CHAPTER 3 Optimizing the Orientation Factor KAPPA-SQUARED for More Accurate
FRET Measurements
B. Wieb van der Meera, Daniel M. van der Meerb, and Steven S. Vogelc
a
Department of Physics and Astronomy, Western Kentucky University, 1906 College
Heights Blvd. #1077, Bowling Green, KY 42101-1077, USA, [email protected]
b
TelaPoint, 9500 Ormsby Station Road, Suite 402, Louisville, KY 40223,
[email protected]
c
Laboratory of Molecular Physiology, National Institute on Alcohol Abuse and
Alcoholism, National Institutes of Health, 5625 Fishers Lane, Room TS-06F: MSC 9411,
Bethesda MD 20892-9413, USA, [email protected]
3.1 Two-thirds or not two-thirds?
Two-thirds or not two-thirds? That is the question. Kappa-squared can vary
between 0 and 4, but when the orientations of donor and acceptor dipoles randomize
quickly its value is 2/3. Many authors of FRET papers adopt this assumption. However,
there is strong evidence that kappa-squared is definitely not equal to 2/3 in many cases.
For example, in reference 1 a series of DNA conjugates in which a donor (stilbene
dicarboxamide) and an acceptor (perylenedicarboxamide) are covalently attached to
opposite sites of an A:T base pair duplex domain consisting of 4-12 base pairs yielding a
FRET efficiency that is strongly nonlinear with varying distance. For 7-9 base pairs the
efficiency drops to almost zero consistent with a near-zero value of kappa-squared,
whereas for 5 and 10 base pairs the efficiency reaches a maximum consistent with a
kappa-squared value of 1 [1]. In another example [2], a Cy3-donor and a Cy5-acceptor
are attached to the 5’ termini of duplex DNA via a 3-carbon linker to the 5’-phosphate so
that they are predominantly stacked onto the ends of the helix in the manner of an
additional base pair [2]. The transition dipoles are essentially perpendicular to the helical
axis, and the periodicity is of the order of 5 base pairs. As a result, kappa-squared
changes dramatically with donor-acceptor distance, approaching zero at 13 and 18 base
pairs. The graph of FRET efficiency versus donor-acceptor distance looks like the graph of
the height of a bouncing ball versus time. In reality there is some motional averaging so
that for none of the base pair choices the efficiency dips to zero, but there are clear
maxima and minima in the efficiency-versus-distance-curve at predictable donoracceptor distances. The error from assuming kappa-squared = 2/3 is about 25 % at 13
base pairs [2]. In a third example [3], a Cy3-donor and a Cy5-acceptor are rigidly
attached to DNA in such a way that the dipoles are essentially parallel to the axis of the
DNA molecule. In two cases a configuration of collinear donor and acceptor dipoles was
engineered: one with Cy3 and Cy5 on the same B-DNA strand and separated by 3 helical
turns (sample 1 in reference 3) and the other with Cy3 and Cy5 on opposite strands and
separated by 2.5 turns (sample 2 in reference 3). In both cases the two oscillating dipoles
are expected to be collinear for which configuration kappa-squared reaches its maximum
value of 4. The experimentally obtained kappa-squared values (from the known
16
distance, the measured efficiency and the known Förster distance, R0  6.34 2 
nanometers) was 3.2 for sample 1 and 3.5 for sample 2, indicating the near-parallel
alignment of the dipoles with the line connecting donor and acceptor [3]. These three
examples of having a kappa-squared different from 2/3 are for traditional donors and

acceptors attached to DNA. Furthermore, the 2/3-assumption also fails in FRET
experiments using fluorescent proteins as donors and acceptors rotate slowly relative to
the excited state lifetime and the inverse transfer rate.
3.2 Relevant Questions
Kappa-squared needs attention, but this orientation factor does not need to be
disorienting! A few relevant questions must be asked. Do the orientations of donor and
acceptor change during the time transfer takes place, that is, during a time equal to the
inverse rate of transfer? And, if they do, how? Do they change rapidly and frequently
during the time transfer takes place? Does the dynamic averaging regime apply or the
static averaging regime or neither? Is the fluorescence polarized? To what extent? Can
depolarization factors be measured? Are simulations available or is structural
information obtainable that may exclude certain orientations? First, we need to know
how to visualize kappa-squared.
3.3. How to visualize Kappa-squared?
Kappa-squared for one pair depends on the direction of the emission transition
moment of the donor, on the orientation of the absorption transition moment of the
acceptor, and on the direction of the line connecting the centers of donor and acceptor.
We can introduce unit vectors: dˆ along the emission transition moment of the donor, aˆ
along the absorption transition moment of the acceptor, and rˆ pointing from the center
of the donor to the center of the acceptor. These three unit vectors are shown in Figure
3.1.



Figure 3.1 The unit vectors , and ;
is along the emission transition moment of
the donor,
along the absorption transition moment of the acceptor, and
points
from the center of the donor to the center of the acceptor.
A diagram like this does not do justice to the 3-dimensional character of kappa-squared.
Holding your two index fingers in front of your face and simulating the donor dipole with
your left index finger and that of the acceptor with the other index finger is much better
than staring at diagrams like the one in Figure 3.1. You can then rotate one hand around
its wrist and figure out how kappa-squared changes, then rotate the other hand and
evaluate the orientation factor again. Rotating the hands around each other also
simulates significant changes in kappa-squared. The angle between dˆ and aˆ is T , that
between dˆ and rˆ is D , and that between aˆ and rˆ is A . There are three common ways
of expressing kappa-squared (  2 ) in angles:

 
2
2


cos


3cos

cos

(3.1)



T
 
D  A

2
(3.2)
 2  sinD sinA cos  2cosD cosA 

  cos  1 3cos 
2
2
2
D

(3.3)

where  is the angle between the projections of dˆ and aˆ on a plane perpendicular to rˆ ,
and  is the
angle between aˆ and the electric dipole field due to the donor at the
location of the acceptor. This electric field is along 3rˆ cosD  dˆ ; a unit vector along this








direction is eˆD  3rˆ cos D  dˆ

1 3cos2  D . The angles appearing in equations 3.1, 3.2,
and 3.3 are illustrated in Figure 3.2.

Figure 3.2 In this illustration of the angles in the equations 3.1, 3.2 and 3.3, T  97.18 ,
D  A  60 ,   120 and   48.59, yielding  2  0.7625 . The unit vectors dˆ , rˆ and
eˆD are in the DR-plane, aˆ and rˆ in the AR-plane, and, eˆD and aˆ are in the EDA-plane.




This diagram
also
illustrates
the
different
planes
formed
by
the
vectors:

 the DR-plane


ˆ
ˆ
through d andrˆ , the
AR-plane through aˆ and
DA-plane through d and aˆ , and the
rˆ , the
EDA-plane through eˆD and aˆ . Note that eˆD lies always in the DR-plane and that, whenever
aˆ is perpendicular to this plane, aˆ is also perpendicular to dˆ , rˆ and eˆD , so that kappasquared
gives 
insight in the distributionof kappa-squared
values.

 is zero. Equation 3.3

ˆ
The highest
possible, 4, can
value
 only be realized if rˆ and d are parallel or anti-parallel
2
yielding cos D 1 and
if, at the same time, aˆ 
and
 eˆD are
 parallel or anti-parallel
2
ensuring that cos  is also equal to 1. On the other hand, whenever aˆ and eˆD are
perpendicular to each other kappa-squared
 zero. Therefore, if we consider all
equals
possible orientations there is a very high
that kappa-squared is low and a much
chance


smaller
is 2/3: if all
 probability that it is high. Accordingly, the isotropicaverage

orientations are equally probable, the average of cos 2  is 1/3 and that of cos2 D is also
equal to 1/3, so that equation 3.3 predicts the isotropically averaged value of kappasquared as 2/3. An expression of kappa-squared in terms of unit vectors and dot
products is also relevant:



 
 2  aˆ  dˆ  3aˆ  rˆ  rˆ  dˆ
2
2 2
2 
 aˆ  eˆD  1 3 rˆ  dˆ 


 
From this expression it is clear that kappa-squared does not change if we:

1. flip the donor transition moment, dˆ  dˆ .
2. flip the acceptor transition moment, aˆ aˆ .


(3.4)
3. let the donor and acceptor trade places, rˆ rˆ .
4. interchange the donor and acceptor transition moments, aˆ  dˆ .
The transition moments can be visualized as rod-like molecular antennas, or even index
fingers. It is instructive to choose
different orientations and evaluate the corresponding
kappa-squared values as is done in Figure 3.3.

Figure 3.3 Examples of donor and acceptor orientations with corresponding
values.
The donor dipole is along the bar in the center of each circle with various examples of
acceptor dipoles, also depicted as bars, along the circumference. For the first circle (on
the left) the donor and acceptor transition moments are parallel to each other and
varies between 4 and 0 depending on the location of the acceptor on the circle. These
values are labeled next to the acceptor. For the second circle the acceptor dipole is along
the electric field caused by the donor at the location of the acceptor with
values
between 1 and 4. On the third and fourth circle (on the right) the orientation factor is
zero for each example as the acceptor dipole is perpendicular to the donor electric field.
For the third circle the acceptor is oriented in the DR-plane, but for the fourth circle the
acceptor is perpendicular to this plane.
3.4. Dynamic averaging regime
One of the most important questions refers to the time scale of the reorientations
exhibited by donor and acceptor compared to the time transfer takes place. If the
transition moments of the donor and acceptor reorient at a rate much larger than the
transfer rate the dynamic averaging regime applies, whereas the static averaging regime
applies if the rate of reorientations for these transition moments is much smaller than
the rate of transfer. Dale et al [4] and Haas et al [5] have shown that in the dynamic
averaging regime fluorescence depolarization data allow one to remove some of the
uncertainty in the FRET distance resulting from kappa-squared. Dale et al emphasize
depolarization because of rapid restricted rotations [4], whereas Haas et al [5] mainly
consider excitation into overlapping transitions as the reason for low polarization values.
However, in the dynamic averaging regime depolarization due to degeneracy or overlap
of transitions is essentially indistinguishable from depolarization resulting from

reorientations. Indeed, it has been shown that the equation for the average kappasquared derived by Dale et al., equation 3.7 below, is the same as the one derived by
Haas et al., if cylindrical symmetry in the transitions is assumed [6]. In the dynamic
averaging regime the kappa-squared value fluctuates around an average value. And, this
average value may be used instead of the kappa-squared value appearing in the FRET
efficiency, if the following assumptions are valid.
1) The donor emission moment, which is along the unit vector dˆ , fluctuates rapidly
around dˆ X (unit vector, called “donor axis”).
2) The acceptor absorption moment, which is along the unit vector aˆ , fluctuates rapidly
around aˆ X (unit vector, called “acceptor axis”).

dˆ depends only on the angle
3)
The
probability
that
the
donor
emission
moment
is
along

DX between dˆ and dˆ X , but not on the azimuthal angle.

4) The probability that the acceptor absorption moment is along aˆ depends only on the
angle AX between aˆ and aˆ X , but not on the azimuthal
 angle.
5) 
The angle
  D between the donor absorption and emission transition moments is
unique if a single transition (for example, S2  S1 ) is 
excited. If, on the other hand,
into overlapping transitions,  D can assume a range of values.
occurs
 excitation
6) The
same is true for  A , that angle between the acceptor absorption and emission
transition moments.

The average kappa-squared will depend on “depolarization factors”, which have the

form:

depolarization factor

3
2
cos2   12   32 cos2   12 F  sind
(3.5),
0
where the brackets indicate an average, defined by the integral, and F   is the
 corresponding distribution function. These factors are larger or equal to -1/2 (when
F   has one sharp peak at 90 degrees) and smaller or equal to 1 (when F   has one
narrow maximum at 0 degrees. The relevant depolarization factors
are:



d  dDX 
3
2
cos2 DX  12
a  dAX 
3
2
cos2 AX  12

(3.5b)
dD 
3
2
cos2 D  12
(3.5c)
dA 
3
2
cos2 A  12
(3.5d)

These depolarization factors are related to anisotropies.
parameters are:

The relevant anisotropy
r0D = “time-zero” value of the donor emission anisotropy

(3.5a)
r0A = “time-zero” value of the acceptor emission anisotropy
rfD = fundamental anisotropy of the donor, in the absence of motion
rfA = fundamental anisotropy of the acceptor, in the absence of motion

Here
“time-zero” must be understood in the context of the dynamic averaging regime,
and a so-called Bürkli-Cherry [7] plot illustrates the concept of “zero-time-anisotropy”.

Figure 3.4 A log-log plot of fluorescence anisotropy versus time after a flash excitation
is also called a Bürkli-Cherry plot [7]. It shows a stepwise decrease of the anisotropy
with time and nicely illustrates that “zero-time-anisotropy” in the context of FRET refers
to the anisotropy value reached after the completion of rotations with frequencies
higher than the average transfer rate.
Figure 3.4 may suggest that time-resolved fluorescence anisotropy measurements are
necessary in order to obtain depolarization factors. However, Corry et al have shown
that steady-state confocal microscopy also enables one to measure such factors, and that
kappa-squared can even be obtained if some knowledge of the relative geometry is
assumed [8].
The relationships between anisotropy parameters and depolarization factors are:



rfD 
2
5
dD
(3.6a)
rfD 
2
5
dD
(3.6b)
r0D  rfD d 2  25 dD d 2
(3.6c)
r0A  rfA a2  25 dA a2
(3.6d)
Note that the experimentally obtained factors d 2 and a 2 are in the range 0 to 1, whereas
d and a are in the range -0.5 to 1, so that d and a can be either positive or negative if

d 2 and a 2 are between 0 and 0.25. These sign ambiguities may be resolved if
independent structural or spectroscopic
is available.
 information






The average value of the orientation factor is [4]:

 2  23  13 a  13 d  d 1 acos2  D  a1 d cos2  A  ad1,12
with
  sin sin cos  2cos cos
D
A
D

2
2
1,1
A
(3.7a)
(3.7b),

where  D is the angle between the donor axis and the line connecting the centers of
donor and acceptor,  A is the angle between this connection line and the acceptor axis,

and  is the angle between the projections of the donor and acceptor axes on a plane
2
perpendicular to the connection line. Dale et al found the maxima and minima of  in
equation 3.7
numerically and presented a contour plot allowing to read the highest and
2
 lowest  value for each combination of depolarization factors [4]. It is also possible to
find these maxima and minima analytically by setting the derivatives of 
 2 with respect
to  D ,  A and  equal to zero, solving the set of three resulting equations [9]. As
shown on the website FRETresearch.org, there are six candidates for maxima and
minima, which are:


 
(3.8a)
 A2  23  23 a  23 d  2ad




 P2  23  13 a  13 d
(3.8b)
 2H  23  13 a  13 d  ad
(3.8c)
 M2  23  16 a  16 d  ad  12 a  d
(3.8d)
 L2  23  16 a  16 d  ad  12 a  d
(3.8e)
T2  19 1 a1 d 
(3.8f)
4
9
1 a1 d1 2a1 2d 

The distributions of transition moments can be visualized as ellipsoids with the symmetry axis
equal to 1 2d for the donor and 1 2a for the acceptor, and any axis perpendicular to the
 axis equal to 1 d for the donor and 1 a for the acceptor. As a result, in the
symmetry
extreme situation where the depolarization factor equals 1, the distribution behaves like
a needle-like “molecular
 antenna”, and in the other extreme where it is -0.5, the



transition dipole distribution resembles a disk-like “antenna”. Figure 3.5 illustrates the
meanings of the six candidates using such ellipsoids and verbal descriptions.
Figure 3.5 Description of the candidates for maxima and minima of the average kappasquared in the dynamic regime, specified by the equations 3.8. They are also candidates
for the most probable kappa-squared in this averaging regime.


Careful comparison (FRETresearch.org) of the magnitudes of one candidate relative to
those of the others in all points of the plane formed by parameter values  12  d 1 and
 12  a 1, leads to the conclusion that there are 9 different regions where the maxima
and minima can be calculated using the expressions for the 6 candidates in equations 3.8a
thru 3.8f. These regions have borders expressed as d  0 , a  0, C  0, E  0, F  0 or

G  0 , where C , E , F and G are defined as:
 








C  a  d  12
(3.9a)
E  3a  3d  5ad 1
(3.9b)
F  2d  3a  2ad 1
(3.9c)
G  2a  3d  2ad 1
(3.9d)

The regions are shown in Figure 3.6.

Figure 3.6 Regions in the a,d-plane showing in each a column with the kappa-squared
maximum indicated by the top letter and the minimum by the bottom letter. Table 3.1
gives details.

The meaning of the symbols and the properties of the different regions are specified in
Table 3.1.
Table 3.1 Maxima and Minima in the Dynamic Averaging Regime








Name
of
Region
Maximum
Minimum
Definition of Region
2
2
 in that  in that
Region
Region
A
1

P 
 A2
 P2
a  0;d  0;F  G  C  0
Yes
A

2

P 

 A2
 P2
a  0;d  0;F  G  C  0
2
All but not T
M 

3

H 

 M2
a  d  0;C  0
2
All but not T

M 

4

T 

 M2
M 

5

L 

 M2
H 

6

A 

 2H
M 

7

A 

 M2
H 

8
L 
 2H
H 

9

T 

 2H









 2H
T2



a  d  0;E  C  0;F  G  0 Yes

a  d  0;F  G  0
2
All but not T
 A2 
a  0;d  0;E  0
2
All but not T

 A2 
a  d  0;E  0
2
All but not T

 L2
a  0;d  0;F  G  0
2
All but not T

 L2


T2 


Are all candidates
valid there?
a  0;d  0;F  G  E  0
Yes


 depolarization factors, for donor and acceptor, are positive the
Note that if both
2
minimum  is  P2 and the maximum is  A2 , as pointed out by Dale et al [4]. The reader
A
A
may wonder why we split up this region in a central  1 -zone and 3 sections of  2
P 
P 

around
it. The reason is that the
6 candidates are not only possible maxima and minima,

but are also potential answers to the question “what is the most probable kappa-squared
value?”, and Figure 3.6 also serves as the starting point in our approach to this question.


3.5 What is the most probable kappa-squared in the dynamic regime?
What is the most probable kappa-squared value? This is an ambiguous question!
If we are trying to find the most probable value leaving all distances, all orientations and
all efficiency values open, we will get one answer, but if we want the most probable  2
at a given efficiency a completely different answer emerges. It is well established that
the probability for kappa-squared of a pair of linear donor and acceptor transition
moments exhibits an infinitely high peak at  2  0 , if we do not specify the
efficiency
[see equation 3.14 below and, for example, references 4, 6 and 10]. However, if we have
a nonzero efficiency in an actual experimental situation, the most probably kappa2
squared value cannot be equal to zero,
 because   0 means that the efficiency equals
zero also. The explanation for this apparent paradox is that considering all possible FRET
situations while varying both distance and orientations independently is completely
different from focusing on a certain FRET
 efficiency where distance and orientation are
linked via the so-called “relative distance”:
 "Relative Distance"
16
Actual FRET Distance
 32  2 
2
2
Distance assuming   3
(3.10)
2
Figure 3.7 illustrates this probability subtlety in the case a  d 1 when  can be
of
calculated from equation 3.3. The area of the square formed by all possible values
2
cosD and cos between 0 and 1 represent the total probability of 1 that  has any
value between 0 and 4, 0 for cos  0 and 4 for cos
. We can divide up
  cosD 1

the square in say a hundred strips by drawing the 101 curves cos   2 1 cos2 D 

 the square choosing  2 equal to 0x4/100, 1x4/100, …, 99x4/100,
inside
and 100x4/100.


This way the area of each strip represents the probability that kappa-squared has a value
2
between the  for the lower boundary and that of the
upper boundary. Such a division
has been initiated 
in the left hand panel of Figure 3.7. The very first strip between the 0curve and the 0.04-curve has by far the largest area and successive strips rapidly
decrease
 in area, indicating that the most probable value for the orientation factor is 0.
However, nothing is said about the distance or the efficiency. Over that very first strip
the relative distance is 0 at the lower boundary but 0.626 at the higher boundary,
2
whereas the maximum relative distance is 1.35. As a result the very first strip in 
represents 46% of all distance choices.

Figure 3.7 Pair of diagrams illustrating that the question “what is the most probable
kappa-squared value?” is ambiguous. This example refers to the dynamic regime and
. The diagram on the left is for unknown efficiency with
as the most
probable value. The diagram on the right corresponds to having a known efficiency and
thus a link between the orientation factor and relative distance resulting in
being
the most probable kappa-squared value.
It seems more appropriate, therefore, to translate the combination of a measured
efficiency and an independently obtained Förster distance to a distance with an
orientational uncertainty specified by equation 3.10. In terms of the example of Figure
3.7, this means we should divide up the square by drawing 101 curves
cos 


   1 cos  
2
3
6
2
D
inside the square by choosing  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 strip
represents the probability that the relative distance has a value between the  for the
lower boundary and that of the upper boundary. 
This is indicated
 in the right hand
panel of Figure 3.7, where careful analysis shows that the strip straddling the left upper


16
2
corner of the square has the biggest area, corresponding to   32   1.07 and to   1

. In general, if we are interested in the most probable average orientation factor in the
dynamic regime for any FRET situation at non-specified efficiency, for a desired
combination of a and d , we should look for the frequency distribution in
the orientation

2
factor, p  [6], and the most probable value considering all possible distances and
orientations is the one where this distribution has its highest peak. On the other hand, if
 the
we want
kappa-squared value corresponding to the most probable distance in a
given FRET situation at a measured efficiency at a certain a,d -pair, we should consider

the frequency distribution of the relative distance, Q [6], and the most probable
relative distance at a given efficiency is the one where Q has its highest peak, PEAK .
 is
In the latter case, the most probable kappa-squared



2
3
 PEAK  .
6
p 2  14 Q   5 ,


and, because of the  5 -factor in this proportionality relation the maximum of Q may
differ dramatically from the maximum of p , as in the example of Figure 3.7.
An algorithm to find the most probable kappa-squared in the second
case (at a given

efficiency) is briefly as follows (see FRETresearch.org):

1) Choose the a,d -pair that best describes the depolarization properties of the actual
system (see equations 3.6c and 3.6d, and the explanations near these. Note that
axial symmetry is assumed. If axial symmetry cannot be justified see [5]).
2) Because  12  a 1 and  12  d 1, this pair must lie within one of the regions

shown in Figure 3.6 and Table 3.1. Use this Figure or Table to decide which region
A A
H 
(  1 ,  2 ,…, or  9 ) applies. Choose B , the number of bins, and vary the bin
P  P   T 
number i from 1 to B , obtaining bins with relative-distance-values between
2
min  i 1max  min  B and min  i max  min  B , with min 32  min
1 6 and

2
16
   max 32 

max  .
 n = 1 to B
3) For
set Qn to 0. This Qn is the n th component of a B -dimensional

end become a histogram approximating

array, which will in the
the frequency
distribution of the relative distance, Q.

3
 4) Choose
 
N , and in so doing
l and m from 1 to N ,
 pick N points, varying j , 
calculating cosD   j 1 N 1, cosA  l 1 N 1,    m 1 N 1
Substitute these into eq. 
3.7 to calculate  2 -values and from there relative the lower
distance-values   3 
2 16
  with
. Compare each
and upper
2



boundary of each bin. Place each  in the appropriate bin by adding 1 to each


Qn whenever  > min  n 1
max  min  B and   min  n max  min  B
n  B.
with 1

3
5) Normalize Q by dividing each
component
by N . As a result, the sum of all Qn
values will
become 1, signifying that the probability

 
  that Q has any value equals
3
 1. (For n = 1 to B , Qn  Qn N ).



We have examined graphs of Q obtained with this
algorithm for a large number of
points
plane formed by the depolarization factors, a and d , varying these
in the

between -0.5 and 1. Results for the most probable kappa-squared are shown in Figure
3.8.



Figure 3.8 Map indicating the most probable kappa-squared in the dynamic regime,
where this most probable value is defined as the kappa-squared for which the relative
distance is the most likely (see text). When one or both of the depolarization factors for
the donor or acceptor are negative  P2 is the best. The region where both depolarization
factors are positive consists of 4 regions labeled H, where  2H is the most probable, 2
regions labeled M where  M2 is the best, and 1 labeled L, where  L2 is the most probable
kappa-squared value. The
border between the H and L regions near the top right corner
is well described by
. The curved border between H and M on top

and L on the bottom, starting at
and ending near
, follows


the trend
; and the one with H and M on the right and L on the left is
described by
.
The definition of the most probable kappa-squared in Figure 3.8 is that value which
corresponds with the highest peak in Q:
most probable 2  23  6peak


(3.11)
Whenever both or one of the depolarization factors is negative, the most probable
kappa-squared is  P2 . In the region where both depolarization factors are positive there
is a rather large central region where it is  L2 , surrounded by four regions with  2H as the
best value and two regions where  M2 is the most probable value. The uncertainty in the
distance
as a result of variations in the orientation factor has two aspects: the most
probable kappa-squared may deviate from 2/3, that is, the location of the peak may


differ from   1, and, the peak may be fairly broad, that is, the 67%-confidence-interval

may have considerable width (the 67%-confindence-interval is the range of  -values
near the peak where the total Q adds up to 67%). It is appropriate to call the first
aspect
(BDE).
 a “Peak-Location-Error” (PLE) and the second a “Broad-Distribution-Error”
2
2
We note that   1 is the relative-distance value that corresponds to   3 , and, thus,

define the PLE as:

 PLE = "Peak - Location - Error" = 1 PEAK  100%
(3.12)
PLE > 0 means that  2  23 overestimates the most probable distance, and PLE < 0
signifies that this assumption underestimates the distance at the peak. Figure 3.9 shows
examples of distributions and PLE-values.

Figure 3.9 Examples of frequency distributions of the relative distance illustrating the
definition of the Systematic Error. The graph on the left is for a  d  0.73 where the
main peak corresponds to  L2 with a relative distance smaller than 1 so that PLE is
positive (this Q has a secondary maximum corresponding to  2H ). The distribution in the
2
2
2
center is for a  1, d  0.5 or d  1, a  0.5 showing one
 peak matching  H   M  3
yielding   1 and PLE
with a peak at   1.07
 = 0. The graph on the right is for
2
corresponding
to  H  1 and a negative PLE.








Our definition of the “Broad-Distribution-Error” is:
BDE = "Broad - Distribution- Error" = 12 67% - confidence- interval = UL  LL   50% (3.13),

where  LL is the lower and UL is the upper limit of the 67%-Confidence-Intervale (CI) for
Q. In some cases the peak is near the center of the confidence interval, but relativedistance-distributions can also be highly asymmetric with the peak at the upper or lower
limit of this interval.
Figure 3.10 shows examples.



Figure 3.10 Examples of frequency distributions of the relative distance illustrating the
definition of the Random Error. In each the 67%-Confidence-Intervale (CI) is shaded dark
grey and runs from  LL , the lower limit of the CI, to UL , the upper limit of the CI. The
three graphs have the same scale in  and Q . The one in the center is relatively broad
and low. The other two are narrow and high, actually extremely high, as both go to
infinity at one point on the interval. The graph on the left is for a  0.5, d  0 or


d  0.5, a  0 with its peak at UL
and a BDE of about 1%. The distribution in the center

is for a  d 1 with its peak near the average of  LL and UL , and a BDE of about 24%.
The graph on the right is for a  1, d  0 or d  1, a  0 with a
peak at UL , and a BDE of
about 7%.






Figure 3.11 shows lines of equal “Peak-Location-Error” in the a,d-plane.

Figure 3.11 Lines of constant “Peak-Location-Error” are shown with the value of WPE
given next to the lines in %. At the red curves the relative-distance-frequencydistribution has two equally high peaks. These curves are borders between regions
where the most probable kappa-squared is calculated differently as indicated in Figure
3.8. At one side of a red curve one of the peaks is highest and on the other side the
other peak is highest. As a result, the WPE changes discontinuously when a red line is
crossed. The green lines are also borders between regions where the most probably
kappa-squared is calculated differently but with a continuous change in the value of the
peak and of WPE.
Near a  d 1 and a  d   12 the PLE is negative, but in the majority of points the WPE is
positive with the most probable distance smaller than the one at  2  23 . A very high
positive SE of about 30% occurs near a  d  0.96 , close to the red line. On the red line
 Q has two
equally high peaks. The red line is the border between two regions where
 peak is calculated differently. As a result, PLE changes discontinuously at this border.

The most dramatic change is
at a  d  0.96 where the PLE is -6%, corresponding to  2H ,
at the side where the factors are slightly higher than 0.96, and +30%, corresponding to

 L2 , at the side where the depolarization factors are slightly smaller than 0.96. The green




lines are also borders between regions where the peak is calculated differently, but with
a continuous change in PLE. A large discontinuous change in PLE also implies a fairly
broad distance-distribution, and, therefore, a relatively large BDE. Results for BDE are
shown in Figure 3.12.






Figure 3.12 Regions with lower and higher “Broad-Distribution-Erro (BDE), defined in eq.
3.13. A high BDE corresponds to a broad Q, and low BDE values indicate graphs for
Q with a narrow peak. On the green lines BDE equals 5%. The long green line
connects 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
, 0.33,0.33 and
0.31,0.5. The short green
 line passes
 through0.5,0.25
is smaller than
0.25,0.5
. In the region

 between
 the green
 lines BDE
 5% reaching 0%
at 
the long green
a  d  0 . At
0.5,0.5
 BDE = 8%.
In between
 line and the red lines
BDE varies between 5% (on green) to 10% (on 
red). At 1,0.5
 and 0.5,1 BDE is about

12% and on the short red curves near these points RE equals 10%. At 1,1 BDE = 24%




and BDE decreases with decreasing a and/or d reaching BDE = 10% on the 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.


This
diagram shows
data
for
the
CI,
the


 67%-Confidence-Interval,

obtained with our
program for finding Q (available on the website) at any choice for the depolarization





factors a and d . To run this program one must choose an a and a d , a value for B (the
number of bins, that is, the number of bars in the histogram-approximation for the Q function) and a value for N (a measure for how many times the relative distance is

3
using
equations 3.7 and 3.10; the number
evaluated
 of points

 is N ). After locating the
peak (allowing one to confirm the results of Figure 3.8) the CI is obtained by moving

away from the peak in both directions while adding the Q -values of the bars in the
histogram until 0.67 has been reached. Near the axes, a  0 or d  0 , the peak is
extremely asymmetric with the relative-distance at the peak, PEAK , coinciding with the
lower limit of the CI,  LL , at positive a or d , and
 matching the upper limit, UL , at
a
d
negative or , as shown in Figure 3.10. Away from
axes, say for a  0.1,d  0.1
 these
or a  0.1,d  0.1 or a  0.1,d  0.1 or a  0.1,d  0.1, the peak is more symmetric

and PEAK is close to the center
of the
CI. A completely different problem arises near the




red borders
shown in Figure 3.11. At the red borders the Q -function


 has two peaks that
a  d  0.96 the Q -function has a peak
are exactly of
For

equal height.
example, at
2
2
 corresponding with    H  0.948 ( PEAK = 1.06) and an equally high peak for

 center of the CI should be chosen at
 2   L2  0.065 ( PEAK = 0.68). In such cases the
 the CI should 
the average of the two PEAK -values, and
be built up from there. Examples

of graphs for
the frequency distributions, Q , versus the relative distance,  , are shown
in Figure 3.13.




Figure 3.13 Examples of frequency distributions for the relative distance with the 67%Confidence Interval (CI) indicated in each as a dark grey shaded area between red lines.
All graphs have the same vertical scale and the same horizontal scale. The width of each
box refers to a relative distance of 1.4, and the height of each box is 9.5 in Q-units. For
most choices of a and d a distribution with one dominant peak is found but for
parameter choices near the red lines in figure 3.11 more than one equally pronounced
peaks may occur as is shown in the right bottom corner for a,d = 0.81,0.95 or
0.95,0.81
. Data
for these plots are shown in Table 3.2. The reader will be able to
generate his or her own graphs for any choice of a and d by visiting the website
(FRETresearch.org).












Table 3.2 Data for Figure 3.13a (see FRETresearch.org)
2
2
minimum maximum LL
PEAK UL
QPEAK b
 minimum
 maximum
a
d
or
or
a
d
1
1
0
1
0.953
1.070
0.953 0.953 1.013 
2






 2 1  1 
1
5
2
2
0.894
1.110
0.958 1.066 1.108 5.613
4
4
1
1
5
0.891
1.038
0.986 1.038 1.038 
3
6
  2 0
1
1
1
17
 2  2
0.794
1.134
0.930 0.994 
1.056 6.602
6
12
1
0
2
0
1.201
0.829 0.949 1.069  c
 1  2 
1
4
1  0
0.891
1.122
0.891 0.891 
1.026 
3
3

1
1
8
0.794
1.260
0.887 0.994 1.101 4.295
6
3
 1  2

1
1
1
11
0.891
1.184
0.929 0.976 
1.020 10.077
2
3
6
 2

1  1
0
4
0
1.348
0.809 1.051 1.294 2.091

0.08
3.379
0.702
1.311
0.702 0.848 1.006 2.691
 0.81
 0.95 
a
= 300 and
B = 100, but when a or d = 0, data have been calculated analytically
 N 
(see FRETresearch.org)
b
The frequency distribution of kappa-squared is proportional to that for the relative
distance [6] according to the relation, p 2  14 Q   5 , with   32  2 
56
c
Mathematically one can show that the Q value at the peak is  , but numerical values
depend on B and N



3.6 Optimistic, 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 or no
error, and there is a “conservative” method based on depolarization factors resulting in a
minimum and maximum kappa-squared (with corresponding minimum and maximum
distances) without the ability to pinpoint the most probable kappa-squared in this range.
The optimistic method is that of Haas et al [5] and Steinberg et al [11], and the
conservative approach is that of Dale et al [4]. This classification is an oversimplification,
of course, as both the first group and the second group of authors have provided a
detailed and versatile discussion of errors resulting from the orientation factor.
Nevertheless, neither group has pointed out that there are at least two different aspects
associated with the kappa-squared-induced-error: the PLE, introduced in equation 3.12,
and the BDE, introduced in equation 3.13. For lack of a better name we would like to call
the procedure introduced in the previous section the “practical” approach. Table 3.2
compares the “optimistic”, “conservative” and “practical” approach for a range of cases.
Table 3.3 Comparison of the Optimistic, Conservative and Practical Approach to the error in the
distance due to kappa-squared.
c
2
2
a
d  minimum  maximum minimum maximum Optimistic Conservative Practical Approach
a
b
or or
Approach
Approach
 LL
PEAK UL
a
d
  
1
1
 

0

4

0
1.348
  1 0.17
  0.674  0.674 0.836
1.070
1.348
  1 0.15
 1.038
0.926 
0.978
 0.147
1.026
  1 0
 1.007  0.116 0.891
1.026
1
2
1
2
1
3
11
6
0.891
1.184
1
0
1
3
4
3
0.891
1.122

1.201


  1
0
0.891
1
  0.601 0.601 0.889 0.953 1.105
1
0
2
0
   - 2 
2
2
  1 0
  1 0
0
0
1
1
1
1
1
3
3


 1 
1
5
  1 0
  0.965  0.074 1.026 1.038 1.038
0
0.891 1.038
3
6
-2


 1
1
1
5
  1 0.13  1.002  0.108 0.930 1.070 1.070
0.894 1.110
2
4
4
-
- 2 


a
Haas et al [5] and Steinberg et al [11]
 point out that
 only if the polarization values for the donor
 

and acceptor polarization are relatively small   1 corresponds to the actual distance. However,
 are not small except for a  d  0 , so that
chosen here these 
polarization values
  for
 the examples

these examples other than a  d  0 , should be considered to lie outside the comfort zone of the
“optimistic approach”. The width of Q at half maximum can be taken as a measure of the error in

the distance [5, 11]. The peak of Q is infinitely high, however, for a,d or d,a = (1,0), (1,-0.5),

and (0,-0.5), so that
the
error
in
the
relative
distance
is
0
for
these
cases
if one adopts such a

measure for error. Dos Remedios
and Moens pointed out that in the great majority of FRET

distances for which crystallographic
distances are known the assumption that kappa-squared equals



to 2/3 is reasonable [12].
b
Dale et al indicate that a maximum and minimum kappa-squared can be obtained without any
recommendation for estimating the most probably distance in that range [4]. Consistent with this
conclusion is to state that relative distance is given by:   12 max  min   12 max  min  as
calculated in this column. It must be pointed out that the examples for which a,d or d,a = (1,0),
(1,-0.5), and (-0.5,-0.5) are actually in regions Dale et al specifically warned to avoid [4]. If the
depolarization factors are much closer to (0,0) the 
uncertainties are much less extreme, see for
example [13]. For instance, If 0.38<kappa-squared<1.25, the error in the distance is lower than 


10% [14].
c
This approach is introduced in the previous section.

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 3.8 and its definition (equation
3.12). However, the discontinuous jump in the systematic error near a,d = (0.96,0.96)
may cause serious problems as a value of 0.96 is experimentally almost undistinguishable
from 1, and thus a slight uncertainty in the depolarization factors near this value may
cause the PLE to shift from -6% to +30%. For such high values of a and d the BDE is also

high (see Figure 3.12). Comparing the confidence interval for a,d = (1,1) to that for



(0.81,0.95) in Figure 3.13 illustrates a problem related to the discontinuity in the PLE: a
relatively minor variation in the depolarization factors may cause this interval to shift
from one that is centered around  = 1 to one that is centered around 0.85. In reference
6 it was assumed that the case a  d 1 was the worst-case scenario. This is a logical
assumption as a  d  0 is the best-case scenario and the kappa-squared-induced-error
gets worse and worse when one moves away from a  d  0 . After reference 6 was

published it became possible
 to generate plots of the frequency distribution of distances
and kappa-squared
with a few keystrokes on a computer. So, now we must set the

record straight: a  d 1 is NOT the worst
case scenario as far as the orientationinduced-error is concerned in the dynamic regime, it appears that
a  d  0.985ad 1.012 , one of the red lines Figure 3.11, is the worst-case-scenario in
this regime.


3.7 Smart Simulations are Superior
It is imperative to be keenly aware of the assumptions underlying any method one
wants to apply. For example, in references 1 and 2 the depolarization factors are not
given but should be positive. Therefore, the “practical approach” would suggest that the
best kappa-squared value should be  2H ,  L2 , or  M2 . However, in the practical approach
it is assumed that nothing is known about cos D , cos A , or , the variables appearing
in equation 3.7. In the spirit of information theory it is assumed in this approach that all
values of these “hidden variables” are equally probable when no information about them
 

is available. Nevertheless, for the systems in reference
 1 and 2 information about the


relative orientation of donor and acceptor IS available: the transition dipoles are
essentially perpendicular to the axis of a helix and the angle between the dipoles should
depend on the pitch of the helix. This is actually an example of the case where the
transfer depolarization is known, within limits, as introduced and analyzed by Dale et al
[4], and in which case equations for kappa-squared have been derived [9]. The geometry
of the donor-acceptor pair in reference 2 suggests that the best kappa-squared value
should be a hybrid between  2H and  P2 ,  2HP  13 2  a  d  adcos2 T (equation 4.38 in
reference 9). Any available information that allows one to exclude certain donoracceptor orientations will help to narrow down the range of possible kappa-squared
values. Simulations can be a powerful tool in this exclusion process. A case in point is

 
the molecular dynamics simulations performed by Lillo et al [15]. Following the
“conservative approach” these authors found for donors and acceptors at specific sites in
a PGK (phosphoglycerate kinase) a fairly large range of possible kappa-squared values
and corresponding donor-acceptor distances, but they noticed that some of the values
for  D ,  A and  appearing in equation 3.7 were inconsistent with the crystal structure
of PGK and the excluded volume of the probes at the known sites in PGK. They
performed molecular dynamic simulations of kappa-squared utilizing equation 3.7,
measured
 depolarization factors, the crystal structure of PGK and the known locations of
 
the donors and acceptors, and found the most probable values of  D ,  A and ,
resulting in an improved kappa-squared value and more precise donor-acceptor
distances [15]. In the same spirit, Borst et al built structural models of the FRET based
calcium sensor YC3.60 and noticed that minor structural changes induced
by slightly
 
rotating the fluorescent protein around a flexible linker while keeping the same average
distance between donor and acceptor gave rise to any value of kappa-squared between 0
and 3, but that a five fold change in orientation factor (from 0.5 to 2.5) only brings about
a 1.3-fold increase in critical distance indicating that the FRET process in YC3.60 is mainly
distance dependent [16]. Gustiananda et al [17] presented FRET results from an intrinsic
tryptophan donor to a dansyl acceptor attached to the N terminus in model peptides
containing the second deca-repeat of the prion protein repeat system from marsupal
possum. They used simulations for finding the best kappa-squared in this system, and
extended their molecular dynamics simulations out to 22 ns to help ensure adequate
sampling of the dansyl and trypotophan ring rotations. They found good agreement of
the simulated kappa-squared value with 2/3 except at the lowest temperatures [17].
Deplazes et al performed molecular dynamics simulations of FRET from Alexafluor488
donors to Alexafluor568 acceptors [18]. In their system the isotropic dynamic condition
was met, meaning that all possible orientations of the transition moments of donor and
acceptor and of the line connecting their centers are equally probable and sampled
within a time short compared to the inverse transfer rate. The frequency distribution of
kappa-squared from the simulation data showed excellent agreement with the
theoretical distribution, which is given as [6]:


 2
p 2  


2
1
3
1
2

ln 2  3

0  2  1
 2  3

2
ln

 1   4
2
2
2
3     1 
(3.14)
Their results show that even in their simple situation, simulations lasting longer than 200
ns would be required to accurately sample the fluorophore separations and kappa
squared if only a single donor-acceptor pair had been included. Many aspects of FRET
were simulated in this study including frequency distributions of relevant angles, donoracceptor distance and FRET efficiency. As expected, very low correlation was found
between donor-acceptor distance and orientation factor [18]. VanBeek et al did find
such a correlation in a molecular dynamics simulation of a coumarin-donor and an eosinacceptor both attached to HEWL (hen egg-white lysozyme) [19]. In the dynamic regime it
is implicitly assumed that kappa-squared is independent of the donor-acceptor distance.
(In the static regime an indirect correlation between distance and kappa-squared is
expected as discussed near equation 3.24, below). The correlation between orientation
and distance in the molecular dynamics study of Vanbeek et al is quite strong and
involves both the sign and the magnitude of kappa (   cosT  3cosA cosD , the
square of which is given in equation 3.1, where the angles are defined as well). This
correlation is illustrated in Figure 3.14, which is a modification of Figure 6 in reference

19, graciously made available for this chapter by Dr. Krueger. An additional advantage of
molecular dynamics simulations is that no assumptions about timescales need to be
made whereas in the interpretations of FRET experiments the results do depend on
whether the system in is the dynamic or static regime.
Figure 3.14 Modification of Figure 6 of reference 19: a scatter plot of the donor-acceptor
distance against kappa showing the correlation between this distance, kappa and the
FRET efficiency. The color code on the right is for the efficiency. This graph has been
prepared by Dr. Brent Krueger.
Note that the FRET efficiency also shows a relationship with kappa and the donoracceptor distance in this illustration. The kappa-squared concept is based on the ideal
dipole approximation which is known to fail when molecules get “too close” to each
other. Muñoz-Losa et al performed molecular dynamics simulations to find out how “too
close” should be defined *20+. They showed that the ideal dipole approximation
performs well down to about a 2 nm separation between donor and acceptor for the
most common fluorescent probes, provided the molecules sample an isotropic set of
relative orientations. If the probe motions are restricted, however, this approximation
performs poorly even beyond 5 nm. In the case of such restricted motion, FRET
practitioners should not only worry about kappa-squared, but also about the failure of
the ideal dipole approximation [20]. In a more recent paper from the same lab an
improved construction of experimental observables from molecular dynamics sampling
has been proposed [21]. Hoefling et al have introduced a similar analysis [22].
3.8 Static Kappa Squared.
The initial steps of a FRET experiment involves the absorption of a photon by a donor
fluorophore. Absorption of a photon is rapid, typically occurring within a femto-second (1
X 10-15 s), and results in the elevation of a ground-state electron into a myriad of
potential electronic and vibrational excited states. Over the next few hundred femtoseconds this array of potential excited-state electronic and vibrational energy sub-level
are consolidated into the lowest energy singlet excited state, as a result of vibrational
energy loss due to subsequent kinetic interactions between the excited fluorophore and
surrounding molecules. Fluorophores in general spend from picoseconds to ten's of
nanoseconds in this relatively long-lived lowest singlet excited state before eventually
transitioning back to a ground state sub-level. With their return to a ground state, excess
excited-state energy will be either emitted as a photon (donor fluorescence), transferred
to a nearby acceptor (FRET), or excited-state energy will be lost by some other nonradiative mechanisms. To understand the factors that can influence the probability of
energy transfer by FRET, one must understand the types of events that can occur while a
fluorophore is in its excited state. Vis-à-vis kappa squared, the main factors that must be
considered is to what extent donor and acceptor fluorophores can move relative to each
other while in the excited state, to determine specifically how fluorophore motion may
influence the position of an acceptor relative to the orientation of the donor emission
dipole, and how it may influence the orientation of the acceptor absorption dipole
relative to the orientation of the donors excited-state electric field.
We begin our consideration of the impact of molecular motion during the excited state
on FRET by considering how the rate of FRET is influenced by the separation distance
between donor and acceptor (rDA), as well as by the orientation of donor and acceptor
dipoles relative to each other (2). The rate of energy transfer by FRET, kT, is dependent
on the inverse sixth-power of the separation between donor and acceptor [23, 24] and is
given by:
6
1  R0 
(3.15)
kT 
 
 0D rDA 
where  0 D is the fluorescence lifetime of the donor molecule and R0 is the Förster
distance, the separation
 at which 50% of the donor excitation events result in energy
transfer to the acceptor. Furthermore, the R0 value used for any specific donor-acceptor
FRET pair always assumes that the dipole-dipole coupling orientation factor (2) will have
a value of 2/3, but in reality can have any value from 0-4 in biological experiments, and
can be expressed as [6,11]:
k 2 = (1+ 3x 2 )z 2
(3.16)
This is equation 3.3 with the abbreviations z = cosw and x = cosqD , where  D is the
angle between the donor emission dipole orientation and the donor-acceptor separation
vector, and  is the angle between the donor electric field vector at the acceptor
location and the acceptor absorption dipole orientation. For a typical donor fluorophore
with a fluorescence lifetime of 3 ns, its excited state may last up to 5-times its lifetime, or
approximately 15 ns. It is therefore reasonable to consider: 1. If the separation distance
RDA can change during this period, 2. If the position of the acceptor relative to the donor
emission dipole orientation, and therefore the value of  D , can change during the
excited state, 3. If the orientation of the donor emission dipole changes and thus the
value of  D , and finally 4. If the angle between the donor electric field vector at the
acceptor location and the acceptor absorption dipole orientation () changes in this 15
ns period. Changes in RDA and/or in  D can be caused by significant lateral motion of the
acceptor fluorophore relative to the position of the donor fluorophore. Thus, our first
consideration should be how far can a fluorophore move by diffusion in 15 ns? Diffusion
is a function of the mass of the molecule, its hydrodynamic shape, temperature, as well
as the viscosity of the buffer. Assuming a temperature of 20°C and a buffer viscosity like
water, a small fluorophore may have a diffusion coefficient between 100 - 1000 µm2/s,
while a larger fluorophore like GFP will have a diffusion coefficient of 70 µm 2/s. Under
these conditions one might expect that a free fluorophore could diffuse a distance
between 1.4 - 5.5 nm during a 15 ns excited state. Clearly, such motion could influence
the effective value of RDA and  D in a FRET experiment. In practice, however, most donor
fluorophores will return to the ground state in a much shorter time span, with a median
value of ln2·  0 D , or in this instance ~2 ns, effectively limiting the distance that most
molecules can diffuse to 0.5 – 2.0 nm. Furthermore, when one considers that the
viscosity of cell cytoplasm is much higher than water, and that fluorophores used in
biological FRET experiments are typically coupled to much larger molecules such as
protein complexes or nucleic acids with much smaller diffusion coefficients, it is typically
assumed that lateral motion during the excited state will be so limited that it will not be
responsible for any alterations in the rDA or  D values for a specific pair of molecules
tagged with donor and acceptor fluorophores.
In addition to lateral motion, another type of motion that must be considered is
molecular rotation. Specifically we will consider if donor and acceptor fluorophores can
rotate during the excited state and if so, the impact of rotational motion on the values of
 D ,  and thus on FRET. Molecular rotation is typically parameterized by a rotational
correlation time (τrot), the average time that it takes a molecule to rotate 1 radian around
a specific axis. For spherical molecules the rotational correlation time will be the same
around all 3 axis. Non-spherical shaped molecules can have different rotational
correlation times for each axis. Rotational correlation times of fluorescent molecules can
be measured experimentally by monitoring the decay of fluorescence anisotropy as a
function of time after a transient excitation pulse [25]. In the absence of homo-FRET, the
decay of fluorescence anisotropy is primarily caused by molecular rotation. By fitting the
anisotropy decay to a model with 3 exponentials, the decay constant for each
exponential can be estimated. These decay constants are a measure of the rotational
correlation times around each independent axis [25].
i=3
r(t) = r0 × å ai × e
-t
t roti
(3.17)
i=1
Where r(t) is the time-dependent decay in fluorescence anisotropy, 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 decay component, and τrot-i is the rotational
correlation time of the ith decay component. In practice, differences in rotational
correlation times for the 3 axis for most fluorophores are hard to experimentally
distinguish, and more typically the anisotropy decay of a fluorophore will be fit to a
model with a single exponential assuming that the rotational correlation time is
approximately the same in each rotational direction [26].
r(t) = r0 × e
-t
t rot
(3.18)
Here the value of τrot is a function of the solution viscosity (η), temperature (T), and the
volume of the rotating molecule (V) given by [24]:
t rot =
hV
RT
(3.19)
Where R is the gas constant. For example, small fluorophores, like Fluorescein (332.31
g/mol), will have a rotational correlation time of ~140 ps in water at room temperature,
while a large 28,000 Da fluorophore like Venus (a yellow GFP derivative) has a rotational
correlation time of ~ 15 ns under the same conditions [25], presumably because the
volume of Venus is approximately 100 times greater than Fluorescein. When a
population or randomly oriented fluorophores (isotropic) are photo-selected using a
linearly polarized light source, the highest anisotropy value theoretically possible
(fundamental anisotropy) is 0.4 with 1-photon excitation, and 0.57 with 2-photon
excitation [25]. In practice, other factors can reduce the value of the initial anisotropy
value at time = 0. Thus, the limiting anisotropy measured in a time-resolved anisotropy
measurement is usually smaller that the fundamental anisotropy expected from theory.
With time, measured anisotropy values for fluorophores in solution that are free to
rotate in any direction will decrease as a single exponential with an asymptote at 0. This
value indicates the point where all remaining molecules in the excited state are randomly
oriented. The speed of this Orientational Randominization is parameterized by the
rotational correlation time. For a system decaying as a single exponential this occurs at ~
5X the rotational correlation time. Thus, for a small molecule like Fluorescein, nearcomplete orientational randominization can occur within 700 ps, well within the excited
state lifetime of Fluorescein (4.1 ns). In contrast, the orientation of Venus under the
same conditions will require 75 ns, much longer than its lifetime of 3 ns). As mentioned
above, most donor fluorophores in a FRET experiment will return to the ground state in a
much shorter time span, with a median value of ln2·  0 D , (for Venus 2 ns). With a
rotational correlation time of 15 ns, free Venus is only expected to rotate 11.3° in 2 ns.
Furthermore, Venus will rotate even slower when attached to another protein, or if
situated in the more viscous cytoplasm found in cells. Thus, Venus is not expected to
rotate much during its excited state. In contrast, a small fluorophore like Fluorescein will
be able to rotate during its excited state. Thus, when considering the value of κ 2 to use in
a FRET experiment it is important to note that the values  D ,  are expected to be
average values of many possible angles when small fluorophores are used as FRET donors
and acceptors, while the values for  D ,  are expected to be static for any particular
donor - acceptor pair comprised of fluorescent protein donors and acceptors. At this
point it is worth noting that the 2/3 value for 2, so ubiquitously used in FRET
experiments, is based on two assumptions: 1. That  D and  have random values (i.e.
they come from isotropic distributions), and 2. That the values of  D and  are changing
rapidly relative to the fluorescence lifetime (dynamic). From the above calculations it
should be clear that these assumptions (isotropic-dynamic) might be valid for some FRET
experiments using small fluorophores like Fluorescein that can rotate rapidly, but are not
valid for FRET experiments using fluorescent proteins as donors and acceptors because
they hardly rotate at all during their fluorescent lifetimes (static).
What κ2 value should be used in a FRET experiment if one assumes that the values of  D
and  are randomly selected from isotropic populations but the donor and acceptors are
in the static regime, i.e. they are hardly rotating during the excited state lifetime of the
donor? Steinberg et al [11] have shown that in the static regime k 2 for an isotropic
population varies with separation distance in a sigmoid fashion, starting at essentially
zero at very low distance, eventually leveling off at a value of 2/3 at very large distances
[11]. Recently, Monte-Carlo simulations were used to address this same issue [27]. This
study confirmed Steinberg’s finding that no single value of κ2 can be used to predict the
energy transfer behavior of a static population, rather it was found that a κ2 value must
be calculated from the random values of  D and  on a FRET pair by FRET pair basis for
each pair in the population. What emerged from this study is that even for a population
that has a homogeneous separation distance that strongly favors energy transfer by FRET
(RDA < R0), because the most probable value of 2 for an isotropic population is zero [6], a
large fraction of FRET pairs in a population will not transfer energy by FRET, and the
population behavior will be heterogeneous with some FRET pairs having very efficient
transfer and some having none at all (2=0). Vis-à-vis FLIM measurements of donor
lifetimes from an isotropic static population of donors and acceptors; a simple singleexponential lifetime decay is expected only if the RDA value is much larger than the R0
value (~ no FRET). In this case, the simple lifetime decay would be the same as the decay
of donor alone. If the RDA value is short enough to support a significant amount of FRET, a
multi-exponential lifetime decay is expected even when only a single fixed RDA value is
present in the population. In this instance, the average FRET efficiency calculated from
the multi-exponential decay should be shorter than the lifetime of donors alone.
Is there experimental evidence for static-FRET behavior in experiments with fluorescent
protein donors and acceptors? Specifically, for FRET in the isotropic static regime we
expect to observe 1. A complex multi-exponential donor lifetime decay, even for a
homogenous population of FRET pairs, and 2. A large fraction of FRET pairs in the
population should fail to transfer energy by FRET because of the prevalence of low 2
values expected in an isotropic population and the absence of rotational motion during
the excited state, even when separation distances between donors and acceptors are
short. In figure 3.15A three different DNA constructs are depicted each engineered to
express in cells a Cerulean [28]. FRET donor (a blue GFP derivative) covalently attached
to a Venus [29] acceptor (a yellow GFP derivative) via either a 5-, 17-, or 32-amino acid
linker. These constructs are called C5V, C17V and C32V respectively [30, 31]. As a
negative FRET control, a single point mutation as introduced into Venus at Y67C to form
‘Amber’ a protein that is thought to have the same structure as Venus, but can not form
the Venus fluorophore and does not act as a dark absorber [32]. This Amber mutation
was then used to create three more constructs; C5A, C17A, and C32A. While the
Cerulean lifetime decay of C5A, C17A and C32A are indistinguishable (Fig 3.15B), the
lifetime decays of C5V, C17V and C32V were all faster than the Cerulean-Amber
constructs, with C5V having the fastest decay, and C32V having the slowest. Using these
Cerulean lifetime decays in the presence and absence of acceptor (Venus), C5V with its
short 5 amino-acid linker had the highest average FRET efficiency (43±2%), the FRET
efficiency of C17V was intermediate (38±3%), and C32V, with the longest linker
separating the donor from the acceptor had a 31±2% FRET efficiency [30]. Note that
C5V, C17V, and C32V all have complex lifetime decays that are clearly not single
exponential, even though every expressed molecule in the population should have one
Cerulean donor covalently attached to one Venus acceptor. These complex multiexponential fluorescence lifetime decays for donor covalently attached to acceptors
suggest that the underlying distribution of FRET efficiencies in these populations is
heterogeneous. While this complex decay behavior is consistent with the first prediction
for FRET in the static isotropic regime, awkward is the observation that the lifetime
decays of the three corresponding Cerulean-Amber constructs also failed to decay as a
purely single exponential as theory predicts for donor-only constructs. This might arise
from more complicated photo-physics for fluorescent protein fluorophores, perhaps
indicating multiple excited states for these fluorophores. While such complicated donoralone decay behavior is problematic, it is quite typical for lifetime decays of isolated
fluorescent proteins and has been observed in experiments measuring FRET between
spectral variants of many different fluorescent proteins [33]. Regardless, to test the
second prediction of FRET in the static-isotropic regime an analysis method is needed
that can account for the complex decay behavior of the donor-alone. To look for a
fraction of molecules in a population that does not undergo energy transfer the data
plotted in figure 3.15B was transformed and re-plotted as the Time-Resolved FRET
Efficiency (TRE, Fig 3.15C) This transformation involves calculating the time dependent
change in FRET efficiency normalized to the fluorescence lifetime decay of the acceptor:
TRE ( t ) =
I D ( t ) - I DA ( t )
I D (t )
(3.20)
Where ID(t) is the fluorescence lifetime decay of the donor-alone, and IDA(t) is the
fluorescence lifetime decay of the donor in the presence of acceptor. Note, that ID(t)
does not have to have a single exponential decay, it could just as well have a more
complex decay resulting from the sum of multiple excited states. Similarly, IDA(t) can also
have a complex lifetime decay resulting from the sum of multiple decay components but
including a component, or components representing energy transfer by FRET from the
donor (or multiple donor excited states) to an acceptor (or multiple acceptors). If every
donor or donor excited state undergoes FRET, the TRE curves will start at a value of zero
at time zero and eventually asymptote at a TRE value of 1. In contrast, if some donors, or
donor excited states never transfer energy by FRET, as predicted for energy transfer in
the static-isotropic regime, the TRE curve will still start at a value of zero at time zero but
appear to asymptote to a TRE value that is less than 1. This difference represents the
fraction of molecules in the population that do not transfer energy by FRET. In figure
3.15C we can see that the TRE curves for the decay data presented in panel B for C5V (&
C5A), C17V (& C17A), and C32V (& C32A) all seem to asymptote to a value that is
between 0.71-0.73 indicating that for these constructs approximately 27-29% of the
donors do not transfer energy by FRET (or any other additional mechanism). This type of
behavior is consistent with the predictions of FRET in the isotropic-static regime.
Figure 3.15 A: Cartoons depicting the FRET-positive protein constructs C5V, C17V, C32V,
and their FRET-negative analogs, C5A, C17A, C32A, where C stands for Cerulean (donor),
V for Venus (acceptor), A for ‘Amber’ (VenusY67C), a non-absorbing Venus with a single
point mutation that prevents chromophore formation. The number between C and V,
and C and A denotes the number of amino acids in the linker connecting them. B: Donor
fluorescence intensity, IDA, versus time after donor excitation in the presence of energy
transfer to Venus for C5V, C17V and C32V, and intensity, ID, versus time in the absence of
energy-transfer for C5A, C17A and C32A. C: Experimental TRE versus time for C5V, C17V,
C32V, compared to C5A, C17A, C32A. D: Theoretical TRE versus time based on equation
3.21 with choices for the relative distance that yield a strong resemblance with the
experimental curves in panel C.
The main advantage of TRE analysis over directly examining fluorescence lifetime decay
curves is that TRE analysis facilitates discriminating between population FRET behavior in
the dynamic and static regimes. If all donor-acceptor pairs in the sample behave similarly
and are expected to have the same overall efficiency, the TRE curve will be 1 minus a
single exponential. In contrast, if a distribution of efficiency values is present in the
system, a sharp deviation of this trend will be seen. It is expected that a singleexponential TRE curve could be a signature for the dynamic regime, whereas the static
regime may be characterized by a more complex TRE curve appearing to asymptote to a
value less than one. In the static isotropic regime, theory predicts that the TRE curve
should follow the following trend:
1
1
0
0
TRE = 1- ò dx ò dz e
æ
ö
-z 2 ç1+3x 2 ÷y
è
ø
= 1-
p
2
1
1
0
0
ò dx ò dz
erf
(
(1+ 3x ) y )
(1+ 3x ) y
2
2
(3.21),
where x and z are introduced in equation 3.16, erf denotes the error function and y is
given by:
6
 R  t
y   0 
rDA   0D
3
2
(3.22),
where t is the time, t 0D is the average Donor lifetime in the absence of transfer, R0 is
the Förster distance
when k 2 = 23 . The rDA values estimated by TRE analysis assuming a
static isotropic regime for C5V, C17V and C32V (5.0, 5.3, and 5.5 nm respectively) are
lower than the RDA values estimated from the average efficiency and fluorescence
lifetime decay analysis assuming a dynamic isotropic regime (5.7, 5.9, and 6.2 nm
respectively). This is expected because a large fraction of the FRET pairs in an isotropic
static regime population will have k 2 values close to zero. It is clear that the
experimental TRE data (Fig 3.15C) are not in perfect agreement with the theoretical TRE
results based on eq. 3.21 (Compare Figs 3.15 C & D). While the basis of these small
discrepancies is not known, we speculate that fluctuations in the separation distance
between donors and acceptors, or deviations from a purely isotropic distribution of  D
and  angles, which are not taken into account in eq. 3.21, may explain this discrepancy.
Regardless, it is quite remarkable that with only one adjustable parameter,
3R06  2 0D rDA6 , 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 the
major reason for why the time-resolved-efficiency for C5V, C17V, and C32V deviates
dramatically from a single exponential rising from 0 to 1.
If it is known that that a population of FRET pairs are in the isotropic static regime, with a
few assumptions it is also possible to estimate the donor-acceptor distance from
experimentally measured k 2 values using as our starting point an estimate of the
average kappa-squared in the static regime introduced by Steinberg et al [11]:
E 
3
2
3
2
 2 R06
6
 2 R06  rDA
(3.23)
The brackets in this equation denote an average, R0 is the Förster distance when 2=2/3,
and rDA is the
donor-acceptor distance. Steinberg et al have shown, in a graph, that k 2
varies with distance in a sigmoid fashion in the static regime, starting at essentially zero
at very low distance, then rising slowly until about rDA  25 R0 , where k 2 starts to
increase more strongly with increasing distance until about rDA  75 R0 , where k 2 begins
to level off reaching 2/3 at very large distances [11]. Between rDA  25 R0 and rDA  75 R0

equal to 2 r R  2  [11]. For
k 2 varies linearly with distance and is approximately
3
DA
0
5
example, the distances between the Cerulean and
Venus fluorophores in C5V, C17V and
C32V most likely fall in this range between 0.4 R0 and
1.4 R0 (2.2-7.7
nm). Substituting
2
2
2
equation for u :
k = 3 ( u - 5 ) (with u  rDA R0 ) into (3.23) yields the following


u6 =
1- E
( u - 25 )
E
(3.24)

Solving this equation numerically using the measured average efficiencies and R0 = 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 estimates derived from TRE
analysis (5.0, 5.3, and 5.5 nm respectively)
With regard to FRET in the static regime, it is important to realize that it is possible to be
in the static regime even when the FRET donor and acceptor used in an experiment are
small fluorophores like Fluorescein. Clearly, experimental factors such as high viscosity
or short rigid linkers can restrain the motion of a small fluorophore. Similarly, if
fluorescent protein donors and acceptors are attached to interacting proteins via a short
rigid linker, the values of  D and , and thus 2 may be fixed and identical for every FRET
pair in the population. If this is the case, FLIM-FRET analysis will reveal a simple
exponential decay that is faster than the lifetime decay of the donor alone, and TRE
curves will asymptote from zero to a value of one. In this case we are still in the static
regime, but the isotropic assumption is no longer valid.
3.9 Beyond Regimes
It is possible that the average rate of transfer is of the same order of magnitude as
a dominant rate of rotation for the donor or acceptor. In that case the system is neither
in the dynamic regime nor in the static regime. Analysis is still possible by building
mathematical models based on the idea that a system of donors and acceptors
undergoing translational and/or rotational motion during the transfer time (inverse of
the average transfer rate) can be described as a collection of states with transitions
between them [34]. These states can be visualized as snapshots: at a certain moment a
donor is excited and has a particular orientation while the acceptor has another
orientation. This donor-acceptor pair is then in a D*A state. A little later the donor or
acceptor changes its orientation, that is, a rotational transition to another D*A state has
occurred. FRET corresponds to a transition to a DA* state. A systematic description of
such time developments implies selecting a representative set of orientation states,
evaluating kappa-squared values, identifying transfer rates and rates of rotation. This
approach leads to a matrix equation for which the eigenvectors and eigenvalues must be
found, so that intensities and anisotropies can be calculated [34]. The following example
illustrates this method. A donor and acceptor are at a fixed distance, rDA , from each
other. The acceptor’s absorption moment has an isotropic degeneracy. The donor’s
emission moment is linear and can only have two orientation states: parallel to the line
“connection” line (line connecting the centers of donor and acceptor) or perpendicular to

6
it. The rate of rotation of this moment is 1  R . The FRET rate is 32  2 1
0D  , where  0D is
the fluorescence lifetime of the donor in the absence of FRET and  is the relative
distance ( rDA divided by the Förster distance if kappa-squared would be equal to 2/3). In
this example  2 equals either 43 
or 13 , 43 when the donor is in the “parallel”
state with its

1
moment parallel to the connection line, and 3 when the donor
is in the “perpendicular”
state with its moment perpendicular to this line. IDA , the fluorescence intensity of the

 in the presence
donor
acceptor
after excitation with a very short pulse of light, is
 of

proportional to y //  y  . Here y // is the fraction of the donors with its moment parallel to

the connection line and y  is the fraction with this moment perpendicular to the

connection line. For ID , the fluorescence intensity of the donor in the absence of FRET,
1
1
y //  y
  2 at all times,
 but for IDA , y //  y   2 only at time zero when the system is
excited by the flash,
 whereas at later times y //  y  until they both decay to zero at times
much larger
than  0D . The rate equation for this example is:
 
1
1
y // 
 1

d y //   0D   R 1 83 
R


 
 
1
2
 1
 1
dt y   

R
0D   R 1 3  y  

(3.25),
where the differentiation is with respect to the time t , and   43 6 1
0D  R . The timeresolved-efficiency, TRE (defined in equation 3.20), can be calculated from the1 solution
of 3.25 in terms of the two eigenvectors with the initial condition y //  y   2 , and for
this example reads (see FRETresearch.org for details):
 

t  5
2 

  t 1 5   1  2 
1    R 1 3   1   1 
1 

1 
e
e  R  3
(3.26).
TRE  1 2 1
 2 
1
2 
2 
1

1






The special cases for this example are:
no FRET with   0 and TRE  0
the static regime with  ,  R   , while   R remains at
2  6 1 t
 1  6 1 t
3
4
6 1
0D , yielding
0D
TRE  TRE STATIC  1 12 e
 12 e 2 0D

the 
dynamic regime with   0 ,  R  0 , while   R remains at



 5  6
 1 t
yielding TRE  TRE DYNAMIC  1 e 4 0D




3
4
6 1
0D ,

3.10 Conclusion

In FRET situations where the transition moments of donor and acceptor are
isotropically degenerate or reorient rapidly and completely within a time comparable to
the inverse transfer rate, one can be certain that kappa-squared equals 2/3. Often this
simplification is not warranted. However, we have indicated which methods can be
utilized to diagnose the potential problems caused by the orientation factor, which
alternative value can be used if the experimental conditions allow one to find an average
kappa-squared value, and what can be done in cases where an average value is poorly
defined.
Acknowledgements
We wish to acknowledge Dr. Paul Blank for stimulating discussions on strategies for
fitting TRE decays to characterize separation distance in the static isotropic regime, Dr.
Brent Krueger for designing Figure 3.14 especially for us, allowing us to use it in this
chapter, and for stimulating discussions about kappa-squared and molecular dynamics
simulations. We are indebted to Dr. Phil Womble for writing a program allowing us, with
help from Sandeep Kothapalli, to obtain some preliminary data for the preparation of
Figure 3.8. We thank Sarah Witten Rogers for valuable help in calculating frequency
distributions for the relative distance. S.S.V. was supported by the intramural program of
the National Institute of Health, National Institute on Alcohol Abuse and Alcoholism,
Bethesda, MD 20892.
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