Journal of Mathematical Biology

, Volume 4, Issue 1, pp 1–20 | Cite as

Effect of the rate of oxygen consumption on muscle respiration

  • B. A. Taylor
  • J. D. Murray
Article

Summary

An important role of myoglobin in red muscle is to facilitate the diffusion of oxygen for metabolism. We consider a model for muscle respiration in which the oxygen consumption is of a MichaelisMenten form. The resulting mathematical model is solved in two different ways with two different boundary conditions. The first uses the singular perturbation method of Murray (1974), while the second, which gives another justication of the simpler procedure, is a direct numerical computation of the full problem.

The oxygen tension and saturation are often small. For realistic values of the Michaelis-Menten constant the oxygen tension, the saturation and the radius of the region in which the oxygen tension is negligibly small can be calculated using the constant consumption model of Murray (1974), with corrected boundary conditions (those for a Stefan problem), which in certain circumstances markedly affect the solution.

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References

  1. [1]
    Bârzu, O., Satre, M.: Determination of oxygen affinity of respiratory systems using oxyhaemoglobin as oxygen donor. Anal. Biochem. 36, 428–433 (1970).Google Scholar
  2. [2]
    Chance, B.: Reaction of oxygen with the respiration chain in cells and tissues. J. Gen. Physiol 49 (no 1, part 2), 163–188 (1965).Google Scholar
  3. [3]
    Chance, B., Schoener, B., Schindler, F.: Intracellular oxidation reduction states. Oxygen in the animal organism I. U. B. symposium series (Dickens, F., Neil, E., eds.), p. 367–393. 1966.Google Scholar
  4. [4]
    Crank, J.: The Mathematics of Diffusion, p. 99–120. Oxford: University Press. 1971.Google Scholar
  5. [5]
    Crank, J., Gupta, R.: A moving boundery value problem arising from the diffusion of oxygen in an absorbing tissue. J. Inst. Maths. Applic. 10, 19–33 (1972).Google Scholar
  6. [6]
    Degn, H., Wahlrab, H.: Measurement of steady-state value of respiratory rate and oxidation levels of respiratory pigments at low oxygen tensions. A new technique. Biochim. Biophys. Acta 245, 347–355 (1971).Google Scholar
  7. [7]
    Jacquez, J. A., Kutchai, H., Daniels, E.: Haemoglobin-facilitated diffusion of oxygen: interfacial and thickness effects. Respir. Physiol. 15, 166–181 (1972).Google Scholar
  8. [8]
    Keller, H. B.: Elliptic boundary-value problems suggested by non-linear diffusion processes. Arch. Rational Mech. Anal. 350, 363–381 (1969).Google Scholar
  9. [9]
    Keller, H. B.: Numerical methods for the solutions of two-point boundary value problems, p. 162–172. Waltham, Mass.: Blaisdell. 1968.Google Scholar
  10. [10]
    Kreuzer, F., Hoofd, L. J. C.: Facilitated diffusion of oxygen in the presence of haemoglobin. Respir. Physiol. 8, 280–302 (1970).Google Scholar
  11. [11]
    Kutchai, H., Jacquez, J. A., Mather, F. J.: Non-equilibrium facilitated oxygen transport in haemoglobin solution. Biophys. J. 10, 38–54 (1970).Google Scholar
  12. [12]
    Mitchell, P. J., Murray, J. D.: Facilitated diffusion: The problem of boundary conditions. Biophysik 9, 177–190 (1973).Google Scholar
  13. [13]
    Murray, J. D.: A simple method for obtaining approximate solutions for a class of diffusionkinetics enzyme problems. 1. General class and illustrative examples. Math. Biosc. 2, 370–411 (1968).Google Scholar
  14. [14]
    Murray, J. D.: A simple method for obtaining approximate solutions for a class of diffusion- kinetics enzyme problems. 2. Further examples and non-symmetric problems Math. Biosc. 3 115–133 (1969).Google Scholar
  15. [15]
    Murray, J. D.: On the molecular mechanism of facilitated diffusion of oxygen by myoglobin and haemoglobin. Proc. Roy. Lond. B173, 95–110 (1971).Google Scholar
  16. [16]
    Murray, J. D., Wyman, J.: Facilitated diffusion: The case of carbon monoxide. J. Biol. Chem. 246, 5903–5906 (1971).Google Scholar
  17. [17]
    Murray, J. D.: On the role of myoglobin in muscle respiration. J. Theor. Biol. 47, 115–126 (1974).Google Scholar
  18. [18]
    Oshino, R., Oshino, N., Tamura, M., Kobilinsky, L., Chance, B.: A sensitive bacterial luminescence problem for O2 in biochemical systems. Biochim. Biophys. Acta B273, 5–17 (1972).Google Scholar
  19. [19]
    Riveros-Moreno, V., Wittenberg, J. B.: The self diffusion coefficients of myoglobin and haemoglobin in concentrated solution. J. Biol. Chem. 247, 895–901 (1972).Google Scholar
  20. [20]
    Scholander, P. F.: Oxygen transport through haemoglobin solutions. Science 131, 585–590 (1960).Google Scholar
  21. [21]
    Smith, K. A., Meldon, J. M., Cotton, C. K.: An analysis of carrier facilitated transport. A. I. Ch. E Journal 19, 102–111 (1973).Google Scholar
  22. [22]
    Taylor, B. A.: D. Phil. Thesis, Oxford University, 1976.Google Scholar
  23. [23]
    Wittenberg, B. A., Wittenberg, J. B., Caldwell, P. R. B.: Role of myoglobin in the oxygen supply to red skeletal muscle. J. Biol. Chem. 250, 9038–9043 (1975).Google Scholar
  24. [24]
    Wittenberg, J. B.: Oxygen transport: a new function proposed for myoglobin. Biol. Bull. 117, 402 (1959).Google Scholar
  25. [25]
    Wittenberg, J. B.: The molecular mechanism of haemoglobin facilitated oxygen diffusion. J. Biol. Chem. 241, 104–114 (1966).Google Scholar
  26. [26]
    Wittenberg, J. B.: Myoglobin facilitated oxygen diffusion: role of myoglobin in oxygen entry into muscle. Physiol. Rev. 50, 559–638 (1970).Google Scholar
  27. [27]
    Wyman, J.: Facilitated diffusion and the possible role of myoglobin as a transport mechanism. J. Biol. Chem. 241, 115–121 (1966).Google Scholar
  28. [28]
    Wyman, J.: Private Communication, 1972.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • B. A. Taylor
    • 1
  • J. D. Murray
    • 2
  1. 1.Oxford
  2. 2.Mathematical InstituteOxfordGreat Britain

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