1cos 2 z 2 : s ²cos 2 z1cos 2 x1 :
The donor intensity and anisotropy in the static averaging regime follow from Eqs. Ž26. and Ž27. by taking the limit: k Rr⌬ k T ª 0 at which ª 1:
²cos 4 z1 : s ²cos 4 z 2 : s
Ž 25.
Static averaging regime: ID is proportional to eyŽ k Dqk Ty⌬ k T . Ž 1 q ey2 ⌬ k T t . t
s ²cos 2 z 2 cos 2 x 2 : 1 s , 15
Static averaging regime:
Ž 31.
rD s
2 . 5
Ž 32.
the donor intensity, ID s IV q 2 IH , and anisotropy, rD , are derived from Eqs. Ž23. and Ž24.:
This specific example illustrates the following general points:
ID is proportional to ey t Ž 1 q ey t .
1. The donor anisotropy in the presence and absence of acceptor indicates which averaging regime applies: if this anisotropy is virtually constant in time, then, most likely, the static averaging regime applies; if the donor anisotropy in the absence of acceptor is essentially identical to that in the presence of acceptor, the dynamic averaging regime applies. 2. Anisotropies give information about kappasquared. In the dynamic averaging regime this information allows one to assess the uncertainty in the FRET distance due to kappa-squared ŽHaas et al., 1978; Dale et al., 1979; Steinberg et al., 1983, see also above.. Outside of this averaging regime the combination of time-resolved donor Žand acceptor. intensities and anisotropies gives information about the distribution of kappa-squared values, and should allow for a more precise determination of the donor᎐acceptor distance. 3. Outside the dynamic and static averaging regimes time-resolved donor Žand acceptor. anisotropies give information about the orientational distributions and rotational motions of the donors and acceptors in an unique and interesting way. For example, such anisotropies should be capable of determining the angular size of rotational jumps occurring in such systems Žvan der Meer et al., 1993..
Ž 26.
2 3 3 1 y sin 2 ␥ q sin 2 ␥ 5 4 4
rD s
½
q
3 2 3 sin ␥ q 1 y sin 2 ␥ ey t 4 4
ž
/
5
r 1 q ey t 4 Ž 27.
with
s
ž
SQ y ⌬ k T y k R SQ y ⌬ k T q k R
2
/
.
Ž 28.
The donor intensity and anisotropy in the dynamic averaging regime follow from Eqs. Ž26. and Ž27. by taking the limit: ⌬ k T rk R ª 0 at which ª 0 yielding: Dynamic averaging regime: ID is proportional to eyŽ k Dqk T .
t
Ž 29.
Dynamic averaging regime: rD s
2 3 3 1 y sin 2 ␥ q sin 2 ␥ ey2 k R t . 5 4 4
ž
/
Ž 30.
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
cos⌰i1cos⌰A1 q cos⌰i2 cos⌰A2
Appendix A Substituting V12 s V22 s 13 Žl y d ., V32 s 13 Žl q 2 d ., W12 s W22 s 13 Žl y a., and W32 s 13 Ž1 q 2 a., into Eq. Ž9. transforms this equation into: 2 s
1 Ž 1 q 2 d . Ž 1 q 2 a. Tp p q Ž 1 y a. Tp m 4 9 q
195
s cos⌰D i y cos⌰i3 cos⌰A
rule 6.
Evaluating the square in the definition of Tp m yields: 2
Tp m s 9cos 2 ⌰D
1 Ž 1 y d . Ž 1 q 2 a. Tm p q Ž 1 y a. Tm m 4 9
Ý cos 2 ⌰A j js1
Ž A1.
2
y 6cos⌰D
2
Ý cos⌰ 3 j cos⌰A j q Ý cos 2 ⌰ 3 j . js1
js1
Ž A4.
where we have utilized the abbreviations given in Eqs. ŽA2. and ŽA3.. Tp p ' Ž 3cos⌰D cos⌰A y cos⌰T .
2
2
Tp m '
Ý Ž3cos⌰D cos⌰A j y cos⌰ 3 j .
Ž A2.
2
Tp m s 9cos 2 ⌰D y 9cos 2 ⌰D cos 2 ⌰A
js1
y 6cos⌰D Ž cos⌰D y cos⌰D cos⌰A .
2
Tm p '
Ý Ž3cos⌰D i cos⌰A y cos⌰i3 .
2
q1 y cos 2 ⌰ T s 1 q 3cos 2 ⌰D y Tp p .
is1 2
Tm m '
Using rule 1 for the first term, rule 6 for the second term and rule 4 for the third term, simplifies Tp m as follows:
2
Ý Ý Ž3cos⌰D i cos⌰A j y cos⌰i j .
2
Ž A3.
Ž A5.
Similarly, we find:
is1 js1
Tm p s 1 q 3cos 2 ⌰A y Tp p . The following projection rules allow us to simplify Tp m , Tm p and Tp p : cos 2 ⌰A1 q cos 2 ⌰A2 s 1 y cos 2 ⌰A
Evaluating the square in the definition of Tm m yields:
rule 1 2
cos 2 ⌰D1 q cos 2 ⌰D 2 s 1 y cos 2 ⌰D
rule 2
2
Tm m s 9 Ý cos 2 ⌰D i is1
Ý cos 2 ⌰A j js1
2
cos ⌰ 1 j q cos ⌰ 2 j s 1 y cos ⌰ 3 j 2
2
2
rule 3
rule 4
2
y 6 Ý cos⌰D i is1 2
cos 2 ⌰i1 q cos 2 ⌰i2 s 1 y cos 2 ⌰i3
Ž A6.
qÝ
Ý cos⌰ 3 j cos⌰A j js1
2
Ý cos 2 ⌰i j .
Ž A7.
is1 js1
cos⌰ 1 j cos⌰D1 q cos⌰ 2 j cos⌰D 2 s cos⌰A j y cos⌰ 3 j cos⌰D
rule 5
Employing rules 1 and 2 for the first term, rule 6 for the second term and rule 4 for the third term reduces Eq. ŽA7. to:
B.W. ¨ an der Meer r Re¨ iews in Molecular Biotechnology 82 (2002) 181᎐196
196
Tm m s 9 Ž 1 y cos 2 ⌰D .Ž 1 y cos 2 ⌰A .
References
2
y6 Ý cos⌰D i Ž cos⌰D i y cos⌰i3 cos⌰A . is1 2
q Ý Ž 1 y cos 2 ⌰i3 . s 9 Ž 1 y cos 2 ⌰D . is1 2
= Ž 1 y cos 2 ⌰A . y 6 Ý cos 2 ⌰D i
Ž A8.
is1 2
q6 Ý cos⌰D i cos⌰i3 cos⌰A is1 2
q Ý Ž 1 y cos 2 ⌰i3 . . is1
Using rule 2 in the 2nd term following the last equal sign in Eq. ŽA8., rule 5 in the next term and rule 3 in the last term, simplifies Tm m to: Tm m s 9 Ž 1 y cos 2 ⌰D .Ž 1 y cos 2 ⌰A . y 6 q 6cos 2 ⌰D q 6cos 2 ⌰A q 1 q cos 2 ⌰A s 4 y 3cos 2 ⌰D y 3cos 2 ⌰A q Tp p .
Ž A9.
Substituting Tp m s 1 q 3cos 2 ⌰D y Tp p , Tm p s 1 q 3cos 2 ⌰A y Tp p and Tp p s 4 y 3cos 2 ⌰D y 3cos 2 ⌰A q Tp p into Eq. ŽA1., and rearranging terms yields: 2 s 23 y 13 dy 13 aq dŽ1 y a.cos 2 ⌰D q aŽ1 y d .cos 2 ⌰A q adTp p , which is identical to Eq. Ž8..
Acknowledgements Support by NASA NCCW60 is gratefully acknowledged.
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