Advertisement

Applications in Chemistry

  • Peter Schuster
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

In chemistry the master equation is the best suited and most commonly used tool to model stochasticity in reaction kinetics. We review the common elementary reactions in mass action kinetics and discuss Michaelis–Menten kinetics as an example of combining several elementary steps into an overall reaction. Multistep reactions or reaction networks are considered and a formal mathematical theory that provides tools for the derivation of general properties of networks is presented. Then we digress into theory and empirical determination of rate parameters. The chemical master equation is introduced as the most popular tool for modeling stochasticity in chemical reactions, and the single reaction step approach is extended to reaction networks. The chemical Langevin equation is discussed as an alternative to the master equation: it has a number of convenient features but is not always applicable. Then, a selection of one-step reactions is presented for which the master equation can be solved exactly. The exact solutions are also used to illustrate the relation between the mathematical approach and the recorded data. A separate chapter deals with correlation functions, fluctuation spectroscopy, single molecule data, and stochastic modeling. Deterministic and stochastic parts of solutions can be separated by means of size expansions. Most reaction mechanisms are not accessible to the analytical approach and therefore we present a simulation method that is exact within the concept of the chemical master equation, and apply it to some selected examples of chemical reactions.

References

  1. 2.
    Abramowitz, M., Segun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1965)zbMATHGoogle Scholar
  2. 4.
    Acton, F.S.: Numerical Methods That Work. Harper & Row, New York (1970)zbMATHGoogle Scholar
  3. 5.
    Acton, F.S.: Numerical Methods That (Usually) Work, fourth printing edn. Mathematical Association of America, Washington, DC (1990)Google Scholar
  4. 7.
    Al-Soufi, W., Reija, B., Novo, M., Kelekyan, S., Kühnemuth, R., Seidel, C.A.M.: Fluorescence correlation sprctroscopy, a tool to inverstigate supramolecular dynamics: Inclusion complexes of pyronines with cyclodextrin. J. Am. Chem. Soc. 127, 8775–8784 (2005)CrossRefGoogle Scholar
  5. 11.
    Anderson, D.F.: Incorporating postleap checks in tau-leaping. J. Chem. Phys. 128, e 054103 (2008)Google Scholar
  6. 12.
    Anderson, D.F., Craciun, G., Kurtz, T.G.: Product-form stationary distributions for deficiency zero chemical reaction networks. Bull. Math. Biol. 72, 1947–1970 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 13.
    Anderson, D.F., Ganguly, A., Kurtz, T.G.: Error analysis of tau-leap simulation methods. Ann. Appl. Probab. 6, 2226–2262 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 19.
    Aragón, S.R., Pecora, R.: Fluorescence correlation spectroscopy and Brownian rotational diffusion. Biopolymers 14, 119–138 (1975)CrossRefGoogle Scholar
  9. 20.
    Arányi, P., Tóth, J.: A full stochastic description of the Michaelis-Menten reaction for small systems. Acta Biochim. et Biophys. Acad. Sci. Hung. 12, 375–388 (1977)Google Scholar
  10. 21.
    Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists, fifth edn. Harcourt Academic Press, San Diego (2001)zbMATHGoogle Scholar
  11. 23.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998). Second corrected printing 2003Google Scholar
  12. 24.
    Arnold, L., Bleckert, G., Schenk-Hoppé, K.R.: The stochastic brusselator: Parametric noise destroys hopf bifurcation. In: Crauel, H., Gundlach, M. (eds.) Stochastic Dynamics, chap. 4, pp. 71–92. Springer, New York (1999)CrossRefGoogle Scholar
  13. 26.
    Arslan, E., Laurenzi, I.J.: Kinetics of autocatalysis in small systems. J. Chem. Phys. 128, e 015101 (2008)Google Scholar
  14. 27.
    Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algortihms and Analysis. Springer, New York (2007)zbMATHGoogle Scholar
  15. 28.
    Aster, R.C., Borchers, B., Thurber, C.H.: Parameter Estimation and Inverse Problems, 2nd edn. Academic Press, Elsevier, Singapore (2013)zbMATHGoogle Scholar
  16. 30.
    Atkins, P.W., Friedman, R.S. (eds.): Molecular Quantum Mechanics, fifth edn. Oxford University Press, Oxford (2010)Google Scholar
  17. 34.
    Bar-Eli, K., Noyes, R.M.: Detailed calculations of multiple steady states during oxidation of cerous ion by bromate in a stirred flow reactor. J. Phys. Chem. 82, 1352–1359 (1978)CrossRefGoogle Scholar
  18. 35.
    Bartholomay, A.F.: On the linear birth and death processes of biology as Markoff chains. Bull. Math. Biophys. 20, 97–118 (1958)MathSciNetCrossRefGoogle Scholar
  19. 37.
    Bartholomay, A.F.: Stochastic models for chemical reactions: II. The unimolecular rate constant. Bull. Math. Biophys. 21, 363–373 (1959)MathSciNetCrossRefGoogle Scholar
  20. 38.
    Bartholomay, A.F.: A stochastic approach to statistical kinetics with applications to enzyme kinetics. Biochemistry 1, 223–230 (1962)CrossRefGoogle Scholar
  21. 39.
    Bartlett, M.S.: Stochastic processes or the statistics of change. J. R. Stat. Soc. C 2, 44–64 (1953)Google Scholar
  22. 40.
    Bazley, N.W., Montroll, E.W., Rubin, R.J., Shuler, K.E.: Studies in nonequilibrium rate processes: III. The vibrational relaxation of a system of anharmonic oscillators. J. Chem. Phys. 28, 700–704 (1958). Erratum: J.Chem.Phys., 29:1185–1186Google Scholar
  23. 41.
    Berg, J.M., Tymoczko, J.L., Stryer, L.: Biochemistry, fifth edn. W. H. Freeman and Company, New York (2002)Google Scholar
  24. 42.
    Berg, J.M., Tymoczko, J.L., Stryer, L.: Biochemistry, seventh edn. W. H. Freeman and Company, New York (2012)Google Scholar
  25. 46.
    Berry, R.S., Rice, S.A., Ross, J.: Physical Chemistry, 2nd edn. Oxford University Press, New York (2000)Google Scholar
  26. 47.
    Berry, R.S., Rice, S.A., Ross, J.: Physical and Chemical Kinetics, 2nd edn. Oxford University Press, New York (2002)Google Scholar
  27. 52.
    Binnig, G., Quate, C.F., Gerber, C.: Atomic force microscopy. Phys. Rev. Lett. 56, 930–933 (1986)ADSCrossRefGoogle Scholar
  28. 54.
    Björck, Å.: Numerical Methods for Least Square Problems. Other Titles in Applied Mathematics. SIAM Society for Industrial & Applied Mathematics, Philadelphia (1996)zbMATHCrossRefGoogle Scholar
  29. 55.
    Bloomfield, V.A., Benbasat, J.A.: Inelastic light-scattering study of macromolecular reaction kinetics. I: The reactions A\(\rightleftharpoons \) B and 2A\(\rightleftharpoons \) A2. Macromolecules 4, 609–613 (1971)ADSCrossRefGoogle Scholar
  30. 59.
    Born, M., Oppenheimer, R.: Zur Quantentheorie der Moleküle. Annalen der Physik 84, 457–484 (1927). In GermanADSzbMATHCrossRefGoogle Scholar
  31. 60.
    Börsch, A., Simon, P. (eds.): Carl Friedrich Gauß: Abhandlungen zur Methode der kleinsten Quadrate. P. Stankiewicz, Berlin (1887). In GermanGoogle Scholar
  32. 62.
    Box, G.E.P., Muller, M.E.: A note on the generation of random normal deviates. Ann. Math. Stat. 29, 610–611 (1958)zbMATHCrossRefGoogle Scholar
  33. 65.
    Briggs, G.E., Haldane, J.B.S.: A note on the kinetics of enzyme action. Biochem. J. 19, 338–339 (1925)CrossRefGoogle Scholar
  34. 73.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: Efficient step size selection for the tau-leaping simulation method. J. Chem. Phys. 124, 044,109 (2004)Google Scholar
  35. 76.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: Adaptive explicit-implicit tau-leaping method with automatic tau selection. J. Chem. Phys. 126, e224,101 (2007)CrossRefGoogle Scholar
  36. 78.
    Cassandras, C.G., Lygeros, J. (eds.): Stochastic Hybrid Systems. Control of Engineering Series. CRC Press, Taylor & Francis Group, Boca Raton (2007)zbMATHGoogle Scholar
  37. 79.
    Castets, V., Dulos, E., Boissonade, J., De Kepper, P.: Exprimental evidence of a sustained standing Turing-type nonequilibrium xhemical pattern. Phys. Rev. Lett. 64, 2953–2956 (1990)ADSCrossRefGoogle Scholar
  38. 80.
    Chang, C., Gzyl, H.: Parameter estimation in superposition of decaying exponentials. Appl. Math. Comput. 96, 101–116 (1998)MathSciNetzbMATHGoogle Scholar
  39. 82.
    Child, M.S.: Molecular Collision Theory. Dover Publications, Mineola (1996). Originally publisher: Academic Press, London (1974)Google Scholar
  40. 87.
    Cook, M., Soloveichik, D., Winfree, E., Bruck, J.: Programmability of chemical reaction networks. In: Condon, A., Harel, D., Kok, J.N., Salomaa, A., Winfree, E. (eds.) Algorithimc Bioprocesses, Natural Computing Series, vol. XX, pp. 543–584. Springer, Berlin (2009)Google Scholar
  41. 93.
    Craciun, G., Tang, Y., Feinberg, M.: Understanding bistability in complex enzyme-driven reaction networks. Proc. Natl. Acad. Sci. USA 103, 8697–8702 (2006)ADSzbMATHCrossRefGoogle Scholar
  42. 98.
    Dalla Valle, J.M.: Note on the Heaviside expansion formula. Proc. Natl. Acad. Sci. USA 17, 678–684 (1931)ADSzbMATHCrossRefGoogle Scholar
  43. 99.
    Darvey, I.G., Ninham, B.W.: Stochastic models for second-order chemical reaction kinetics. Time course of reactions. J. Chem. Phys. 46, 1626–1645 (1967)Google Scholar
  44. 100.
    Darvey, I.G., Ninham, B.W., Staff, P.J.: Stochastic models for second-order chemical reaction kinetics. The equilibirum state. J. Chem. Phys. 45, 2145–2155 (1966)ADSCrossRefGoogle Scholar
  45. 101.
    Darvey, I.G., Staff, P.J.: Stochastic approach to first-order chemical reaction kinetics. J. Chem. Phys. 44, 990–997 (1966)ADSMathSciNetCrossRefGoogle Scholar
  46. 103.
    DeKepper, P., Epstein, I.R., Kustin, K.: Bistability in the oxidatiion of arsenite by iodate in a stirred flow reactor. J. Am. Chem. Soc. 103, 6121–6127 (1981)CrossRefGoogle Scholar
  47. 104.
    Delbrück, M.: Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124 (1940)ADSCrossRefGoogle Scholar
  48. 106.
    Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986)zbMATHCrossRefGoogle Scholar
  49. 111.
    Djermoune, E.H., Tomczak, M.: Statistical analysis of the Kumaresan-Tufts and matrix pencil methods in estimating damped sinusoids. In: Hlawatsch, F., Matz, G., Rupp, M., Wistawel, B. (eds.) Proceedings of the XII. European Signal Processing Conference, vol. II, pp. 1261–1264. Technische Universität Wien, Wien (2004)Google Scholar
  50. 115.
    Doob, J.L.: Topics in the theory of Markoff chains. Trans. Am. Math. Soc. 52, 37–64 (1942)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 116.
    Doob, J.L.: Markoff chains – Denumerable case. Trans. Am. Math. Soc. 58, 455–473 (1945)MathSciNetzbMATHGoogle Scholar
  52. 118.
    Dushman, S.: The reaction between iodic and hydroiodic acid. J. Phys. Chem. 8, 453–482 (1903)CrossRefGoogle Scholar
  53. 121.
    Edelson, D., Field, R.J., Noyes, R.M.: Mechanistic details of the Belousov-Zhabotinskii oscillations. Int. J. Chem. Kinet. 7, 417–423 (1975)CrossRefGoogle Scholar
  54. 125.
    Edman, L., Földes-Papp, Z., Wennmalm, S., Rigler, R.: The fluctuating enzyme: A single moleculae approach. Chem. Phys. 247, 11–22 (1999)ADSCrossRefGoogle Scholar
  55. 126.
    Edman, L., Rigler, R.: Memory landscapes of single-enzyme molecules. Proc. Natl. Acad. Sci. USA 97, 8266–8271 (2000)ADSCrossRefGoogle Scholar
  56. 128.
    Ehrenberg, M., Rigler, R.: Rotational Brownian motion and fluorescence intensity fluctuations. Chem. Phys. 4, 390–401 (1974)ADSCrossRefGoogle Scholar
  57. 129.
    Ehrenfest, P., Ehrenfest, T.: Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem. Z. Phys. 8, 311–314 (1907)zbMATHGoogle Scholar
  58. 137.
    Elson, E., Magde, D.: Fluorescence correlation spectroscopy. I. Conceptual basis and theory. Biopolymers 13, 1–27 (1974)Google Scholar
  59. 138.
    Engl, H.W., Flamm, C., Kügler, P., Lu, J., Müller, S., Schuster, P.: Inverse problems in systems biology. Inverse Prob. 25, 123,014 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 139.
    Engl, H.W., Hanke, M., Neubauuer, A.: Regularization of Inverse Problems. Kluwer Academic, Boston (1996)CrossRefGoogle Scholar
  61. 140.
    Érdi, P., Lente, G.: Stochastic Chemical Kinetics. Theory and (Mostly) Systems Biological Applications. Understanding Complex Systems. Springer, Berlin (2014)Google Scholar
  62. 148.
    Eyring, H.: The activated complex in chemical reactions. J. Chem. Phys. 3, 107–115 (1935)ADSCrossRefGoogle Scholar
  63. 151.
    Feinberg, M.: Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49, 187–194 (1972)MathSciNetCrossRefGoogle Scholar
  64. 152.
    Feinberg, M.: Mathematical aspects of mass action kinetics. In: Lapidus, L., Amundson, N.R. (eds.) Chemical Reactor Theory – A Review, pp. 1–78. Prentice Hall, Englewood Cliffs (1977)Google Scholar
  65. 153.
    Feinberg, M.: Lectures on Chemical Reaction Networks. Chemical Engineering & Mathematics. The Ohio State University, Columbus (1979)Google Scholar
  66. 154.
    Feinberg, M.: Chemical oscillations, multiple equilibria, and reaction network structure. In: Stewart, W.E., Ray, W.H., Conley, C.C. (eds.) Dynamics and Modelling of Reactive Systems, pp. 59–130. Academic Press, New York (1980)CrossRefGoogle Scholar
  67. 155.
    Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors – II. Multiple steady states for networks of deficiency one. Chem. Eng. Sci. 43, 1–25 (1988)Google Scholar
  68. 156.
    Feller, W.: On the integro-differential equations of purely discontinuous Markoff processes. Trans. Am. Math. Soc. 48, 488–515 (1940)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 163.
    Fernández-Ramos, A., Miller, J.A., Klippenstein, S.J., Truhlar, D.G.: Modeling the kinetics of bimolecular reactions. Chem. Rev. 106, 4518–4584 (2006)CrossRefGoogle Scholar
  70. 164.
    Fersht, A.: Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding. W. H. Fremman and Company, New York (1999)Google Scholar
  71. 166.
    Field, R.J., Körös, E., Noyes, R.M.: Oscillations in chemical systems. II. Thorough analysis of temporal oscillations in the bromate-cerium-malonic acid system. J. Am. Chem. Soc. 94, 8649–8664 (1972)Google Scholar
  72. 167.
    Field, R.J., Noyes, R.M.: Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60, 1877–1884 (1974)Google Scholar
  73. 168.
    Firth, C.J.M., Bray, D.: Stochastic simulation of cell signalling pathways. In: Bower, J.M., Bolouri, H. (eds.) Computational Modeling of Genetic and Biochemical Networks, pp. 263–286. MIT Press, Cambridge (2000)Google Scholar
  74. 183.
    Föllner, H.H., Geiseler, W.: A model of bistability in an open homogeneous chemical reaction system. Naturwissenschaften 64, 384 (1977)ADSCrossRefGoogle Scholar
  75. 187.
    Frauenfelder, H., Sligar, S.G., Wolynes, P.G.: The eenergy landscape and motions of proteins. Science 254, 1598–1603 (1991)ADSCrossRefGoogle Scholar
  76. 188.
    Freire, J.G., Field, R.J., Gallas, J.A.C.: Relative abundance and structure of chaotic behavior: The nonpolynomial belousov-zhabotinsky reaction kinetics. J. Chem. Phys. 131, e044,105 (2009)CrossRefGoogle Scholar
  77. 190.
    Gadgil, C., Lee, C.H., Othmer, H.G.: A stochastic analysis of first-order reaction networks. Bull. Math. Biol. 67, 901–946 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 194.
    Gardiner, C.W.: Stochastic Methods. A Handbook for the Natural Sciences and Social Sciences, fourth edn. Springer Series in Synergetics. Springer, Berlin (2009)Google Scholar
  79. 198.
    Geisler, W., Föllner, H.H.: Three steady state situation in an open chemical reaction system. I. Biophys. Chem. 6, 107–115 (1977)CrossRefGoogle Scholar
  80. 202.
    Gibbs, J.W.: Elementary Principles in Statistical Mechanics. Charles Scribner’s Sons, New York (1902). Reprinted 1981 by Ox Bow Press, Woodbridge, CTGoogle Scholar
  81. 203.
    Gibbs, J.W.: The Scientific Papers of J. Willard Gibbs, vol.I, Thermodynamics. Dover Publications, New York (1961)Google Scholar
  82. 204.
    Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104, 1876–1889 (2000)CrossRefGoogle Scholar
  83. 206.
    Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phys. 22, 403–434 (1976)ADSMathSciNetCrossRefGoogle Scholar
  84. 207.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977)CrossRefGoogle Scholar
  85. 208.
    Gillespie, D.T.: Markov Processes: An Introduction for Physical Scientists. Academic Press, San Diego (1992)zbMATHGoogle Scholar
  86. 209.
    Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188, 404–425 (1992)ADSCrossRefGoogle Scholar
  87. 211.
    Gillespie, D.T.: The chemical Langevin equation. J. Chem. Phys. 113, 297–306 (2000)ADSCrossRefGoogle Scholar
  88. 212.
    Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115 (4), 1716–1733 (2001)ADSCrossRefGoogle Scholar
  89. 213.
    Gillespie, D.T.: Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem. 58, 35–55 (2007)ADSCrossRefGoogle Scholar
  90. 216.
    Goel, N.S., Richter-Dyn, N.: Stochastic Models in Biology. Academic Press, New York (1974)Google Scholar
  91. 217.
    Goutsias, J., Jenkinson, G.: Markovian dynamics on complex reaction networks. Phys. Rep. 529, 199–264 (2013)ADSMathSciNetCrossRefGoogle Scholar
  92. 226.
    Gunawardena, J.: Chemical reaction network theory for in-silico biologists. Tech. rep., Bauer Center for Genomics Research at Harvard University, Cambridge, MA (2003)Google Scholar
  93. 227.
    Györgyi, L., Field, R.J.: A three-variable model of deterministic chaos in the belousov-zhabotinsky reaction. Nature 355, 808–810 (1992)ADSCrossRefGoogle Scholar
  94. 230.
    Hamill, O.P., Marty, A., Neher, E., Sakmann, B., Sigworth, F.J.: Improved patch-clamp techniques for high-resolution current recording from cels and cell-free mambrane patches. Pflügers Archiv. Eur. J. Physiol. 391, 85–100 (1981)CrossRefGoogle Scholar
  95. 237.
    Hanna, A., Saul, A., Showalter, K.: Detailed studies of propagating frints in the iodate oxidation of arsenous acid. J. Am. Chem. Soc. 104, 3838–3844 (1982)CrossRefGoogle Scholar
  96. 238.
    Hansma, H.G., Kasuya, K., Oroudjev, E.: Atomic force microscopy imaging and pulling of nucleic acids. Curr. Op. Struct. Biol. 14, 380–385 (2004)CrossRefGoogle Scholar
  97. 243.
    Hatzakis, N.S., Wei, L., Jorgensen, S.K., Kunding, A.H., Bolinger, P.Y., Ehrlich, N., Makarov, I., Skjot, M., Svendsen, A., Hedegård, P., Stamou, D.: Single enzyme studies reveal the existence of discrete states for monomeric enzymes and how they are ”selected” upon allosteric regulation. J. Am. Chem. Soc. 134, 9296–9302 (2012)CrossRefGoogle Scholar
  98. 247.
    Hazeltine, E.L., Rawlings, J.B.: Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117, 6959–6969 (2002)ADSCrossRefGoogle Scholar
  99. 251.
    Higham, D.J.: Modeling and somulationg chemical reactions. SIAM Rev. 50, 347–368 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  100. 252.
    Hinshelwood, C.N.: On the theory of unimolecular reactions. Proc. R. Soc. Lond. A 113, 230–233 (1926)ADSCrossRefGoogle Scholar
  101. 254.
    Hirschfeld, T.: Optical microscopic observation of small molecules. Appl. Opt. 15, 2965–2966 (1976)ADSCrossRefGoogle Scholar
  102. 262.
    Horn, F.: Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49, 172–186 (1972)MathSciNetCrossRefGoogle Scholar
  103. 263.
    Horn, F., Jackson, R.: General mass action kinetics. Arch. Ration. Mech. Anal. 47, 81–116 (1972)MathSciNetCrossRefGoogle Scholar
  104. 265.
    Houston, P.L.: Chemical Kinetics and Reaction Dynamics. The McGraw-Hill Companies, New York (2001)Google Scholar
  105. 266.
    Hu, J., Lygeros, J., Sastry, S.: Towards a theory of stochastic hybrid systems. In: Lynch, N., Krogh, B. (eds.) Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 1790, pp. 160–173. Springer, Berlin (2000)CrossRefGoogle Scholar
  106. 267.
    Hu, Y., Li, T.: Highly accurate tau-leaping methiods with random corrections. J. Chem. Phys. 130, e124,109 (2009)CrossRefGoogle Scholar
  107. 268.
    Hua, Y., Sarkar, T.K.: Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans. Acoust. Speech Signal Process. 38, 814–824 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 271.
    Ishida, K.: Stochastic model for bimolecular reaction. J. Chem. Phys. 41, 2472–2478 (1964)ADSCrossRefGoogle Scholar
  109. 274.
    Jachimowski, C.J., McQuarrie, D.A., Russell, M.E.: A stochastic approach to enzyme-substrate reactions. Biochemistry 3, 1732–1736 (1964)CrossRefGoogle Scholar
  110. 278.
    Jahnke, T., Huisinga, W.: Solving the chemical master equation for monomolecular reaction systems analytically. J. Math. Biol. 54, 1–26 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  111. 291.
    Kassel, L.S.: Studies in homogeneous gas reactions I. J. Phys. Chem. 32, 225–242 (1928)CrossRefGoogle Scholar
  112. 292.
    Kendall, D.G.: An artificial realization of a simple “birth-and-death” process. J. R. Stat. Soc. B 12, 116–119 (1950)zbMATHGoogle Scholar
  113. 301.
    Kim, S.K.: Mean first passage time for a random walker and its application to chemical knietics. J. Chem. Phys. 28, 1057–1067 (1958)ADSCrossRefGoogle Scholar
  114. 310.
    Kolmogorov, A.N.: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104, 415–458 (1931)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 314.
    Koroborov, V.I., Ochkov, V.F.: Chemical Kinetics with Mathcad and Maple. Springer, Wien (2011)Google Scholar
  116. 315.
    Koshland, D.E.: Application of a theory of enzyme specificity to protein synthesis. Proc. Natl. Acad. Sci. USA 44, 98–104 (1958)Google Scholar
  117. 316.
    Kou, S.C., Cherayil, B.J., Min, W., English, B.P., Xie, X.S.: Single-molecule Michaelis-Menten equations. J. Phys. Chem. B 109, 19,068–19,081 (2005)Google Scholar
  118. 318.
    Krichevsky, O., Bonnet, G.: Fluorescence correlation spectroscopy: The technique and its applications. Rep. Prog. Phys. 65, 251–297 (2002)ADSCrossRefGoogle Scholar
  119. 319.
    Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966)ADSzbMATHCrossRefGoogle Scholar
  120. 320.
    Kügler, P., Gaubitzer, E., Müller, S.: Perameter identification for chemical reaction systems using sparsity enforcing regularization A case study for the chlorite–iodide reaction. J. Phys. Chem. A 113, 2775–2785 (2009)CrossRefGoogle Scholar
  121. 321.
    Kulzer, F., Orrit, M.: Single-molecule optics. Annu. Rev. Phys. Chem. 55, 585–611 (2004)ADSCrossRefGoogle Scholar
  122. 322.
    Kumaresan, R., Tufts, D.W.: Estimating the parameters of exponentially damped sinusoids and pole-zero modeiling in noise. IEEE Trans. Acoust. Speech Signal Process. 30, 833–840 (1982)CrossRefGoogle Scholar
  123. 323.
    Laidler, K.J.: Chemical Kinetics, 3rd edn. Addison Wesley, Boston (1987)Google Scholar
  124. 324.
    Laidler, K.J., King, M.C.: The development of transition-state theory. J. Phys. Chem. 87, 2657–2664 (1983)CrossRefGoogle Scholar
  125. 329.
    Laurenzi, I.J.: An analytical solution of the stochastic master equation for reversible bimolecular reaction kinetics. J. Chem. Phys. 113, 3315–3322 (2000)ADSCrossRefGoogle Scholar
  126. 332.
    Le Novère, N., Shimizu, T.S.: StochSim: Modeling of stochastic biomolecular processes. Bioinformatics 17, 575–576 (2001)CrossRefGoogle Scholar
  127. 335.
    Lefever, R., Nicolis, G., Borckmans, P.: The Brusselator: It does oscillate all the same. J. Chem. Soc. Faraday Trans. 1, 1013–1023 (1988)CrossRefGoogle Scholar
  128. 336.
    Legendre, A.M.: Nouvelles méthodes pour la détermination des orbites des comètes. F. Didot, Paris (1805). In FrenchGoogle Scholar
  129. 337.
    Lerch, H.P., Rigler, R., Mikhailov, A.S.: Functional conformational motions in the turnover cycle of cholesterol oxidase. Proc. Natl. Acad. Sci. USA 102, 10,807–10,812 (2005)Google Scholar
  130. 341.
    Li, H., Cao, Y., Petzold, L.R., Gillespie, D.T.: Algortihms and software for stochastic simulation of biochemical reacting systems. Biotechnol. Prog. 24, 56–61 (2008)CrossRefGoogle Scholar
  131. 342.
    Li, P.T.X., Bustamante, C., Tinoco, Jr., I.: Real-time control of the energy landscape by force directs the folding of RNA molecules. Proc. Natl. Acad. Sci. USA 104, 7039–7044 (2007)ADSCrossRefGoogle Scholar
  132. 343.
    Li, T.: Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul. 6, 417–436 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 345.
    Liao, D., Galajda, P., Riehn, R., Ilic, R., Puchalla, J.L., Yu, H.G., Craighead, H.G., Austin, R.H.: Single molecule correlation spectroscopy ib continuous flow mixers with zero-mode waveguides. Opt. Express 16, 10,077–10,090 (2008)Google Scholar
  134. 347.
    Lin, H., Truhlar, D.G.: QM/MM: What have we learned, where are we, and where do we go from here? Theor. Chem. Acc. 117, 185–199 (2007)CrossRefGoogle Scholar
  135. 348.
    Lin, S.H., Lau, K.H., Richardson, W., Volk, L., Eyring, H.: Stochastic model of unimolecular reactions and the RRKM theory. Proc. Natl. Acad. Sci. USA 69, 2778–2782 (1972)ADSCrossRefGoogle Scholar
  136. 351.
    Lindemann, F.A.: Discussion on the radiation theory on chemical action. Trans. Farad. Soc. 17, 598–606 (1922)CrossRefGoogle Scholar
  137. 355.
    Lu, H.P., Xun, L., Xie, X.S.: Single-molecule enzyme dynamics. Science 282, 1877–1882 (1998)ADSCrossRefGoogle Scholar
  138. 356.
    Lu, J., Engl, H.W., Rainer Machné, Schuster, P.: Inverse bifurcation analysis of a model for the mammalian G1/S regulatory module. Lect. Notes Comput. Sci. 4414, 168184 (2007)Google Scholar
  139. 357.
    Lu, J., Engl, H.W., Schuster, P.: Inverse bifurcation analysis: Application to simple gene systems. ABM – Algorithms Mol. Biol. 1, e11 (2006)Google Scholar
  140. 363.
    Magde, D., Elson, E., Webb, W.W.: Thermodynamic fluctuations in a reating system – Measurement by fluorescence correlation spectroscopy. Phys. Rev. Lett. 29, 705–708 (1972)ADSCrossRefGoogle Scholar
  141. 368.
    Marcus, R.A.: Unimolecular dissociations and free radical recombination reactions. J. Chem. Phys. 20, 359–364 (1952)ADSCrossRefGoogle Scholar
  142. 369.
    Marcus, R.A.: Vibrational nonadiabaticity and tunneling effects in thranition state theory. J. Chem. Phys. 83, 204–207 (1979)CrossRefGoogle Scholar
  143. 370.
    Marcus, R.A.: Unimolecular reactions, rates and quantum state distributions of products. Philos. Trans. R. Soc. Lond. A 332, 283–296 (1990)ADSCrossRefGoogle Scholar
  144. 371.
    Marcus, R.A., Rice, O.K.: The kinetics of the recombination of methyl radical and iodine atoms. J. Phys. Colloid Chem. 55, 894–908 (1951)CrossRefGoogle Scholar
  145. 373.
    Marx, D., Jürg Hutter: Ab initio Molecular Dynamics. Basic Theory and Advanced Methods. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  146. 381.
    McQuarrie, D.A.: Kinetics of small systems. I. J. Chem. Phys. 38, 433–436 (1962)ADSCrossRefGoogle Scholar
  147. 382.
    McQuarrie, D.A.: Stochastic approach to chemical kinetics. J. Appl. Probab. 4, 413–478 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  148. 384.
    McQuarrie, D.A., Jachimowski, C.J., Russell, M.E.: Kinetics of small systems. II. J. Chem. Phys. 40, 2914–2921 (1964)ADSCrossRefGoogle Scholar
  149. 387.
    Medina, M.Ángel., Schwille, P.: Fluorescence correlation spectroscopy for the detection and study of single molecules in biology. BioEssays 24, 758–764 (2002)CrossRefGoogle Scholar
  150. 389.
    Meinhardt, H.: Models of Biological Pattern Formation. Academic Press, London (1982)Google Scholar
  151. 395.
    Messiah, A.: Quantum Mechanics, vol. II. North-Holland Publishing, Amsterdam (1970). Translated from the French by J. PotterGoogle Scholar
  152. 397.
    Michaelis, L., Menten, M.L.: The kinetics of the inversion effect. Biochem. Z. 49, 333–369 (1913)Google Scholar
  153. 402.
    Moerner, W.E., Kador, L.: Optical detection and spectroscopy of single molecules in a solid. Phys. Rev. Lett. 62, 2535–2538 (1989)ADSCrossRefGoogle Scholar
  154. 403.
    Monod, J., Wyman, J., Changeaux, J.P.: On the natur of allosteric transitions: A plausible model. J. Mol. Biol. 12, 88–118 (1965)CrossRefGoogle Scholar
  155. 405.
    Montroll, E.W., Shuler, K.E.: Studies in nonequilibrium rate processes: I. The relaxation of a system of harmonic oscillators. J. Chem. Phys. 26, 454–464 (1956)MathSciNetGoogle Scholar
  156. 413.
    Motulsky, H.J., Christopoulos, A.: Fitting Models to Biological Data Using Linear and Nonlinear Regression. A Practical Guide to Curve Fitting. GraphPad Software Inc., San Diego (2003)zbMATHGoogle Scholar
  157. 416.
    Müller, S., Regensburger, G.: Generalized mass action systems: Complex balanding equilibria and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J. Appl. Math. 72, 1926–1947 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  158. 420.
    Neher, E., Sakmann, B.: Single-cheannel currents recorded from membrane of denervated frog muscle fibres. Nature 260, 799–802 (1976)ADSCrossRefGoogle Scholar
  159. 421.
    Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems. Wiley, New York (1977)zbMATHGoogle Scholar
  160. 429.
    Noyes, R.M., Field, R.J., Körös, E.: Oscillations in chemical systems. I. Detailed mechanism in a system showing temporal oscillations. J. Am. Chem. Soc. 94, 1394–1395 (1972)Google Scholar
  161. 432.
    Olbregts, J.: Termolecular reaction of nitrogen monoxide and oxygen. A still unsolved problem. Int. J. Chem. Kinetics 17, 835–848 (1985)CrossRefGoogle Scholar
  162. 433.
    Onuchic, J.N., Luthey-Schulten, Z., Wolynes, P.G.: Theory of protein folding: The energy landscape perspective. Annu. Rev. Phys. Chem. 48, 545–600 (1997)ADSCrossRefGoogle Scholar
  163. 434.
    Orrit, M., Bernard, J.: Single pentacene molecules detected by fluorescence exitation in a p-terphenyl crystal. Phys. Rev. Lett. 65, 2716–2719 (1990)ADSCrossRefGoogle Scholar
  164. 435.
    Oster, G.F., Perelson, A.S.: Chemical reaction dynamics. Part I: Geometrical structure. Arch. Ration. Mech. Anal. 55, 230–274 (1974)MathSciNetGoogle Scholar
  165. 439.
    Paschotta, R.: Field Guide to Laser Puls Generation. SPIE Press, Bellingham (2008)CrossRefGoogle Scholar
  166. 440.
    Patrick, R., Golden, D.M.: Third-order rate constants of atmospheric importance. Int. J. Chem. Kinetics 15, 1189–1227 (1983)CrossRefGoogle Scholar
  167. 449.
    Peterman, E.J.G., Sosa, H., Moerner, W.E.: Single-molecule flourescence spectrocopy and microscopy of biomolecular motors. Annu. Rev. Phys. Chem. 55, 79–96 (2004)ADSCrossRefGoogle Scholar
  168. 451.
    Phillipson, P.E., Schuster, P.: Modeling by Nonlinear Differential Equations. Dissipative and Conservative Processes, World Scientific Series on Nonlinear Science A, vol. 69. World Scientific, Singapore (2009)Google Scholar
  169. 453.
    Plass, W.R., Cooks, R.G.: A model for energy transfer in inelasitc molecular collisions applicable at steady state and non-steady state and for an arbitrary distribution of collision energies. J. Am. Soc. Mass Spectrom. 14, 1348–1359 (2003)CrossRefGoogle Scholar
  170. 461.
    Provencher, S.W., Dovi, V.G.: Direct analysis of continuous relaxation spectra. J. Biophys. Biochem. Methods 1, 313–318 (1979)CrossRefGoogle Scholar
  171. 462.
    Qian, H., Elson, E.L.: Single-molecule enzymology: Stochastic Michaelis-Menten kinetics. Biophys. Chem. 101–102, 565–576 (2002)CrossRefGoogle Scholar
  172. 464.
    Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T.: Stiffness in stochastic chemically reacting systems: The implicit τ-leaping method. J. Chem. Phys. 119, 12,784–12,794 (2003)Google Scholar
  173. 465.
    Rice, O.K., Ramsperger, H.C.: Theories of unimolecular gas reactions at low pressures. J. Am. Chem. Soc. 49, 1617–1629 (1927)CrossRefGoogle Scholar
  174. 466.
    Rigler, R., Mets, U., Widengren, J., Kask, P.: Fluorescence correlation spectroscopy with high count rate and low-background-analysis of translational diffusion. Eur. Biophys. J. 22, 169–175 (1993)CrossRefGoogle Scholar
  175. 468.
    Risken, H.: TheFokker-Planck Equation. Methods of Solution and Applications, 2nd edn. Springer, Berlin (1989)Google Scholar
  176. 470.
    Roebuck, J.R.: The rate of the reaction between arsenious acid and iodine in acid solution, the rate of the reverse reaction, and the equilibrium between them. J. Phys. Chem. 6, 365–398 (1901)CrossRefGoogle Scholar
  177. 471.
    Rotman, B.: Measurement of activity of single molecules of β-d-galactosidase. Proc. Natl. Acad. Sci. USA 47, 1981–1991 (1961)ADSCrossRefGoogle Scholar
  178. 472.
    Sagués, F., Epstein, I.R.: Nonlinear chemical dynamics. J. Chem. Soc. Dalton Trans. 2003, 1201–1217 (2003)CrossRefGoogle Scholar
  179. 473.
    Salis, H., Kaznessis, Y.: Accurate hybrid stochastic simulation of a system of coupled chemicel or biochemical reactions. J. Chem. Phys. 122, e054,103 (2005)CrossRefGoogle Scholar
  180. 474.
    Sanft, K.R., Wu, S., Roh, M., Fu, J., Lim, R.K., Petzold, L.R.: StochKit2: Software for discrete stochastic simulation of biochemical systems with events. Bioinformatics 27, 2457–2458 (2011)CrossRefGoogle Scholar
  181. 476.
    Scatchard, G.: The attractions of proteins for smal molecules and ions. Ann. New York Acad. Sci. 51, 660–672 (1949)ADSCrossRefGoogle Scholar
  182. 487.
    Schwabl, F.: Quantum Mechanics, 4th edn. Springer, Berlin (2007)zbMATHGoogle Scholar
  183. 488.
    Schwarz, G.: Kinetic analysis by chemical relaxation methods. Rev. Mod. Phys. 40, 206–218 (1968)ADSCrossRefGoogle Scholar
  184. 489.
    Seber, G.A., Lee, A.J.: Linear Regression Analysis. Wile Series in Probabiity and Statistics. Wiley-Intersceince, Hoboken (2003)zbMATHCrossRefGoogle Scholar
  185. 490.
    Sehl, M., Alekseyenko, A.V., Lange, K.L.: Accurate stochastic simulation via the step anticipation τ-leaping (SAL) algorithm. J. Comp.,Biol. 16, 1195–1208 (2009)Google Scholar
  186. 494.
    Senn, H.M., Thiel, W.: QM/MM Methods for biological systems. Top. Curr. Chem. 268, 173–290 (2007)CrossRefGoogle Scholar
  187. 495.
    Senn, H.M., Thiel, W.: QM/MM Methods for biomolecular systems. Angew. Chem. Int. Ed. 48, 1198–1229 (2009)CrossRefGoogle Scholar
  188. 496.
    Seydel, R.: Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos, Interdisciplinary Applied Mathematics, vol. 5, 2nd edn. Springer, New York (1994)Google Scholar
  189. 499.
    Shapiro, B.E., Levchenko, A., World, E.M.M.B.J., Mjolsness, E.D.: Cellerator: Extending a computer algebra system to include biochemical arrows for signal transduction simulations. Bioinformatics 19, 677–678 (2003)CrossRefGoogle Scholar
  190. 501.
    Shuler, K.E.: Studies in nonequilibrium rate processes: II. The relaxation of vibrational nonequilibrium distributions in chemical reactions and shock waves. J. Phys. Chem. 61, 849–856 (1957)Google Scholar
  191. 504.
    Stauffer, P.H.: Flux flummoxed: A proposal for consistent usage. Ground Water 44, 125–128 (2006)CrossRefGoogle Scholar
  192. 511.
    Strang, G.: Linear Algebra and its Applications, 3rd edn. Brooks Cole Publishing Co, Salt Lake City (1988)zbMATHGoogle Scholar
  193. 524.
    Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics, Philadelphia (2005)zbMATHCrossRefGoogle Scholar
  194. 526.
    Taylor, H.M., Karlin, S.: An Introduction to Stochastic Modeling, 3rd edn. Academic press, San Diego (1998)zbMATHGoogle Scholar
  195. 529.
    Thomas, G.B., Finney, R.L.: Calculus and Analytic Geometry, 9th edn. Addison-Wesley, Reading (1996)zbMATHGoogle Scholar
  196. 531.
    Tolman, R.C.: The Principle of Statistical Mechanics. Oxford University Press, Oxford (1938)zbMATHGoogle Scholar
  197. 532.
    Tsukahara, H., Ishida, T., Mayumi, M.: Gas-phase oxidation of nitric oxide: Chemical kinetics and rate constant. Nitric Oxide Biol. Chem. 3, 191–198 (1999)CrossRefGoogle Scholar
  198. 533.
    Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237 (641), 37–72 (1952)ADSMathSciNetCrossRefGoogle Scholar
  199. 538.
    van den Bos, A.: Parameter Estimation for Scientists and Engineers. Wiley, Hoboken (2007)zbMATHGoogle Scholar
  200. 540.
    van Kampen, N.G.: A power series expansion of the master equation. Can. J. Phys. 39, 551–567 (1961)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  201. 541.
    van Kampen, N.G.: The expansion of the master equation. Adv. Chem. Phys. 34, 245–309 (1976)Google Scholar
  202. 543.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 3rd edn. Elsevier, Amsterdam (2007)zbMATHGoogle Scholar
  203. 544.
    van Oijen, A.M., Blainey, P.C., Crampton, D.J., Richardson, C.C., Ellenberger, T., Xie, X.S.: Single-moleucles kinetics of λ exconuclease reveal base dependence and dynamic disorder. Science 301, 1235–1238 (2003)ADSCrossRefGoogle Scholar
  204. 545.
    van Slyke, D.D., Cullen, G.E.: The mode of action of urease and of enzymes in general. J. Biol. Chem. 19, 141–180 (1914)Google Scholar
  205. 560.
    Waage, P., Guldberg, C.M.: Studies concerning affinity. J. Chem. Educ. 63, 1044–1047 (1986). English translation by Henry I. AbrashGoogle Scholar
  206. 561.
    Walter, N.G.: Single molecule detection, analysis, and manipulation. In: Meyers, R.A. (ed.) Encyclopedia of Analytical Chemistry, pp. 1–10. Wiley, Hoboken (2008)Google Scholar
  207. 569.
    Widengren, J., Mets, Ülo., Rigler, R.: Photodynamic properties of green fluorescent proteins investigated by fluoresecence correlation spectroscopy. Chem. Phys. 250, 171–186 (1999)Google Scholar
  208. 570.
    Wilheim, T.: The smallest chemical rwaction system with bistability. BMC Syst. Biol. 3, e90 (2009)CrossRefGoogle Scholar
  209. 571.
    Wilheim, T., Heinrich, R.: Smallest chemical rwaction system with Hopf bifurcation. J. Math. Chem. 17, 1–14 (1995)MathSciNetCrossRefGoogle Scholar
  210. 572.
    Wilkinson, D.J.: Stochastic modeling for quatitative description of heterogeneous biological systems. Nat. Rev. Genet. 10, 122–133 (2009)CrossRefGoogle Scholar
  211. 573.
    Wilkinson, D.J.: Stochastic Modelling for Systems Biology, 2nd edn. Chapman & Hall/CRC Press – Taylor and Francis Group, Boca Raton (2012)zbMATHGoogle Scholar
  212. 576.
    Winzor, D.J., Jackson, C.M.: Interpretation of the temperature dependence of equilibrium and rate contants. J. Mol. Recognit. 19, 389–407 (2006)CrossRefGoogle Scholar
  213. 577.
    Wolberg, J.: Data Analysis Using the Method of Least Squares. Extracting the Most Information from Experiments. Springer, Berlin (2006)Google Scholar
  214. 581.
    Yang, Y., Rathinam, M.: Tau leaping of stiff stochastical chemical systems via local central limit approximation. J. Comp. Phys. 242, 581–606 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  215. 582.
    Yashonath, S.: Relaxation time of chemical reactions from network thermodynamics. J. Phys. Chem. 85, 1808–1810 (1981)CrossRefGoogle Scholar
  216. 584.
    Zhang, W.K., Zhang, X.: Single molecule mechanochemistry of macromolecules. Prog. Polym. Sci. 28, 1271–1295 (2003)CrossRefGoogle Scholar
  217. 585.
    Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, San Diego (1998)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Peter Schuster
    • 1
  1. 1.Institut für Theoretische ChemieUniversität WienWienAustria

Personalised recommendations