When single-molecule fluorescence localization techniques are pushed to their lower limits in attempts to measure ever-shorter distances, measurement errors become important to understand. Here we describe the non-Gaussian distribution of measured distances that is the key to proper interpretation of distance measurements. We test it on single-molecule high-resolution colocalization data for a known distance, 10 nm, and find that it gives the correct result, whereas interpretation of the same data with a Gaussian distribution gives a result that is systematically too large

Abramowitz

Balci

Barlow

Churchman

Gordon

Hagerman

Henrik Flyvbjerg

James A. Spudich

L. Stirling Churchman

Thompson

PubMed

Churchman, L.S.

Flyvbjerg, H.

Spudich, J.A.

Online Research Database In Technology

A non-Gaussian distribution quantifies distances measured with fluorescence localization techniques

Crossref

A Non-Gaussian Distribution Quantifies Distances Measured with Fluorescence Localization Techniques

AbstractWhen single-molecule fluorescence localization techniques are pushed to their lower limits in attempts to measure ever-shorter distances, measurement errors become important to understand. Here we describe the non-Gaussian distribution of measured distances that is the key to proper interpretation of distance measurements. We test it on single-molecule high-resolution colocalization data for a known distance, 10nm, and find that it gives the correct result, whereas interpretation of the same data with a Gaussian distribution gives a result that is systematically too large

Stirling Churchman, L.

Flyvbjerg, Henrik

Spudich, James A.

Elsevier - Publisher Connector 

A Non-Gaussian Distribution Quantiﬁes Distances Measured
with Fluorescence Localization Techniques
L. Stirling Churchman,*y Henrik Flyvbjerg,z and James A. Spudich*
*Department of Biochemistry and yDepartment of Physics, Stanford University School of Medicine, Stanford, California 94305;
and zBiosystems Department and Danish Polymer Centre, Risø National Laboratory, DK-4000 Roskilde, Denmark
ABSTRACT When single-molecule ﬂuorescence localization techniques are pushed to their lower limits in attempts to
measure ever-shorter distances, measurement errors become important to understand. Here we describe the non-Gaussian
distribution of measured distances that is the key to proper interpretation of distance measurements. We test it on single-
molecule high-resolution colocalization data for a known distance, 10 nm, and ﬁnd that it gives the correct result, whereas
interpretation of the same data with a Gaussian distribution gives a result that is systematically too large.
INTRODUCTION
Many single-molecule experiments with biological macro-
molecules aim to probe the molecule’s structural conforma-
tions and follow their temporal evolution. This goal is often
pursued by measuring intramolecular distances. In recent
years, several single-molecule ﬂuorescence-localization
techniques have been developed to this end. Examples are
SHRIMP (single-molecule high-resolution imaging with
photo bleaching), NALMS (nanometer-localized multiple
single-molecule ﬂuorescence), and SHREC (single-molecule
high-resolution colocalization) (1–4). Tomeasure ever-smaller
distances, these techniques are pushed to their limits with the
unavoidable consequence that errors are signiﬁcant.
These single-molecule techniques share two crucial steps.
First, the positions in the image plane of two ﬂuorophores are
determined. The second step involves a calculation to de-
termine the Euclidian distance between these two-dimen-
sional (2D) vector positions. Although this calculation is simple,
its error analysis is demanding and has generally not been
correctly applied.
The vector positions of the ﬂuorophores—call them x1
/
and x2
/—are generally determined by ﬁtting the ﬂuoro-
phore’s photon count distribution with Gaussian distribu-
tions and using the centers of these Gaussians functions as
positions (5). With a ﬁnite signal/noise ratio, the results for
x1
/ and x2
/ are not exact. They are approximations that differ
from the unknown true positions by a Gaussian distributed
amount. The cause of these errors and other factors affecting
nanometer localization measurements have been analyzed by
Thompson et al. (5), whereas in this work we discuss the
additional challenges facing the analysis of distance data.
Consequently, the vector difference between positions,
x1
/ x2/; is distributed in the same manner with a variance
that is the sum of the variances on the distributions for x1
/ and
x2
/: Thus, a critical question is: with the x1
/ x2/ Gaussian
distributed in 2D, how is the Euclidean distance, jx1/ x2/j;
frequently of experimental interest, distributed? Since the
distance is a nonnegative number, it follows that it cannot be
Gaussian distributed. The proper distribution, discussed in
this article, allows for an accurate analysis of data derived
from the recent high precision single molecule assays.
RESULTS AND DISCUSSION
Fig. 1 shows an example of distance distributions and
how they depend on the relative importance of the error on
position measurements. Measured positions of two ﬂuoro-
phores separated by exactly 10 nm were Monte Carlo
simulated, and the Euclidean distance between them was
calculated. This was done a large number of times, and the
distances measured in this manner were binned and plotted
to obtain their distribution. When the signal/noise ratio (m/s)
in distance measurements is good, the distribution is ap-
proximately Gaussian (Fig. 1 A). However as the signal/
noise ratio decreases, the distribution becomes increasingly
skewed as it broadens (Fig. 1, B–D), and the position of its
maximum differs increasingly from the true distance be-
tween the ﬂuorophores (Fig. 1, dotted line). If this were a real
experiment and the binned data were ﬁtted with a Gaussian
whose central value were interpreted as the distance between
the ﬂuorophores, this distance would be grossly overes-
timated in Fig. 1 D. Even in Fig. 1 A, where the distribution
looks reassuringly Gaussian, a 5% systematic overestimation
would be introduced with a Gaussian approximation, as
discussed below. This overestimation would be 13% and
32% for Fig. 1, B and C, respectively. Consequently, Fig. 1,
A–C, represents the range of signal/noise ratios found in
current single molecule localization experiments (1–4).
The derivation of the function for proper treatment of
data sets of distances in the plane goes as follows. Two
different ﬂuorophores, 1 and 2, with true 2D positions x
1
ðtrueÞ!
and x
2
ðtrueÞ! then give rise to experimentally recorded positions
x1
/ and x2
/ with Gaussian probability distributions
Submitted May 3, 2005, and accepted for publication October 7, 2005.
Address reprint requests to James A. Spudich, E-mail: jspudich@stanford.
edu.
 2006 by the Biophysical Society
0006-3495/06/01/668/04 $2.00 doi: 10.1529/biophysj.105.065599
668 Biophysical Journal Volume 90 January 2006 668–671
pðxi/Þ ¼ ð1=2ps2i Þexpðð xi/ xðtrueÞi
!
Þ2=2s2i Þ: (1)
Here, s2i is the variance in the ﬂuorophores’ location
stemming from errors Gaussian distributed in the plane and
i ¼ 1; 2 (5). Consequently, r~¼ x1/ x2/ is Gaussian distrib-
uted about ~m ¼ x
1
ðtrueÞ! x
2
ðtrueÞ! with variance s2 ¼ s211s22:
We wish to estimate m ¼ j~mj ¼ jx
1
ðtrueÞ! x
2
ðtrueÞ!j from mea-
surements of r ¼ jr~j: To this end, we write the Gaussian prob-
ability distribution for r~ as a function of r and the angle f
between r~ and ~m;
pðr~Þ ¼ pðr;uÞ ¼ ð1=2ps2Þexpððr~~mÞ2=2s2Þ
¼ ð1=2ps2Þexpððm21 r2  2r m cosuÞ=2s2Þ (2)
and integrate over a circle in the r~plane with radius r (Fig. 2,
black dotted line) to obtain the probability distribution, p2D(r),
on the r axis in 2D,
p2DðrÞ ¼ r
Z 2p
0
du pðr;uÞ ¼
r
2ps
2
 
exp m
21 r2
2s
2
 Z 2p
0
du exp
rm
s
2cosu
 
:
(3)
The last integral is the modiﬁed Bessel function of integer
order zero, I0, so we have the result
p2DðrÞ ¼ r
s
2
 
exp m
21 r2
2s
2
 
I0ðrm=s2Þ: (4)
The solid curves in Fig. 1, A–D, show graphs of this function
for ﬁxed m and increasing values of s. They describe the
binned Monte Carlo simulated data of apparent distances
precisely. This non-Gaussian distribution of distance mea-
surements is not only the case for two dimensions. The dis-
tributions that replace p2D(r) in one or three dimensions are,
respectively,
p1DðrÞ ¼
ﬃﬃﬃ
2
p
r
1
s
exp m
21 r2
2s
2
 
cosh
mr
s
2
 
(5)
and
p3DðrÞ ¼
ﬃﬃﬃ
2
p
r
r
sm
exp m
21 r2
2s
2
 
sinh
rm
s
2
 
: (6)
Fig. 2 shows how the qualitative features in Eq. 4 arise.
The distribution in shades of gray represents the Gaussian
probability distribution for the vector difference between
the two observed positions, assuming the ﬁrst point is at the
origin and the second is at a distance m along the x axis.
The three circles represent three different values of r along
the distance axis. The corresponding probability distribution
for apparent distances, p2D(r), is obtained by integrating this
Gaussian over all points a distance, r, from the origin, i.e.,
over circles centered on the origin. The three circles shown
illustrate how the qualitative features of the apparent distance
distribution come about. For the small blue circle the density
of gray is almost constant across the circle. Consequently,
the integrated probability for such a circle is proportional to
its perimeter, and hence to its radius. Therefore p2D(r) in Eq.
4 increases from zero value at zero distance in a manner that
FIGURE 1 Distribution of apparent distances for a given true distance,
m ¼ 10 nm (dotted line), for four different values for the root mean-square
deviation, s, of the 2D Gaussian distribution of measured end-to-end vector
differences. (Circles) Binned results from Monte Carlo simulations.
(Curves) Graphs of p2D(r) given in Eq. 4. The signal/noise ratios m/s in
panels A–D are 3.33, 2, 1.25, and 0.667, respectively. A Gaussian analysis of
these distributions would result in values that differ from the simulated
value, m, by 5% (A), 13% (B), 32% (C), and 112% (D). Note that although m
is kept ﬁxed at 10 nm here for comparison to Fig. 3, simulations of different
values of m would yield the same functional forms assuming that the signal/
noise ratio were kept the same, because p2D(r) really depends only on r=m
and s=m.
FIGURE 2 A diagram to explain how a Gaussian distribution of apparent
end-to-end vector differences integrates to the distribution of Euclidean
distances given in Eq. 4.
Quantiﬁcation of Nanometer Distances 669
Biophysical Journal 90(2) 668–671
is proportional to distance for small distances, i.e., in a linear
manner.
The black circle with radius m passes through the point
(m, 0), which is where the vector difference has its probabil-
ity maximum. However, the apparent distance distribution,
p2D(r), has its maximum at a larger distance, because a circle
with larger radius (e.g., the red circle) has a longer section
passing through densely gray regions of high probability,
even though it does not pass through the point of maximal
probability. For the same reason, yet longer apparent dis-
tances do have lower probability, but their probability den-
sity does not decrease as fast as a Gaussian function because
their larger circles cut through larger parts of the Gaussian
distributed vector difference. This is also the reason the
apparent distance distribution decreases more slowly at large
values of r than it increases at small values of r, i.e., why it is
skewed.
Fig. 3 shows a histogram of dsDNA apparent end-to-end
distances measured using SHREC. The materials and meth-
ods involved to collect these data are as described (4). The
distribution of these dsDNA end-to-end distance measure-
ments (Fig. 3) is non-Gaussian. A maximum likelihood ﬁt to
the data with p2D(r) in Eq. 4 (Fig. 3, solid line) results in an
estimate for the end-to-end distance, m, of 10 6 1 nm and
m/s of 1.3. This estimate for m is in excellent agreement with
the expected end-to-end distance of a 30 bp dsDNA mol-
ecule, assuming a 3.4 A˚ rise per base and a persistence length
of 50 nm (6). A maximum likelihood ﬁt with a Gaussian
function is included for comparison (Fig. 3, dotted line) and
yields the estimate for m of 14 6 0.5 nm. Although least-
squares ﬁtting to the histogram of data in Fig. 3 is inap-
propriate because of the low count in some bins, we did
compute x2 of our maximum-likelihood ﬁts after they had
been done for the beneﬁt of readers who ﬁnd this quantity
easier to judge than the statistical support of a ﬁt. We found
a reduced x2 of 1.7 of the p2D(r) ﬁt and 2.3 for the Gaussian
ﬁt. The Gaussian function, which above was mathematically
proven to be the incorrect model for this experimental data
set, expectedly ﬁts the data less well and provides an esti-
mator that is inaccurate. The correct measure of quality of
our ﬁts is the ‘‘goodness of ﬁt’’ associated with all max-
imum-likelihood ﬁts. For the p2D(r) ﬁt, we found the
goodness of ﬁt to be 68%. Details of the data analysis are
provided in Appendix A.
Even when s2 m2, in which case p2D(r) in Eq. 4 can be
approximated by a Gaussian function, the mean of this
Gaussian is not a good estimate for m. The maximum of
p2D(r) is not located at m, but approximately at r¼ m(11 s2/
(2m2)) (See Appendix B for details). So naively ﬁtting data
like those in Figs. 1 and 3 with a Gaussian function may
result in an acceptable ﬁt, depending on the noise in the data.
Yet it will not yield the correct value for m, but a systematic
overestimation by a relative amount, s2/(2m2).
The p2D(r) function (Eq. 4) appears to be sufﬁcient to
explain the 2D distance data sets arising from recent single-
molecule ﬂuorescent localization experiments. A number of
researchers have ﬁtted their data sets with Gaussian or log-
normal functions with a constant background added to yield
results closer to the data distribution’s maximum (1–3). In
these types of experiments it is unclear what would cause a
uniform background in distance measurements. The skew-
ness of the p2D(r) distribution may provide an explanation
for what previously has been perceived as a background.
CONCLUSION
It is natural to assume that a distribution of errors is Gaussian
when it appears Gaussian by eye. However we conclude that
by applying a Gaussian ﬁt, one commits systematic errors on
distance measurements with single-molecule ﬂuorescence lo-
calization techniques. These techniques have overcome many
technological hurdles to measure ever-shorter distances with
high accuracy. The correct analysis of such data, as described
here, ensures that the hard earned precision is not lost.
APPENDIX A
The maximum likelihood ﬁt was calculated in the following manner.
Assuming a model such that P(xi, a) describes the experimental data, one can
calculate the probability that for a given set of estimators, fajg, a particular
data point (xi) occurs. Multiplying the probabilities of each data point gives
the probability that the entire data set occurs with this set of estimators. The
total probability is called the likelihood and is mathematically deﬁned by the
following equation,
Lðx1; . . . ; xN; aÞ ¼
YN
i
Pðxi; aÞ:
The ﬁt is found by numerically maximizing the likelihood (L) via varying the
estimators, fajg(7).
FIGURE 3 Histogram of apparent end-to-end distance measurements of
dsDNA molecules (N ¼ 112) determined by SHREC. The solid line
represents the solution found when a maximum likelihood ﬁt was performed
with p2D(r) given in Eq. 4. Goodness of ﬁt is 68%. For comparison, the ﬁt to
the data with a Gaussian function is also shown (dotted line).
670 Churchman et al.
Biophysical Journal 90(2) 668–671
Errors on the found estimators can be found by investigating the shape of
the likelihood function along the axis of an estimator. Due to the central limit
theorem, the probability distribution for each estimator, aj, is a Gaussian
function. To determine the error of an estimator, one needs to simply ﬁnd the
root mean-square deviation, s, of the Gaussian likelihood function. In this
article, errors were reported as being at s from the maximum or at 68%
conﬁdence limits. In the online supplemental materials, we give a MATLAB
(The MathWorks, Natick, MA) script that will automatically ﬁt p2D(r) in Eq.
4 to a data set and ﬁnd the errors associated with the ﬁt.
To test the signiﬁcance of the maximum likelihood ﬁt of p2D(r) in Eq. 4 to
the experimental data, a data set of the same size as the experimental data set
(N ¼ 112) was Monte Carlo simulated using the ﬁt parameters faˆjg as input
(as done to make the histograms in Fig. 1). The likelihood, L(x1, . . ., xN, aˆ),
was calculated and compared against the likelihood of the experimental
data set. This was repeated a large number of times. The fraction of simu-
lated data sets with a lower likelihood was 0.68, so the statistical support for
the hypothesis that p2D(r) is the correct theory for the data is 68%.
APPENDIX B
With our knowledge of p2D(r) (Eq. 1), we can estimate the size of the
systematic error that one commits if one interprets distance measurement
data with a Gaussian function. The asymptotic expansion of the modiﬁed
Bessel function is
I0ðzÞ ¼ ðez=
ﬃﬃﬃﬃﬃﬃﬃﬃ
2pz
p
Þ 11 ð1=8zÞ1 ð9=2ð8zÞ2Þ1 . . . ;
so for s2  m2 we have a good approximation in
p2DðrÞ  1ﬃﬃﬃﬃﬃﬃ
2p
p
s
ﬃﬃﬃ
r
m
r
exp ðr  mÞ
2
2s
2
 
;
which looks similar to a Gaussian (8). However the factor
ﬃﬃﬃﬃﬃﬃﬃﬃ
r=m
p
is what
makes p2D(r) differ from a Gaussian function. This factor shifts the
maximum of p2D(r) to approximately r ¼ m(1 1 s2/(2m2)). (Here we have
ignored higher powers of s2/m2).
SUPPLEMENTARY MATERIAL
An online supplement to this article can be found by visiting
BJ Online at http://www.biophysj.org.
This collaboration grew out of Steven Block’s wonderful workshop on
single-molecule biophysics at the Aspen Center for Physics, January 2–8,
2005. We thank Alexander R. Dunn, David Altman, Zev Bryant, and
Benjamin J. Spink for their critical comments of the manuscript.
This work was supported by National Institutes of Health grant GM33289
(to J.A.S.).
REFERENCES
1. Gordon, M. P., T. Ha, and P. R. Selvin. 2004. Single-molecule high-
resolution imaging with photobleaching. Proc. Natl. Acad. Sci. USA.
101:6462–6465.
2. Qu, X., D. Wu, L. Mets, and N. F. Scherer. 2004. Nanometer-localized
multiple single-molecule ﬂuorescence microscopy. Proc. Natl. Acad.
Sci. USA. 101:11298–11303.
3. Balci, H., T. Ha, H. L. Sweeney, and P. R. Selvin. 2005. Interhead
distance measurements in myosin VI via SHRImP support a simpliﬁed
hand-over-hand model. Biophys. J. 89:413–417.
4. Churchman, L. S., Z. Okten, R. S. Rock, J. F. Dawson, and J. A.
Spudich. 2005. Single molecule high-resolution colocalization of Cy3
and Cy5 attached to macromolecules measures intramolecular distances
through time. Proc. Natl. Acad. Sci. USA. 102:1419–1423.
5. Thompson, R. E., D. R. Larson, and W. W. Webb. 2002. Precise
nanometer localization analysis for individual ﬂuorescent probes.
Biophys. J. 82:2775–2783.
6. Hagerman, P. J. 1988. Flexibility of DNA. Annu. Rev. Biophys. Biophys.
Chem. 17:265–286.
7. Barlow, R. J. 1989. Statistics: A Guide to the Use of Statistical Methods
in the Physical Sciences. John Wiley and Sons, New York.
8. Abramowitz, M., and I. A. Stegun. 1972. Handbook of Mathematical
Functions. Dover Publications, New York.
Quantiﬁcation of Nanometer Distances 671
Biophysical Journal 90(2) 668–671


A Non-Gaussian Distribution Quantifies Distances Measured with Fluorescence Localization Techniques 

file:///data/remote/core/dit/data/elsevier/pdf/74a/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAwNjM0OTUwNjcyMjQ1Nw==.pdf

A Non-Gaussian Distribution Quantifies Distances Measured with Fluorescence Localization Techniques

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology

Crossref

Elsevier - Publisher Connector

Elsevier - Publisher Connector