Nonbias-Limited Tracking of Spherical Particles, Enabling Nanometer Resolution at Low Magnification

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1 Nonbias-Limited Tracking of Spherical Particles, Enabling Nanometer Resolution at Low Magnification Marijn T. J. van Loenhout, Jacob Kerssemakers, Iwijn De Vlaminck, and Cees Dekker Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands

2 Supporting Material: S1: Center of Mass () Centroid or Center-of-mass tracking is commonly used in tracking and serves as a reference algorithm (1-3). The algorithm is computationally efficient, but is very sensitive to a non-zero-background in the image, which can be especially prominent in bright-field imaging microscopy. A first step, therefore, is to subtract the image median, I med, and take the absolute value I I I. Then, the center-off-mass x-position is given by: ij ' ij med n n n n ' ' (1) X i I I com ij ij i 1 j 1 i 1 j 1 Were i is an integer pixel x-coordinate in an n by n image. The y-position is calculated analogously. S2: Cross-correlation () Cross-correlation is generally the preferred method for tracking a non-diffraction-limited object and is used here as a second reference algorithm (2, 4-6). This method is based on selecting a band of pixel rows (for the x-direction) from the image and cross correlating its average intensity profile with its own mirror profile. In this work, we average a band of width 0.2n around the image center line, for an n by n image: 0.6n 1 P() i Iij (2) 0.2 n j 0.4n For computational efficiency the cross-correlation C xx of P(i) with its mirror image P(-i) is performed in Fourier space C IFFT(FFT( P( i)) FFT( P(- i))) (3) xx If the pattern is off the symmetry axis by a distance x, the correlation curve will exhibit a peak at position 2 x. Following common practice, we perform a 5-point parabolic fit around this peak to obtain a sub-pixel position, 2 x. The relative center position finally is x 2 x 2

3 S3: A simple estimate of resolution Here we describe a simple analytical expression to estimate the resolution of positional information in an image, irrespective of how this information is extracted. Methods to estimate this accuracy have been proposed, either by simulating the full optical path, or by determining the Cramer-Rao lower bound of a specific image with respect to translational motion (8, 31, 32). While the first method requires careful optical evaluation of the imaging system (which is often difficult to achieve), the latter requires elaborate analysis especially when the shape of the object is not a priori known. Here we propose a simple alternative method that allows a quick, order-of magnitude estimate of the obtainable resolution which only requires a single image. Such a quick estimate can be performed online while tuning an optical system, thus allowing easy optimization of illumination levels, magnification, etc. We assume an image consists of nn N pixels, mapping out an image pattern and background noise. Next, we note that, for any tracking algorithm, the part of the image with the steepest intensity gradient will contribute most to the accuracy, where, by contrast, areas with the weakest gradient will contribute little information on any translation. Likewise, noise at the position of steep intensity gradients will perturb the tracking most. Thus, instead of considering all pixels we simplify our problem by only considering the contributing pixels, N c, defined by setting a threshold to detect the regions of high intensity gradient. Moreover, we assume that a one-pixel translation will change the value of a contributing pixel over the signal range S, defined as the difference between maximum and minimum intensity of the pattern. A sub-pixel motion of (in pixel units) will analogously cause a change of S. Note that, although in doing this we oversimplify the information content of the image, the effects of ignoring both noise and signal contributions outside the steep parts will partly cancel each other out. The smallest detectable change in intensity will be on the order the background noise. If we take a 95 percent confidence level, the smallest detectable motion is then simply 4 S. Since we have N c of such measurements, we arrive at a simple estimate for the image resolution x, x (4) S N c Where a least-significant-bit error of 0.5 bit was added. In practice, representative values of the various parameters should be obtained from an experimental image. We assume a typical image pattern, having continuous, noisy edges with the pattern of interest roughly centered. The noise relevant for x-directed motion is taken by differentiating the two top and two bottom pixel rows yielding a standard deviation σ, for the background intensity as a measure for the noise: std( P( j) P( j 1)) /(2) (5) i i Where std denotes the standard deviation. The average signal range is simply estimated as the difference of the image maximum and minimum intensity, S ( I max - Imin). To determine the number of edges in the x-direction, we apply a threshold to the image. Using Eq.(1), we may

4 evaluate and compare various images intended for tracking, such as the typical defocusing rings of beads used in magnetic tweezers experiments. We note that this approach not only allows estimating translational precision, but also potentially for motions in the z-direction (defocusing changes) and rotation (angular changes). For such motions, it is necessary to identify the average change of pixel pattern upon one unit of motion (a defocus step, or a degree of rotation) and the relevant edges.

5 S4: A 4.5 SNR=9, Z-Resolution B 0.6 SNR=9, X-Resolution C 2.0 SNR =38, Z-Resolution D SNR=38, X-Resolution 0.1 Experimental data showing tracked z- and x-positions for 4 immobilized beads while displacing the objective in the z-direction. A, z-positions for 4 immobilized beads (different colors) tracked at SNR=9. The results for the, and algorithms, from top to bottom respectively, are offset 400 nm for clarity. B, x-positions for 4 immobilized beads (different colors) tracked at SNR=9. The results for the, and algorithms, from top to bottom respectively, are offset 160 nm for clarity. C, z-positions for 4 immobilized beads (different colors) tracked at SNR=38. The results for the, and algorithms, from top to bottom respectively, are offset 400 nm for clarity. D, x- Positions for 4 immobilized beads (different colors) tracked at SNR=38. The results for the, and algorithms, from top to bottom respectively, are offset 160 nm for clarity.

6 Supporting References 1. Carter, B. C., G. T. Shubeita, and S. P. Gross Tracking single particles: a userfriendly quantitative evaluation. Phys. Biol. 2: Cheezum, M. K., W. F. Walker, and W. H. Guilford Quantitative comparison of algorithms for tracking single fluorescent particles. Biophys. J. 81: Berglund, A. J., M. D. McMahon, J. J. McClelland, and J. A. Liddle Fast, biasfree algorithm for tracking single particles with variable size and shape. Opt. Express 16: Gelles, J., B. J. Schnapp, and M. P. Sheetz Tracking kinesin-driven movements with nanometre-scale precision. Nature 331: Otto, O., F. Czerwinski, J. L. Gornall, G. Stober, L. B. Oddershede, R. Seidel, and U. F. Keyser Real-time particle tracking at 10,000 fps using optical fiber illumination. Opt. Express 18: Gosse, C., and V. Croquette Magnetic tweezers: Micromanipulation and force measurement at the molecular level. Biophys. J. 82: