# American Institute of Mathematical Sciences

doi: 10.3934/fods.2020007

## Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry

 1 Mathematics Dept., University of Tennessee, Knoxville, TN 37996, USA 2 Center for Biological Physics, Arizona State University, Tempe, AZ 85281, USA

* Corresponding author: vmaroula@utk.edu

Microscopy imaging of plant cells allows the elaborate analysis of sub-cellular motions of organelles. The large video data set can be efficiently analyzed by automated algorithms. We develop a novel, data-oriented algorithm, which can track organelle movements and reconstruct their trajectories on stacks of image data. Our method proceeds with three steps: (ⅰ) identification, (ⅱ) localization, and (ⅲ) linking. This method combines topological data analysis and Ensemble Kalman Filtering, and does not assume a specific motion model. Application of this method on simulated data sets shows an agreement with ground truth. We also successfully test our method on real microscopy data.

Citation: Le Yin, Ioannis Sgouralis, Vasileios Maroulas. Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry. Foundations of Data Science, doi: 10.3934/fods.2020007
##### References:
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Cai and A. Nebenführ, A multicolored set of in vivo organelle markers for co-localization studies in Arabidopsis and other plants, The Plant Journal, 51 (2007), 1126-1136.  doi: 10.1111/j.1365-313X.2007.03212.x.  Google Scholar [40] T. D. Pollard and J. A. Cooper, Actin, a central player in cell shape and movement, Science, 326 (2009), 1208-1212.  doi: 10.1126/science.1175862.  Google Scholar [41] G. Ren, V. Maroulas and I. Schizas, Distributed spatio-temporal association and tracking of multiple targets using multiple sensors, IEEE Transactions on Aerospace and Electronic Systems, 51 (2015), 2570-2589.  doi: 10.1109/TAES.2015.140042.  Google Scholar [42] G. Ren, V. Maroulas and I. D. Schizas, Exploiting sensor mobility and covariance sparsity for distributed tracking of multiple sparse targets, EURASIP Journal on Advances in Signal Processing, 2016 (2016), 53. doi: 10.1186/s13634-016-0354-y.  Google Scholar [43] I. F. Sbalzarini and P. Koumoutsakos, Feature point tracking and trajectory analysis for video imaging in cell biology, Journal of Structural Biology, 151 (2005), 182-195.  doi: 10.1016/j.jsb.2005.06.002.  Google Scholar [44] I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899.  doi: 10.1137/16M1095755.  Google Scholar [45] D. M. Shotton, Video-enhanced light microscopy and its applications in cell biology, J. Cell Sci., 89 (1988), 129-150.   Google Scholar [46] G. Singh, F. Mémoli and G. E. Carlsson, Topological methods for the analysis of high dimensional data sets and 3d object recognition, in SPBG, (2007), 91–100. Google Scholar [47] I. Smal, K. Draegestein, N. Galjart, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in dynamic fluorescence microscopy images: Application to microtubule growth analysis, IEEE Transactions on Medical Imaging, 27 (2008), 789-804.  doi: 10.1109/TMI.2008.916964.  Google Scholar [48] I. Smal, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in molecular cell biology, in Nonlinear Statistical Signal Processing Workshop, 2006 IEEE, IEEE, (2006), 129–132. doi: 10.1109/NSSPW.2006.4378836.  Google Scholar [49] D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera, JOSA A, 10 (1993), 1014-1023.  doi: 10.1364/JOSAA.10.001014.  Google Scholar [50] E. H. Spanier, Algebraic Topology, vol. 55, Springer Science & Business Media, 1989. Google Scholar [51] I. A. Sparkes, Motoring around the plant cell: Insights from plant myosins, Biochem Soc. Trans., 38 (2010), 833–838. doi: 10.1042/BST0380833.  Google Scholar [52] L. D. Stone, R. L. Streit, T. L. Corwin and K. L. Bell, Bayesian Multiple Target Tracking, Artech House, 2013. Google Scholar [53] M. Tavakoli, S. Jazani, I. Sgouralis, O. M. Shafraz, S. Sivasankar, B. Donaphon, M. Levitus and S. Pressé, Pitching single-focus confocal data analysis one photon at a time with bayesian nonparametrics, Phys. Rev. X, 10 (2020), 011021. doi: 10.1103/PhysRevX.10.011021.  Google Scholar [54] J. K. Vick and A. Nebenführ, Putting on the breaks: Regulating organelle movements in plant cells, Journal of Integrative Plant Biology, 54 (2012), 868-874.  doi: 10.1111/j.1744-7909.2012.01180.x.  Google Scholar

show all references

##### References:
 [1] P. Bendich, S. P. Chin, J. Clark, J. Desena, J. Harer, E. Munch, A. Newman, D. Porter, D. Rouse and N. Strawn et al., Topological and statistical behavior classifiers for tracking applications, IEEE Transactions on Aerospace and Electronic Systems, 52 (2016), 2644-2661.   Google Scholar [2] G. Bishop, An Introduction to the Kalman Filter, Technical report, TR 95-041, Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3175, Monday, 2006. Google Scholar [3] S. S. Blackman, Multiple-target Tracking with Radar Applications, Dedham, MA, Artech House, Inc., 1986. Google Scholar [4] S. S. v. Braun and E. Schleiff, Movement of endosymbiotic organelles, Current Protein and Peptide Science, 8 (2007), 426-438.   Google Scholar [5] G. Cai, L. Parrotta and M. Cresti, Organelle trafficking, the cytoskeleton, and pollen tube growth, Journal of Integrative Plant biology, 57 (2015), 63-78.   Google Scholar [6] G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.  doi: 10.1090/S0273-0979-09-01249-X.  Google Scholar [7] G. Casella and R. L. Berger, Statistical Inference, The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.  Google Scholar [8] N. Chenouard, I. Bloch and J.-C. Olivo-Marin, Multiple hypothesis tracking for cluttered biological image sequences, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2736-3750.   Google Scholar [9] D. A. Collings, J. D. Harper, J. Marc, R. L. Overall and R. T. Mullen, Life in the fast lane: Actin-based motility of plant peroxisomes, Canadian Journal of Botany, 80 (2002), 430-441.   Google Scholar [10] G. Danuser, Computer vision in cell biology, Cell, 147 (2011), 973-978.   Google Scholar [11] J. Derksen, Pollen tubes: A Model system for plant cell growth, Botanica Acta, 109 (1996), 341-345.   Google Scholar [12] H. Edelsbrunner and J. L. Harer, Computational Topology. An Introduction, American Mathematical Soc., Providence, RI, 2010.  Google Scholar [13] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.  Google Scholar [14] A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, Texts in Statistical Science Series. CRC Press, Boca Raton, FL, 2014.   Google Scholar [15] R. Gutierrez, J. J. Lindeboom, A. R. Paredez, A. M. C. Emons and D. W. Ehrhardt, Arabidopsis cortical microtubules position cellulose synthase delivery to the plasma membrane and interact with cellulose synthase trafficking compartments, Nature Cell Biology, 11 (2009), 797. Google Scholar [16] T. Hamada, M. Tominaga, T. Fukaya, M. Nakamura, A. Nakano, Y. Watanabe, T. Hashimoto and T. I. Baskin, Rna processing bodies, peroxisomes, golgi bodies, mitochondria, and endoplasmic reticulum tubule junctions frequently pause at cortical microtubules, Plant and Cell Physiology, 53 (2012), 699-708.   Google Scholar [17] B. Herman and D. F. Albertini, A time-lapse video image intensification analysis of cytoplasmic organelle movements during endosome translocation, The Journal of Cell Biology, 98 (1984), 565-576.   Google Scholar [18] M. Hirsch, R. J. Wareham, M. L. Martin-Fernandez, M. P. Hobson and D. J. Rolfe, A stochastic model for electron multiplication charge-coupled devices–from theory to practice, PloS One, 8 (2013), e53671. Google Scholar [19] B. Huang, M. Bates and X. Zhuang, Super-resolution fluorescence microscopy, Annual Review of Biochemistry, 78 (2009), 993-1016.   Google Scholar [20] F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes et al., Video-rate nanoscopy using sCMOS camera–specific single-molecule localization algorithms, Nature Methods, 10 (2013), 653. Google Scholar [21] A. Jaiswal, W. J. Godinez, M. J. Lehmann and K. Rohr, Direct combination of multi-scale detection and multi-frame association for tracking of virus particles in microscopy image data, in 2016 IEEE 13th International Symposium on Biomedical Imaging (ISBI), IEEE, 2016,976–979. Google Scholar [22] J. Janesick and T. Elliott, History and advancement of large array scientific ccd imagers, in Astronomical CCD Observing and Reduction Techniques, 23 (1992), 1. Google Scholar [23] S. Jazani, I. Sgouralis and S. Pressé, A method for single molecule tracking using a conventional single-focus confocal setup, The Journal of Chemical Physics, 150 (2019), 114108. Google Scholar [24] S. Jazani, I. Sgouralis, O. M. Shafraz, M. Levitus, S. Sivasankar and S. Pressé, An alternative framework for fluorescence correlation spectroscopy, Nature Communications, 10. Google Scholar [25] K. Kang, V. Maroulas, I. Schizas and F. Bao, Improved distributed particle filters for tracking in a wireless sensor network, Comput. Statist. Data Anal., 117 (2018), 90-108.  doi: 10.1016/j.csda.2017.07.009.  Google Scholar [26] K. Kang, V. Maroulas, I. D. Schizas and E. Blasch, A multilevel homotopy MCMC sequential Monte Carlo filter for multi-target tracking, in Information Fusion (FUSION), 2016 19th International Conference on, IEEE, (2016), 2015–2021. Google Scholar [27] A. Lee, K. Tsekouras, C. Calderon, C. Bustamante and S. Pressé, Unraveling the thousand word picture: An introduction to super-resolution data analysis, Chemical Reviews, 117 (2017), 7276-7330.   Google Scholar [28] J. W. Lichtman and J.-A. Conchello, Fluorescence microscopy, Nature Methods, 2 (2005), 910. Google Scholar [29] I. Lichtscheidl and I. Foissner, Video microscopy of dynamic plant cell organelles: Principles of the technique and practical application, Journal of Microscopy, 181 (1996), 117-128.   Google Scholar [30] S. Liu, M. J. Mlodzianoski, Z. Hu, Y. Ren, K. McElmurry, D. M. Suter and F. Huang, sCMOSnoise-correction algorithm for microscopy images, Nature Methods, 14 (2017), 760. Google Scholar [31] C. W. Lloyd, The plant cytoskeleton: The impact of fluorescence microscopy, Annual Review of Plant Physiology, 38 (1987), 119-137.  doi: 10.1146/annurev.pp.38.060187.001003.  Google Scholar [32] D. C. Logan and C. J. Leaver, Mitochondria-targeted GFP highlights the heterogeneity of mitochondrial shape, size and movement within living plant cells, Journal of Experimental Botany, 51 (2000), 865-871.  doi: 10.1093/jexbot/51.346.865.  Google Scholar [33] V. Maroulas and A. Nebenführ, Tracking rapid intracellular movements: A Bayesian random set approach, Ann. Appl. Stat., 9 (2015), 926-949.  doi: 10.1214/15-AOAS819.  Google Scholar [34] V. Maroulas and P. Stinis, Improved particle filters for multi-target tracking, J. Comput. Phys., 231 (2012), 602-611.  doi: 10.1016/j.jcp.2011.09.023.  Google Scholar [35] J. Munkres, Topology, Pearson Education, 2014. Google Scholar [36] A. Nebenführ, Identifying subcellular protein localization with fluorescent protein fusions after transient expression in onion epidermal cells, in Plant Cell Morphogenesis, Springer, (2014), 77–85. Google Scholar [37] A. Nebenführ and R. Dixit, Kinesins and myosins: Molecular motors that coordinate cellular functions in plants, Annual Review of Plant Biology, 69 (2018), 329-361.   Google Scholar [38] A. Nebenführ, L. A. Gallagher, T. G. Dunahay, J. A. Frohlick, A. M. Mazurkiewicz, J. B. Meehl and L. A. Staehelin, Stop-and-go movements of plant golgi stacks are mediated by the acto-myosin system, Plant Physiology, 121 (1999), 1127-1141.   Google Scholar [39] B. K. Nelson, X. Cai and A. Nebenführ, A multicolored set of in vivo organelle markers for co-localization studies in Arabidopsis and other plants, The Plant Journal, 51 (2007), 1126-1136.  doi: 10.1111/j.1365-313X.2007.03212.x.  Google Scholar [40] T. D. Pollard and J. A. Cooper, Actin, a central player in cell shape and movement, Science, 326 (2009), 1208-1212.  doi: 10.1126/science.1175862.  Google Scholar [41] G. Ren, V. Maroulas and I. Schizas, Distributed spatio-temporal association and tracking of multiple targets using multiple sensors, IEEE Transactions on Aerospace and Electronic Systems, 51 (2015), 2570-2589.  doi: 10.1109/TAES.2015.140042.  Google Scholar [42] G. Ren, V. Maroulas and I. D. Schizas, Exploiting sensor mobility and covariance sparsity for distributed tracking of multiple sparse targets, EURASIP Journal on Advances in Signal Processing, 2016 (2016), 53. doi: 10.1186/s13634-016-0354-y.  Google Scholar [43] I. F. Sbalzarini and P. Koumoutsakos, Feature point tracking and trajectory analysis for video imaging in cell biology, Journal of Structural Biology, 151 (2005), 182-195.  doi: 10.1016/j.jsb.2005.06.002.  Google Scholar [44] I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899.  doi: 10.1137/16M1095755.  Google Scholar [45] D. M. Shotton, Video-enhanced light microscopy and its applications in cell biology, J. Cell Sci., 89 (1988), 129-150.   Google Scholar [46] G. Singh, F. Mémoli and G. E. Carlsson, Topological methods for the analysis of high dimensional data sets and 3d object recognition, in SPBG, (2007), 91–100. Google Scholar [47] I. Smal, K. Draegestein, N. Galjart, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in dynamic fluorescence microscopy images: Application to microtubule growth analysis, IEEE Transactions on Medical Imaging, 27 (2008), 789-804.  doi: 10.1109/TMI.2008.916964.  Google Scholar [48] I. Smal, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in molecular cell biology, in Nonlinear Statistical Signal Processing Workshop, 2006 IEEE, IEEE, (2006), 129–132. doi: 10.1109/NSSPW.2006.4378836.  Google Scholar [49] D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera, JOSA A, 10 (1993), 1014-1023.  doi: 10.1364/JOSAA.10.001014.  Google Scholar [50] E. H. Spanier, Algebraic Topology, vol. 55, Springer Science & Business Media, 1989. Google Scholar [51] I. A. Sparkes, Motoring around the plant cell: Insights from plant myosins, Biochem Soc. Trans., 38 (2010), 833–838. doi: 10.1042/BST0380833.  Google Scholar [52] L. D. Stone, R. L. Streit, T. L. Corwin and K. L. Bell, Bayesian Multiple Target Tracking, Artech House, 2013. Google Scholar [53] M. Tavakoli, S. Jazani, I. Sgouralis, O. M. Shafraz, S. Sivasankar, B. Donaphon, M. Levitus and S. Pressé, Pitching single-focus confocal data analysis one photon at a time with bayesian nonparametrics, Phys. Rev. X, 10 (2020), 011021. doi: 10.1103/PhysRevX.10.011021.  Google Scholar [54] J. K. Vick and A. Nebenführ, Putting on the breaks: Regulating organelle movements in plant cells, Journal of Integrative Plant Biology, 54 (2012), 868-874.  doi: 10.1111/j.1744-7909.2012.01180.x.  Google Scholar
The motion of organelles, during an experiment starting at $t_1 = 0$ ending at $t_N = T$, is identified at discrete times $t_n$ (dots). For simplicity, space is represented with one dimension, although real datasets are two dimensional. The black dots represent the locations of organelles at different time levels. $\tilde {\mathcal{R}}$ is the set contains the locations of all black dots
Here, $\bar x^j$ is the position of an organelle producing the images shown (gray), $\bar f_{n, +}$ and $\bar f_{n, -}$ illustrate 1-level forward displacement and backward displacement of $\bar x^j$, respectively. For clarity, the image produced by the organelle are shown as multi-peaked and space as 1D
The relations of forward fields and backward fields are indicated here. (a) shows the approach depiction of forward displacement fields, (b) shows the approach depiction of backward displacement fields. For clarity, time marches forward in (a) and backward in (b)
(a) shows $\tilde {\mathcal{R}}$ as black dots and ${\mathcal{R}}$ as gray lines; (b) shows ${\mathcal{R}}$, ${\mathcal{T}}_n$ in Eq. (12) and $P_ {\mathcal{R}}^{-1}( {\mathcal{T}}_n)$ as blue segments; (c) shows ${\mathcal{R}}$, ${\mathcal{T}}_n$, $P_ {\mathcal{R}}^{-1}( {\mathcal{T}}_n)$, $\tilde R$ and reconstructed discrete trajectories. For visualization purpose, space is shown in 1D
Case I: The frame size is 320 by 320 pixels. Trajectories of 20 organelles are in red spanning from time $t = 0\;$s to $t = 99\;$s. Their motion is described by a diffusion process containing both a diffusion and a drift term. The starting distance of any two adjacent organelles at $t = 0\;$s is 10 pixels
Case I: Four histograms of mean error of each frame. Each one compares estimated forward and backward displacement with ground truth along $x$-axis and $y$-axis, respectively
Case I: Linking result of all trajectories in red. The accuracy rate is 100%
Case I: Positions of organelles over time after adding perturbation $U(-\epsilon, \epsilon)$ when $\epsilon = 0, \ 1, \ 1.5, \ 2, \ 2.5, \ 3, \ 3.5, \ 4\;$pixels, respectively. If $\epsilon$ increases, it is more difficult to detect trajectories, especially, when $\epsilon = 4\;$ pixels, there are no clear patterns for all trajectories to be reconstructed
Case II: The left shows the rough detection result, the right show the locations after correction. The red dots in the left penal and blue pentagons in the right penal are the original locations before Bayesian identification. The red pentagons in the right panel are the fitted location after Bayesian identification
Case II: Estimated displacement fields of 17th frame using EnKF. Panel (a) shows the estimated displacement fields for the entire focal plane. Panel (b) shows the enlarged area of $[140,230]\times[260,350]$ in Penal (a)
Case II: Trajectories reconstruction result
Case II: Four specific sets of trajectory reconstructions vs ground truth. Each panel shows reconstructions versus one true trajectory. The upper left is amplified from the area $[290,380]\times[40,130]$ in Fig. 11; the upper right is amplified from the area $[40,190]\times[190,340]$ in Fig. 11; the bottom left is amplified from the area $[80,210]\times[200,330]$ in Fig. 11; the bottom right is amplified from the area $[230,380]\times[200,350]$ in Fig. 11;
Case III: Panel (a) is the first frame of the video. Panel (b) exhibits all estimated trajectories in red. Panel (c) further shows each estimated trajectory in different colors
Case I: Table of detection result
 $\epsilon$ (in pixels) total $> 10\; s$ $=100\%$ $\geq 90\%$ $\geq 50\%$ $\epsilon=1$ 20 20 20 20 20 $\epsilon=1.5$ 20 20 20 20 20 $\epsilon=2$ 20 20 20 20 20 $\epsilon=2.5$ 24 24 14 17 20 $\epsilon=3$ 32 27 9 14 17 $\epsilon=3.5$ 33 30 5 10 15 $\epsilon=4$ 53 40 1 4 14
 $\epsilon$ (in pixels) total $> 10\; s$ $=100\%$ $\geq 90\%$ $\geq 50\%$ $\epsilon=1$ 20 20 20 20 20 $\epsilon=1.5$ 20 20 20 20 20 $\epsilon=2$ 20 20 20 20 20 $\epsilon=2.5$ 24 24 14 17 20 $\epsilon=3$ 32 27 9 14 17 $\epsilon=3.5$ 33 30 5 10 15 $\epsilon=4$ 53 40 1 4 14
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