Propagation of Fluctuations in Biochemical Systems, I: Linear SSC Networks

Abstract

We investigate the propagation of random fluctuations through biochemical networks in which the number of molecules of each species is large enough so that the concentrations are well modeled by differential equations. We study the effect of network topology on the emergent properties of the reaction system by characterizing the behavior of variance as fluctuations propagate down chains and studying the effect of side chains and feedback loops. We also investigate the asymptotic behavior of the system as one reaction becomes fast relative to the others.

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References

  1. Anderson, D.F., 2005. Stochastic perturbations of biochemical reaction systems, Duke University Thesis.

  2. Anderson, D.F., Mattingly, J.C., 2007. Propagation of fluctuations in biochemical systems II: nonlinear chains, submitted.

  3. Ball, K., Kurtz, T., Popovic, L., Rempala, G., 2006. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab. 16(4), 1925–1961.

    Article  MATH  MathSciNet  Google Scholar 

  4. Crabtree, B., Newsholme, E.A., 1985. A quantitative approach to metabolic control. In: Current Topics in Cellular Regulation, pp. 21–76. Academic, San Diego.

    Google Scholar 

  5. Delbruck, M., 1940. Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124.

    Article  Google Scholar 

  6. Feinberg, M., 1979. Lectures on chemical reaction networks; delivered at the Mathematics Research Center, Univ. Wisconsin–Madison.

  7. Feinberg, M., 1987. Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems, Review article 25. Chem. Eng. Sci. 42, 2229–2268.

    Article  Google Scholar 

  8. Gadgil, C., Othmer, H., Lee, C.H., 2005. A stochastic analysis of chemical first-order reaction networks. Bull. Math. Biol. 67, 901–946.

    Article  MathSciNet  Google Scholar 

  9. Gans, P.J. (1960). Open first-order stochastic processes. J. Chem. Phys. 33(3), 691.

    Article  MathSciNet  Google Scholar 

  10. Gillespie, D.T., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434.

    Article  MathSciNet  Google Scholar 

  11. Heinrich, R., Rapoport, T.A., 1974. A linear steady-state treatment of enzymatic chains General properties control and effector strength. Eur. J. Biochem. 42, 89–95.

    Article  Google Scholar 

  12. Horn, F.J.M., Jackson, R., 1972. General mass action kinetics. Arch. Rat. Mech. Anal. 47, 81–116.

    Article  MathSciNet  Google Scholar 

  13. Kacser, H., Burns, J.A., 1973. The control of flux. Symp. Soc. Exp. Biol., 27, 65–104.

    Google Scholar 

  14. Kurtz, T., 1972. The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57(7), 2976–2978.

    Article  Google Scholar 

  15. Nijhout, F., Reed, M., Budu, P., Ulrich, C., 2004. A mathematical model of the folate cycle—new insights into folate homeostasis. J. Biol. Chem. 279(53), 55008–55016.

    Article  Google Scholar 

  16. Nijhout, F., Reed, M., James, J., Anderson, D., Mattingly, J.C., Ulrich, C., 2006a. Long-range allosteric interactions between the folate and methionine cycles stabilize DNA methylation reaction rate. Epigenetics 1, 81–87.

    Article  Google Scholar 

  17. Nijhout, H.F., Reed, M.C., Shane, B., Gregory, J.F., Ulrich, C.M., 2006b. In silico experimentation with a model of hepatic mitochondrial folate metabolism. Theor. Biol. Med. Model. 3, 40–56.

    Article  Google Scholar 

  18. Oksendal, B., 2003. Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin.

    Google Scholar 

  19. Reed, M., Nijhout, F., Sparks, R., Ulrich, C., 2004. A mathematical model of the methionine cycle. J. Theor. Biol. 226, 33–43.

    Article  MathSciNet  Google Scholar 

  20. Reed, M.C., Nijhout, H.F., Neuhouser, M.L., Gregory, J.F., III, Shane, B., James, S.J., Boynton, A., Ulrich, C.M., 2006. A mathematical model gives insights into nutritional and genetic aspects of folate-mediated one-carbon metabolism. J. Nutr. 136, 2653–2661.

    Google Scholar 

  21. Westerhoff, H.V., Chen, Y.-D., 1984. How do enzyme activities control metabolite concentrations? An additional theorem in the theory of metabolic control. Eur. J. Biochem. 142, 425–430.

    Article  Google Scholar 

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Correspondence to Michael C. Reed.

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Anderson, D.F., Mattingly, J.C., Nijhout, H.F. et al. Propagation of Fluctuations in Biochemical Systems, I: Linear SSC Networks. Bull. Math. Biol. 69, 1791–1813 (2007). https://doi.org/10.1007/s11538-007-9192-2

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Keywords

  • Biochemical systems
  • Fluctuations
  • Stochastic differential equations