Accurate estimation of the axial position of a molecule using a single lateral image remains a challenge in fluorescent single particle tracking. Here, a principled algorithm for the Bayesian estimation of the axial position of a molecule in three-dimensional astigmatism-based particle tracking is proposed. This technique uses the data from a calibration image set to derive the position without assuming a functional form for the abberated defocusing curve. Using a calibration image set from forty 57 nm beads, the axial position is calculated, and the error associated with position estimation is discussed. This method is compared to previously published algorithms.

Light microscopy
using fluorescent probes has led to numerous discoveries in biology and has
become a standard tool for biological scientists. In the last several decades,
techniques that monitor the position of a single fluorescent molecule, herein
referred to as single particle tracking (SPT), have found applications from
quantifying the motion of single molecular motors to analyzing the spatial
distribution of membrane associated complexes in cells; see, for example,
[

While SPT in two dimensions is somewhat
straightforward, three-dimensional (3D) localization of a fluorescent molecule
from a single lateral-section image has remained difficult. Some of these
methods use information from two different focal planes [

One class of popular techniques uses a cylindrical
optic to separate the axial focus of the two lateral dimensions creating an
elliptical PSF. These astigmatism-based techniques are straightforward to
implement by adding a single optical element in the imaging path. However, the
estimation of the axial position from these techniques is not straightforward.
Indeed, a number of different techniques have been proposed for estimation of
the axial position. All of these techniques use a
user-defined parameterization of the defocusing curve which is then fit to a
calibration data set [

An alternative methodology is to use the calibration
data as representative of the population of all possible molecular images from
different heights. This data set can then be thought of as an estimate of the
probability distribution that relates parameters of the measured image to the
axial position. The approach presented here uses Bayesian estimation to
calculate the expectation value of the axial position given this probability
distribution and the measured lateral widths of an elliptical image. This
procedure generates a likelihood function of the axial position given the data,
which allows for a statistical understanding of the uncertainty in the
localization. Information theoretical has been previously used to discuss the
limits of lateral particle localization [

Experimental measurements were carried out on a
Nikon (Tokyo, Japan) TE2000 inverted microscope. Fluorescence excitation was achieved by
focusing a 473 nm diode laser (Dragon Lasers (ChangChun, China)) onto the back aperture of a 100x
magnification, 1.49 N.A. objective lens (Nikon (Tokyo, Japan)). Emission light was projected
onto an EMCCD camera (Andor Technology, Belfast, Northern Ireland) using a 2.5x projection lens, yielding a pixel
size of approximately 80 nm in the recorded image. To create an astigmatism, a
1 m focal length cylindrical lens was placed between the projection lens and
the camera, which resulted in a focal separation of ~100 nm between the

For each field of fluorescent molecules to be
analyzed, a series of images was recorded at different axial positions. For
these experiments, 57 nm Dragon green fluorescent beads (Bangs Labs) were
adhered to a coverglass surface in a fluid cell, similar to that described
previously [

To calculate the width of a fluorescent spot in the
two lateral dimensions, each bead image was fit to a two-dimensional elliptical
Gaussian using a Levenberg-Marquardt algorithm to minimize the chi-square:

The probability function

We wish to estimate the most likely

Because this approach is derived from knowledge of the
probability distribution, we can estimate the error on

Figure

Variation in the lateral bead profile with axial
position. (a) Five images of the same 57 nm beam taken at different axial
positions demonstrate the astigmatism. (b) The

By binning the data in Figure

Calibration probability distributions. (a) 2D
plots of

We chose four points in the

Localization results for points A–D from Figure

Point | SD (nm) | ||||
---|---|---|---|---|---|

A | 441 | 450 | 388–474 | 31 | 2.9 |

B | 243 | 236 | 176– 343 | 60 | 19 |

C | 106 | 113 | 62–176 | 41 | 7.0 |

Posterior probability distributions for points A–D
from Figure

While the posterior distributions for points A–C are
easily interpreted (see Figure

Several techniques have been used previously for axial
estimation which rely on a parameterized fit of the microscope defocusing curve
[

To compare the Bayesian estimation described here to
this algorithm, we fit our data with the same function used in [

Distance plots using the method of Huang et al.
[

This fit-based technique has several drawbacks. First,
the defocusing curved used in the fit is chosen by hand, and corrections for
aberrations are added ad hoc as high-order polynomial terms. Systematic
deviations of the data from this curve will give rise to systematic errors in
calculating

In order to compare these two methods, we used two
different metrics. Figure

Comparison of the Bayesian and distance
estimation methods. (a) Residual errors calculated from a jackknife analysis of
the calibration data (see text, Bayesian method: red circles, distance method:
black squares). Error bars are standard deviations. (b)

The Bayesian method does slightly worse at the very
lowest and highest

We additionally examined the difference between

Our Bayesian method of axial position estimation offers two main advantages over previous techniques. First, it allows a probabilistic estimate of the localization error not afforded by previously described techniques. Second, it facilitates the identification of spurious data points that might otherwise be included in data analysis.

While the algorithm presented here describes a
Bayesian method for estimating axial particle position, it does not attempt to
correct for the effect of optical aberrations on particle localization. We find
that using a high-numerical-aperture objective, a
large systematic error is induced in axial localization above the focal plane
[

Single-particle tracking has become an important technique in numerous fields of biology. The recent implementation of these techniques for super-resolution 3D imaging has the potential to extend our knowledge of cellular processes on the nanometer level. The axial-localization technique presented here should prove useful as these imaging methods become more quantitative and less descriptive.

This work was partly funded by NIH Grant P50 GM07150. The author would like to thank Yi Deng and Greg Stephens for fruitful discussions.