Supplementary Information: Visualizing the entire DNA from a chromosome in a single frame

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1 Supplementary Information: Visualizing the entire DNA from a chromosome in a single frame C. Freitag, C. Noble, J. Fritzsche, F.Persson, M. Reiter-Schad, A. N. Nilsson, A. Graneli, T. Ambjörnsson, K. U. Mir and J. O. Tegenfeldt Extracting kymographs from experiments in meandering nanochannels The output of an experiment is a movie of stained DNA in the meandering channel. The time-average of such a movie is a meander image as illustrated in Figure 4 (Top, Left) in the main text. Our approach for extracting the experimental signal along the meander is a three step procedure: 1. To prepare for step 2 below, we first rotate the meander image so that the meander s linear parts are vertical. To that end we use the Sobel edge-detection algorithm to turn the meander image into a blackwhite image i.e. a map consisting of 1 s and 0 s where 1 s represented edges, i.e. points with a maximal gradient. Once the edges have been identified, we apply and identify peaks in the Hough transform of the black-white image, which provide the rotation angle of the image. The meander image is then rotated. We utilize functions from Matlab s Image Processing Toolbox for this first step. 2. The second step is to overlay a parameterized (based on the nanolithography mask) meander on top of the rotated meander image (coregistration). A meander is characterized by the distance D between linear parts, and the length L of the linear parts. Also the bends has a length b. The meander is divided into identical, but translated, pieces labeled by an integer n (n = 1,..., N). Each one of these pieces is divided into four parts: linear parts going up or down, and the two bends. The linear part of the meander, moving in the positive vertical directions, is parameterized according to: x = 2nD n [1, N] y = b + tl t [0, 1] (1) 1

2 n=1 n=2 n=n L b D Figure S1: Parameterization of the overlaid meander, used when creating kymographs from experimental movies of stained DNA molecules. The meander is divided into geometrically identical pieces labeled by an integer n (n = 1,..., N). These identical pieces are characterized by: D = the distance between linear parts, L = the length of the linear parts, and b = the length of the meander bends. The bends are further characterized by a power-law exponent m; we here use m = 2. For the upper bend we use a power-law functional form: x = 2nD + D 2 + L 2 t y = [1 ( Lt D )2m )b + L + b t [ D/L, D/L] (2) where the larger the value for the power-law exponent m, the sharper is the bend. We use m = 2 throughout this study. For linear part of the meander moving in the negative vertical directions we use: For the lower bend we use: x = (2n + 1)D n [1, N] y = b tl t [0, 1] (3) x = 2nD + 3D 2 + L t n [1, N] 2 y = ( Lt D )2m b t [ D/L, D/L] (4) 2

3 x = 2nD n [1, N] y = b + tl t [0, 1] (5) Our algorithm first estimates the value of D by finding the distance between peaks in the vertical sums of the meander intensity map. We then perform a global minimization of the meander overlap score Q (see below) to find the best values for L and D. We choose Q simply as Q = 1/( x I x) where the sum is over pixels along the meander contour, and I x is the measured intensity at pixel x. We use the Matlab function fminsearch for performing the minimization. As input initial guess to this function we assume that the meander has its center of mass located at the center of the image. The initial guess of L/D is estimated from the nanolithography mask. We run two separate optimizations with flip-flopped initial guesses (one where the meander has its first bend in the top left corner and one where this bend is in the lower left corner). The result of this second step is illustrated in Figure 4 (Top, Middle) in the main text. 3. Once the parameterized meander has been placed on top of the meander image (time-averaged movie) we turn back to the original movie. The third and final step in our method is to walk along the parameterized meander contour, time-frame by time-frame, in the movie. In this procedure a 7 pixel wide window is used. The output is a intensity profile at different times, a kymograph, as illustrated in Figure 4 (Middle) in the main text. Simple method for detecting repetitive regions in a barcode In this section we provide a simple method for finding repetitive regions in theoretical (noise-free) barcodes. Method description Consider a barcode, B(x), sampled at x = 1,..., N pixels, see Figure S2. Our method for detecting repetitive regions is a three step procedure: 1. First, from the original barcode, B(x), we cut out shorter barcodes B s = B s (x s, N s ) with start position x s and of size N s, see Figure S2. The quantity N s is the typical size for an expected repetitive region. In our analysis of the S. pombe barcodes we choose 5 kbps N s 120 kbps and use a step-size of 1 kbps. The start position includes all allowed positions: 1 x s N N s

4 xs x s +N s 1 repetitive region B(x) cut out barcode, B s x=1 x=n pixels, x Figure S2: Illustration of our repetitive region finding method. From a barcode, B(x), consisting of N pixels, we cut out a short barcode of size N s pixels, starting at position x s. This short barcode is circularly permutated pixels and the Pearson cross-correlations between B s and B c are calculated for all allowed. For a repetitive region, the number of -values for which we have the cross-correlation = 1 (perfect match) equals the number of repeats in the short barcode. Thus, by iterating over x s and N s we can identify the start position of a repetitive region and its size. 2. Next, we calculate the Pearson cross-correlation C = C(x s, N s, ) between B s and a circularly permutated version, B c (shifted by pixels) of B s. The shift is iterated to yield all allowed circular permutations of B s. For a perfect match between B s and B c we have C = 1. Therefore, in the absence of noise, we would get n number of C = 1 values for a region with n repeats (if the cut-out region has the correct size), see the red box in Figure S2. For a given start position, x s, there is an optimal choice, ˆNs (x s ), for the size of the cut-out barcode at a given position x s. In order to quantify whether there is a repetitive region starting at x s we then simply count the number hits, H(x s ), i.e. of cross correlation values which satisfies C > C threshold for N s (x s ) = ˆN s (x s ). We here use a threshold value C threshold = Note that a region with no repeat will have H(x s ) = 1 (since = 0 gives C = 1). 3. Finally, we introduce a simple criteria to define whether there is a repetitive region with n repeats, starting at position x s : If H(x s ) αn were α is a significance level (we use α = 0.5) we deem the region starting at position x s a repetitive region with n repeats. Analysis of S. pombe theory barcodes We applied our approach is applied to all three S. pombe chromosome barcodes; chromosome 1 has length 5.57 Mbps, chromosome 2 is 4.54 Mbp long 4

5 and chromosome 3 has length 2.45 Mbps. In order to validate the method, we inserted a mock barcode of size 10 5 basepairs with 40 repeats starting at basepair into the S. pombe chromosome 3 barcode, thus creating a mock version of chromosome 3, see Figure S3. We then applied our 1 S. Pombe, chromosome 3 (with mock region) probability profile position (basepairs) x 10 6 Figure S3: Mock barcode (within the vertical dashed black lines) with a repetitive region of size basepairs with 40 repeats inserted into the middle of the barcode of S. pombe, chromosome 3. method from the previous subsection for all four barcodes (including the mock one), and calculated the number of hits, H(x s ), see Figure S4. In the mock barcode there is a sharp peak in H(x s ) at the correct position along the barcode and no further peaks. Also, the correct size of mock repetitive region was obtained (not shown). In the three S. pombe barcodes we found no corresponding repetitive regions as defined through the criteria introduced in the previous subsection. The horizontal line in Figure S4 5

6 40 35 chromosome 1 chromosome 2 chromosome 3 chromosome 3 (with mock region) 30 H(x s ) = no of "hits" x s = start position for repetitive region (bps) x 10 6 Figure S4: Number of hits, H(x s ), for the three S. pombe barcodes and for a mock barcode created by inserting a repetitive region into the chromosome 3 barcode, see Figure S3. Note that our method correctly identifies both the position and size of the mock repetitive region and that the original barcodes contain no repetitive regions. shows our choice of cut-off for deeming a region repetitive (see step 3. in the method description). Repetitive regions in experimental barcodes Whereas our simple method works well for theory barcodes, we point out that if attempting to detect repetitive regions in experimental barcodes certain care would be needed. First, all experiments are noisy with respect to intensities. Our method can, however, potentially be adapted to such noisy barcodes simply by lowering the cross-correlation threshold, C threshold. Second, and more severe, is that experimental barcodes may be subject to horizontal local stretchings, due to for instance, nano channel impurities. Such local stretchings may potentially throw our method off, see Ref. [1] for an illustrative example and how one can potentially deal with such a scenario. We leave the development of robust repetitive finding methods in experimental barcodes as a future challenge. 6

7 References [1] D. Yankov, E. Keogh, J. Medina, B. Chiu, and Z. Zordan, Detecting time series motifs under uniform scaling. In Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining, (2007). 7