TOWARDS FAST 3D NANOPARTICLE LOCALIZATION
FOR STUDYING MOLECULAR DYNAMICS IN LIVING CELLS
Stefan Sokoll
1,2
, Klaus T¨onnies
2
and Martin Heine
1
1
Research Group Molecular Physiology, Leibniz Institute for Neurobiology, Magdeburg, Germany
2
Research Group Computer Vision, Otto-von-Guericke University, Magdeburg, Germany
Keywords:
3D particle localization, Nanometer sub-resolution, Live cell imaging, Confocal fluorescence microscopy.
Abstract:
Studying molecular dynamics is crucial for understanding biological processes in living cells. In principle, this
is achieved by attaching fluorescent particles to molecules of interest and their detection using fluorescence
microscopy. These analysis require fast optical techniques with at least 20Hz frame rate and a resolution
below the diffraction limit in all three spatial dimensions. Current approaches basically rely on determining
the correlation between features of the particle’s 2D point spread function (PSF) and the focal distance to the
center of the particle. However, they are still unsuitable for the application to live cell imaging where the
refractive index mismatch is present. This mismatch leads to non-stationary optical properties of the particles
on which the algorithms rely, necessitating a calibration procedure prior to every experiment. However, this is
almost unfeasible to particles attached to living cells.
We established a spinning disk confocal setup and employ Quantum dots (QD) as fluorescence particles. Cor-
responding models of the axial PSF features that define the distance to the center of the particle are developed
and analyzed in the presence of the refractive index mismatch. We present this analysis as the base for the
future development of a 3D localization technique applicable to living cells.
1 INTRODUCTION
Neurons are polarized cells that communicate via
synapses with each other. Each synapse consists of a
pre- and post-synaptic side, with the flow of informa-
tion going from the former to the latter. Basically, on
the arrival of a signaling action potential at a chem-
ical synapse, the interplay of different molecules at
the pre-synapse may lead to a release of neurotrans-
mitters into the synaptic cleft (see Figure 1). After ab-
sorbance by receptors at the post-synaptic side, neu-
rotransmitters may be again convertedinto a signaling
potential.
Figure 1: Signal transmission at a chemical synapse.
The probability of successful signal transmission
between the pre- and post-synaptic side is liable to
strong variation, which is called synaptic plasticity. It
is believed to be fundamental for learning and mem-
ory. Many different molecules participate in synaptic
transmission, e.g. adhesion molecules, receptors, and
voltage-gated ion channels. In order to be successful,
their arrangement, number, and density seems to be
very important. In particular, molecular dynamics at
the active zone of the pre-synapse, which describes
the area of possible neurotransmitter release, are of
special interest.
A well established technique for studying molec-
ular dynamics in living cells is the attachment of flu-
orescent particles at molecules of interest and their
detection using fluorescence microscopy. This pro-
cedure involves the acquisition of time-series images,
either solely two-dimensional or taken at different fo-
cal positions combined to three-dimensional stacks.
The trajectories of single particles are then computed
from the particular position in each acquisition (see
Figure 2).
Studying molecular dynamics requires the optical
techniques to perform with at least 20Hz frame rate
210
Sokoll S., Tönnies K. and Heine M..
TOWARDS FAST 3D NANOPARTICLE LOCALIZATION FOR STUDYING MOLECULAR DYNAMICS IN LIVING CELLS.
DOI: 10.5220/0003166502100215
In Proceedings of the International Conference on Bioinformatics Models, Methods and Algorithms (BIOINFORMATICS-2011), pages 210-215
ISBN: 978-989-8425-36-2
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
Figure 2: Microscopic 3D image of a QD (in red) and its
preceding trajectory (in blue), together with synapses (in
green), allowing colocalization analysis.
and a resolution at the molecular level in all three
spatial dimensions. For biological applications, 20Hz
frame rate are necessary to distinguish different types
of molecular diffusion by their diffusion coefficients.
High spatial resolution in all three dimensions is es-
pecially required as the active zone is usually not big-
ger than 100-500nm in size and does not necessar-
ily reside in plane with the focal plane of the micro-
scope. To date, various approaches have been devel-
oped, but these techniques are only suitable in the ab-
sence of the refractive index mismatch, which is the
deviation of the refractive indices between the objec-
tive’s immersion medium and the observed biologi-
cal medium. As this mismatch is unavoidable in live
cell imaging, the optical properties of the particles on
which the algorithms rely, cannot be expected to be
stationary along the optical axis (Hell et al., 1993).
Thus, a calibration has to be performed prior to every
experiment, which is almost unfeasible to particles at-
tached to living cells and remains a major challenge.
In this paper, we present a model for the optical
properties of Quantum dot (QD) nanoparticles, im-
aged with a spinning disk confocal setup, and analyze
it in the presence of the refractive index mismatch.
We first highlight recent developments in the field of
single nanoparticle localization followed by the pre-
sentation of our axial model. Furthermore, we ana-
lyze the model qualitatively and finally conclude that
the influence of the refractive index mismatch has to
be considered.
2 RELATED WORK
In confocal fluorescence microscopy, the spatial reso-
lution is limited to 200-300nm in the lateral and about
500-700nm in the axial direction by the diffraction of
light (Huang et al., 2009). Thus, the image of every
sharp spot on the object appears with a blurred inten-
sity profile in the image plane of the microscope. This
is referred to as the point spread function (PSF) of the
microscope and defines the minimal distance of two
light sources such that they can still be resolved in the
image.
Localization microscopyis a reasonabletechnique
to overcome this limitation ((Schmidt et al., 1996),
(Kubitscheck et al., 2000), (Cheezum et al., 2001)).
For two-dimensional images, this technique basically
comprises the fitting of a Gaussian functionto the PSF
of a particle (see Figure 3). Thus, at least the center
Figure 3: Principle of localization microscopy, leading to
sub-pixel localization accuracy.
of a particle can be determined with higher accuracy
than the resolution, leading to an accuracy δx approx-
imated by
δx =
s
N
, (1)
where s is the standard deviation of the PSF and N the
number of collected photons (Thompson et al., 2002).
By analyzing image stacks, this technique is in
principle extendable to 3D. The axial center of the
particle is defined by the minimum standard devia-
tion of the Gaussian fits in all slices ((van Oijen et al.,
1998), (Sch¨utz et al., 2000), (Ragan et al., 2006)).
However, such approaches show only a poor time res-
olution due to the necessity to acquire multiple im-
ages for one stack, leading to deformations of the PSF
induced by particle movement (see Figure 4). Re-
Figure 4: (a) shows the symmetric 3D PSF of an immobile
particle. Due to particle movement, the PSF shown in (b) is
deformed.
TOWARDS FAST 3D NANOPARTICLE LOCALIZATION FOR STUDYING MOLECULAR DYNAMICS IN LIVING
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211
Figure 5: Axial dependency of 2D PSF features. (a) Illustrates the computation of the 2D PSF features of a 3D particle. The
resulting approximations to the Gaussian intensity and the parabolic width curve are presented in (b) and (c), respectively. The
measurements, together with the standard deviations, are shown in red and the approximations to the measurements in blue.
The parabola becomes asymmetric (d), if the refractive indexes differ. Knowing the particular curve, allows the computation
of the ambiguous distance to the center of the particle from a single measurement.
cently developed techniques allow the computation of
the axial position from one image slice by defocusing
the image ((Speidel et al., 2003), (Wu et al., 2005)) or
introducing an astigmatism into the light path ((Kao
and Verkman, 1994), (Holtzer et al., 2007)). The ba-
sic principle amounts to determine the correlation be-
tween features of the 2D PSF and the focal distance to
the center of the particle. However, due to the refrac-
tive index mismatch this correlation varies with the
depth of the particle within the biological medium and
needs to be adjusted for each experiment. This makes
current algorithms unsuitable for live cell imaging as
particles attached to molecules of living cells cannot
be immobilized for calibration.
3 AXIAL MODEL
The presented techniques were mostly carried out on
widefield microscopes. We established a spinning
disk confocal setup
1
, to combine the advantages of
improved spatial resolution as well as the ability to
image particles that reside deeper within the medium
(Inou´e, 2006). Decreased signal intensity is the down-
side of this choice, leading to less accurate localiza-
tion results (see Equation 1). However, in the fu-
ture, this tradeoff allows us to apply the technique
to thicker brain slices, which are biologically more
relevant than primary cultured neurons. As fluores-
cence particles, we use QDs, which are semiconduc-
tor nanoparticles exhibiting excellent photostability
and narrow emission spectra (Toomre and Pawley,
2006).
Initially, we analyzed axial scans of QDs that were
1
In detail, our setup comprises a Nano Focusing
Piezo Stage (PRIOR Scientific), an Olympus Microscope
(BX51WI), a Yokogawa spinning disk (CSU-X1), and an
Andor iXionEM+897 CCD camera.
immobilized below the cover slip such that the refrac-
tive index mismatch had no influence. Here we found
two features of the PSF that vary with the axial dis-
tance of the measurement to the center of the particle.
These are the intensity I and the width σ of the fluo-
rescent peaks at each axial position (see Figure 5(a)).
They are computed from the height and the full width
at half maximum (FWHM) of the Gaussian fits, re-
spectively. As it is depicted in Figures 5(b-c), the
axial intensity distribution follows a Gaussian func-
tion and the width of the fluorescence peaks matches
a parabola.
Considering the refractive index mismatch by im-
mobilizing QDs at different depths below the cover
slip, the parabola of the width becomes asymmet-
ric, where the asymmetry correlates with the depth
of the particle within the observed medium (see Fig-
ure 5(d)). The Gaussian function of the intensity is
also slightly skewed for higher particle depths. How-
ever, we found the differences to be neglectable and
for simplicity of the model, we retain a symmetric in-
tensity distribution at the moment.
Based on our experimental observations, we iden-
tified approximations to both curves. The correlation
of the intensity is modeled as a Gaussian function
given by
I(z) = o + h·e
z
2
2c
2
, (2)
where o, h and c are the offset, height and standard
deviation, respectively. The axial dependency of the
PSF’s width is described in (Niedrig, 1993) and ex-
tending the presented formula to consider the asym-
metry introduced by the refractive index mismatch,
we get
σ(z) = σ
0
·
r
1+
z
d
2
+ m·z, (3)
where d is the focal depth, σ
0
the FWHM in focus and
m represents the skewness of the parabola. Solving
BIOINFORMATICS 2011 - International Conference on Bioinformatics Models, Methods and Algorithms
212
Figure 6: (a-c) Present the axial localization accuracy dis-
tribution, the correctness of the axial model and the axial
dependency of the lateral position, respectively. All graphs
belong to measurements at 5µm depth. (a) and (c) show
measurements of individual particles and (b) provides the
average result of all particles at this depth. Illustrative ex-
amples that explain how the given curves are computed are
presented next to each graph.
these equations for z, we can compute the axial posi-
tion of the particle from only one slice applying either
the Gaussian or the parabolic approximation to the in-
tensity or the width, respectively. However, so far the
model cannot distinguish if the particle is above or
below the focal plane, as it is depicted in Figure 5(d).
4 QUALITATIVE ANALYSIS OF
THE MODEL
After introduction of the axial model we now provide
further properties that have to be considered for the
development of a fast 3D localization algorithm. We
present only qualitative results because the final accu-
racy of the localization depends mainly on the imag-
ing conditions
2
.
First, we have analyzed the dependency of the ax-
ial localization accuracy on the distance of the im-
age acquisition to the center of the particle (see Fig-
ure 6(a)). It becomes apparent that the accuracy dis-
tributions of the intensity I and the width σ proceed
contrary to each other. The intensity model provides
higher accuracy when the particle is off-focus while
the width model has best accuracy close to the center
position. The shape of these accuracy distributions
is typical for all depths, but the localization accuracy
worsens with increasing depth. However, the contrary
course of the distributions supports the development
of a fusion strategy, possibly leading to a constant lo-
calization accuracy valid at all axial scan positions.
Next to the axial localization accuracy, the cor-
rectness of the approximation to the PSF has to be
determined. Figure 6(b) shows the systematic errors
of the axial model, computed from the constant devi-
ation of the model to the mean measurements at each
axial scan position. It reveals that the measured data
twists around the approximation, which is valid for
all depths and is consistent with earlier findings from
(Sch¨utz et al., 2000). Of course, in principle a more
accurate PSF could be computed, however, the cor-
rect computation requires the knowledge about opti-
cal properties of the setup that are generally not acces-
sible as (Speidel et al., 2003) already pointed out. We
consider a more accurate model as a possible future
improvement to the accuracy of the localization algo-
rithm. For now we accept that error, which is only
in the order of 1/4 of the localization accuracy and
is expected to be reducible via fusion of the models.
Furthermore, we found the systematic error to remain
constant for all tested depths.
To compute the 3D position of a particle, the de-
pendencyof the lateral position on the axial scan posi-
tion is also required. Figure 6(c) shows that the lateral
position varies with different measurements along the
axial axis. This hinders the computation of the lat-
eral position independently of the axial position and
has to be further investigated. However, at least these
curve shapes, presented in Figure 6(c), are generally
valid for all particles and depths. Finally, we found
that the lateral localization accuracy is best at the cen-
ter of the particle, which agrees with Equation 1 and
slightly improves with increasing depth.
2
Current data was acquired by imaging of QDs emitting
655nm light with an oil objective of magnification 100x.
QDs were illuminated by 488nm light and consecutively
scanned for 50ms with z-stacks covering 4µm and a step
size of 100nm. 50 z-stacks were acquired of each particle.
TOWARDS FAST 3D NANOPARTICLE LOCALIZATION FOR STUDYING MOLECULAR DYNAMICS IN LIVING
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Figure 7: Variation of axial model parameterization for particles at different depths. (b) shows width curve examples for each
depth with the measured data in red and the parabolic approximations in a dedicated color. (a) and (c) visualize the PSF for
particles at depth 0µm and 15µm, respectively.
5 INFLUENCE OF THE
REFRACTIVE INDEX
MISMATCH
The presented axial model considers the refractive in-
dex mismatch and is valid independently of the par-
ticle’s depth in the medium. However, the parame-
ters of the model change with varying depths in the
medium.
To prove that the consideration of this variation
is necessary, we performed calibration measurements
at different depths with QDs immobilized in Mowiol
and at monolayers of HEK cells
3
. The test set com-
prises always 40 particles at depths of 0, 5, 10 and
15µm in Mowiol and at 0 and 5µm at HEK cells. Each
particle was axially scanned 50 times. Figure 7 de-
picts example data from width measurements at dif-
ferent depths in Mowiol. The different depths are
color coded. It becomes apparent that the curve is
symmetric only at zero depth and turns asymmetric
for higher depths.
Analysis of the curve parameters revealed that the
standard deviation c of the Gaussian and the gradient
m of the parabola increase with depth. They uniquely
define a certain depth. Figure 8 depicts the different
curves when only these classifying parameters are ad-
justed to the depth and mean values are taken for the
remaining ones.
Using wrong parameters for the model will lead
to the computation of incorrect axial positions of the
particle. For instance, let’s assume that a particle was
acquired at depth m but is mistakenly expected to
be at depth 10µm. Given that, Figure 9 shows the
axial position error for each slice in the stack. It is
apparent that in this example the error is roughly half
as large as the axial localization accuracy presented in
3
Mowiol is a transparent embedding medium for cell
preparations. HEK cells are Human Embryonic Kidney
cells, widely used in cell biology research due to their sim-
ple handling and fast growth.
Figure 8: The mean intensity and width curves at different
particle depths if only the Gaussian standard deviation c,
respectively the gradient m, are adjusted. These findings
are valid for Mowiol and HEK cell preparations.
Figure 6(a) even though only the classifying parame-
ters are varied. Hence, the error cannot be neglected
and this proves that for live cell imaging, where the re-
fractive index mismatch cannot be avoided, calibrat-
ing the model parameters is indispensable.
Figure 9: Illustrated are the axial position errors for each
model in case one mistakenly expects a particle at depth
10µm instead of its correct position at 5µm.
6 CONCLUSIONS
We presented a model that facilitates the fast 3D local-
ization of nanoparticles in living cells, using a spin-
ning disk confocal setup. We performed qualitative
analyzes and described its dependency on the parti-
cle’s depth in the medium, proving that the refrac-
tive index mismatch has to be considered for live cell
imaging applications.
We propose two major objectives for future work.
BIOINFORMATICS 2011 - International Conference on Bioinformatics Models, Methods and Algorithms
214
First, we have to develop a localization algorithm
that takes the properties of our model into account.
Here, major challenges are the ambiguity of the axial
model, which we like to address by the introduction
of a cylindric lens in the light path as well as a fusion
strategy for the intensity and width models to increase
the localization accuracy. Second, and far more im-
portant, we have to solve the problem that the model
parameters cannot be adjusted via calibration. Here,
we focus on the estimation of the parameters during
each experiment by formulation as an expectation-
maximization problem.
ACKNOWLEDGEMENTS
This work is supported by grants from the DFG (HE
3604/2-1).
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