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  • THB JOURNAL OF BIOLOGICAL CHEMISTRY Vol. 241, No. 13, Issue of July 10, pp. 3008-3022, 1966

    Printed in U.S.A.

    A Theoretical Study of Hepatic Glycogen Metabolism* (Received for publication, November 2, 1965)

    WAYNE P. LONDONt

    From the Division of Mathematical Biology of The Department of Medicine of The Harvard Medical School at the Peter Bent Brigham Hospital, Boston, Massachusetts

    SUMMARY

    Hepatic glycogen metabolism is investigated with the aid of a mathematical model under the assumptions of steady state conditions for glucose 6-phosphate, glucose -phos- phate, and uridine diphosphate glucose, and constant con- centrations of adenosine triphosphate, uridine triphosphate, inorganic phosphate, and pyrophosphate. The model, which includes a constant rate of gluconeogenesis, contains 32 in- put parameters, and is based on enzymatic reaction mech- anisms and kinetic data in vitro pertaining to six reactions which interconvert blood glucose and liver glycogen. Cal- culations in the model have been programmed for the IBM- 1620 digital computer.

    The predicted rates of net glucose production and net glycogen synthesis and the steady state concentrations of glucose-6-P and UDP-glucose compare favorably with ex- perimental observations. The glucose-l-P concentration is shown to be near the equilibrium concentration in the phos- phoglucomutase reaction. The predicted glucose threshold for net glycogen synthesis is 90 mg/100 ml, and the predicted glucose threshold for net glucose production is 149 mg/100 ml.

    The results indicate that the hepatic glycogen system is well stabilized in that a 100-fold change in the glucose con- centration, from 5 to 500 mg/100 ml, produces about a 2-fold change in the steady state concentrations of the components and in most enzymatic rates. If the inorganic phosphate concentration is taken as a decreasing function of the glucose concentration, the steady state intermediates are further stabilized but more net glucose production and glycogen synthesis result.

    It is shown that the rate of gluconeogenesis accounts for the separate hepatic thresholds for net glucose production and net glycogen synthesis and for the latter threshold occur- ring within the normal glucose concentration range.

    The results imply that mass action effects are more im- portant than the dependence of glycogen synthetase on glucose-6-P in determining the steady state concentration of UDP-glucose.

    In the model, phosphoglucomutase and UDP-glucose pyro-

    * This work has been supported by United States Public Health Service Training Grants GM-984-02 and -03 through Harvard Medical School.

    tPresent address, Office of Mathematical Research, National Institute of Arthritis and Metabolic Diseases, National Institutes of Health, United States Public Health Service, Bethesda, Mary- land 20014.

    phosphorylase are near equilibrium, and the amount of either enzyme is not critical. The amount of glucokinase, glucose 6-phosphatase, glycogen synthetase, or phosphorylase is critical even though these enzymes may not be "rate- limiting" on the basis of maximal velocity data.

    The necessity of considering UDP-glucose pyrophospho- rylase as a reversible reaction is established on the basis of maximal velocity data and from the point of view of glyco. gen synthetase being an effective control point in the system.

    It is shown that the dependence of glycogen synthetase on glucose-6-P severely limits enzymatic activity but that it allows the enzymatic rate to vary more extensively with changes in the glucose-6-P and UDP-glucose concentrations.

    The results suggest that the assumption that the reactions occur in a homogeneous phase without compartments is not a good one, and the possibility of two pools of glucose-6-P and glucose-l-P is suggested.

    A simulation of fasting or diabetes produces results con- sistent with experimental observation. A simulation of the early effects of glucocorticoid administration raises the ques- tion of increased glucose production under conditions of a nonincreasing glucose-6-P concentration.

    The liver plays an important role in the regulation of the glu- cose concentration in the blood; it stores glucose as glycogen and it produces glucose anew by gluconeogenesis (1, 2).

    The present study considers these hepatic functions in the context of a mathematical model of hepatic glycogen metabolism based on the kinetics of the reactions which interconvert blood glucose and liver glycogen. The study illustrates the value of using kinetic data in vitro in predicting physiological charac- teristics of the system in vivo. Several factors which influence the system are examined, the assumption that the reactions occur in a homogeneous phase without compartments is evaluated, and certain abnormal metabolic states are successfully simulated.

    There have recently appeared related studies and models of the glycolytic pathway which are based on kinetic data (3, 4).

    METHOD

    Assumptions and Description of Model-The reactions consid- ered in the model are depicted in Fig. 1.1 Since the liver cellmem-

    1 In figures, tables, and equations the following nonstandard abbreviations appear: G, glucose, P, inorganic phosphate, PP,

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  • 3009Issue of July 10, 1966

    brane is freely permeable to glucose (5, 6), this membrane is neglected in the model and the blood and liver glucose concen- trations are assumed equal.

    A constant production of glucose-6-P by gluconeogenesis is signified in Fig. 1 as GNEO. Although GNEO is used to simu- late gluconeogenesis, the term reflects the net effect of the gly- colytic and oxidative pathways which utilize glucose-6-P and the gluconeogenic pathway which produces glucose-6-P. The con- stant rate2 of GNEO is taken as 1 X 10- 4 M per min, which cor- responds approximately to the value obtained for the total incor- poration of labeled pyruvate into glucose and glycogen in liver slices (7).

    The branching and debranching of glycogen are ignored in the model. This assumption is compatible with the observations of Stetten and Stetten (8) that the ratio of unbranched to branched residues is at least 12 and that active glycogen turnover occurs at the periphery of the molecule. Other reactions involving UDP- glucose in transformations to UDP-galactose and UDP-glucu- ronic acid are excluded for simplicity and because they are not directly concerned with glycogen storage.

    The rates of the enzymatic reactions are specified in terms of kinetic data in vitro obtained from the literature, and interpreted in terms of the individual reaction mechanisms using the Briggs- Haldane kinetic theory (9). The enzymatic rate equations were derived directly from the reaction mechanisms by making the steady state assumption for the enzyme substrate complexes (10, 11); some of the derivations were checked by the King-Altman technique (12). The enzymatic mechanisms and rate equations appear in Table I.

    The rate equation for glucokinase is a modified Briggs-Haldane equation where V1 is the maximum velocity, and the kinetic con- stant of one substrate was determined in the presence of a high concentration of the other substrate. The empirical equation is consistent with the data of Salas et al. (13), the poorly understood inhibition by ADP being omitted. In the normal well fed ani- mal, glucokinase accounts for at least 80% of hepatic phospho- rylation (13-16), and the contribution of hexokinase, which would be constant over the physiological glucose concentration, is neglected.

    The mechanism for glucose 6-phosphatase proposed by Hass and Byrne (17) is used; the slight inhibition by physiological concentrations of glucose and inorganic phosphate can be neg- lected (17, 18), and the empirical rate equation is a typical Briggs- laldane equation.

    The mechanism for rabbit muscle phosphoglucomutase of Najjar and Pullman (19) is incorporated. The kinetic constants in the empirical rate equation were calculated from the data of Bodansky (20) under initial conditions for the reaction in the presence of excess cofactor. The equilibrium constant calcu- lated from these data is 12.7.

    The mechanism of yeast UDP-glucose pyrophosphorylase proposed by Munch-Petersen (21) with an equilibrium constant of unity is accepted.3 The kinetic constant of one substrate was

    PYrophosphate, AMP, 5'-adenosine monophosphate, and Glyc, glycogen. For clarity in the equations, brackets denoting con- centrations are omitted.

    The specific gravity of the liver is assumed to be unity, and rates of change of amount per liver weight at 37° are taken as rates of change of concentration in units of molar concentration per min.

    ' Very recent data for liver enzyme are consistent with this mechanism (22).

    FIG. 1. The reactions of hepatic glycogen metabolism. 1, glucokinase; 2, glucose 6-phosphatase; S, phosphoglucomutase; 4, UDP-glucose pyrophosphorylase; 5, glycogen synthetase (de- pendent on glucose-6-P); 6, phosphorylase. The diffusion of glucose across the liver cell membrane is neglected in the model. GNEO represents glucose-6-P production by gluconeogenisis.

    determined with excess of the other substrate and in the absence of the products. Since there are no determinations of the disso- ciation constants of UTP and UDP-glucose in the literature, these constants are assumed large; they do not appear in the rate equation.

    The assumed mechanism of glycogen synthetase is based on observations with the muscle enzyme that glucose-6-P affects the kinetic constant for UDP-glucose but not the maximum velocity cons