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A Computational Model for Glycogenolysis in Skeletal Muscle
MELISSA J. LAMBETH1and MARTIN J. KUSHMERICK1,2,3
1Department of Bioengineering, 2Department of Radiology, and 3Department of Physiology and Biophysics,
University of Washington, Seattle, WA
(Received 23 August 2001; accepted 10 May 2002)
Abstract—A dynamic model of the glycogenolytic pathway to
lactate in skeletal muscle was constructed with mammalian
kinetic parameters obtained from the literature. Energetic buff-
ers relevant to muscle were included. The model design fea-
tures stoichiometric constraints, mass balance, and fully revers-
ible thermodynamics as defined by the Haldane relation. We
employed a novel method of validating the thermodynamics of
the model by allowing the closed system to come to equilib-
rium; the combined mass action ratio of the pathway equaled
the product of the individual enzymes’ equilibrium constants.
Adding features physiologically relevant to muscle—a fixed
glycogen concentration, efflux of lactate, and coupling to an
ATPase—allowed for a steady-state flux far from equilibrium.
The main result of our analysis is that coupling of the glyco-
genolytic network to the ATPase transformed the entire com-
plex into an ATPase driven system. This steady-state system
was most sensitive to the external ATPase activity and not to
internal pathway mechanisms. The control distribution among
the internal pathway enzymes—although small compared to
control by ATPase—depended on the flux level and fraction of
glycogen phosphorylase a. This model of muscle glycogenoly-
sis thus has unique features compared to models developed for
other cell types. © 2002 Biomedical Engineering Society.
关DOI: 10.1114/1.1492813兴
Keywords—Metabolism, Flux analysis, Enzyme kinetics.
INTRODUCTION
Glycolysis was the first metabolic pathway to develop
in evolution. It is centrally important to intermediary
metabolism in all cells. The kinetic mechanisms of its
component enzymes in many cell types have been well
characterized. Glycolysis has fascinating nonlinear and
oscillatory properties of feedback inhibition that warrant
quantitative description. Much of the modeling effort has
been focused on erythrocytes,29,39 yeast,61 and T. brucei.3
One common result from all of this work seems to be
that regulation of flux in this pathway occurs by separate
mechanisms in the different cell types. It may be a gen-
eral principle that regulation of flux in the pathway de-
pends more on cell properties and reactions extrinsic to
the pathway itself than on the basic properties of the
enzymes in the pathway. The work presented here is
motivated by questions and issues relevant both to mod-
eling of metabolic networks per se as well as to under-
standing the role of glycogenolysis and glycolytic fluxes
in muscle energy metabolism.
Modeling of glycolysis in muscle has not been ex-
tended since initial efforts by Garfinkel et al. in 1968,22
for a number of possible reasons: 共1兲the large dynamic
range of fluxes 关⬃0.6– 60 mMATP/min 共Ref. 12兲兴;共2兲
large changes in metabolite concentrations 共lactate, inor-
ganic phosphate, and phosphocreatine兲due to muscle
properties and intensity and duration of activity; 共3兲un-
certainty in the kinetic function and substrate concentra-
tion describing the glycogen phosphorylase reaction; and
共4兲lack of consensus regarding the control mechanisms
that regulate flux.10,12,15
Our approach is to develop a parsimonious model of
glycogenolysis to enable future tests of conflicting hy-
potheses concerning controls of its flux in particular
muscle types and to investigate how glycogenolytic and
glycolytic fluxes are regulated when connected to other
key reactions in muscle energy balance, such as the cre-
atine and adenylate kinases and ATPases. We developed
this model of muscle glycogenolysis to address three
questions:
共1兲Is there a value in making all the reactions bio-
chemically reversible in the equations? We explore the
consequences of the use of irreversible reaction steps in
metabolic modeling. Equation simplification by irrevers-
ibility limits the possible steady states allowed in the
system. More importantly, it is incorrect in principle due
to microscopic reversibility.13 We explored a reaction
scheme in which all component reaction steps were fully
reversible 共despite the fact that some have a very favor-
able unidirectional thermodynamic driving force兲.We
will show that a detailed mathematical analysis of a
reversible scheme proves to be valuable because it al-
lows the use of thermodynamics as a tool to validate the
model.
共2兲What are the consequences of coupling glyco-
genolysis to other reactions within the cell on the regu-
lation of its pathway flux? The stoichiometry and con-
served moieties of the pathway and its functional
relations with other intracellular processes, e.g., ATPases,
Annals of Biomedical Engineering, Vol. 30, pp. 808–827, 2002 0090-6964/2002/30共6兲/808/20/$15.00
Printed in the USA. All rights reserved. Copyright © 2002 Biomedical Engineering Society
808
creatine and adenylate kinases, and mitochondrial oxida-
tive phosphorylation, provide constraints in modeling by
reducing the overall degrees of freedom. The degree of
complexity may be reduced by addition of stoichiometri-
cally coupled reactions to the main pathway. When the
rates of the ATPase reaction and ATP synthesis are rapid,
creatine kinase and adenylate kinase are necessary for
effective buffering of the ATP/ADP/AMP ratios. The
consequence of this buffering is that different ATPase
activities change the ATP/ADP/AMP ratio relatively little
as compared to the large increase in Pi. Thus, depending
on cell type, the presence of these buffers will affect
differently the flux and metabolite distribution within
glycolysis. These results may lead to a general principle
that models of particular networks, even if the kinetic
parameters in the model are similar, will have different
behaviors and possibly different overall regulations de-
pending on the environment in which the network of
glycolytic and glycogenolytic reactions operates.
共3兲What aspects of the network of reactions are able
to account for the ability to achieve and sustain a wide
range of fluxes? Muscle activity is able to change the
relevant metabolic fluxes over a large range, up to 100-
fold depending on the intensity and type of activity.12
Existing models in other cell types do not attempt to
account for a significant dynamic range, nor do the mod-
els account for a wide range of steady states observed
experimentally in muscles.29,39,61 We coupled ATPase
flux with glycolytic and glycogenolytic fluxes. We find
this coupling produces additional stoichiometric con-
straints, which have the effect of simplifying the total
system.
METHODS
Development of the Model
The abundant information on glycogenolytic enzyme
kinetics in the literature was compiled into a consensus
list of substrate and product concentrations and kinetic
parameters for each enzyme 共Tables 1, 2, and 3兲. The
model consisted of the rate equations for 12 reactions
from glycogen to lactate in skeletal muscle 共Fig. 1兲. The
basic glycogenolytic model describes the complete path-
way from glycogen to lactate plus the inclusion of ade-
nylate kinase and creatine kinase. To this basic model
共which is a closed system of the glycolytic network兲,we
added additional reactions important for muscle energet-
ics and determined the components necessary for the
model to reach steady state in both a closed and in an
open system.
The glycolytic pathway, represented by this model, is
purposely considered to be an isolated pathway that does
not interact with other metabolites, reactions, or path-
ways. The behavior of this model, therefore, reflects how
the glycogenolytic system will behave without extrinsic
signaling, without free transport of monocarboxylates by
the blood supply, without glucose uptake and phospho-
rylation via hexokinase, and without additional redox
reactions such as glycerol-3-phosphate dehydrogenase
and mitochondrial oxidative phosphorylation. The pri-
mary reason for these choices is to characterize the path-
way itself as a functional unit.
Several features in the description of the pathway are
different from existing glycolytic models available for
other cellular systems. Inorganic phosphate is described
as a variable in the model due to its dramatic range of
concentrations 共2–30 mM兲in skeletal muscle.11 NAD/
NADH is also allowed to vary, consistent with the cho-
sen absence of interaction with other cellular systems.
The glycolytic model is a continuum model that is
based on standard differential equation form, with the
derivatives of the metabolite concentrations 共fluxes兲for-
mulated by the velocity expressions for the enzymes 共see
the Appendix兲. Descriptions of the fluxes are in
Michaelis–Menten form with substrate–product, allos-
teric, or rapid equilibrium bi–bi kinetics. The decisions
on the type of kinetic equation used were based on
known behavior of the enzyme in vitro and on available
parameter values in the literature; values and functions
used in the model are given in Table 1 and in the Ap-
pendix. Ordered Michaelis–Menten kinetics were used as
a default form in bi–bi reactions, when altered affinity
binding constants could not be determined from the lit-
erature, as defined by Eq. 共1兲:
V⫽
Vmax fAB
KaKb
⫺Vmax rPQ
KpKq
1⫹A
Ka
⫹B
Kb
⫹AB
KaKb
⫹P
Kp
⫹Q
Kq
⫹PQ
KpKq
.共1兲
It differs from a Cleland, or random bi–bi reaction, by
the absence of an altered affinity for the second substrate
共or product in the reverse direction兲after the first sub-
strate is bound, effectively resulting in an ordered bi–bi
rapid equilibrium reaction as defined by Segel.56
In order to determine whether rapid-equilibrium as-
sumptions are suitable for dynamic metabolic modeling,
we included the full reversible enzyme kinetic equation
for all enzyme reactions in the pathway. Kinetic controls
via the near-equilibrium enzymes are often omitted from
models of glycolysis, especially phosphoglucomutase
and enolase, but we will show these reactions have im-
portant influences on the dynamic behavior. The reverse
direction Vmax values were calculated using the Haldane
expression 共see below in the section on thermodynamic
constraints兲.
We define, for the purposes of this paper, the redox
potential, NAD/NADH, as ⬃R共note this is the inverse
of the common nomenclature in the mitochondria兲and
809Modeling of Skeletal Muscle Glycogenolysis
TABLE 1. Model kinetic parameters.
Literature
values Model value
Enzyme Parameter (mM
unless noted) Species Reference (mM unless
noted)
Glycogen Phosphorylase
AK
eq 0.42
(unitless) 28 0.42 (unitless)
K
GLYf 1.7 Rabbit 23 1.7
K
P
i
4234
K
i
GLY 2232
K
iP
i
4.7 23 4.7
K
GLYb0.15 23 0.15
K
G
1
P
2.7 23 2.7
K
iG
1
P
10.1 23 10.1
Glycogen Phosphorylase
BK
eq 0.42
(unitless) 28 0.42 (unitless)
K
P
i
0.2 Rabbit 19 0.2
K
ip
i
4.6 19 4.6
K
i
GLYf15 19 15
K
G
1
P
1.5 19 1.5
K
iG
1
P
7.4 19 7.4
K
i
GLYb4.4 19 4.4
K
AMP
⬘1.9e-6 Rabbit 14 1.9e-6
nH
1.75 14 1.75
Phosphoglucomutase
K
eq 16.62
(unitless) 28 16.62 (unitless)
K
G
1
P
0.063 Rabbit 30 0.063
K
G
6
P
0.03 30 0.03
Phosphoglucoisomerase
K
eq 0.45
(unitless) 28 0.45 (unitless)
K
G
6
P
0.48 Mouse 49 0.48
K
F
6
P
0.031 Rabbit 31 0.119
Phosphofructokinase
K
eq 242 (unitless) 28 242 (unitless)
K
F
6
P
0.18 Rabbit 40 0.18
K
F
6
P
⬘20 40 20
K
ATP 0.08 40 0.08
K
ATP
⬘0.25 40 0.25
K
FBP 4.02 Rabbit 36 4.02
810 M. J. LAMBETH and M. J. KUSHMERICK
TABLE 1. „Continued….
K
FBP
⬘¯4.02
K
ADP 2.71 36 2.7
K
ADP
⬘¯2.7
K
i
ATP 0.87 40 0.87
K
a
AMP ¯0.06
d
0.01
(unitless) 40 0.01 (unitless)
e
¯0.01 (unitless)
L
o13 (unitless) 40 13 (unitless)
Aldolase
K
eq 9.5e-5 (M⫺1) 28 9.5e-5 (M⫺1)
K
FBP 0.05 0.05
K
DHAP 2.1 Rabbit 45 2
K
GAP 1.1 45 1
Triose Phosphate Isomerase
K
eq 0.052
(unitless) 28 0.052 (unitless)
K
GAP 0.32 Rabbit 34 0.32
K
DHAP 0.61 Human 16 0.61
Glyceraldehyde-3-Phosphate
Dehydrogenase
K
eq 0.089
(unitless) 28 0.089 (unitless)
K
GAP 0.0025 63 0.0025
K
NAD 0.09 63 0.09
K
P
i
0.29 Rabbit 21 0.29
K
13BPG 0.0008 63 0.0008
K
NADH 0.0033 63 0.0033
Phosphoglycerate Kinase
K
eq 57,109
(unitless) 28 57109 (unitless)
K
13BPG 0.0022 Rabbit 33 0.002
K
ADP 0.05 Pig 38 0.008
K
3PG 1.2 33 1.2
K
ATP 0.36 Rabbit 20 0.35
Phosphoglycerate Mutase
K
eq 0.18
(unitless) 28 0.18 (unitless)
K
3PG 0.2 Chicken 52 0.2
K
2PG 0.014 52 0.014
Enolase
K
eq 0.49
(unitless) 28 0.49 (unitless)
K
2PG 0.12 Rat 51 0.1
K
PEP 0.37 51 0.37
811Modeling of Skeletal Muscle Glycogenolysis
the phosphorylation potential, ATP/(ADP•Pi兲,as⬃P.
We chose this definition of energetic driving force for
our purposes 共instead of using a stoichiometric relation,
e.g., Atkinson’s energy charge兲due to the availability of
inorganic phosphate measurements, the need to include
Pidue to its large changes, and the undersaturation of
adenylate kinase by AMP at all physiological concentra-
tions in skeletal muscle.
Simplification of the Model
The model contains no external effectors; any inhibi-
tors or activators in the equations are metabolites vari-
able elsewhere in the model. For this reason, citrate,
fructose-2,6-diphosphate and calcium ion are not present
in the equations. An additional simplification is the as-
sumption that all cationic-bound species have the same
affinity for the enzyme and can be described generically
TABLE 2. Enzyme activities.a
Activity Conc.
V
max
Enzyme (U/mg) (mg/ml) Species (M/min)
GPP
B
25 0.6⫻2 Rabbit 0.03
GPP
A
25 0.4⫻2 Rabbit 0.02
PGLM 800 0.6 Pig 0.48
PGI 1100 0.8 Pig 0.88
PFK 160 0.35 Rabbit 0.056
ALD 16 6.5 Rabbit 0.104
TPI 6000 2 Rabbit 12.0
GAPDH 115 11 Rabbit 1.265
PGK 800 1.4 Pig 1.12
PGM 1400 0.8 Rabbit 1.12
EN 80 2.4 Rabbit 0.192
PK 450 3.2 Rabbit 1.44
LDH 600 3.2 Pig 1.92
ADK 2200 0.4 Pig 0.88
CK 100 5 Pig 0.5
aSee Ref. 55.
TABLE 1. „Continued….
Pyruvate Kinase
K
eq 10304 28 10304 (unitless)
(unitless)
K
PEP 0.08 Rat 27 0.08
K
ADP 0.3 27 0.3
K
PYR 7.05 Rabbit 18 7.05
K
ATP 0.82–1.13 18 1.13
Lactate Dehydrogenase
K
eq 16,198
(unitless) 28 16198 (unitless)
K
PYR 0.6 Pig 5 0.335
K
NADH 0.008 57 0.002
K
LAC 17 Mouse 41 17
K
NAD 0.253 Rabbit 66 0.849
Creatine Kinase
K
eq 233 (unitless) Rabbit 54 233 (unitless)
K
PCr 1.11 54 1.11
K
i
ATP 3.5 54 3.5
K
i
ADP 0.135 54 0.135
K
i
PCr 3.9 54 3.9
K
Cr 3.8 54 3.8
Adenylate Kinase
K
eq 2.21
(unitless) Rabbit 17 2.21 (unitless)
K
AMP 0.32 Human 62 0.32
K
ATP 0.27 62 0.27
K
ADP 0.35 62 0.35
812 M. J. LAMBETH and M. J. KUSHMERICK
in one pool as ⌺ATP, ⌺ADP, ⌺AMP, ⌺Pi, etc. The
model thus exists implicitly at constant pH, fixed con-
centration of Mg⫹⫹ and other ions.
Table 1 lists the in vitro literature values for each
parameter describing the kinetics of the model reactions,
as well as the precise values we used for our calcula-
tions. Values from sources—at physiological pH 共7.0兲
and temperature (37°C) when available—of mammalian
skeletal muscle origin were utilized. In addition, if infor-
mation could be obtained all from the same source, we
chose to use those values, as they would be more con-
sistent relative to one another. Vmax values were calcu-
lated in M/min 共molar concentration in cellular water per
minute兲from the known specific activity of the enzyme
in U/mg multiplied by the approximate concentration of
the enzyme in the cell by mg/ml, as shown in Table 2.
We obtained the values for the enzyme concentrations
and activities from Scopes et al.,55 measured in skeletal
muscle extracts and in purified enzymes from the same
source. The reverse direction Vmax values were calculated
by the Haldane expression. Initial concentrations of the
variable metabolites have been defined by averaging
three sources of biopsy data from rat and human resting
skeletal muscle 共Table 3兲; 13BPG and 2PG were esti-
mated by the order of magnitude reported for other cell
types.
Constraints by Mass Conservation
Constraints on the system are in the form of mass
relations and the assumptions cited above. The total
adenosine nucleotides is a fixed number such
that 8.2 mM ATP⫹0.0013 mM ATP⫹1e-5 mM AMP
⫽8.21301 mM; this quantity is consistent with measured
values of human skeletal muscle ATP content in intrac-
ellular water.11,24 Likewise, phosphocreatine and creatine
are constrained to sum to 40 mM.24,44 0.5
M NADH
⫹0.5 mM NAD equals a fixed amount of 0.5005 mM
共with a starting ⬃Rratio of 1000兲.37,53 These are the
simplest of the constraints by conserved moieties.
Correct conservation of mass within the model was
proven for both open and closed systems by calculating
the total phosphate using the following equation:
⌺P⫽G1P⫹G6P⫹F6P⫹共2FDP兲⫹DHAP⫹GAP
⫹共2⫻1,3BPG兲⫹3PG⫹2PG⫹PEP⫹共2ATP兲
⫹ADP⫹PCr⫹Pi.共2兲
The number of sites available for phosphate transfer de-
termines the stoichiometry in Eq. 共2兲. Thus, AMP is not
included in this definition of total phosphate but it would
be added for a tally of soluble phosphate content of
metabolites. The structure of the model results in a con-
FIGURE 1. Schematic representation of all reactions used in
the model.
TABLE 3. Averaged resting concentrations used for initial
conditions.a,b
Metabolite Concentration (mM)
GLY 112
G1P 0.0589
G6P 0.75
F6P 0.228
FBP 0.0723
DHAP 0.0764
GAP 0.0355
13BPG 0.065*
3PG 0.052
2PG 0.005*
PEP 0.0194
PYR 0.0994
LAC 1.3
ATP 8.2
ADP 0.013
AMP 2e-5
P
i
4.1
PCr 34.67
NAD 0.5
aSee Refs. 2, 8, 24, and 53.
bConcentration unreported for skeletal muscle. Estimated values
are based upon orders of magnitude seen in other cell types.
813Modeling of Skeletal Muscle Glycogenolysis
stant sum of all substrates and products for the closed
system. The open system is defined via an ATPase term,
which is coupled to the ADP, ATP, and Piderivatives;
the sum of the concentrations in Eq. 共2兲also remains
constant with a steady-state flux through the pathway
共see the Results兲.
In the same manner, the total mass of oxidized and
reduced intermediates must be maintained by the moiety:
total oxidized⫽NAD⫹1,3BPG⫹3PG⫹2PG⫹PEP
⫹PYR,
total reduced⫽NADH⫹GAP⫹LAC. 共3兲
Thermodynamic Constraints
The reverse Vmax parameter values are calculated
based upon the Haldane relation:
Vmax reverse⫽Vmax forwardKmreverse
KmforwardKeq .共4兲
Because all equations are reversible when defined in this
manner the metabolites can be allowed to equilibrate in
the closed system to test the validity of the model ther-
modynamically. By constructing the model with built-in
Haldane relations, no adjustment/optimization of the pa-
rameters will violate the thermodynamics of any reaction
or of the entire network.
The combined equilibrium constant is calculated by
using the total products over reactants, assuming that
glycogenn⫺1/glycogennis unity. By multiplying each re-
action’s product/reactant ratio 共⌫兲 together, all intermedi-
ates cancel out except for the expression in Eq. 共5兲:
Keqcombo⬅LAC2
冉
ATP
ADPPi
冊
3.共5兲
For conditions other than the standard state, the product
of the terms (⌫Keqobs) for each reaction 共i兲, equals
Keqcombo :
Keqcombo⫽
兿
i⫽1
12
Keqithermo
m⫽
兿
i⫽1
12
共⌫Keqiobs
m兲,
where
m⫽
再
1, GP→ALD
2, GAPDH→LDH. 共6兲
Power maccounts for the branching of the six-carbon
intermediates into two three-carbon intermediates at al-
dolase. Note that the directionality of triose-phosphate
isomerase towards the production of glyceraldehyde
3-phosphate is used in Eqs. 共5兲and 共6兲.
Numerical Integration
The model was constructed of algebraic and differen-
tial equations using the SAAMII 共SAAM Institute, Se-
attle, WA兲software program and numerically integrated
using a Rosenbrock semi-implicit method with a relative
error of 0.00001 and an absolute error of 0.000001. The
numerical results were identical when checked by inte-
gration with GEPASI 共http://www.gepasi.org/兲. Subse-
quent sensitivity analysis was performed by translating
the model into XSIM 共National Simulation Resource,
Seattle, WA兲and numerically integrated using an implicit
Runga–Kutta method of variable order 共RADAU兲with
an error of less than 0.001%.
RESULTS
Thermodynamic Validation
The model used for this analysis is as depicted in Fig.
1, without an ATPase; without ATPase the network is a
closed system. This closed system of 12 enzymes does
produce a steady state, the one at equilibrium where the
forward and reverse fluxes are equal and net flux asymp-
totes to zero. We take advantage of the drive towards
equilibrium to give a method of validation for our model.
Because the Haldane relation has described each reaction
reversibly, the mass action ratios 共products/reactants of
each reaction兲of the closed system’s stationary state
should come to equilibrium. The distribution of mass
should be defined as the product of the individual equi-
librium constants.
We set the initial concentrations of the metabolites to
standard conditions 共1M兲prior to integration, as these
are the conditions under which the equilibrium constants
are measured. We integrated the model until a stationary
state occurred 共approximately 5000 min in simulation
time兲. This condition defines the second term in Eq. 共6兲,
兿
i⫽1
12 Keqithermo
m. We find that the model achieves numerical
values consistent with the mass action ratios compared to
the equilibrium constants applied to the model, uphold-
ing Eq. 共6兲. The overall Keq (
兿
i⫽1
12 Keqithermo
m) has a value
of 1.5572e19 as reported for glycogen→lactate
共Johnson,28 as tabulated in Mahler and Cordes35兲while
the model for standard state results in a Keqcombo
⫽1.5597e19. The resulting values are indistinguishable
within the accuracy of the experimental results for
Keqthermo 共⌬Gomeasured to two significant digits兲and
the numerical error of the integration. The compilation of
the inputted Keqthermo values is independent of the calcu-
lation, i.e., no adjustment or fitting of the model was
814 M. J. LAMBETH and M. J. KUSHMERICK
performed to achieve these results. Because of the rela-
tion expressed in Eq. 共6兲, the mass in the system will be
distributed with the same resulting product/substrate ra-
tios regardless of initial conditions. This was achieved
with our model using the physiological initial conditions
from Table 3. This analysis validates the model and its
parameters on the basis of their thermodynamic proper-
ties.
Figure 2 displays the time course for the approach to
equilibrium of each contributor to the combined equilib-
rium constant of Eq. 共5兲. The large decrease in inorganic
phosphate, displayed as 1/(Pi
3) in Eq. 共5兲, dominates the
Keqcombo term, causing it to rise as it approaches equilib-
rium, while ATP and lactate remain at the same order of
magnitude for the time course. The progress curves for
the network of reactions were integrated by the model
equations. These results show a depletion of the large
stores of glycogen toward lactate until the inorganic
phosphate, a cosubstrate in the glycogen phosphorylase
reaction, reaches a concentration low enough that the
entire system reaches equilibrium. Because Piis also
used at the GAPDH step, the decrease in its concentra-
tion results in 共1兲an increase of the triose phosphates,
which in turn slows down PFK and aldolase; 共2兲a de-
crease in substrate 共e.g., 1,3BPG兲available for the ki-
nases such that ATP cannot be synthesized; and 共3兲a
lower NADH concentration that slows down the catalysis
of pyruvate to lactate through LDH. The latter finding is
exacerbated by pyruvate kinase, which has a greater dif-
ference between the observed free energy and the ⌬G°
than LDH. Thus, more mass builds up at pyruvate than
lactate at equilibrium.
The results from this equilibrium analysis, while
trivial on one level, provide an opportunity that should
be of general interest for modeling. Naturally, the model
should arrive at equilibrium because it is a closed system
of fully reversible equations. The convergence of the
mass to the theoretical mass action ratios is as expected
due to the imposed Haldane relation on the reverse Vmax
parameter. But, in fact, these equalities 共or lack thereof兲
also provide a useful and novel test of the validity of the
equations.
Inorganic phosphate depletion through glycogen phos-
phorylase prevents the closed model from achieving a
steady state except at equilibrium. This property makes it
impossible to compute sensitivities. It also suggests that
the analysis of glycolysis as an isolated system is fruit-
less, for it is only when features are added to the path-
way to maintain the mass balance far from equilibrium
that realistic behavior is simulated.
Forcing the Model to Steady State
One method of forcing the system into a steady state
is to create a closed loop—an efflux of lactate from the
system equal to 2
3ATPase and an influx into the system
at glycogen of 1
3ATPase; the coefficients are determined
by the stoichiometry of the system of reactions. Then,
the rate of carbohydrate utilization is tightly regulated
due to the forced addition and subtraction of mass from
the pathway at the beginning and end. The problem with
this approach is the inherent and fixed coupling of the
flux to the ATPase; this was not analyzed further.
Instead, we found that specification of three condi-
tions to the model was needed in order to achieve a
physiologically realistic steady-state system—共1兲holding
the glycogen constant, 共2兲allowing an efflux from lactate
into an infinite sink, and 共3兲adding an ATPase. The
branched structure of glycogen, the localization of the
phosphorylase enzyme within the glycogen particle, and
the effective concentration of glucosyl moieties at about
100 mM justify the constant apparent glucosyl concen-
tration at the enzyme. Next, we added an output from
lactate that is dependent on the lactate concentration. By
using a mass action coefficient 共with an arbitrary value
of 0.2兲, we are in effect simulating a flushing of a frac-
tion of the lactate pool from the cytosol as would occur
by the monocarboxylate transporter into the extracellular
fluid and blood stream, but here in a nonsaturating,
concentration-dependent manner. This condition is not
meant to represent the physiological lactate efflux. The
first two conditions, as illustrated by the lactate efflux in
Fig. 3, are not sufficient by themselves for creating a
steady-state system due to a continuous utilization of Pi
with no reactions contributing to Piproduction 共except
FIGURE 2. Approach of metabolites towards equilibrium. Ini-
tial conditions are standard „1M…for all metabolites.
K
eqcombo
„MÀ1…is multiplied by 10À19 for visual purposes.
815Modeling of Skeletal Muscle Glycogenolysis
for the small reversible flux of GP and GAPDH兲.
The essential requirement for the model to realize a
steady-state flux is the addition of an ATPase. The ATP
utilization is defined by a simple mass action flux,
VATPase⫽kATP, where kis a rate constant that can be
assigned different values to represent various levels of
ATP hydrolysis appropriate for graded muscle activity.
Small values of krepresent a muscle at rest and large
values represent a muscle actively contracting at higher
ATPase activity. The addition of an ATPase causes a
balanced stoichiometry, and thus steady state; the ATPase
must consume ATP at a rate three times faster than the
breakdown of one glucosyl unit to lactate. With these
constraints on the input to glycogen and efflux of lactate,
steady states were achieved, but the flux through
the system became a function of the ATPase flux, as
we describe in the next two paragraphs and results in
Table 4.
We chose to examine the system under different en-
ergetic requirements. We studied the system driven with
graded ATPase flux values, approximately 0.6, 6, and 60
mM/min, to represent three energetic steady states of the
muscle—resting, moderate exercise, and maximal exer-
cise conditions 共Table 4兲. The physiological appropriate-
ness of the energetic steady states was based on the
resultant orders of magnitude of the phosphorylation po-
tential compared to published NMR data.10,11 The ratio
of phosphorylase a:bwas fixed across all three states as
40%:60 共a fraction of isozyme distribution observed dur-
ing muscle activity55兲. Like a real physiological system,
the flux through the system adjusts to the ATPase even
though the input 共fixed glycogen兲and output 共lactate
efflux兲are not directly coupled to the ATP sink. The
ATPase reaction is coupled to the system by require-
ments of moiety conservation at the sites involving ATP,
ADP, and Picoupling.
This property is important for subsequent control
analysis of the system. Although the network of glyco-
genolytic and glycolytic enzymes are poised far from
equilibrium when coupled to the ATPase, a steady-state
systemic flux can occur only with continuous ATP hy-
drolysis and synthesis. Table 4 shows the 1:1 relationship
between the resultant systemic flux and each ATPase
level in addition to the steady-state values of key me-
tabolites. From the fact that steady-state concentrations
are arrived at for the varied energetic demands that
closely mimic observed values for skeletal muscles, we
can assume that the assigned kinetic parameter values are
appropriate for the model. Note that in the actual fluxes
reported, the metabolite concentrations and enzyme ac-
tivity used represent generic consensus values in the lit-
erature. Specific muscle cell types may vary in these
properties and so will the characteristics of their specific
fluxes and distributions of mass; these are matters for
future analyses.
Sensitivity Analysis of the Model in Induced Steady State
For a sensitivity analysis of the model, each enzyme
Vmax and metabolite initial value in the model was per-
turbed by 1%. The sensitivity run calculated the sensi-
tivity Sp
yof the variable (y) of interest to the parameter
(p) being tested by the following equation:
Sp
y⫽
ln y共p,t兲
ln p.共7兲
FIGURE 3. Demonstration of steady-state lactate efflux cre-
ated with addition of the ATPase to the model. Inset: log–log
plot of lactate flux vs ATPase coefficient.
TABLE 4. Values for the three physiological states. Rest
corresponds to a fixed ATPase value of 0.6 mMÕmin, moderate
exercise has an ATPase value of 6.1 mMÕmin, and maximal
exercise has an ATPase rate of 58.4 mMÕmin. Refer to the text
for more details.
Moderate Maximal
Physiological Rest exercise exercise
State (mM) (mM) (mM)
ATPase
coefficient 0.075 0.75 7.5
ATP flux
(mM/min) 0.6 6.1 58.4
ATP 8.2 8.2 7.8
ADP 0.0078 0.032 0.415
AMP 3.3E-6 5.7E-5 0.010
P
i
0.64 7.1 31.5
PCr 32.8 28.6 3.0
NAD/NADH 513 12 1
816 M. J. LAMBETH and M. J. KUSHMERICK
For this sensitivity analysis, the function of interest
y(p,t) was either the steady-state metabolite concentra-
tion or steady-state flux as defined by the efflux from
lactate. The parameter pbeing perturbed was an initial
condition for one of the variable metabolites in the
model or the Vmax parameter of a given enzyme step.
Equation 共7兲is equivalent to the response coefficient as
defined by metabolic control analysis theory.
As seen from Fig. 4, a shift occurs in the enzyme
control of the pathway 共12 enzymes in the pathway, Fig.
1兲from rest, where enzymes far from equilibrium have
control, to maximal muscle activity conditions, where
enzymes with low Vmax also exert control. Not included
in Fig. 4 is the control of the ATPase, which naturally
has the largest impact on the flux 共95%–99% of control兲
over the conditions studied. The general trend is for the
increase in flux of the system to reduce the influence of
the ATPase 共driving force兲on the system; but the control
by ATPase always remains dominant over the 100-fold
range of fluxes analyzed.
At resting flux, with respect to the enzymes within the
glycogenolytic and glycolytic pathway, GP and PK are
the predominant controls. As the rate of ATP utilization
increases, enzymes with lower Vmax 共and thus, less over-
all flux capacity兲such as PFK and PGLM start to share
more of the control. At maximal muscle activity, the
control coefficients in the pathway have increased rela-
tive to the ATPase 共summing to 0.05 at their maximal兲.
Thus, they are always much smaller than the control
coefficient for the ATPase, and even at high fluxes the
system remains ATPase driven. Interestingly, glycogen
phosphorylase activity gains importance in the network
as flux increases with contractile activity.
Sensitivity analysis of the initial conditions of the
metabolite concentrations showed a surprising homoge-
neity in sensitivity of steady-state concentration values to
initial conditions across all energetic states; this sensitiv-
ity was overwhelmingly high for all metabolites towards
ATP and PCr 共detailed results not shown兲. The sensitivity
of the resultant flux toward the metabolites, however,
showed differences among the energetic state, as seen in
Fig. 5. ATP and ADP remained at similar levels for the
FIGURE 4. Flux control of pathway with efflux, ATPase, CK,
and ADK omitted. Sensitivities were calculated based upon
1% perturbations of the
V
max values for each reaction and
the resultant change in steady-state systemic flux. „a…Phos-
phorylase
a
set at 40%; „b…phosphorylase
a
set at 1%. Zero
values are plotted on the graphs as 1e-6 for illustrative pur-
poses.
FIGURE 5. Flux sensitivity plot of initial conditions of key
metabolites. Sensitivities were determined by perturbing
starting concentrations by 1% and calculating the resulting
change on the overall steady-state systemic flux. Glycogen
is a parameter, not an initial condition in this plot. Zero val-
ues are plotted on the graph as 1e-6 for illustrative pur-
poses.
817Modeling of Skeletal Muscle Glycogenolysis
different fluxes while Pi, PCr, and glycogen 共a fixed
value兲gradually increase their share in control as fluxes
increase. AMP sensitivity was negligible for all states.
The sensitivity of intermediates and the systemic flux
towards NADH perturbations 共and thus redox potential
by the conserved moiety兲was zero at all flux levels,
repeating the observations made by Richard et al.50 re-
garding redox being completely driven by phosphoryla-
tion potential.
Because the previous analysis was performed at a
phosphorylase a:bratio of 40:60, we wanted to investi-
gate the implications of changing the isozyme fractions
to the physiological ratio observed under basal condi-
tions. Therefore, the sensitivity analysis was repeated for
the case of phosphorylase a:b⫽1%:99% for the three
ATPase values of 0.6, 6, and 60 mM/min 关Fig. 4共b兲兴. The
system is only sensitive to glycogen phosphorylase and
phosphofructokinase at resting and moderate exercise
conditions; this reflects the greater dependence of the
pathway on AMP-dependent reactions to achieve the
same flux. Yet, the most surprising result was the system
showing the same distributed control pattern over the
maximal flux, as seen in Fig. 4共a兲. The fact that the
control pattern is the same for 1% and 40% phosphory-
lase areflects a physiologically observed phenomenon—
the dephosphorylation back to the bform while maximal
muscle activity is still maintained.9The differences in the
1% and 40% phosphorylase aanalyses at rest and mod-
erate control illustrate the design of the system to shift
the sensitivity away from ⬃Pand towards external ef-
fectors 共e.g., AMP兲.
Fractional Factorial Design Analysis
We also performed a two-level fractional factorial de-
sign analysis on the metabolite values. By toggling vari-
ables in a predetermined pattern of assigned high and
low states 共as determined by maximal and minimal val-
ues found physiologically兲and observing the difference
in the resultant yield, one can determine the most impor-
tant factor involved in the yield. For this analysis, the
outcome, or yield, of interest is the systemic flux in the
pathway as it depends on changes in metabolite concen-
trations known to occur physiologically 共Table 5兲. A two-
level factorial design requires 2nruns, where n⫽the
number of variables being investigated, to analyze the
effects on a system, while a two-level fractional factorial
design takes advantage of redundancy in the patterns to
simplify analysis. In the analysis performed, 16 runs
(24) were needed for eight variables instead of 256
(28)—the notation 28-4 is used to signify the fractional
factorial design used. Readers interested in the details of
these calculations are referred to Box et al.6The defini-
tion of a main effect is the size of the difference between
the average yield for the runs at the high value minus the
average yield for the runs performed at the low value:
y
¯
⫹⫺y
¯
⫺. The analysis used a series of 16 simulations
based upon initial conditions of eight variables that vary
significantly over the course of muscle activity 共Pi,
AMP, ADP, PCr, NADH, GLY, G6P, LAC兲; the metabo-
lite values used are given in Table 5. The values chosen
are typical for different levels of muscle activity and
were based on the range of values reported from resting
to active muscle.10 ATP was not included among
these variables due to its near-constant intracellular
concentration.
ADP emerged as the dominant main effect on the flux
共10⫻greater than the next main effect value兲, with PCr
and AMP as secondary main effects 共Table 6兲. The domi-
nant contributor to the largest mixed effect term was the
combination of ADP and Pi. This result, of course, is
congruent with preexisting knowledge of the ⬃Pcontrol
on the system. It differs, however, from the conventional
sensitivity analysis where ADP played a minor role in
control in comparison to ATP.
DISCUSSION
Is There a Value in Making All the Reactions
Biochemically Reversible in the Equations?
Some of the glycolysis models in the literature have
violated the thermodynamic principle of microscopic re-
versibility by parameter fitting that does not maintain the
Haldane relation or by fitting the equilibrium constants
of the reaction. We constructed a fully reversible series
of equations to represent the glycogenolytic system in
skeletal muscle so that an accurate measure of the de-
viation of the model results from the observed Keq could
be made. Cornish-Bowden and Cardenas have delineated
how irreversibility versus reversibility of a reaction, even
one far from equilibrium, alters the control of a pathway
if feedback inhibition is not built into the system.13 In
the model at hand, feedback inhibition through ATP is
present at phosphofructokinase, but not at other steps
typically considered unidirectional, such as pyruvate ki-
nase. For these reasons, we argue the necessity of includ-
TABLE 5. Values used for 2
IV
8-4 fractional factorial design
analysis.
Low (⫺)conc. High(⫹) conc.
Variable (mM) (mM)
P
i
140
AMP 1.0E-05 0.05
ADP 0.01 0.5
NADH 5.0E-04 0.01
PCr 3 34.7
GLY 10 100
G6P 0.75 6.7
LAC 1.3 30
818 M. J. LAMBETH and M. J. KUSHMERICK
ing the reverse direction in each of our kinetic equations,
despite the added complexity and additional parameters
needed, and we recommend this practice in general. Such
an approach proved useful here because thermodynamics
can be employed as a useful tool for validation of the
model structure. As the equations are structured, any
combination of kinetic values would arrive at the correct
Keqcombo as long as the Haldane relation is used. How-
ever, only physiologically appropriate kinetic parameter
values will result in reasonable fluxes and accurate dy-
namics. This property was later tested by the ability of
the model to simulate steady-state fluxes over a large
range of fluxes. A secondary benefit is the prediction of
metabolite influences on nonlinear behavior once physi-
ological conditions are imposed. For example, the large
control of ADP and Piand relative unimportance of
lactate on the system is highlighted by examining the
dynamics of each variable that contributes to the com-
bined Keq of the pathway as the closed system ap-
proaches equilibrium 共Fig. 2兲. This is indeed seen in the
subsequent fractional factorial design analysis, with ADP
main effects and ADP:Pimixed effects dominating the
system 共Table 6兲.
What are the Consequences of the Regulation
to Glycogenolytic Flux by Coupling to Other
Reactions Within the Cell?
The bulk of glycolytic modeling has been performed
in simpler cell types 共erythrocyte, yeast兲. Despite the
common emphasis on the minimal glycolysis network
per se in these models, enormous differences in meta-
bolic regulation among different cell types occurs de-
pending on which additional pathway structures are in-
cluded. In a detailed red blood cell model by Mulquiney
and Kuchel,39 the 2,3DPG shunt modulating hemoglobin
binding is essential to the functionality of oxygen bind-
ing. A thorough metabolic control analysis on the model
showed primary control of the glycolytic flux occurred
via 2,3DPG and nonglycolytic ATPase. In yeast, Teusink
et al.61 determined that additional pathways, like the
glycerol-3-phosphate dehydrogenase shunt, are necessary
to reproduce steady-state data for glycolysis. The com-
mon feature of these results suggests a common prin-
ciple: the regulation of glycolytic flux is controlled dif-
ferently in different cells because the boundary
conditions and connectivity with reactions extrinsic to
the main pathway are different.
TABLE 6. Design matrix for 2
IV
8-4 fractional factorial design analysis.
Run P
i
AMP ADP NADH PCr GLY G6P LAC
1⫺ ⫺ ⫺ ⫹ ⫹⫹⫺⫹
2⫹ ⫺ ⫺ ⫺ ⫺⫹⫹⫹
3⫺ ⫹ ⫺ ⫺ ⫹⫺⫹⫹
4⫹ ⫹ ⫺ ⫹ ⫺⫺⫺⫹
5⫺ ⫺ ⫹ ⫹ ⫺⫺⫹⫹
6⫹ ⫺ ⫹ ⫺ ⫹⫺⫺⫹
7⫺ ⫹ ⫹ ⫺ ⫺⫹⫺⫹
8⫹ ⫹ ⫹ ⫹ ⫹⫹⫹⫹
9⫹ ⫹ ⫹ ⫺ ⫺⫺⫹⫺
10 ⫺ ⫹ ⫹ ⫹ ⫹⫺⫺⫺
11 ⫹ ⫺ ⫹ ⫹ ⫺⫹⫺⫺
12 ⫺ ⫺ ⫹ ⫺ ⫹⫹⫹⫺
13 ⫹ ⫹ ⫺ ⫺ ⫹⫹⫺⫺
14 ⫺ ⫹ ⫺ ⫹ ⫺⫹⫹⫺
15 ⫹ ⫺ ⫺ ⫹ ⫹⫺⫹⫺
16 ⫺ ⫺ ⫺ ⫺ ⫺⫺⫺⫺
Main
effects 7.64E-
07 1.65E-
05 1.59E-
04 1.72E-
06 4.24E-
06 5.03E-
06 2.78E-
06 4.33E-
06
17 ⫹ ⫹ ⫹ ⫹ ⫹⫺⫺⫹
18 ⫹ ⫹ ⫹ ⫹ ⫹⫺⫹⫺
19 ⫹ ⫹ ⫹ ⫹ ⫹⫹⫺⫺
20 ⫹ ⫹ ⫹ ⫹ ⫹⫹⫹⫹
Mixed
effects ADP:P
i
5.2e-4 PCr:LAC
3.4e-6 AMP:G6P
1.1e-6 NADH:GLY
1.0e-6
Largest mixed effect consists of ADP:P
i
, AMP:G6P, GLY:NADH, PCr:LAC
819Modeling of Skeletal Muscle Glycogenolysis
We asked what effect the boundary conditions and
connectivity unique to skeletal muscle—the necessary
energetic buffering reactions, large range of metabolite
concentrations, and large range of ATPase flux
magnitudes—would have on regulation of the flux. The
most readily apparent feature of energetic buffering is
the damping of transients in ATP due to the capacitance
of the phosphate pools 共phosphocreatine for creatine ki-
nase and ADP for adenylate kinase兲. Besides the buffer-
ing holding the concentration of ATP constant, the com-
bined action of cellular ATPases and buffer reactions
renders changes in the concentration of ADP to the range
of tens to hundreds of micromolar, AMP to the range of
nanomolar to micromolar, and inorganic phosphate to
tens of millimolar. The wide physiological range of these
three metabolites then act to increase the glycogenolytic
flux as substrate 共ADP, Pi兲and activators 共AMP兲in the
system.
The ATPase coupling to glycolysis is unique among
the models of glycolysis for other cell types. As with
yeast and red blood cell, the importance of the ATPase in
driving the system emerged with control analysis. How-
ever, while the red blood cell model showed control
coefficients for the ATPase from 0.67 to 0.886 for the
glycolytic fluxes,39 the skeletal muscle shows larger
domination 共0.95–0.99兲by the energetic demands of the
ATPase. As outlined by Hofmeyr25 and further com-
mented upon by Teusink,60 a highly efficient metabolic
network should be controlled from the demand, not the
supply side. This argument also renders meaningless any
physiological discussion of rate-limiting steps within a
pathway such as glycolysis because such an argument
omits the fact that reactions external to the pathway exert
⬎0.95 of the control strength. The historical fallacy of
referring to rate-limiting steps in glycolysis without in-
tegration with other intracellular processes is a point
addressed by Hofmeyr and Cornish-Bowden.26
What Aspects of the Network of Reactions are Able
to Account for the Ability to Achieve and Sustain
a Wide Range of Fluxes?
We attempted to understand the regulation of the path-
way at different rates of ATP utilization through sensi-
tivity analysis and fractional factorial design analysis.
The sensitivity analysis for three different physiologi-
cally relevant fluxes demonstrates a shift in control
within the enzymes for glycolysis as fluxes increase to-
wards maximal muscle activity with glycogen phospho-
rylase becoming the predominant control within the path-
way itself under the highest flux. However, under the
maximal flux conditions, control is more distributed
among pathway enzymes, including near-equilibrium
steps often omitted from computational models 共phos-
phoglucomutase, aldolase, enolase兲. When the phospho-
rylase ratio is altered to a deactivated basal state, glyco-
gen phosphorylase and phosphofructokinase become the
singular source of flux control within the pathway for
resting and moderate exercise, highlighting the depen-
dence on AMP at these steps. These interesting features
of control within the reactions of glycolysis are minor
compared to the control exerted by ATPase.
The results of our factorial design demonstrates that
exploring the complete domain of concentration values
observed in vivo can elicit a different interpretation of
primary controllers in an intact system. The two forms of
analysis provide different information. The sensitivity
analysis allows a straightforward method of altering each
variable and observing the consequence on the system
when each variable is perturbed in a constant proportion
共1%兲. The available range that a variable can achieve in
vivo, however, is not proportional to the order of mag-
nitude of its concentration. For example, ADP in basal
conditions exists at a relatively low concentration com-
pared to other metabolites in the system (⬃0.01 mM). A
sensitivity analysis that perturbs ADP by 1% does not
show significant control in the system. However, ADP
changes ⬎50-fold over the course of muscle activity.
The fractional factorial design is useful in exploring the
upper and lower bounds of variable values, as they exist
in vivo. When we toggle between the basal and fatigue
levels of ADP 共0.01–0.5 mM兲, a larger difference in the
resulting flux relative to other metabolites becomes ob-
vious. The conventional sensitivity analysis, however, is
useful in examining the control exerted by enzyme ac-
tivities as enzyme activity is constant for the model ana-
lyzed here.
The results of the fractional factorial design and the
sensitivity analyses indicate the following: 共1兲The pri-
mary function of the phosphorylase isozyme intraconver-
sion is not necessary to achieve maximal fluxes, but
rather to shift the distribution of pathway control at low
and moderate fluxes. By making the low and moderate
fluxes controlled by the AMP-dependent enzymes, a sig-
naling mechanism can easily be elicited due to the con-
nection of the adenylate kinase buffering to the network
as described in the prior section of this discussion. 共2兲
The apparently conflicting results of the two forms of
analyses actually indicate the common theme of phos-
phorylation potential control of the system. The network
structure of the ATPase with the pathway drives the sys-
tem and contributes to the ATP, ADP, and Piconcentra-
tions which then, through substrate effects, lead to in-
crease or reduction of the flux. Steady states at the
different flux rates are sustainable due to the balanced
stoichiometry of the system.
Despite the success of the present model, the physi-
ological reality can be more complex. Conflicting results
in the human muscle for the transition from rest to maxi-
mal glycolytic fluxes 共significant delay in turning on
820 M. J. LAMBETH and M. J. KUSHMERICK
glycogenolysis15 versus rapid onset43兲demonstrate that
the details of glycolytic flux dynamics are not fully un-
derstood. Although the work presented here focuses on
steady-state fluxes, the kinetic model is capable of dis-
tinguishing hypotheses of transient mechanisms.
Limitations of the Model
Ideally, the kinetic parameters listed in Table 1 would
be obtained from in vivo observations under identical
conditions and from the same species. Unfortunately, no
such data set exists in the literature. This information is
needed for a correct analysis of any particular muscle
cell type. Despite the complexity and numerous param-
eters that are imposed on the model, Eq. 共1兲is itself a
simplification from altered binding affinity for bi–bi re-
actions. This simplification resulted from a lack of infor-
mation on random binding patterns, especially in the
gluconeogenic direction. Thus, the model is limited in
the accurate mathematical representation as well as the
possible parameter values used. However, the appropri-
ateness of using multiple species to create a consensus is
justified by an interspecies similarity in binding constants
across mammals.1,7 Additionally, although interspecies
enzyme concentrations vary, the proportionality of en-
zyme concentrations remains fairly constant.4,47
ACKNOWLEDGMENTS
This work was supported by a graduate fellowship to
one of the authors 共M.J.L.兲from the Whitaker Founda-
tion, and by NIH Grant Nos. AR AR45184 awarded to
K. Conley and AR AR36281 and AR AR41928 awarded
to one of the authors 共M.J.K.兲. Paolo Vicini gave invalu-
able assistance with the use of the SAAM II software.
Zheng Li gave invaluable assistance with the use of the
XSIM software tool. Kevin Conley provided insight into
the use of fractional factorial sensitivity analysis. Bryant
Chase, Kevin Conley, and Greg Crowther patiently read
early versions of the paper and made valuable contribu-
tions.
APPENDIX: MODEL EQUATIONS
Differential Equations
GLY⬘⫽⫺fluxGP ,
G1P⬘⫽fluxGP⫺VPGLM ,
G6P⬘⫽VPGLM⫺VPGI ,
F6P⬘⫽VPGI⫺VPFK ,
FBP⬘⫽VPFK⫺VALD ,
DHAP⬘⫽VALD⫹VTPI ,
GAP⬘⫽VALD⫺VTPI⫺VGAPDH ,
13BPG⬘⫽VGAPDH⫺VPGK ,
3PG⬘⫽VPGK⫺VPGM ,
2PG⫽VPGM⫺VENOL ,
PEP⬘⫽VENOL⫺VPK ,
PYR⬘⫽VPK⫺VLDH ,
LAC⬘⫽VLDH⫺output,
Pi
⬘⫽⫺fluxGP⫺VGAPDH⫹VATPase ,
ADP⬘⫽VPFK⫺VPGK⫺VPK⫹2VADK⫹VCK⫹VATPase ,
ATP⬘⫽⫺VPFK⫹VPGK⫹VPK⫺VADK⫺VCK⫺VATPase ,
AMP⬘⫽⫺VADK ,
PCr⬘⫽VCK ,
Cr⬘⫽⫺VCK ,
NADH⫽VGAPDH⫺VLDH ,
NAD⬘⫽⫺VGAPDH⫹VLDH .
Glycogen Phosphorylase
The user designates the fraction of isozymes Aand B
of glycogen phosphorylase, instead of the inclusion of a
function describing phosphorylase kinase activation in
the model. Detailed allosteric kinetics of the enzyme
available in the literature do not include phosphate as a
substrate in the reaction equations, and therefore, are not
utilized in our model.32 As a result, a rapid-equilibrium
bi–bi equation is used instead. Isozyme Bdiffers from A
by AMP dependence for catalytic activity, described by
an essential cooperative activator term:56
Glycogenn⫹⌺Pi↔Glycogenn⫺1⫹Glucose-1-P
821Modeling of Skeletal Muscle Glycogenolysis
VGPa⫽
Vmax f
冉
GLY Pi
KiGLYfKPi
冊
⫺Vmax r
冉
GLY G1P
KGLYb•KiG1P
冊
冉
1⫹GLY
KiGLYf
⫹Pi
KiPi
⫹GLY
KiGLYb
⫹G1P
KiG1P
⫹GLY Pi
KGLYfKiPi
⫹GLY G1P
KGLYbKiG1P
冊
,
Vmax r⫽Vmax fKGLYbKiG1P
KiGLYfKPiKeqGP ,
VGPb⫽
Vmax f
冉
GLY Pi
KiGLYfKPi
冊
⫺Vmax r
冉
GLY G1P
KiGLYb•KG1P
冊
冉
1⫹GLY
KiGLYf
⫹Pi
KiPi
⫹GLY
KiGLYb
⫹G1P
KiG1P
⫹GLY Pi
KiGLYfKPi
⫹GLY G1P
KiGLYb•KG1P
冊
冉
AMPnH
Kamp
⬘
冊
冉
1⫹AMPnH
KAMP
⬘
冊
,
Vmax r⫽Vmax fKiGLYbKG1P
KGLYfKiPiKeqGP ,
fluxGP⫽fracaVGPa⫹fracbVGPb.
Phosphoglucomutase and Phosphoglucoisomerase
Due to the standard reporting of the PGI Keq in the
gluconeogenic direction, the Haldane relation defines the
Vmax finstead of the Vmax r.
Glucose-1-P↔Glucose-6-P
VPGLM⫽
冉
Vmax fG1P
KG1P
冊
⫺
冉
Vmax rG6P
KG6P
冊
1⫹G1P
KG1P
⫹G6P
KG6P
,
Vmax r⫽Vmax fKG6P
KG1PKeqPGLM .
Glucose-6-P↔Fructose-6-P
VPGI⫽
冉
Vmax fG6P
KG6P
冊
⫺
冉
Vmax rF6P
KF6P
冊
1⫹G6P
KG6P
⫹F6P
KF6P
,
Vmax f⫽Vmax rKG6PKeqPGI
KF6P.
Phosphofructokinase
We do not include a major PFK inhibitor, citrate,
because studies of the effects on skeletal muscle PFK at
a physiological range of citrate concentration showed
very little inhibition,46 nor do we include pH effects of
PFK within the scope of this paper. Several authors have
modeled the skeletal muscle PFK kinetics based on in
vitro assays, resulting in widely disparate descriptions of
the dynamic behavior.59,65 Due to our requirements that
the equation be thermodynamically valid and reversible,
we used a reversible two-substrate allosteric equation of
the form derived by Popova and Selkov,48 with allosteric
term Ldefined as described by Nagata et al.40 The L
term includes AMP activation and ATP inhibition. The
notation K⬘represents altered binding properties due to
the active state of the enzyme.
Fructose-6-P⫹⌺ATP↔Fructose-1,6-P⫹⌺ADP
VPFK⫽
冉
Vmax f
冉
ATP F6P
KATP KF6P
冊
⫺Vmax r
冉
ADP FBP
KADPKFBP
冊
⌬
冊
⫻
冉
1⫹
␣
L
冉
⌬⬘
⌬
冊
3
1⫹L
冉
⌬⬘
⌬
冊
4
冊
,
⌬⫽
冉
1⫹F6P
KF6P
冊
•
冉
1⫹ATP
KATP
冊
⫹ADP
KADP
⫹FBP
KFBP
⫻
冉
1⫹ADP
KADP
冊
,
822 M. J. LAMBETH and M. J. KUSHMERICK
⌬⬘⫽
冉
1⫹F6P
KF6P
⬘
冊
•
冉
1⫹ATP
KATP
⬘
冊
⫹ADP
KADP
⬘⫹FBP
KFBP
⬘
⫻
冉
1⫹ADP
KADP
⬘
冊
,
␣
⫽KF6PKATP
KF6P
⬘KATP
⬘,
L⫽Lo
冋
冉
1⫹ATP
KiATP
1⫹dATP
KiATP
冊
•
冉
1⫹eAMP
KaAMP
1⫹AMP
KaAMP
冊
册
4
,
Vmax r⫽Vmax fKADPKFBP
KATP KF6P.
Aldolase and Triose Phosphate Isomerase
Fructose-1,6-P↔Glyceraldehyde-3-P
⫹Dihydroxyacetone-P
VALD⫽
冉
Vmax fFBP
KFBP
冊
⫺
冉
Vmax rDHAP GAP
KDHAPKGAP
冊
1⫹FBP
KFBP
⫹DHAP
KDHAP
⫹GAP
KGAP
,
Vmax r⫽Vmax fKDHAPKGAP
KFBPKeqALD .
The forward direction of TPI is defined as producing
dihydroxyacetone phosphate.
Glyceraldehyde-3-P↔Dihydroxyacetone-P
VTPI⫽
冉
Vmax fGAP
KGAP
冊
⫺
冉
Vmax rDHAP
KDHAP
冊
1⫹GAP
KGAP
⫹DHAP
KDHAP
,
Vmax r⫽Vmax fKDHAP
KGAPKeqTPI .
Glyceraldehyde-3-Phosphate Dehydrogenase (GAPDH)
Although various sources have attributed the kinetics
to random ter–bi rapid equilibrium,21,58 Orsi and Cleland
produced data to support an ordered mechanism,42 reduc-
ing the complexity of the denominator by limiting pos-
sible enzyme complexes.
Glyceraldehyde-3-P⫹NAD
⫹⌺Pi↔1,3-bisphosphoglycerate⫹NADH
VGAPDH
⫽
冉
Vmax fGAP NAD Pi
KGAPKNADKPi
冊
⫺
冉
Vmax r13BPG NADH
K13BPGKNADH
冊
DGAPDH ,
DGAPDH⫽1⫹GAP
KGAP
⫹NAD
KNAD
⫹Pi
KPi
⫹GAP NAD
KGAPKNAD
⫹GAP NAD Pi
KGAPKNADKPi
⫹13DPG
K13DPG
⫹NADH
KNADH
⫹13BPG NADH
K13BPGKNADH ,
Vmax r⫽Vmax fK13BPGKNADH
KGAPKNADKPiKeqGAPDH .
Phosphoglycerate Kinase
Experimental data for the altered dissociation con-
stants of PGK due to the binding of one substrate/
product are not available. Therefore, the reaction equa-
tion was constructed as a rapid equilibrium ordered bi–bi
equation as described in Eq. 共1兲. Like PGI, this reaction
is usually reported with the forward direction towards
gluconeogenesis; in keeping with this definition, the
Haldane relation defines the Vmax f:
1,3-Bisphosphoglycerate⫹⌺ADP↔3-Phosphoglycerate
⫹⌺ATP
823Modeling of Skeletal Muscle Glycogenolysis
VPGK⫽
Vmax f
冉
13BPG ADP
K13BPGKADP
冊
⫺Vmax r
冉
3PG ATP
K3PGKATP
冊
1⫹13BPG
K13BPG
⫹ADP
KADP
⫹13BPG ADP
K13BPGKADP
⫹3PG
K3PG
⫹ATP
KATP
⫹3PG ATP
K3PGKATP
,
Vmax f⫽Vmax rK13BPGKADPKeqPGK
K3PGKAT P .
Phosphoglyceromutase and Enolase
3-Phosphoglycerate↔2-Phosphoglycerate
VPGM⫽
冉
Vmax f3PG
K3PG
冊
⫺
冉
Vmax r2PG
K2PG
冊
1⫹3PG
K3PG
⫹2PG
K2PG
,
Vmax r⫽Vmax fK2PG
K3PGKeqPGM .
2-Phosphoglycerate↔Phosphoenolpyruvate
VENOL⫽
冉
Vmax f2PG
K2PG
冊
⫺
冉
Vmax rPEP
KPEP
冊
1⫹2PG
K2PG
⫹PEP
KPEP
,
Vmax r⫽Vmax fKPEP
K2PGKeqENOL .
Pyruvate Kinase (PK)
Pyruvate kinase is often considered an allosteric en-
zyme in glycolysis; however, the muscle isozyme repeat-
edly shows a lack of cooperativity, and thus has been
assigned a Hill coefficient of 1.26 ATP has been reported
to competitively inhibit ADP in the forward direction,
however, these data were produced under initial rate con-
ditions. We assume that the steady-state equation with
the presence of the product in the denominator appropri-
ately describes the kinetics. The equation for PK is in
rapid equilibrium bi–bi form; several sources describe
the reaction as random, however, insufficient data are
available for the secondary binding constants in the glu-
coneogenesis direction:
Phosphoenolpyruvate⫹⌺ADP↔Pyruvate⫹⌺ATP
VPK⫽
冉
Vmax fPEP ADP
KPEPKADP
冊
⫺
冉
Vmax rPYR ATP
KPYRKATP
冊
1⫹PEP
KPEP
⫹ADP
KADP
⫹PEP ADP
KPEPKADP
⫹PYR
KPYR
⫹ATP
KATP
⫹PYR ATP
KPYRKATP
,
Vmax r⫽Vmax fKATP KPYR
KPEPKADPKeqPK .
Lactate Dehydrogenase (LDH)
Pyruvate⫹NADH↔Lactate⫹NAD
VLDH⫽
冉
Vmax fPYR NADH
KPYRKNADH
冊
⫺
冉
Vmax rLAC NAD
KLACKNAD
冊
1⫹PYR
KPYR
⫹NADH
KNADH
⫹PYR NADH
KPYRKNADH
⫹LAC
KLAC
⫹NAD
KNAD
⫹LAC NAD
KLACKNAD
,
824 M. J. LAMBETH and M. J. KUSHMERICK
Vmax r⫽Vmax fKLACKNAD
KPYRKNADHKeqLDH .
ATP Reactions: Creatine Kinase, Adenylate Kinase,
ATPase
The equation for creatine kinase represents a Cleland
共random ordered兲bi–bi reaction without dead-end prod-
ucts. The equation and parameters are from Shimerlik
and Cleland as used in a published model by Vicini and
Kushmerick.64
Phosphocreatine⫹⌺ADP↔Creatine⫹⌺ATP
VCK
⫽
Vmax rATP Cr
KiATP KCr
⫺Vmax fADP PCr
KiADPKPCr
1⫹ADP
KiADP
⫹PCr
KiPCr
⫹ADP PCr
KiADP•KPCr
⫹ATP
KiATP
⫹ATP Cr
KiATP KCr
,
Vmax f⫽Vmax rKiATP KCrKeqCK
KiADPKPCr .
⌺ATP⫹⌺AMP↔2⌺ADP
VADK
⫽
冉
Vmax fATP AMP
KATP KAMP
冊
⫺
冉
Vmax rADP2
KADP
2
冊
1⫹ATP
KATP
⫹AMP
KAMP
⫹ATP AMP
KATP KAMP
⫹2ADP
KADP
⫹ADP2
KADP
2
,
Vmax r⫽Vmax fKADP
2
KATP KAMPKeqADK .
ATPase is defined as a simple mass action flux:
⌺ATP→⌺ADP⫹⌺Pi,
VATPase⫽kATP.
NOMENCLATURE
GLY 共glycogen兲
G1P 共glucose 1-phosphate兲
G6P 共glucose 6-phosphate兲
F6P 共fructose 6-phosphate兲
FBP 共fructose 1,6-bisphosphate兲
DHAP 共1,3-dihydroxyacetone phosphate兲
GAP 共glyceraldehyde 3-phosphate兲
1,3BPG 共glycerate 1,3-bisphosphate兲
3PG 共glycerate 3-phosphate兲
2PG 共glycerate 2-phosphate兲
PEP 共phosphoenolpyruvate兲
PYR 共pyruvate兲
LAC 共lactate兲
Pi共inorganic phosphate兲
PCr 共phosphocreatine兲
Cr 共creatine兲
GP 共glycogen phosphorylase, E.C. 2.4.1.1兲
PGLM 共phosphoglucomutase E.C. 5.4.2.2兲
PGI 共phosphoglucose isomerase, E.C. 5.3.1.9兲
PFK 共6-phosphofructokinase, E.C. 2.7.1.11兲
ALD 共fructose-bisphosphate aldolase, E.C. 4.1.2.13兲
TPI 共triose-phosphate isomerase, E.C. 5.3.1.1兲
GAPDH 共glyceraldehyde-3-phosphate dehydrogenase,
E.C. 1.2.1.12兲
PGK 共phosphoglycerate kinase, E.C. 2.7.2.3兲
PGM 共phosphoglycerate mutase, E.C. 5.4.2.1兲
ENOL 共phosphopyruvate hydratase, E.C. 4.2.1.11兲
PK 共pyruvate kinase, E.C. 2.7.1.40兲
LDH 共L-lactate dehydrogenase, E.C. 1.1.1.28兲
CK 共creatine kinase, E.C. 2.7.3.2兲
ADK 共adenylate kinase, E.C. 2.7.4.3兲
ATPase 共myosin ATPase, E.C. 3.6.4.1兲
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827Modeling of Skeletal Muscle Glycogenolysis
