Stochastic modeling in nanoscale biophysics: subdiffusion within proteins. (English) Zbl 1400.62272

Summary: Advances in nanotechnology have allowed scientists to study biological processes on an unprecedented nanoscale molecule-by-molecule basis, opening the door to addressing many important biological problems. A phenomenon observed in recent nanoscale single-molecule biophysics experiments is subdiffusion, which largely departs from the classical Brownian diffusion theory. In this paper, by incorporating fractional Gaussian noise into the generalized Langevin equation, we formulate a model to describe subdiffusion. We conduct a detailed analysis of the model, including (i) a spectral analysis of the stochastic integro-differential equations introduced in the model and (ii) a microscopic derivation of the model from a system of interacting particles. In addition to its analytical tractability and clear physical underpinning, the model is capable of explaining data collected in fluorescence studies on single protein molecules. Excellent agreement between the model prediction and the single-molecule experimental data is seen.


62P10 Applications of statistics to biology and medical sciences; meta analysis
60J60 Diffusion processes
60H99 Stochastic analysis
92C40 Biochemistry, molecular biology
Full Text: DOI arXiv


[1] Adler, R., Feldman, R. and Taqqu, M. (1998)., A Practical Guide to Heavytails : Statistical Techniques for Analyzing Heavy-Tailed Distributions . Birkhäuser, Boston. · Zbl 0901.00010
[2] Alòs, E., Mazet, O. and Nualart, D. (2000). Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2., Stochastic Process. Appl. 86 121-139. · Zbl 1028.60047
[3] Asbury, C., Fehr, A. and Block, S. M. (2003). Kinesin moves by an asymmetric hand-over-hand mechanism., Science 302 2130-2134.
[4] Bouchaud, J. and Georges, A. (1990). Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications., Phys. Rep. 195 127-293.
[5] Carmona, P. and Coutin, L. (2000). Intégrale stochastique pour le mouvement brownien fractionnaire., C. R. Acad. Sci. Paris 330 231-236. · Zbl 0951.60042
[6] Champeney, D. C. (1987)., A Handbook of Fourier Theorems . Cambridge Univ. Press. · Zbl 0632.42001
[7] Chandler, D. (1987)., Introduction to Modern Statistical Mechanics . Oxford Univ. Press, New York.
[8] Corben, H. C. and Stehle, P. (1995)., Classical Mechanics . Dover Publications, New York. · Zbl 0041.32101
[9] Crovella, M. and Bestavros, A. (1996). Self-similarity in world wide web traffic: Evidence and possible causes., Performance Evaluation Review 24 160-169.
[10] Dai, W. and Heyde, C. C. (1996). Ito’s formula with respect to fractional Brownian motion and its application., J. Appl. Math. Stochast. Anal. 9 439-448. · Zbl 0867.60029
[11] Doetsch, G. (1974)., Introduction to the Theory and Application of the Laplace Transformation . Springer, New York. · Zbl 0278.44001
[12] Duncan, T. E., Hu, Y. and Pasik-Duncan, B. (2000). Stochastic calculus for fractional Brownian motion I. Theory., SIAM J. Control Optim. 38 582-612. · Zbl 0947.60061
[13] Embrechts, P. and Maejima, M. (2002)., Selfsimilar Processes . Princeton Univ. Press. · Zbl 1008.60003
[14] English, B., Min, W., van Oijen, A. M., Lee, K. T., Luo, G., Sun, H., Cherayil, B. J., Kou, S. C. and Xie, X. S. (2006). Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited., Nature Chemical Biology 2 87-94.
[15] Erdélyi, A. et al. (1953)., High Transcendental Functions 3 . McGraw-Hill, New York. · Zbl 0051.30303
[16] Glynn, P. and Zeevi, A. (2000). On the maximum workload of a queue fed by fractional Brownian motion., Ann. Appl. Probab. 10 1084-1099. · Zbl 1073.60089
[17] Gray, H. and Winkler, J. (1996). Electron transfer in proteins., Annu. Rev. Biochem. 65 537-561.
[18] Gripenberg, G. and Norros, I. (1996). On the prediction of fractional Brownian motion., J. Appl. Probab. 33 400-410. JSTOR: · Zbl 0861.60049
[19] Heath, D., Resnick, S. and Samorodnitsky, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream., Ann. Appl. Probab. 7 1021-1057. · Zbl 0905.60070
[20] Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time., J. Appl. Probab. 36 1234-1239. · Zbl 1102.62345
[21] Hill, T. (1986)., An Introduction to Statistical Thermodynamics . Dover, New York.
[22] Karlin, S. and Taylor, H. (1981)., A Second Course in Stochastic Processes . Academic Press, New York. · Zbl 0469.60001
[23] Klafter, J., Shlesinger, M. and Zumofen, G. (1996). Beyond Brownian motion., Physics Today 49 33-39.
[24] Konstantopoulos, T. and Lin, S. J. (1996). Fractional Brownian approximations of queueing networks., Stochastic Networks. Lecture Notes in Statist. 117 257-273. Springer, New York. · Zbl 0856.60097
[25] Kou, S. C., Cherayil, B., Min, W., English, B. and Xie, X. S. (2005). Single-molecule Michaelis-Menten equations., J. Phys. Chem. B 109 19068-19081.
[26] Kou, S. C. and Xie, X. S. (2004). Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule., Phys. Rev. Lett. 93 180603(1)-180603(4).
[27] Kou, S. C., Xie, X. S. and Liu, J. S. (2005). Bayesian analysis of single-molecule experimental data (with discussion)., J. Roy. Statist. Soc. Ser. C 54 469-506. JSTOR: · Zbl 05188696
[28] Kou, S. C. (2007). Stochastic networks in nanoscale biophysics: Modeling enzymatic reaction of a single protein., J. Amer. Statist. Assoc. · Zbl 1205.62172
[29] Kupferman, R. (2004). Fractional kinetics in Kac-Zwanzig heat bath models., J. Statist. Phys. 114 291-326. · Zbl 1060.82034
[30] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1994). On the self-similar nature of Ethernet traffic (Extended Version)., IEEE/ACM Trans. Networking 2 1-15.
[31] Lin, S. J. (1995). Stochastic analysis of fractional Brownian motions., Stochast. Stochast. Rep. 55 121-140. · Zbl 0886.60076
[32] Lu, H. P., Xun, L. and Xie, X. S. (1998). Single-molecule enzymatic dynamics., Science 282 1877-1882.
[33] Mandelbrot, B. (1997)., Fractals and Scaling in Finance . Springer, New York. · Zbl 1005.91001
[34] Mandelbrot, B. and Van Ness, J. (1968). Fractional Brownian motions, fractional noises and applications., SIAM Rev. 10 422-437. JSTOR: · Zbl 0179.47801
[35] Mikosch, T. and Norvaisa, R. (2000). Stochastic integral equations without probability., Bernoulli 6 401-434. · Zbl 0963.60060
[36] Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Probab. 12 23-68. · Zbl 1021.60076
[37] Min, W., English, B., Luo, G., Cherayil, B., Kou, S. C. and Xie, X. S. (2005). Fluctuating enzymes: Lessons from single-molecule studies., Accounts of Chemical Research 38 923-931.
[38] Min, W., Luo, G., Cherayil, B., Kou, S. C. and Xie, X. S. (2005). Observation of a power law memory kernel for fluctuations within a single protein molecule., Phys. Rev. Lett. 94 198302(1)-198302(4).
[39] Moerner, W. (2002). A dozen years of single-molecule spectroscopy in physics, chemistry, and biophysics., J. Phys. Chem. B 106 910-927.
[40] Moser, C., Keske, J., Warncke, K., Farid, R. and Dutton, P. (1992). Nature of biological electron transfer., Nature 355 796-802. · Zbl 0132.31801
[41] Mukamel, S. (1995)., Principle of Nonlinear Optical Spectroscopy . Oxford Univ. Press, New York.
[42] Nie, S. and Zare, R. (1997). Optical detection of single molecules., Ann. Rev. Biophys. Biomol. Struct. 26 567-596.
[43] Nualart, D. (2006)., The Malliavin Calculus and Related Topics ( Probability and Its Applications ). Springer, New York. · Zbl 1099.60003
[44] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion., Probab. Theory Related Fields 118 251-291. · Zbl 0970.60058
[45] Pipiras, V. and Taqqu, M. S. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli 7 873-897. · Zbl 1003.60055
[46] Reif, F. (1965)., Fundamentals of Statistical and Thermal Physics . McGraw-Hill, New York.
[47] Risken, H. (1989)., The Fokker-Planck Equation : Methods of Solution and Applications . Springer, Berlin. · Zbl 0665.60084
[48] Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion., Math. Finance 7 95-105. · Zbl 0884.90045
[49] Samorodnitsky, G. and Taqqu, M. (1994)., Stable Non-Gaussian Random Processes . Chapman and Hall, New York. · Zbl 0925.60027
[50] Shiryaev, A. N. (1998). On arbitrage and replication for fractal models. Research Report No. 2, 1998, MaPhySto, Univ., Aarhus.
[51] Sokolov, I., Klafter, J. and Blumen, A. (2002). Fractional kinetics., Physics Today 55 48-54.
[52] Tamarat, P., Maali, A., Lounis, B. and Orrit, M. (2000). Ten years of single-molecule spectroscopy., J. Phys. Chem. A 104 1-16.
[53] Taqqu, M. S. (1986). Sojourn in an elliptical domain., Stochastic Process. Appl. 21 319-326. · Zbl 0598.60037
[54] Van Kampen, N. G. (2001)., Stochastic Processes in Physics and Chemistry . North-Holland, Amsterdam. · Zbl 0974.60020
[55] Wang, K. G. and Tokuyama, M. (1999). Nonequilibrium statistical description of anomalous diffusion., Phys. A 265 341-351.
[56] Weiss, S. (2000). Measuring conformational dynamics of biomolecules by single molecule fluorescence spectroscopy., Nature Struct. Biol. 7 724-729.
[57] Whitt, W. (2002)., Stochastic-Process Limits . Springer, New York. · Zbl 0993.60001
[58] Xie, X. S. and Lu, H. P. (1999). Single-molecule enzymology., J. Bio. Chem. 274 15967-15970.
[59] Xie, X. S. and Trautman, J. K. (1998). Optical studies of single molecules at room temperature., Ann. Rev. Phys. Chem. 49 441-480.
[60] Yang, H., Luo, G., Karnchanaphanurach, P., Louise, T.-M., Rech, I., Cova, S., Xun, L. and Xie, X. S. (2003). Protein conformational dynamics probed by single-molecule electron transfer., Science 302 262-266.
[61] Zhuang, X., Kim, H., Pereira, M., Babcock, H., Walter, N. and Chu, S. (2002). Correlating structural dynamics and function in single ribozyme molecules., Science 296 1473-1476. · Zbl 0598.00019
[62] Zwanzig, R. (2001)., Nonequilibrium Statistical Mechanics . Oxford Univ. Press, New York. · Zbl 1267.82001
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