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
Bârzu, O., Satre, M.: Determination of oxygen affinity of respiratory systems using oxyhaemoglobin as oxygen donor. Anal. Biochem. 36, 428–433 (1970).
Chance, B.: Reaction of oxygen with the respiration chain in cells and tissues. J. Gen. Physiol 49 (no 1, part 2), 163–188 (1965).
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.
Crank, J.: The Mathematics of Diffusion, p. 99–120. Oxford: University Press. 1971.
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).
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).
Jacquez, J. A., Kutchai, H., Daniels, E.: Haemoglobin-facilitated diffusion of oxygen: interfacial and thickness effects. Respir. Physiol. 15, 166–181 (1972).
Keller, H. B.: Elliptic boundary-value problems suggested by non-linear diffusion processes. Arch. Rational Mech. Anal. 350, 363–381 (1969).
Keller, H. B.: Numerical methods for the solutions of two-point boundary value problems, p. 162–172. Waltham, Mass.: Blaisdell. 1968.
Kreuzer, F., Hoofd, L. J. C.: Facilitated diffusion of oxygen in the presence of haemoglobin. Respir. Physiol. 8, 280–302 (1970).
Kutchai, H., Jacquez, J. A., Mather, F. J.: Non-equilibrium facilitated oxygen transport in haemoglobin solution. Biophys. J. 10, 38–54 (1970).
Mitchell, P. J., Murray, J. D.: Facilitated diffusion: The problem of boundary conditions. Biophysik 9, 177–190 (1973).
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).
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).
Murray, J. D.: On the molecular mechanism of facilitated diffusion of oxygen by myoglobin and haemoglobin. Proc. Roy. Lond. B173, 95–110 (1971).
Murray, J. D., Wyman, J.: Facilitated diffusion: The case of carbon monoxide. J. Biol. Chem. 246, 5903–5906 (1971).
Murray, J. D.: On the role of myoglobin in muscle respiration. J. Theor. Biol. 47, 115–126 (1974).
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).
Riveros-Moreno, V., Wittenberg, J. B.: The self diffusion coefficients of myoglobin and haemoglobin in concentrated solution. J. Biol. Chem. 247, 895–901 (1972).
Scholander, P. F.: Oxygen transport through haemoglobin solutions. Science 131, 585–590 (1960).
Smith, K. A., Meldon, J. M., Cotton, C. K.: An analysis of carrier facilitated transport. A. I. Ch. E Journal 19, 102–111 (1973).
Taylor, B. A.: D. Phil. Thesis, Oxford University, 1976.
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).
Wittenberg, J. B.: Oxygen transport: a new function proposed for myoglobin. Biol. Bull. 117, 402 (1959).
Wittenberg, J. B.: The molecular mechanism of haemoglobin facilitated oxygen diffusion. J. Biol. Chem. 241, 104–114 (1966).
Wittenberg, J. B.: Myoglobin facilitated oxygen diffusion: role of myoglobin in oxygen entry into muscle. Physiol. Rev. 50, 559–638 (1970).
Wyman, J.: Facilitated diffusion and the possible role of myoglobin as a transport mechanism. J. Biol. Chem. 241, 115–121 (1966).
Wyman, J.: Private Communication, 1972.
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B. A. T. would like to thank the Science Research Council of the United Kingdom for their financial support.
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Taylor, B.A., Murray, J.D. Effect of the rate of oxygen consumption on muscle respiration. J. Math. Biology 4, 1–20 (1977). https://doi.org/10.1007/BF00276348
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DOI: https://doi.org/10.1007/BF00276348