THB JOURNAL OF BIOLOGICAL CHEMISTRYVol. 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 Schoolat the Peter Bent Brigham Hospital, Boston, Massachusetts

SUMMARY

Hepatic glycogen metabolism is investigated with the aidof a mathematical model under the assumptions of steadystate 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, whichincludes 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 reactionswhich 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 netglycogen synthesis and the steady state concentrations ofglucose-6-P and UDP-glucose compare favorably with ex-perimental observations. The glucose-l-P concentration isshown to be near the equilibrium concentration in the phos-phoglucomutase reaction. The predicted glucose thresholdfor net glycogen synthesis is 90 mg/100 ml, and the predictedglucose threshold for net glucose production is 149 mg/100ml.

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

It is shown that the rate of gluconeogenesis accounts forthe separate hepatic thresholds for net glucose productionand 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 onglucose-6-P in determining the steady state concentrationof UDP-glucose.

In the model, phosphoglucomutase and UDP-glucose pyro-

* This work has been supported by United States Public HealthService Training Grants GM-984-02 and -03 through HarvardMedical School.

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

phosphorylase are near equilibrium, and the amount ofeither enzyme is not critical. The amount of glucokinase,glucose 6-phosphatase, glycogen synthetase, or phosphorylaseis 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 basisof 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 onglucose-6-P severely limits enzymatic activity but that itallows the enzymatic rate to vary more extensively withchanges in the glucose-6-P and UDP-glucose concentrations.

The results suggest that the assumption that the reactionsoccur in a homogeneous phase without compartments is nota good one, and the possibility of two pools of glucose-6-Pand glucose-l-P is suggested.

A simulation of fasting or diabetes produces results con-sistent with experimental observation. A simulation of theearly effects of glucocorticoid administration raises the ques-tion of increased glucose production under conditions of anonincreasing 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 andit produces glucose anew by gluconeogenesis (1, 2).

The present study considers these hepatic functions in thecontext of a mathematical model of hepatic glycogen metabolismbased on the kinetics of the reactions which interconvert bloodglucose and liver glycogen. The study illustrates the valueof using kinetic data in vitro in predicting physiological charac-teristics of the system in vivo. Several factors which influencethe system are examined, the assumption that the reactions occurin a homogeneous phase without compartments is evaluated, andcertain abnormal metabolic states are successfully simulated.

There have recently appeared related studies and models ofthe 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 nonstandardabbreviations appear: G, glucose, P, inorganic phosphate, PP,

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

A constant production of glucose-6-P by gluconeogenesis issignified 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 thegluconeogenic 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 liverslices (7).

The branching and debranching of glycogen are ignored in themodel. This assumption is compatible with the observations ofStetten and Stetten (8) that the ratio of unbranched to branchedresidues is at least 12 and that active glycogen turnover occurs atthe 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 notdirectly concerned with glycogen storage.

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

The rate equation for glucokinase is a modified Briggs-Haldaneequation where V1 is the maximum velocity, and the kinetic con-stant of one substrate was determined in the presence of a highconcentration of the other substrate. The empirical equation isconsistent with the data of Salas et al. (13), the poorly understoodinhibition 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 wouldbe constant over the physiological glucose concentration, isneglected.

The mechanism for glucose 6-phosphatase proposed by Hassand Byrne (17) is used; the slight inhibition by physiologicalconcentrations 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 ofNajjar and Pullman (19) is incorporated. The kinetic constantsin the empirical rate equation were calculated from the data ofBodansky (20) under initial conditions for the reaction in thepresence of excess cofactor. The equilibrium constant calcu-lated from these data is 12.7.

The mechanism of yeast UDP-glucose pyrophosphorylaseproposed by Munch-Petersen (21) with an equilibrium constantof 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, andrates of change of amount per liver weight at 37° are taken as ratesof change of concentration in units of molar concentration per min.

' Very recent data for liver enzyme are consistent with thismechanism (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 ofglucose 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 absenceof the products. Since there are no determinations of the disso-ciation constants of UTP and UDP-glucose in the literature, theseconstants are assumed large; they do not appear in the rateequation.

The assumed mechanism of glycogen synthetase is based onobservations with the muscle enzyme that glucose-6-P affects thekinetic constant for UDP-glucose but not the maximum velocityconstant (23-25). This implies that glucose-6-P combines withthe enzyme first. Although the liver enzyme is strongly depend-ent on glucose-6-P (26, 27), the possibility of catalysis withoutglucose-6-P is included in order to simulate conditions with vary-ing degrees of glucose-6-P dependence. The mechanism as-sumes excess glycogen and no effect of UDP. (In Table I, K51is the dissociation constant of glucose-6-P; K 52 is the kinetic con-stant of UDP-glucose for the activated enzyme; Ks3 is the kineticconstant of UDP-glucose for the unactivated enzyme.)

Because of the paucity of data on hepatic phosphorylase ac-tivity and because it is not clear how the inorganic phosphateconcentration varies with the glucose concentration, the rate ofphosphorylase in the basic model is maintained nearly constantas a first approximation. In order to simulate a changing phos-phate concentration, a reaction mechanism based on data ob-tained with muscle phosphorylase b (28, 29) is used. The mecha-nism assumes dependence on 5'-AMP, inhibition by glucose-6-P,and excess glycogen. The observation that 5'-AMP does notaffect the maximum velocity (28) implies that this activatorcombines with the enzyme before the substrates. This mecha-nism predicts that the observed kinetic constants for Pi, glucose-1-P, glucose-6-P, and 5'-AMP may vary with changing concentra-tions of substrate, activator, or inhibitor, but that true kineticconstants can be obtained which are independent of the concentra-tion of any substance. These latter constants were calculated fromthe databy techniques outlined inthe Appendix. The equilibriumconstant calculated from these data (at pH 7.5) is 3.1.

Since phosphorylase activity is held nearly constant, the modeldoes not depend on this reaction mechanism and rate equation;only in the study with a changing phosphate concentration (see"Results") is the rate equation of significance. The smallchanges in phosphorylase that do occur result from the slightinhibition by glucose-6-P, since the phosphate and 5'-AMP con-centrations are held constant and the very low glucose-l-P con-centration does not produce a significant back reaction. De-pendence of phosphorylase on 5'-AMP and inhibition byglucose-6-P are devices used here to simulate slight changes in

4 UDPG 5BLOODI LIVER I

G TF GG-6-P= G-I-P+ lyc

2GN EO6

GNEO

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TABLE I

Enzymatic mechanisms and rate equations

Glucokinase

ATP + G + E - G-6-P + ADP + E,

V1-GV = K + G(K'Tp/ATP + 1)

Glucose 6-phosphatase

G-6-P + 20 + E2 - G + P + E2

V2-G-6-P2 = K-P + G-6-P

Phosphoglucomutase

G-6-P + E3 - P E3 + G-1,6-di-P = G-1-P + E3 - P

(V3, KGI.P G-6-P) (2KG-6P-G-1-P)

v (KGepG-1-P) + (K6_.pG-6-1-P) + (KG-6-P) (KG K)

3 G-- V3 K G-6-P V 2 KG-1-P *,-G--

UDP-glucose pyrophosphorylase

UTP + G-1-P + E4 = UDPG + PP + E4

V41 V42((UTP G-1-P) - (PP UDPG/Kq))V4 (K.-.V 4 2 UTP) + (KU4DPG V4iPP/Keq) + V4 2(KUrT + UTP) G-1-P + V41(Kpp + PP).UDPG/Ke(

4 PP*UDPG V(4 1\2 KPP K4UDPG

q UTP G-1-P V2 KTP KG-I-P

Glycogen synthetase

1G-6-P + E5s C51

2 3C5l + UDPG ( ' C52 , C5, + products

4 5Es + UDPG Cs - E5 + products

(G-6-P + (k5sk3) [Ks2'K 52]/K 53) UDPGK52 (G-6-P + K5s) + ([K 5slK52/K53] + G-6-P) UDPG

Phosphorylase

AMP + Es Z C61

2G-6-P + E6 ;zi C62

3 4 5C61 + P .___C 63 ' C64 ' C6 + G-1-P

(V61 KG--.pP) - (V62 KP.G-1-P)KG-I.'PP + K}.G-1-P + Kp-KGi-p (KMrp/AMP ([G-6-P/K6G6-p] + 1) + 1)

6 P V62 KP= G-1-P V61 KG-1-P

A symbol represents a compound in the chemical equations (reaction mechanisms), and that same symbol represents the concen-tration of that compound in the mathematical equations (rate equations). See also Footnote 1 in the text.

enzymatic activity; although there is a precedent for these con- following exceptions. The constant concentrations of UTP andtrol mechanisms in muscle, they have not been reported in liver. PPi were chosen to simplify the calculations. The reported

The 32 input constants used in the model appear in Table II; UTP and PPi concentrations coupled with an observed UDP-the values were obtained directly from the literature with the glucose concentration of 3 X 10- 4 M (44) and a physiological

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TABLE II

Kinetic data and constant concentrations used in model

If a value used in the model differs significantly from those in available experimental reports, the latter appear in brackets; thesedeviations are discussed in the text. Except where noted, the values pertain to liver. The units of the velocity constants and GNEOare moles per liter per min; those for the kinetic constants and the constant concentrations are moles per liter.

Kinetic data

V1KG

KATP

V2

K- 6-P

V31V32

KG -6-P

KG-l-PV41

V42

K TP

KG-I-P

KUDPG

KPPV5KslK 52K53K54

d

V61

V62

K6

DAMP

GNEOConstant concentrations

ATPUTPPPiPiAMP

Value used

2.00 X 10- 3a

2.27 X 10- 2

3.00 X 10- 4

1.60 X 10-22.00 X 10- 31.04 X 10- 2

3.77 X 10-28.65 X 10- 5

2.46 X 10- 5

1.065.90 X 10 -

3.00 X 10-31.00 X 10-36.00 X 10- 5 b

1.54 X 10-43.12 X 10-31.50 X 10- 4

4.50 X 10-4

0

1.25 X 10- 3

3.12 X 10- 2

1.00 X 10-38.00 X 10-4

3.60 X 10-4

2.50 X 10- 4

1.00 X 10- 4

2.00 X 10-31.80 X 10-31.60 X 10- 5

3.00 X 10-32.60 X 10- 4

This value was taken from maximal rates of hepatic phos-phorylation which include both glucokinase and hexokinase ac-tivity. Thus hexokinase has been incorporated in the model asglucokinase.

b A value of 9 X 10- 5 M was recently reported for the liver en-zyme (22).

c The observed kinetic constant for glucose-6-P in the glycogensynthetase reaction of 6 X 10- 4

M was obtained at a UDP-glucose

glucose-l-P concentration of 4 X 10 -6 M predict net UTP produc-tion by UDP-glucose pyrophosphorylase. 4 This suggests thatthe reported PPi concentration, which is thought to be very lowowing to an active pyrophosphatase (45), is too high (or that thereported UTP concentration is too low), since the enzyme cata-lyzes UDP-glucose production as part of the glycogen system.Smaller UTP and PPi concentrations could be incorporated inthe model without altering the results because decreasing bothIJTP and PPi together has almost no effect on the UDP-glucose

4The value is based on a glucose-6-P concentration of 5 X 10- 5M(44). Reliable direct estimations of glucose-1-P are unavailable.

Remarks

Calculated from K3qMuscle enzymeMuscle enzymeCalculated from Keq

Brain enzymeBrain enzymeBrain enzymeMuscle enzymeBrain enzyme

[6 X 10-4] ¢

[5.00 X 10-31Calculated from KqMuscle phosphorylase bMuscle phosphorylase bMuscle phosphorylase bMuscle phosphorylase b

[2.70 X 10-4]

[5.00 X 10-5]

Reference

(13, 14, 30-32)

(33)(33)(34, 35)(17, 35, 36)

(37)

(20)(20)

(37)(38)(38)(39)(40)(39)(41)(26)(26)

(37)

(28)(28)(28)(29)

(7)

(42, 43)(42)(42)(42)(42)

concentration about 10 times its kinetic constant. The rate equa-tion for glycogen synthetase (Table I) predicts that the observedconstant is actually (Ks5.K52)/UDP-glucose (assuming K53 is in-finite and K54 is zero, i.e. no catalysis without glucose-6-P).Hence, K51, the dissociation constant of glucose-6-P, is calculatedto be 6 X 10- 3

M. Smaller values, however, have been reported forthe muscle enzyme (24).

d K54 = k/k3 (see Table I).

pyrophosphorylase reaction (see "Results"). However, thesmaller values would be less convenient, mathematically.

The other deviations from experimental data involve the disso-ciation constant for glucose-6-P in the glycogen synthetase reac-tion (K51) and the maximum velocities of phosphorylase (V61 andV62). In order to have adequate synthetase activity while stillassuming strong dependence on low concentrations of glucose-6-P(44), a small dissociation constant for glucose-6-P was chosen.The mechanism of the reaction (Table I) indicates that precisedetermination of this constant is difficult (Table II). Modelswith properties nearly identical with those of the present model

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and which use a larger dissociation constant were also constructedas part of this study. These models either operate at high, non-physiological concentrations of glucose-6-P and UDP-glucose, orassume more total synthetase activity or partial dependence ofthe reaction on glucose-6-P.

The alteration in the maximum velocities of phosphorylase isnecessary to ensure a physiological balance with glycogen syn-thetase; the reported maximum velocities of the two enzymes(Table II) are similar but the low glucose-6-P concentrationseverely limits synthetase activity. The necessary change in thephosphorylase velocities emphasizes the inadequacy of the phos-phorylase reaction scheme which, for example, does not includethe interconversion of active and inactive forms (46). Theactivity of this enzyme is nearly constant in the model. Al-though altering the maximum velocities is probably the simplestway of limiting the activity, changing any of the kinetic constantsor constant concentrations in the rate equation would be equallyeffective. In the model the phosphorylase equation assumes amore important role in the study with a varying phosphate con-centration (see "Results").

If changes in K51, V61, and V62 are not made, glycogen break-down greatly exceeds glycogen synthesis, and the predicted glu-cose concentration at which the rates are equal exceeds 500 mg/100 ml, a value with no physiological meaning. The steadystate concentrations of the intermediates are also several timesthe observed values.

Mathematical Model-The following system of differentialequations expressing the rates of change of the concentrations ofthe components (Fig. 1) in terms of the enzymatic rate equations(Table I) constitutes the mathematical model of the system de-tailed in the previous paragraphs (the vi appear in Table I):

d(G)/dt = V - vl (1)

d(G-6-P)/dt = - V2 - v3 + GNEO (2)

d(G-1-P)/dt = V3 - v4 + V6 (3)

d(UDPG)/dt = v4 - v5 (4)

d(Glyc)/dt = v5 - v6 (5)

If the steady state assumption 5 is made for glucose-6-P, glu-cose-1-P, and UDP-glucose, this system reduces to

d(G)/dt = -d(Glyc)/dt + GNEO (6)

This deduction from the mathematical model confirms the factthat under steady state conditions, net hepatic glucose produc-tion is the sum of net glycogen breakdown and gluconeogenesis.Inspection of the mathematical model also allows the furtherdeduction that with constant concentrations of ATP, UTP, Pi,PPi, and 5'-AMP, a specified glucose concentration determinesthe net rates of glycogen synthesis and glucose production, thesteady state concentrations of the three intermediates, and therates of the six enzymatic reactions. This algebraic calculationsequence (see "Appendix") which produces the steady state solu-tion to Equations 1 to 5 has been programmed for the IBM-1620digital computer in order to save time in- repeating the process

5 Preliminary studies of a simplified version of the model weremade without the steady state assumption by approximating thesolution to the nonlinear differential equations with a numericalapproximation method on the IBM-1620 digital computer. Thesestudies suggested that the transient state is very brief and un-complicated. Expanded studies of this type are being plannedwith an analogue computer.

under a multitude of varying conditions. To shorten the calcu-lations the glucose-6-P concentration is specified and the glucoseconcentration is calculated along with the steady state concen-trations of glucose-1-P and UDP-glucose. The selection of alarge constant UTP concentration and a small constant PPi con-centration (see above) simplifies the calculations and avoids acubic equation. Further simplification is obtained by assumingthat the net rate of glycogen synthesis is small relative to themaximum velocities in the phosphoglucomutase reaction (10-4versus 10-2 M per min). These simplifications introduce about a12% error in the glucose-1-P concentration which occurs at highglucose concentrations.

RESULTS

The predicted net rates of glucose production and glycogensynthesis and the predicted steady state concentrations of glu-cose-6-P, glucose-l-P, and UDP-glucose are shown in Fig. 2.The predicted blood glucose threshold for net glycogen synthesisis about 90 mg/100 ml, and the predicted threshold for net glu-cose uptake is 149 mg/100 ml. The difference in the thresholdsis accounted for by the constant rate of gluconeogenesis; at aparticular threshold the appropriate derivative in Equation 6 iszero, and the derivatives for glucose and glycogen are not zerosimultaneously unless GNEO equals zero. The model predictsthat the steady state concentrations of glucose-6-P, glucose-1-P,and UDP-glucose increase with the glucose concentration.

The predicted rates of the enzymatic reactions are shown inFig. 3. Under steady state conditions the rate of glucose 6-phos-phatase (v2) is the absolute rate of glucose production, and therates of UDP-glucose pyrophosphorylase (v4) and glycogen syn-thetase (v5) are equal. Since the glucose-6-P concentration inthe model rises as the glucose concentration increases (Fig. 2),the rate of glucose 6-phosphatase also increases with the glucoseconcentration.

The predicted enzymatic rates and steady state concentrationsat glucose concentrations of 5 and 500 mg/100 ml are shown inTable III. Over the 100-fold change in blood glucose concen-tration, the concentrations of the intermediates are stabilizedand change only about 2-fold. Glucokinase activity increasesabout 50-fold because of its large kinetic constant for glucose.Glycogen synthetase activity reflects increases in both the con-centrations of glucose-6-P and UDP-glucose.

If the rate of glucose 6-phosphatase is plotted against glucose-6-P, phosphoglucomutase against glucose-6-P, UDP-glucosepyrophosphorylase against glucose-l-P, glycogen synthetaseagainst glucose-6-P or UDP-glucose, and phosphorylase againstglucose-6-P, linear curves result. Thus, with the exception ofglucokinase, which is a nonlinear function of its substrate (Fig. 3),the model predicts that the other enzymatic rates are linearfunctions of their substrates, activators, or inhibitors over theglucose concentration range of 0 to 500 mg/100 ml.

In Table IV the predicted characteristics of the glycogen sys-tem are compared with observed characteristics. The thresholdsfor glucose production and glycogen synthesis depend on manyfactors, in particular, on the rate of gluconeogenesis. Thelimited rates of net glucose production and net glycogen synthesispredicted by the model are discussed later in terms of the homoge-neous phase assumption. The predicted concentrations of glu-cose-6-P and UDP-glucose compare favorably with the observedconcentrations; reliable estimations of glucose-1-P are not avail-able.

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I 00 200 300 400 500

GLUCOSE mg./ OOml.

FIG. 2. The normal glycogen system. The curve marked glycogen synthesis is the net rate of change of glycogen (d(Glyc)/dt), and thecurve marked glucose uptake is the negative rate of change of the glucose concentration (-d(G)/dt) in molarpermin X 104. The steadystate concentrations of glucose-6-P, glucose-l-P, and UDP-glucose are shown in units of 105, 106, and 104 M, respectively.

Contribution of Individual Reactions-In the model gluco-neogensis enhances both glycogen synthesis and glucose produc-tion. The model predicts that gluconeogensis accounts for ofthe net glucose production at 5 mg/100 ml and of net glycogensynthesis at 500 mg/100 ml. In considering the various sourcesof glucose-6-P in the model, gluconeogenesis provided ½ of theglucose-6-P at a glucose concentration of 5 mg/100 ml, ½ at 112ag/100 ml, and T T at 500 mg/100 ml.

If gluconeogenesis is deleted from the model, the predictedthreshold for both glucose production and glycogen synthesis isabout 130 mg/100 ml, which is above the normal blood glucoseconcentration. If gluconeogenesis is increased 3-fold, the thresh-old for glycogen synthesis falls from 90 to 35 mg/100 ml and therate of synthesis increases about 50%; the threshold for glucose

production rises from 149 to 210 mg/100 ml, and the rate of pro-duction increases about 25%. There is also a 15 to 20% increasein the steady state concentrations of the intermediates. Con-versely, net utilization of glucose-6-P by the oxidative and gly-colytic pathways (simulated by a negative GNEO of 1 X 10- 4 Mper min) results in the opposite changes: the thresholds for gly-cogen synthesis and glucose production become 165 and 100mg/100 ml, respectively, and the rates of these processes and thesteady state concentrations of the intermediates are diminished.Thus the model elaborates the important role played by gluco-neogenesis in determining the separate thresholds for hepaticglycogen synthesis and glucose production.

The model predicts that the glycogen system is very sensitiveto changes in the glucokinase reaction. If the amount of enzyme

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ciw

0-Jxw> r

ELLJN2 E

NzI

GLUCOSE mg./ 100 ml.

FIG. 3. Individual enzymatic velocities. v, glucokinase; v2, glucose 6-phosphatase; V3, phosphoglucomutase; 5(v4), glycogen syn-thetase (or UDP-glucose pyrophosphorylase); and v6, phosphorylase. The difference between the upper two curves is d(G)/dt; and thedifference between the lower two curves, d(Glyc)/dt. Under steady state conditions v4 = v5 and v3 = d(Glyc)/dt.

is decreased by 20% (simulated by a 20% decrease in V) thethresholds for glycogen synthesis and glucose production rise to125 and 205 mg/100 ml, respectively, and glucose productionand glycogen breakdown are enhanced. If the kinetic constantfor glucose is increased by 20% (to 429 mg/100 ml), the twothresholds become 107 and 180 mg/100 ml, respectively. Thusthe system is slightly more sensitive to the amount of enzymethan to the kinetic constant for glucose. Since the selected ATPconcentration is nearly 10 times larger than its kinetic constant,decreasing this concentration by 25 % produces minimal changesat glucose concentrations below 250 mg/100 ml, and increasingATP has almost no effect on the system.

If the amount of glucose 6-phosphatase is increased by 50%(simulated by increasing V2), glucose is produced at rates 1.5 to

2 times normal at all glucose concentrations below about 300mg/100 ml. Even though phosphorylase is the "rate-limiting"enzyme in the pathway of glucose production (on the basis ofthe maximal velocity data in Table II), the model predicts thatan increase in glucose 6-phosphatase activity greatly enhancesglucose production.

The model predicts that the phosphoglucomutase reaction isnever far from equilibrium with the equilibrium ratio of glucose6-P to glucose-l-P of 12.7 occurring at 90 mg/100 ml; at 5 mg/100

ml the predicted ratio is 12.3, and at 500 mg/100 ml, 13.8. Onlyfactors which alter the equilibrium constant (Table I), hencecatalysis, in one direction more than in the other, affect the gly-

cogen system; for example, decreasing by half the amount ofenzyme has negligible effect on the system.

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TABLE III

Changes in concentrations and enzymatic rates

Aa B A Ratio, Column B toColumn A

M 1M/min

Glucose-6-P ......................... 4.30 X 10- l 9.10 X 10-6 2.1Glucose-l-P ....................... 3.60 X 10- 6 6.51 X 10- 6 1.8UDP-glucose ....................... 3.39 X 10- 4 6.08 X 10- 4 1.8

Glucokinase (vl) .................... 1.97 X 10- 5 1.01 X 10- 3 51Glucose 6-phosphatase (v2) 3.37 X 10- 4 6.96 X 10- 4 2.1UDP-glucose pyrophosphorylase (4)

or glycogen synthetase (vs) ....... 4.24 X 10- 4 1.0 X 10-a 2.4Phosphorylase (v6) .................. 6.41 X 10- 4 5.93 X 10- 4 0.92

Column A, glucose concentration, 5 mg/100 ml; Column B, glucose concentration, 500 mg/100 ml. Ratio of B:A, 100.

TABLE IV

Predicted and observed characteristics of hepatic glycogen system

Characteristic Predicted Observed Reference Remarks

Blood glucose threshold for net gly- 90 mg/100 mlcogen synthesis

Blood glucose threshold for net glu- 149 mg/100 ml 115-150 mg/100 ml (5, 47, 48) Studies in vivo in rats and dogscose production

Net rate of glycogen synthesis at 400 3.6 X 10- 4 M/min 3-4 X 10- 4 M/min (49) Fasted, anesthetized ratsmg/100 ml

Net rate of glycogen synthesis at 6.5 X 10- 4 M/min 10 X 10- 4 M/min (50) Liver slices: media rich in K+

1000 mg/100 mlNet rate of glycogen synthesis at 500 4.0 X 10 - 4 M/min 0.5-3 X 10- 4 M/min (51) Liver slices: media rich in K+

mg/100 mlNet glucose production at 80 mg/100 1.3 X 10- 4 M/min 3 X 10- 4 M/min (52) Canine study in vivo (liver assumed

ml to be 2.5% of body weight)Net glucose production at 0 mg/100 3 X 10- 4 M/min 9-18 X 10- 4 M/min (53) Liver slice: Pi concentration 10

ml times the concentration in themodel

Absolute rate of glucose production 4.8 X 10- 4 M/min 5.6 X 10- 4 M/min (54) Canine study in vivo (liver assumed(glucose 6-phosphatase) at 107 to be 2.5% of body weight)mg/100 ml

Steady state concentration of glu- 5.6 X 10- 5 M 5.5 X 10- 5 M (44) Rat livercose-6-P (80 mg/100 ml)

Steady state concentration of glu- 4.4 X 10-6 Mcose-l-P (80 mg/100 ml)

Steady state concentration of UDP- 4.0 X 10-4 M 3.0 X 10- 4 M (44) Rat liverglucose (80 mg/100 ml)

UDP-glucose pyrophosphorylase, like phosphoglucomutase,is never far from equilibrium; the minimum ratio of reactantsand products is 0.7 and the equilibrium constant in the model isunity. Only factors which affect the equilibrium constant(Table I) affect the glycogen system; the system is relativelyinsensitive to changes in the amount of enzyme, and changingthe constant concentrations of UTP and PPi together has noeffect.

Both the glucose-6-P and UDP-glucose concentrations increasewith the glucose concentration (Fig. 2); hence, the UDP-glucoseconcentration is an increasing function of the glucose-6-P con-centration. (A plot of UDP-glucose against glucose-6-P revealsan increasing function at all glucose-6-P concentrations corre-sponding to glucose concentrations of at least 5000 mg/100 ml.)1O the basis of the law of mass action, UDP-glucose should

reflect increases in its precursor, glucose-6-P; however, glycogen

synthetase, which utilizes UDP-glucose, is totally dependent inthis model on glucose-6-P, and an increase in this latter compoundcould produce a fall in UDP-glucose. The model predicts thatthe former mechanism predominates.

Glycogen synthetase responds more to changes in UDP-glucosethan to changes in glucose-6-P. For example, if the equilibriumconstant of phosphoglucomutase is arbitrarily taken to favorglucose-l-P formation, glucose-6-P falls but UDP-glucose rises,and there is a net stimulation of glycogen synthesis.

The model predicts two consequences of the dependence ofglycogen synthetase on glucose-6-P. Since both glucose-6-Pand UDP-glucose increase with the glucose concentration, thesynthetase rate rises more than if the enzyme were independentof glucose-6-P (Table III). Secondly, because of the lowintracellular concentration of glucose-6-P, dependence on glu-cose-6-P severely limits synthetase activity.

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If the synthetase enzyme is taken as partially independent ofglucose-6-P by including catalysis without glucose-6-P (a finiteK 53 and K54 greater than zero in Table I), there results a verysignificant increase in glycogen synthesis. Changes in the cata-lytic ability of the independent species (k5/k 3), which are akin tochanges in the amount of enzyme, are of greater consequencethan changes in the kinetic constant (K53 ).

The rate of phosphorylase in the model is maintained nearlyconstant with a constant phosphate concentration of 3 X 10- 3

M.

If the phosphate concentration is varied from 5 X 10-3 M at aglucose concentration of 5 mg/100 ml to 1 X 10- S M at a glucoseconcentration of 500 mg/100 ml (with the normal value occurringat 149 mg/100 ml), the glycogen system becomes a more respon-sive one (Fig. 4). There is more net glucose production andglycogen synthesis, yet the steady state concentrations of the

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en Metabolism Vol. 241, No. 13

intermediates show less variation than normal. Under theseconditions the rate of phosphorylase varies nearly 3-fold overthe glucose concentration range of 5 to 500 mg/100 ml, and thereaction assumes a more dynamic role in the glycogen system.

Stimulations of Abnormal Conditions in Vivo-The model isused to simulate abnormal conditions in vivo by incorporatingthe various enzymatic changes which occur in these conditions.The results reflect the extent of the changes which are, in somecases, arbitrary.

In the fasting and diabetic animal, glucokinase activity fallsabout 50%, glucose 6-phosphatase activity increases 50%, andgluconeogenesis increases (1). The conditions are simulated byimposing a 50% decrease in the maximum velocity of glucokinasea 50% increase in the maximum velocity for glucose 6-phosphatase, a 3-fold increase in the constant rate of gluconeogenesi

GLUCOSE mg./ 100ml.

FIG. 4. The effects of changing the inorganic phosphate concentration. Only the effect of phosphate on phosphorylase is included inthis study. Fig. 2 (broken line) is presented for comparison. The inorganic phosphate concentration is plotted as M X 103, and theglucose-1-P concentration is not shown; otherwise the format is the same as Fig. 2.

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200 300

GLUCOSE mg./ 100l.

400 500

FIG. 5. Fasting or diabetes. To simulate these abnormalities the maximum velocity of glucokinase is decreased 50%, that for glucose6-phosphatase increased 50%, and GNEO increased 3-fold. The effect of hexokinase, which becomes more significant with decreasedglucokinase activity, has not been included. Fig. 2 (broken lines) is presented for comparison.

(GNEO). The model (Fig. 5) predicts that glycogen is syn-thesized only at glucose concentrations greater than 285 mg/100mil. At a glucose concentration of 500 mg/100 ml the rate ofglycogen synthesis is one-eighth of normal. The liver producesglucose at all glucose concentrations below 1300 mg/100 ml.At a glucose concentration of 100 mg/100 ml the rate of glucoseproduction is 4 times normal. The intermediates are decreasedabout 15% at low glucose concentrations and about 30% at highglucose concentrations. These results compare favorably withexperimental observation, since the blood glucose threshold forhepatic glucose production in diabetic dogs reported by Madisonet al. (2) is about 900 mg/100 ml, and lower concentrations ofglucose-6-p and UDP-glucose are also found in fasting and dia-betes (44, 55).

When glucocorticoids are administered to adrenalectomizedanimals the observations within 4 hours include an increasedblood glucose concentration, increased glycogen synthesis, in-creased gluconeogenesis (measured as increased pyruvate incor-poration into glucose), and an increase in total glycogen synthe-tase activity (7, 44, 56). (Glucose 6-phosphatase is not increasedat 4 hours (7).) In the fasted, adrenalectomized animal gluco-corticoids produce lowered glucose-6-P and UDP-glucose con-centrations; in the fed, adrenalectomized animal these inter-mediates remain unaltered or show slight elevation at 6 hours(44). Glucocorticoid administration is simulated by a 4-foldincrease in gluconeogenesis and a 50% increase in the maximumvelocity of glycogen synthetase.

'If these two changes are superimposed on the normal system

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Hepatic Glycogen Metabolism

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GLUCOSE mg./ 100ml.FIG. 6. Glucocorticoid administration to a fed animal. The maximum velocity for glycogen synthetase is increased 50% and GNEO

is increased 4-fold. Fig. 2 (broken lines) is presented for comparison.

(fed animal) (Fig. 6), the model predicts glycogen synthesis atall glucose concentrations and at rates 2 to 3 times normal. Glu-cose production, however, is essentially unchanged, and there isalmost no change in the concentrations of the intermediates ex-cept at high glucose concentrations. If the two changes aresuperimposed on the fasting condition where gluconeogenesis isalready 3 times normal, the model predicts glycogen synthesis atglucose concentrations above 55 mg/100 ml and at rates 4 to 5times the fasting rate. There is a slight decrease in glucose pro-duction and a 10% fall in the intermediates. If glucocorticoidadministration were simulated by a larger than 4-fold increase ingluconeogenesis (or a smaller increase in glycogen synthetase),glucose production would be enhanced and the concentrationsof the intermediates would increase.

DISCUSSION

The model of hepatic glycogen metabolism used in this studypredicts characteristics which are difficult to determine experi-mentally. At glucose concentrations of 0 to 500 mg/100 mi,values are predicted for absolute and net rates of glucose produc-tion and glycogen synthesis and the steady state concentrationsof glucose-6-P, glucose-1-P, and UDP-glucose. The glucose-l-Pconcentration, which is at the lower limit of experimental detec-tion, is shown to be near the equilibrium concentration in thephosphoglucomutase reaction. The predicted characteristiscompare favorably with observed characteristics (Table IV),and it is therefore concluded that with the present system kinetidata in vitro are useful in predicting many physiological chaTr-teristics of a system of biochemical reactions in vivo.

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Several reasonable hypotheses about the system are put forth.The results suggest that mass action effects are more importantthan the dependence of glycogen synthetase on glucose-6-P indetermining the steady state concentration of UDP-glucose.They also suggest that the glycogen system is well stabilized,with a 100-fold change in the glucose concentration producingabout a 2-fold change in the other components. The Pi concen-tration, which affects phosphorylase activity, further stabilizesthe intermediates and produces more net glucose production andglycogen synthesis. Morgan and Parmeggiani (57) have empha-sized the role of Pi in controlling glycogenolysis in heart muscle.

The model points up the important role of gluconeogenesis indetermining the separate hepatic thresholds for glucose produc-tion and glycogen synthesis. Without gluconeogenesis thethreshold for glycogen synthesis would be above the normal bloodglucose concentration range, and continual glycogen breakdownwould be occurring at normal glucose concentrations. Inordi-nately high and or sustained postprandial glucose concentra-tions would be required to supply the necessary store of glycogen.

The model emphasizes the distinction between enzymes whichare reversible in their physiological setting (phosphoglucomutaseand UDP-glucose pyrophosphorylase) as opposed to physiologi-cally irreversible enzymes (the other four). With a reversibleenzyme only factors which affect the equilibrium constant, hencecatalysis in one direction more than the other, affect the glycogensystem. 6 The system is relatively insensitive to changes in theamount of these enzymes. On the other hand, the system is verysensitive to changes in the amount of enzyme in irreversible reac-tions because net glucose production depends on a balance be-tween glucokinase and glucose 6-phosphatase, and net glycogensynthesis, on a balance between glycogen synthetase and phos-phorylase. Even though phosphorylase is the "rate-limiting"enzyme in the pathway of glucose production on the basis ofmaximum velocity data (Table II), changing the amount ofglucose 6-phosphatase definitely affects the glycogen system.In this connection it is interesting to observe that the hormoneswhich act on the glycogen system and which might affect cataly-sis of a reaction symmetrically act at the four irreversible stepsand not on the two reversible reactions.

Even though UDP-glucose pyrophosphorylase always cata-lyzes the net formation of UDP-glucose in the model, the reversereaction must be included. Of the six enzymes of the glycogensystem, the pyrophosphorylase has the largest maximal velocityconstants (Table II), and in the absence of the reverse reaction,the rate of UDP-glucose production even at very low glucose-i-Pconcentrations greatly exceeds the maximum rate of UDP-glucoseutilization by glycogen synthetase. This imbalance precludes asteady state for UDP-glucose. The constant PPi concentrationin the model is low-in keeping with active pyrophosphatase

activity (45)-but its contribution to the reverse reaction is essen-tial. Reversibility of the pyrophosphorylase (actually, depend-ence of the reaction on UDP-glucose) also increases the impor-tance of the glycogen synthetase reaction as a control point. If

Pyrophosphorylase did not depend on UDP-glucose, changes insynthetase would result only in an altered UDP-glucose con-centration. Since pyrophosphorylase depends on UDP-glucose,glueose-l-p and glucose-6-P are affected and the synthetase

'For this reason it is unlikely that the amount of glucose 1,6-diphosphate cofactor in the phosphoglucomutase reaction iscritical.

change is manifest throughout the system. (This argumentneglects possible controls exerted by glycogen.) The reportedfall in both glucose-6-P and UDP-glucose concentrations follow-ing glucocorticoid administration (44) is consistent with a reversi-ble UDP-glucose pyrophosphorylase.

Since the hepatic cell membrane poses no barrier to glucose(5), the response of the glycogen system to the glucose concen-tration must be mediated through the phosphorylation reaction.If the kinetic constant for glucose were orders of magnitude belowthe physiological glucose concentration, as with hexokinase, thephosphorylation rate would be independent of the glucose con-centration (possible inhibition by glucose-6-P being neglected).The larger the kinetic constant for glucose, the more sensitivethe phosphorylation reaction to the glucose concentration.

The kinetic data in Table II are of two types: maximum veloc-ity constants which reflect the total amount of enzyme and ki-netic constants for the substrates (and activators). Generallyspeaking, changes in the maximum velocity produce greaterchanges in the enzymatic rates than do changes in the kineticconstants, because the substrates or activator concentrationsoften attenuate the changes in the kinetic constants. However,where the substrate concentration is much smaller than the ki-netic constant, as with glucose 6-phosphatase, the kinetic con-stant behaves as the inverse of the maximum velocity constant.

One purpose of the study was to evaluate the assumption thatthe reactions occur in a homogeneous phase without compart-ments. For many reasons it appears that this is not a good as-sumption. First, the model predicts that the glucose-6-P concen-tration rises as the glucose concentration rises (Fig. 2), which isconsistent with experimental observation (44, 55). A conse-quence of the homogeneous phase assumption in the model isthat the glucose 6-phosphatase rate (v2) also rises with the glu-cose concentration (Fig. 3). However a perfusion study in vivo(58) suggests that, as one might expect, the absolute rate ofhepatic glucose production (i.e. the rate of glucose 6-phosphatase)falls with an increasing glucose concentration. Second, in Fig. 3the difference between the glucokinase (v1) and glucose 6-phos-phatase (v) curves which is net glucose production is limited bythe increasing phosphatase rate. In fact, according to the pre-dictions of the model, net glucose production is achieved muchless by a rising rate of dephosphorylation than by falling phos-phorylation activity. Third, with the homogeneous phase as-sumption there is cyclic conversion of glucose to glucose-6-Pand back to glucose which results in net ATP utilization and astabilized glucose-6-P concentration at the expense of an unstabi-lized glucose concentration. (The cyclic interconversion ofglucose-l-P and glycogen results in net UTP utilization.)7

The inconsistency between the predicted and observed behav-ior of glucose 6-phosphatase, the limited glucose productionpredicted by the model, and the net utilization of ATP and UTPcould be avoided by assuming a special compartment for glucose6-phosphatase and phosphoglucomutase and two pools of glucose-6-P and glucose-l-P. Glucose 6-phosphatase is a microsomalenzyme (59). Histochemical studies reveal a perinuclear dis-tribution for the enzyme in liver (60) and in the -cell of the

7 The utilization of ATP and UTP required to maintain theglucose-6-P and UDP-glucose concentrations in the absence ofnet glucose uptake or net glycogen synthesis is substantial and farexceeds the rate of gluconeogenesis. From an energetic point ofview these requirements would be unsound.

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pancreas (61). The additional compartment would completelyseparate the glycogen synthetic and degradative pathways.Since gluconeogenesis stimulates both glycogen synthesis andglucose production, the glucose-6-P produced by this routeshould probably be included in both pools. The possibility oftwo hepatic glucose-6-P pools has been suggested by Segal andLopez (62), among others.

An alternative but less likely hypothesis to explain the behav-ior of glucose 6-phosphatase would be the existence of an inhibitorof the enzyme. The inhibitor would have to be potent to over-come the increase in the enzymatic rate produced by the increasein glucose-6-P, and it would also have to increase with the glu-cose concentration. The inhibition by glucose itself is not ade-quate (18) nor does insulin affect the enzyme directly (63). In-organic phosphate inhibits the enzyme slightly (17) but it isunlikely that the phosphate concentration increases with theglucose concentration. Furthermore, in comparison with gluco-kinase (and hexokinase), glucose 6-phosphatase activity is lowowing to the low intracellular glucose-6-P concentration; inhibi-tion of the enzyme would further lower this activity and wouldprobably destroy the balance between hepatic phosphorylationand dephosphorylation and would result in a nonphysiologicalthreshold for glucose production. Even with inhibition of glu-cose 6-phosphatase, net ATP utilization would still result.

The simulation of fasting and diabetes produces results whichare consistent at the semiquantitative level with experimentalobservations. The simulation of the early effects of glucocorti-coids shows no increase in hepatic glucose production. An in-crease in gluconeogenesis larger than the arbitrary 4-fold oneimposed, or a smaller increase in glycogen synthetase, would resultin increased net glucose production and elevated concentrationsof the intermediates. If the liver contributes to the early hyper-glycemia following glucocorticoid administration (64), it is diffi-cult to reconcile the increased hepatic glucose output with theobservations that glucose 6-phosphatase activity is not elevatedat 4 hours (7) and that the glucose-6-P concentration is not in-creased (44). The question of two glucose-6-P pools againarises.

As is true with all theoretical formulations, the present model,which is based on experimental mechanisms and data and whichpredicts many physiological properties, cannot be consideredthe unique model of glycogen metabolism. The model assumessteady state conditions for glucose-6-P, glucose-l-P, and UDP-glucose. Other models can be constructed which assume lesserdependence of glycogen synthetase on glucose-6-P or which pro-vide broader variation in phosphorylase activity, or where gluco-neogenesis and the concentrations of ATP, UTP, and Pi arevaried with the glucose concentration. Nor does this modelinclude hormonal activity that varies with the glucose concentra-tion such as may occur with insulin, epinephrine, and glucagon.Several features of the model in this study have been dictated bythe availability of experimental data. Furthermore, errors inthe input parameters affect the model to different degrees de-pending on the sensitivity of the system to a particular parame-ter. For example, in the glucokinase reaction a 20% change inthe maximum velocity or in the kinetic constant of glucose pro-duces significant changes in the system, whereas a 2-fold changein the kinetic constant for ATP or in the constant ATP concen-tration is of little consequence. These considerations takentogether imply that the predictions of the present study, while

suggesting quantitative trends and patterns, do not necessarilyreflect the absolute amount or degree of change of the systemin vivo.

Acknowledgments-The author is particularly grateful to Dr.Anthony F. Bartholomay for his continual interest, guidance, andencouragement in all phases of this work and in the preparationof the manuscript. He thanks Dr. Joseph B. Alpers for manyhelpful suggestions and fruitful discussions of the biochemistryand Miss Yuling Li for assistance in programming and operatingthe computer.

APPENDIX

Steady State Calculation Sequence-The following calculationsequence uses a glucose-6-P concentration and equations (A-1 toA-5) to calculate net rates of change of glucose and glycogen, theconcentrations of glucose, glucose-1-P, and UDP-glucose, andthe six enzymatic rates.8

d(G)/dt = v2(G-6-P) - v(G) (A-l)

d(G-6-P)/dt = vl(G) - v2(G-6-P) - (A-2)

v3(G-6-P, G-1-P) + GNEO = 0

d(G-1-P)/dt = v3(G-6-P, G-1-P) - v4(G-1-P, UDPG) + A(A3)

v6(G-6-P, G-1-P) = 0

d(UDPG)/dt = v4(G-1-P, UDPG) - v(UDPG, G-6-P) = 0 (A4)

d(Glyc)/dt = v(UDPG, G-6-P) - v6(G-1-P, G-6-P) (A-5)

With UDP-glucose in the steady state, V4 = v5 and UDP-glucosemay be expressed in terms of glucose-6-P and glucose-1-P andsubstituted into the expression for vs, yielding

(A-6)v5 = v5 (G-6-P, G-1-P)

With this substitution Equation A-5 becomes

d(Glyc)/dt = d(Glyc)/dt(G-6-P, G-1-P)

With glucose-l-P and UDP-glucose in the steady state, d(Glyc)/dt = v8 (G-6-P, G-1-P), or

(A-8)G-1-P = G-1-P(d(Glyc)/dt, G-6-P)

Substituting A-8 into A-7 yields

d(Glyc)/dt = d(Glyc)/dt (G-6-P)

and the net rate of glycogen synthesis may be found at a givenglucose-6-P concentration. Glucose-1-P is found from EquationA-8, and v5 is calculated from v6 and d(Glyc)/dt in Equation A-5.UDP-glucose is found from the v5 term in Equation A-4.

Having calculated d(G) /dt from

d(G)/dt = -d(Glyc)/dt + GNEO (A-10)

vI is found from Equation A-1, which is then used to obtain theglucose concentration. It is, of course, possible to start withthe glucose concentration, but the calculations are more cumber-some. ·

Calculation of Kinetic Constants for Phosphorylase-RateEquation A-11 is derived from the enzymatic mechanism in

8 The compounds that participate in each reaction are indicatedin parentheses. Vi(x, y) denotes Vi is a function of x and y.

(A-9)

(A-7)

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(V1.KG- 1-P.P) - (V2*K, ,.G-1-P)(Kp-G-1-P) + (KG--P-P) + (Kp-KaG'-r) (KA (G- + 1) )

Table I. (The superscript and subscripts 6 are omitted.) Kpand KC1p are the kinetic constants for Pi and glucose-1-P,respectively, and KA and KG-6-P are the dissociation constantsfor 5'-AMP and glucose-6-P, respectively; these constants do notdepend on substrate, activator, or inhibitor concentrations.

In the absence of glucose-6-P and one substrate, for example,glucose-l-P, Equation A-11 reduces to

V1. PP -KP (A-12)

where K = Kp (KAMP/AMP + 1). The latter expression im-plies that the observed kinetic constant for Pi, K , decreases with

the activator, 5'-AMP, but that a plot of K against 1/AMP hasslope Kp.KAMP and intercept K ,. K and KAMP (Table II) arefound by this method from the data of Helmrich and Cori (28).

Equation A-12 may be rearranged to

V1. AMP

V = AMP(P + K (A-13)

where

KAMP = () (Kp) KAMP

This latter expression implies that the observed kinetic constantfor 5-AMP (KAMP) decreases with the Pi (or glucose-l-P) con-centration but that a plot of KAMP against 1/P has slopeKvpKAMP. Knowing Kp from above, KAMP may be evaluated.

Using different data (28) KAMP = 3.2 X 10- 4M is found from

Equation A-12 and KAMP = 3.6 X 10- 4 M from Equation A-13.

If the substrate is glucose-1-P instead of Pi, Equation A-13yields K'AMP = (KI.p/G-1-P) KAm , from which KGI,p may be

found from measurements of KAMP at different glucose-1-P con-centrates once KAMP is known.

In the absence of Pi, Equation A-11 rearranges to

V2 AMP

(G-1-P + KG--P) AMP KM (A-14)

G-1-P

where

KG0 t1- KAMP G-6-PAKiMp = G-1-P KG-P P)

from which KG,.p may be evaluated from measurements ofKMP made at constant glucose-1-P and 5'-AMP concentrationsbecause Kcl- and KAMP are known.

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Wayne P. LondonA Theoretical Study of Hepatic Glycogen Metabolism

1966, 241:3008-3022.J. Biol. Chem. 

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