Summary
Cyclic processes in stochastic models of macromolecular biological systems are considered. The diagram solution of the model equations (master equation) gives rise to special functions of the rate constants, called the circuit (or one-way cycle) fluxes. As Hill has shown, these functions are the fundamental theoretical components of the operational fluxes, i.e., of the rates of reaction, of transport, of energy conversion, etc. Evidence recently has been found by Monte Carlo simulations that the circuit fluxes can be interpreted as the frequencies of circuit completions. Making use of the theory of graphs, we prove that this physical interpretation of the circuit fluxes is generally valid.
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References
Berge, C.: The theory of graphs. London: Methuen & Co. 1962
Berge, C.: Graphs and hypergraphs. Amsterdam-London: North-Holland Publishing Comp. 1973
Chizmadjev, Y. A., Aityan, S. K.: Ion transport across sodium channels in biological membranes. J. Theor. Biol. 64, 429–453 (1977)
Cleland, W. W.: Partition analysis and the concept of net rate constants as tools in enzyme kinetics. Biochemistry 14, 3220–3224 (1975)
Cohen, M., Palti, Y., Adelman, W. J.: Ionic dependence of sodium currents in squid axons analyzed in terms of specific ion “channel” interactions. J. Membrane Biol. 24, 201–223 (1975)
Eisenman, G., Sandblom, J., Neher, E.: Interactions in cation permeation through the gramicidin channel. Biophys. J. 22, 307–340 (1978)
Fishman, S. N., Chodorov, B. I., Volkenstein, M. V.: Molecular mechanisms of membrane ionic permeability changes. Biochim. Biophys. Acta 225, 1–10 (1971)
Harary, F.: Graph theory. Reading-Menlo Park-London-Don Mills: Addison-Wesley Publishing Comp. 1972
Heckmann, K.: Single file diffusion. In: Biomembranes, Vol. 3 (K. Kreuzer, J. F. G. Slegers, eds.), pp. 127–153. New York: Plenum Publishing Corp. 1972
Heckmann, K., Vollmerhaus, W., Kutschera, J., Vollmerhaus, E.: Mathematische Modelle für reaktionskinetische Phänomene. Z. Naturforsch. 24a, 664–673 (1969)
Hill, T. L.: Studies in irreversible thermodynamics IV. Diagrammatic representation of steady state fluxes for unimolecular systems. J. Theor. Biol. 10, 442–459 (1966)
Hill, T. L.: Thermodynamics for chemists and biologists. Reading-Menlo Park-London-Don Mills: Addison-Wesley Publishing Comp. 1968
Hill, T. L.: Free energy transduction in biology. New York-San Francisco-London: Academic Press 1977
Hill, T. L., Chen, Y.: On the theory of ion transport across the nerve membrane IV. Noise from the openclose kinetics of K+ channels. Biophys. J. 12, 948–959 (1972)
Hill, T. L., Chen, Y.: Stochastics of cycle completions (fluxes) in biochemical kinetic diagrams. Proc. Nat. Acad. Sci. USA 72, 1291–1295 (1975)
Hill, T. L., Kedem, O.: Studies in irreversible thermodynamics III. Models for steady state and active transport across membranes. J. Theor. Biol. 10, 399–441 (1966)
Hille, B.: Ionic selectivity, saturation, and block in sodium channels: A four barrier model. J. Gen. Physiol. 66, 535–560 (1975)
Hille, B., Schwarz, W.: Potassium channels as single-file pores. J. Gen. Physiol. 72, 409–442 (1978)
Hladky, S. B., Urban, B. W., Haydon, D. A.: Ion movements in pores formed by gramicidin A. In: Membrane transport processes, Vol. 3 (D. F. Stevens, R. W. Tsien, eds.), pp. 89–103. New York: Raven Press 1979
Keizer, J.: On the solutions and the steady state of a master equation. J. Stat. Phys. 6, 67–72 (1972)
King, E. L., Altman, C.: Schematic method of deriving the rate laws for enzyme-catalyzed reactions. J. Phys. Chem. 60, 1375–1378 (1956)
Kohler, H.-H.: A single-file model for potassium transport in squid giant axon. Biophys. J. 19, 125–140 (1977)
Kohler, H.-H., Heckmann, K.: Unidirectional fluxes in saturated single-file pores of biological and artificial membranes I. Pores containing no more than one vacancy. J. Theor. Biol. 79, 381–401 (1979)
Läuger, P.: Ion transport through pores: A rate-theory analysis. Biochim. Biophys. Acta 311, 423–441 (1973)
Papoulis, A.: Probability, random variables, and stochastic processes. International student edition. Tokyo: McGraw-Hill Kogakusha 1965
Porter, B.: Synthesis of dynamical systems. London: Nelson 1969
Schnakenberg, J.: Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys. 48, 571–585 (1976)
Stein, W. D.: An algorithm for writing down flux equations for carrier kinetics, and its application to cotransport. J. Theor. Biol. 62, 467–478 (1976)
Wilbrandt, W.: Carrier diffusion. In: Biomembranes, Vol. 3 (K. Kreuzer, J. F. G. Slegers, eds.), pp. 79–99. New York: Plenum Publishing Corp. 1972
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Kohler, HH., Vollmerhaus, E. The frequency of cyclic processes in biological multistate systems. J. Math. Biology 9, 275–290 (1980). https://doi.org/10.1007/BF00276029
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DOI: https://doi.org/10.1007/BF00276029