Abstract
It is shown on the basis of (1) conservation of mass, (2) positive concentrations, and (3) the principle of detail balancing that periodic reactions cannot occur in a closed system described bylinear differential equations. The matrix,A, of the rate equations must be such that |A|=0,a ij>0 fori≠j,a ii<0, andVAV −1=B, whereV is diagonal andB is symmetric. These properties ofA imply that the latent roots are real and non-positive and that neither catalysis nor inhibition can be described bylinear equations. It is further shown that periodic reactions cannot occur in anopen system for which the matrix associated with the chemical reactions has the above properties and in which thesimple law of diffusion is obeyed. The relation of these results to Onsager's reciprocal relations and to previous work on periodic and cyclic chemical reactions is discussed. The utility of certain of these results for the treatment of isotope kinetics is indicated.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Literature
Aitkens, A. C. 1948.Determinants and Matrices. 5th Edition. New York: Interscience Publishers.
Bonhoeffer, K. F. 1948. “Über Periodischen Chemische Reaktion. 1.”Z. Electrochem.,52, 24–29.
Brauer, A. 1946. “Limits for the Characteristic Roots of a Matrix.”Duke Math. Jour.,13, 387–95.
Bray, W. C. 1921. “Periodic Reactions in Homogeneous Solution and Its Relation to Catalysis.”Jour. Am. Chem. Soc.,43, 1262–67.
Ferrar, W. L. 1941.Algebra. 1st Edition. London: Oxford University Press.
— 1951.Finite Matrices. London: Oxford University Press.
Hearon, J. Z. 1950a. “The Steady State Kinetics of Some Biological Systems: III.”Bull. Math. Biophysics,12, 57–83.
— 1950b. “Some Cellular Diffusion Problems Based on Onsager's Generalization of Fick's Law.”,12, 135–59.
Hearon, J. Z. and W. C. Sangren. 1953. Forthcoming paper.
Hirniak, J. 1911. “Zur Frage der Periodischen Reactionen.”Z. Physik. Chem.,75, 675–80.
Householder, A. S. 1942. “The Tri-Axial Cell.”Bull. Math. Biophysics,4, 159–68.
Lewis, G. N. 1925. “A New Principle of Equilibrium.”Proc. Soc. Nat. Acad. Sci.,11, 179–83.
Lotka, A. J. 1910. “Contributions to the Theory of Periodic Reactions.”Jour. Phys. Chem.,14, 271–74.
— 1920. “Undamped Oscillations Derived from the Law of Mass Action.”Jour. Am. Chem. Soc.,42, 1595–99.
Mayer, J. E. and M. G. Mayer. 1947.Statistical Mechanics. New York: John Wiley & Son.
Moore, M. J. 1949. “Kinetics of Open Systems.”Trans. Faraday Soc.,45, 1–12.
Morales, M. F. 1944. “On a Possible Mechanism for Biological Periodicity”Bull. Math. Biophysics,6, 65–70.
Onsager, L. 1931. “Reciprocal Relations in Irreversible Processes. I.”Phys. Rev.,37, 405–26.
Opatowski, I. 1945. “Chain Processes and Their Biophysical Applications: I. General Theory.”Bull. Math. Biophysics,7, 161–80.
— 1946. “Chain Processes and Their Biophysical Applications: II. Effect of Recovery.”,8, 7–15.
Rashevsky, N. 1938.Mathematical Biophysics. Chicago: University of Chicago Press.
— 1940.Advances and Applications of Mathematical Biology. Chicago: University of Chicago Press.
— 1948a. “On Periodicities in Metabolizing Systems.”Bull. Math. Biophysics,10, 159–74.
— 1948b. “A Note On Biological Periodicities.”,10, 201–04.
— 1948c.Mathematical Biophysics. Rev. Ed., Chicago: University of Chicago Press.
— 1949. “Time Variable Osmotic Pressures Produced by Coupled Reactions as a Possible Cause of Cell Division.”Bull. Math. Biophysics,11, 1–7.
Wegscheider, R. 1901. “Über Simultane Gleichgewichte und die Beziehungen swischen Thermodynamik und Reaktionskinetik.”Z. Physik. Chem.,39, 257–303.
Weinberg, A. M. 1938. “A Case of Biological Periodicity.”Growth,2, 81–92.
— 1939. “A Note On Biological Periodicities.”Bull. Math. Biophysics,1, 19–22.
Author information
Authors and Affiliations
Additional information
A portion of this work was performed while the author was in the Department of Physiology, University of Chicago, and was supported by a grant from the Dr. Wallace C. and Clara A. Abbott Memorial Fund.
Rights and permissions
About this article
Cite this article
Hearon, J.Z. The kinetics of linear systems with special reference to periodic reactions. Bulletin of Mathematical Biophysics 15, 121–141 (1953). https://doi.org/10.1007/BF02476377
Issue Date:
DOI: https://doi.org/10.1007/BF02476377