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Equilibria and Steady States

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Mathematics as a Laboratory Tool

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Abstract

Mathematical models of dynamical systems often take the general form

$$\displaystyle\begin{array}{rcl} \frac{dx_{1}} {dt} & =& f_{1}(x_{1},x_{2},\mathop{\ldots },x_{n}), \\ \frac{dx_{2}} {dt} & =& f_{2}(x_{1},x_{2},\mathop{\ldots },x_{n}), \\ & \cdots & \\ \frac{dx_{n}} {dt} & =& f_{n}(x_{1},x_{2},\mathop{\ldots },x_{n}),{}\end{array}$$
(3.1)

where the x i are the variables and the f i describe the interactions between variables.

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Notes

  1. 1.

    This conclusion is a consequence of the second law of thermodynamics, discussed in Chapter 15.

  2. 2.

    Archibald Vivian Hill (1886–1977), English physiologist, one of the founders of biophysics and operations research.

  3. 3.

    Of course, if we waited a sufficiently long time, we would eventually arrive at an equilibrium state.

  4. 4.

    Leonor Michaelis (1875–1949), German physician and biochemist. Maud Leonora Menten (1879–1960), Canadian physician and biochemist.

  5. 5.

    George Edward Briggs (1893–1985), British botanist. John Burdon Sanderson Haldane (1892–1964), British-born Indian biologist and mathematician.

  6. 6.

    Britton Chance (1913–2010), American biochemist.

  7. 7.

    Bernoulli’s equations for fluid flow predict that the rate of fluid flow out of the cup will be, to a first approximation, proportional to \(\sqrt{2gh}\), where g is the acceleration due to gravity. However, we hypothesize that because of irregularities in the edge of the hole and the nonlaminar nature of water flow through the hole and from the faucet, the flow rate is more likely to be proportional to h [676]. The interested reader should perform this experiment and see whether our hypothesis is correct.

  8. 8.

    Named for Lee Segel and Marshall Slemrod (b. 1944), American applied mathematician.

  9. 9.

    Donald Dexter Van Slyke (1883–1971) and Glenn Ernest Cullen (1890–1940), American biochemists.

  10. 10.

    Oliver Heaviside (1850–1925), English electrical engineer, mathematician, and physicist.

  11. 11.

    Hans Lineweaver (1907–2009), American physical chemist, and Dean Burk (1904–1988), American biochemist.

  12. 12.

    Michael C. Mackey (b. 1942), Canadian–American biomathematician and physiologist; Leon Glass (b. 1943), American mathematical biologist.

  13. 13.

    André Longtin, Canadian neurophysicist (b. 1961); John Milton, (b. 1950) American–Canadian physician–scientist and mathematical biologist.

References

  1. W. E. Boyce and R. C. DiPrima. Elementary differential equations and boundary value problems, 9th ed. John Wiley & Sons, New York, 2005.

    Google Scholar 

  2. F. Brauer and C. Castillo-Chavez. Mathematical models in population biology and epidemiology. Springer, New York, 2001.

    Book  MATH  Google Scholar 

  3. B. Chance. The kinetics of the enzyme–substrate compound of peroxidase. J. Biol. Chem., 151:553–755, 1943.

    Google Scholar 

  4. L. Edelstein-Keshet. Mathematical models in biology. Random House, New York, 1988.

    MATH  Google Scholar 

  5. S. P. Ellner and J. Guckenheimer. Dynamic models in biology. Princeton University Press, Princeton, New Jersey, 2006.

    MATH  Google Scholar 

  6. E. H. Flach and S. Schnell. Use and abuse of the quasi-steady-state approximation. Systems Biology, 153:187–191, 2006.

    Article  Google Scholar 

  7. A. A. Frost and R. G. Pearson. Kinetics and mechanism, 2nd ed. J. Wiley & Sons, New York, 1961.

    Google Scholar 

  8. T. S. Gardner, C. R. Cantor, and J. J. Collins. Construction of a genetic toggle switch in Escherichia coli. Nature, 403:339–342, 2000.

    Google Scholar 

  9. H. L. Grubin and T. R. Govindan. Quantum contributions and violations of the classical law of mass action. VLSI Design, 6:61–64, 1998.

    Article  Google Scholar 

  10. J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  11. J. R. S. Haldane. Enzymes. Longmans, Green and Company, London, 1930.

    Google Scholar 

  12. A. V. Harcourt and W. Esson. On the laws of connexion between the conditions of a chemical change and its amount. Phil. Trans. Roy. Soc. (London), 156:193–221, 1866.

    Google Scholar 

  13. A. V. Hill. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J. Physiol., 40 (Suppl):iv–vii, 1910.

    Google Scholar 

  14. F. Horn and R. Jackson. General mass action kinetics. Springer, New York, 1972.

    Google Scholar 

  15. T. Insperger, J. Milton, and G. Stepan. Acceleration feedback improves balancing against reflex delay. J. Royal Society Interface, 10:20120763, 2013.

    Article  Google Scholar 

  16. A. Longtin, A. Bulsara, and F. Moss. Time interval sequences in bistable systems and the noise induced transmission of information by sensory neurons. Phys. Rev. Lett., 67:656–659, 1991.

    Article  Google Scholar 

  17. A. Longtin and J. G. Milton. Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback. Math. Biosci., 90:183–199, 1988.

    Article  MathSciNet  Google Scholar 

  18. E. W. Lund. Guldberg and Waage and the law of mass action. J. Chem, Ed., 42:548–550, 1965.

    Google Scholar 

  19. M. C. Mackey and U. an der Heiden. The dynamics of recurrent inhibition. J. Math. Biol., 19:211–225, 1984.

    Google Scholar 

  20. M. C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 197:287–289, 1977.

    Article  Google Scholar 

  21. L. Michaelis and M. L. Menten. Die Kinetik der Invertinwirkung. Biochem. Zts., 49:333, 1913.

    Google Scholar 

  22. J. G. Milton, T. Ohira T, J. L. Cabrera, R. M. Fraiser, J. B. Gyorffy, F. K. Ruiz, M. A. Strauss, E. C. Balch, P. J. Marin, and J. L. Alexander. Balancing with vibration: A prelude for “drift and act” balance control. PLoS ONE, 4:e7427, 2009.

    Google Scholar 

  23. J. D. Murray. Mathematical Biology. Springer-Verlag, New York, 1989.

    Book  MATH  Google Scholar 

  24. L. G. Ngo and M. R. Rousweel. A new class of biochemical oscillator models based on competitive binding. Eur. J. Biochem., 245:182–190, 1997.

    Article  Google Scholar 

  25. L. Noethen and S. Walcher. Quasi-steady state in the Michaelis–Menten system. Nonlinear Anal.: Real World Appl., 8:1512–1535, 2007.

    Google Scholar 

  26. A. R. Peacocke and J. N. H. Skerrett. The interaction of aminoacridines with nucleic acids. Trans. Faraday Soc., 52:261–279, 1956.

    Article  Google Scholar 

  27. I. Prigogine and I. Stengers. Order out of chaos: Man’s dialogue with nature. Bantam Books, New York, 1984.

    Google Scholar 

  28. U. Ryde-Pettersson. Oscillation in the photosyntheic Calvin cycle: examination of a mathematical model. Acta Chem. Scand., 46:406–408, 1992.

    Article  Google Scholar 

  29. L. A. Segel. Modeling dynamic phenomena in molecular and cellular biology. Cambridge University Press, New York, 1984.

    MATH  Google Scholar 

  30. L. A. Segel and M. Slemrod. The quasi-steady-state assumption: a case study in perturbation. SIAM Rev., 31:446–477, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Shankar. Basic training in mathematics: A fitness program for science students. Springer, New York, 2006.

    Book  Google Scholar 

  32. R. E. Sheridan and H. A. Lester. Functional stoichiometry at the nicotinic receptor: The photon cross section for phase 1 corresponds to two bis-Q molecules per channel. J. Gen. Phys., 80:499–515, 1982.

    Article  Google Scholar 

  33. S. H. Strogatz. Nonlinear dynamics and chaos. Addison–Wesley, New York, 1994.

    Google Scholar 

  34. J. Wei. Axiomatic treatment of chemical reaction systems. Journal of Chemical Physics, 36:1578–1584, 1962.

    Article  Google Scholar 

  35. J. Wei and C. D. Prater. The structure and analysis of complex reaction schemes. Advances in Catalysis, 13:203–392, 1962.

    Google Scholar 

  36. J. N. Weiss. The Hill equation revisited: uses and misuses. FASEB J., 11:835–841, 1997.

    Google Scholar 

  37. R. Werman. Stoichiometry of GABA–receptor interactions: GABA modulates the glycine receptor. Adv. Exp. Med. Biol., 123:287–301, 1979.

    Article  Google Scholar 

  38. Ya. B. Zedovitch. Higher mathematics for beginners and its applications to physics. MIR Publishers, Moscow, 1973.

    Google Scholar 

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Author information

Authors and Affiliations

  1. W.M. Keck Science Department, The Claremont Colleges, Claremont, CA, USA

    John Milton

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Toru Ohira

Authors
  1. John Milton
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  2. Toru Ohira
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Milton, J., Ohira, T. (2014). Equilibria and Steady States. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_3

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