Abstract
Biophysicists use single particle tracking (SPT) methods to probe the dynamic behavior of individual proteins and lipids in cell membranes. The mean squared displacement (MSD) has proven to be a powerful tool for analyzing the data and drawing conclusions about membrane organization, including features like lipid rafts, protein islands, and confinement zones defined by cytoskeletal barriers. Here, we implement time series analysis as a new analytic tool to analyze further the motion of membrane proteins. The experimental data track the motion of 40 nm gold particles bound to Class I major histocompatibility complex (MHCI) molecules on the membranes of mouse hepatoma cells.
Our first novel result is that the tracks are significantly autocorrelated. Because of this, we developed linear autoregressive models to elucidate the autocorrelations. Estimates of the signal to noise ratio for the models show that the autocorrelated part of the motion is significant. Next, we fit the probability distributions of jump sizes with four different models. The first model is a general Weibull distribution that shows that the motion is characterized by an excess of short jumps as compared to a normal random walk. We also fit the data with a chi distribution which provides a natural estimate of the dimension d of the space in which a random walk is occurring. For the biological data, the estimates satisfy 1<d<2, implying that particle motion is not confined to a line, but also does not occur freely in the plane. The dimension gives a quantitative estimate of the amount of nanometer scale obstruction met by a diffusing molecule. We introduce a new distribution and use the generalized extreme value distribution to show that the biological data also have an excess of long jumps as compared to normal diffusion. These fits provide novel estimates of the microscopic diffusion constant.
Previous MSD analyses of SPT data have provided evidence for nanometer-scale confinement zones that restrict lateral diffusion, supporting the notion that plasma membrane organization is highly structured. Our demonstration that membrane protein motion is autocorrelated and is characterized by an excess of both short and long jumps reinforces the concept that the membrane environment is heterogeneous and dynamic. Autocorrelation analysis and modeling of the jump distributions are powerful new techniques for the analysis of SPT data and the development of more refined models of membrane organization.
The time series analysis also provides several methods of estimating the diffusion constant in addition to the constant provided by the mean squared displacement. The mean squared displacement for most of the biological data shows a power law behavior rather the linear behavior of Brownian motion. In this case, we introduce the notion of an instantaneous diffusion constant. All of the diffusion constants show a strong consistency for most of the biological data.
Article PDF
Similar content being viewed by others
References
Anderson, T.W., Darling, D.A., 1952. Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes. Ann. Math. Stat. 23, 193–212.
Andrews, N.L., Lidke, K.A., Pfeiffer, J.R., Burns, A.R., Wilson, B.S., Oliver, J.M., Lidke, D.S., 2008. Actin restricts FcεRI diffusion and facilitates antigen-induced receptor immobilization. Nat. Cell Biol. 10, 955–963.
Berg, H.C., 1993. Random Walks in Biology. Princeton University Press, Princeton.
Berry, H., 2002. Monte Carlo simulations of enzyme reactions in two dimensions: Fractal kinetics and spatial segregation. Biophys. J. 83(4), 1891–1901.
Bickel, T., 2007. A note on confined diffusion. Phys. A, Stat. Mech. Appl. 377(1), 2007.
Capps, G.G., Pine, S., Edidin, M., Zúñiga, M.C., 2004. Short class I major histocompatibility complex cytoplasmic tails differing in charge detect arbiters of lateral diffusion in the plasma membrane. Biophys. J. 86(5), 2896–2909.
Cherry, R.J., Smith, P.R., Morrison, I.E.G., Fernandez, N., 1998. Mobility of cell surface receptors: A re-evaluation. FEBS Lett. 430(1), 88–91.
Coles, S., 2001. An Introduction to Statistical Modeling of Extreme Values. Springer, New York.
Coscoy, S., Huguet, E., Amblard, F., 2007. Statistical analysis of sets of random walks: How to resolve their generating mechanism. Bull. Math. Biol. 69(8), 2467–2492.
Dahan, M., Levi, S., Luccardini, C., Rostaing, P., Riveau, B., Triller, A., 2003. Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking. Science 302(5644), 442–445.
Daumas, F., Destainville, N., Millot, C., Lopez, A., Dean, D., Salome, L., 2003. Confined diffusion without fences of a G-protein-coupled receptor as revealed by single particle tracking. Biophys. J. 84(1), 356–366.
Destainville, N., Salome, L., 2006. Quantification and correction of systematic errors due to detector time-averaging in single-molecule tracking experiments. Biophys. J. 90(2), L17–L19.
Destainville, N., Saulière, A., Salomé, L., 2008. Comment to the article by Michael J. Saxton: A biological interpretation of transient anomalous subdiffusion. i. qualitative model. Biophys. J. 95(7), 3117–3119.
Dietrich, C., Yang, B., Fujiwara, T., Kusumi, A., Jacobson, K., 2002. Relationship of lipid rafts to transient confinement zones detected by single particle tracking. Biophys. J. 82(1), 274–284.
Echarte, M.M., Bruno, L., Arndt-Jovin, D.J., Jovin, T.M., Pietrasanta, L.I., 2007. Quantitative single particle tracking of NGF-receptor complexes: Transport is bidirectional but biased by longer retrograde run lengths. FEBS Lett. 581(16), 2905–2913.
Edidin, M., 2003a. Lipids on the frontier: a century of cell-membrane bilayers. Nat. Rev. Mol. Cell Biol. 4, 414–418.
Edidin, M., 2003b. The state of lipid rafts: From model membranes to cells. Annu. Rev. Biophys. Biomol. Struct. 32(1), 257–283.
Edidin, M., Zuniga, M.C., Sheetz, M.P., 1994. Truncation mutants define and locate cytoplasmic barriers to lateral mobility of membrane glycoproteins. Proc. Acad. Natl. Sci. 91(8), 3378–3382.
Fujiwara, T., Ritchie, K., Murakoshi, H., Jacobson, K., Kusumi, A., 2002. Phospholipids undergo hop diffusion in compartmentalized cell membrane. Cell Biol. 157(6), 2002.
Ghosh, R.N., Webb, W.W., 1994. Automated detection and tracking of individual and clustered cell surface low density lipoprotein receptor molecules. Biophys. J. 66(5), 1301–1318.
Giepmans, B.N.G., Adams, S.R., Ellisman, M.H., Tsien, R.Y., 2006. The fluorescent toolbox for assessing protein location and function. Science 312(5771), 217–224.
Gorenflo, R., Mainardi, F., 2003. Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk. LNP, vol. 621, pp. 148–166. Springer, Berlin.
Gutenkunsta, R., Newlands, N., Lutcavage, M., Edelstein-Keshet, L., 2006. Inferring resource distributions from Atlantic bluefin tuna movements: An analysis based on net displacement and length of track. J. Theor. Biol. 245(2), 243–257.
Jin, S., Verkman, A.S., 2007. Single particle tracking of complex diffusion in membranes: Simulation and detection of barrier, raft, and interaction phenomena. J. Phys. Chem. B 111(14), 3625–3632.
Jin, S., Haggie, P.M., Verkman, A.S., 2007. Single-particle tracking of membrane protein diffusion in a potential: Simulation, detection, and application to confined diffusion of CFTR Cl- channels. Biophys. J. 93(3), 1079–1088.
Jonsen, I.D., Myers, R.A., Flemming, J.M., 2003. Meta-analysis of animal movement using state-space models. Ecology 84(11), 3055–3063.
Kabouridis, P., 2006. Lipid rafts in T cell receptor signalling (review). Mol. Membr. Biol. 23(9), 49–57.
Kalay, Z., Parris, P.E., Kenkre, V.M., 2008. Effects of disorder in location and size of fence barriers on molecular motion in cell membranes. J. Phys. 20(24), 245105.
Kenkel, N.C., Walker, D.J., 1996. Fractals in the biological sciences. COENOSES 11, 77–100.
Kenkre, V.M., Giuggioli, L., Kalay, Z., 2008. Molecular motion in cell membranes: Analytic study of fence-hindered random walks. Phys. Rev. E 77(5), 051907.
Khan, S., Reynolds, A.M., Morrison, I.E.G., Cherry, R.J., 2005. Stochastic modeling of protein motions within cell membranes. Phys. Rev. E 71(4), 041915.
Kusumi, A., Murakoshi, H., Murase, K., Fujiwara, T., 2005a. Single-molecule imaging of diffusion, recruitment, and activation of signaling molecules in living cells. In: Damjanovich, S. (Ed.), Biophysical Aspects of Transmembrane Signaling. Springer, Heidelberg
Kusumi, A., Ike, H., Nakada, C., Murase, K., Fujiwara, T., 2005b. Single-molecule tracking of membrane molecules: plasma membrane compartmentalization and dynamic assembly of raft-philic signaling molecules. Semin. Immunol. 17(1), 3–21.
Kusumi, A., Nakada, C., Ritchie, K., Murase, K., Suzuki, K., Murakoshi, H., Kasai, R.S., Kondo, J., Fujiwara, T., 2005c. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: High-speed single-molecule tracking of membrane molecules. Annu. Rev. Biophys. Biomol. Struct. 34(1), 351–378.
Lidke, D.S., Arndt-Jovin, D.J., 2004. Imaging takes a quantum leap. Physiology 19(6), 322–325.
Lidke, D.S., Nagy, P., Heintzmann, R., Arndt-Jovin, D.J., Post, J.N., Grecco, H., Jares-Erijman, E.A., Jovin, T.M., 2004. Quantum dot ligands provide new insights into receptor-mediated signal transduction. Nat. Biotechnol. 22, 198–203.
Lidke, D.S., Lidke, K.A., Rieger, B., Jovin, T.M., Arndt-Jovin, D.J., 2005. Reaching out for signals: Filopodia sense EGF and respond by directed retrograde transport of activated receptors. J. Cell Biol. 170(4), 619–626.
Luttich, S.N., Bergman, C.M., Schaefer, J.A., 2000. Caribou movement as a correlated random walk. Oecologia 123(3), 364–374.
Martin, D.S., Forstner, M.B., Käs, J.A., 2002. Apparent subdiffusion inherent to single particle tracking. Biophys. J. 83(4), 2109–2117.
McCulloch, C.E., Cain, M.L., 1989. Analyzing discrete movement data as a correlated random walk. Ecology 70(2), 383–388.
Meilhac, N., Le Guyader, L., Salome, L., Destainville, N., 2006. Detection of confinement and jumps in single-molecule membrane trajectories. Phys. Rev. E 73(1), 011915.
Metzler, R., Klafter, J., 2000. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1–77.
Murase, K., Fujiwara, T., Umemura, Y., Suzuki, K., Iino, R., Yamashita, H., Saito, M., Murakoshi, H., Ritchie, K., Kusumi, A., 2004. Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques. Biophys. J. 86(6), 4075–4093.
Nicolau, D.V. Jr., Hancock, J.F., Burrage, K., 2007. Sources of anomalous diffusion on cell membranes: A Monte Carlo study. Biophys. J. 92(6), 1975–1987.
Oliver, J.M., Pfeiffer, J.R., Surviladze, Z., Steinberg, S.L., Leiderman, K., Sanders, M., Wofsy, C., Zhang, J., Fan, H.Y., Andrews, N., Bunge, S., Boyle, T.J., Kotula, P., Wilson, B.S., 2004. Membrane receptor mapping: the membrane topography of FcεRI signaling. In: Quinn, P.J. (Ed.), Subcellular Biochemistry 37: Membrane Dynamics and Domains, pp. 3–34. Kluwer Academic/Plenum, Dordrecht/New York
Reynolds, A.M., 2005. On the anomalous diffusion characteristics of membrane-bound proteins. Phys. Lett. A 342(5–6), 439–442.
Ritchie, K., Shan, X.-Y., Kondo, J., Iwasawa, K., Fujiwara, T., Kusumi, A., 2005. Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. Biophys. J. 88(3), 2266–2277.
Saxton, M.J., 1994. Anomalous diffusion due to obstacles: A Monte Carlo study. Biophys. J. 66(2), 394–401.
Saxton, M.J., 1996. Anomalous diffusion due to binding: A Monte Carlo study. Biophys. J. 70(3), 1250–1262.
Saxton, M.J., 2001. Anomalous subdiffusion in fluorescence photobleaching recovery: A Monte Carlo study. Biophys. J. 81(4), 2226–2240.
Saxton, M.J., 2007. A biological interpretation of transient anomalous subdiffusion. i. qualitative model. Biophys. J. 92(4), 1178–1191.
Saxton, M.J., 2008. Response to comment by Destainville et al. Biophys. J. 95(7), 3120–3122.
Saxton, M.J., Jacobson, K., 1997. Single particle tracking: Applications to membrane dynamics. Annu. Rev. Biophys. Biomol. Struct. 26, 373–399.
Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance test for normality (complete samples). Biometrika 52(34), 591–611.
Shumway, R.H., Stoffer, D.S., 2006. Time Series Analysis and Its Applications with R Examples. Springer, New York.
Sibert, J.R., Musyl, M.K., Brill, R.W., 2003. Horizontal movements of bigeye tuna (Thunnus obesus) near Hawaii determined by Kalman filter analysis of archival tagging data. Fish. Oceanogr. 12(3), 141–151.
Smith, P.R., Morrison, I.E.G., Wilson, K.M., Fernandez, N., Cherry, R.J., 1999. Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking. Biophys. J. 76(6), 3331–3344.
Sokolov, I.M., 2008. Statistics and the single molecule. Physics 1, 8.
Suzuki, K., Ritchie, K., Kajikawa, E., Fujiwara, T., Kusumi, A., 2005. Rapid hop diffusion of a G-protein-coupled receptor in the plasma membrane as revealed by single-molecule techniques. Biophys. J. 88(5), 3659–3680.
Tang, Q., Edidin, M., 2003. Lowering the barriers to random walks on the cell surface. Biophys. J. 84(1), 400–407.
Tojo, C., Argyrakis, P., 1996. Correlated random walk in continuous space. Phys. Rev. E 54(1), 58–63.
Vrljic, M., Nishimura, S.Y., Brasselet, S., Moerner, W.E., McConnell, H.M., 2002. Translational diffusion of individual class II MHC membrane proteins in cells. Biophys. J. 83(5), 2681–2692.
Wieser, S., Moertelmaier, M., Fuertbauer, E., Stockinger, H., Schutz, G.J., 2007. (Un)confined diffusion of CD59 in the plasma membrane determined by high-resolution single molecule microscopy. Biophys. J. 92(10), 3719–3728.
Wilson, B.S., Steinberg, S.L., Liederman, K., Pfeiffer, J.R., Surviladze, Z., Zhang, J., Samelson, L.E., Yang, L.-H., Kotula, P.G., Oliver, J.M., 2004. Markers for detergent-resistant lipid rafts occupy distinct and dynamic domains in native membranes. Mol. Biol. Cell 15(6), 2580–2592.
Wu, M., Roberts, J.W., Kim, S., Koch, D.L., DeLisa, M.P., 2006. Collective bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique. Appl. Environ. Microbiol. 72(7), 4987–4994.
Ying, W., 2008. Modeling non-stationary extremal events via Bayesian method. PhD thesis, University of New Mexico, Albuquerque, NM, USA.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Ying, W., Huerta, G., Steinberg, S. et al. Time Series Analysis of Particle Tracking Data for Molecular Motion on the Cell Membrane. Bull. Math. Biol. 71, 1967–2024 (2009). https://doi.org/10.1007/s11538-009-9434-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-009-9434-6