(Received for publication, December 5, 1995)
From the
Human erythrocytes are among the simplest of cells. Many of their enzymes have been characterized kinetically using steady-state methods in vitro, and several investigators have assembled this kinetic information into mathematical models of the integrated system. However, despite its relative simplicity, the integrated behavior of erythrocyte metabolism is still complex and not well understood. Errors will inevitably be encountered in any such model because of this complexity; thus, the construction of an integrative model must be considered an iterative process of assessment and refinement. In a previous study, we selected a recent model of erythrocyte metabolism as our starting point and took it through three stages of model assessment and refinement using systematic strategies provided by biochemical systems theory. At each stage deficiencies were diagnosed, putative remedies were identified, and modifications consistent with existing experimental evidence were incorporated into the working model. In this paper we address two issues: the propagation of biochemical signals within the metabolic network, and the accuracy of kinetic representation. The analysis of signal propagation reveals the importance of glutathione peroxidase, transaldolase, and the concentration of total glutathione in determining systemic behavior. It also reveals a highly amplified diversion of flux between the pathways of pentose phosphate and nucleotide metabolism. In determining the range of accurate representation based on alternative kinetic formalisms we discovered large discrepancies. These were identified with the behavior of the model represented within the Michaelis-Menten formalism. This model fails to exhibit a nominal steady state when the activity of glutathione peroxidase is decreased by as little as 9%. Our current understanding, as embodied in this working model, is in need of further refinement, and the results presented in this paper suggest areas of the model where such effort might profitably be concentrated.
The primary physiological function of the red blood cell is to bind oxygen in the lung and transport it to oxygen-consuming cells elsewhere in the body. In addition, there are several metabolic functions that a red blood cell must perform for its own survival. These include generation of metabolic energy (e.g. ATP), generation of reducing agents (e.g. NADH and NADPH), generation of 2,3-diphosphoglycerate, and maintenance of ionic and concentration gradients across the cell membrane (for detailed reviews, see e.g.(1) and (2) ). These functions involve a minimum set of metabolic pathways, including the glycolytic sequence, the pentose phosphate pathway, the 2,3-diphosphoglycerate shunt, and the pathways of nucleotide metabolism; the ability to synthesize RNA, protein, lipid, and purines has been lost. Over a period of 3 decades most of the molecular components that constitute the pathways of erythrocyte metabolism have been experimentally characterized. The kinetic data that have been accumulated are derived for the most part from measurements made in vitro, rather than in vivo, and they have been obtained under various experimental conditions (e.g. pH and temperature) from many independent laboratories. These data have formed the basis for a number of kinetic models dealing with various aspects of the human erythrocyte (e.g. 3-16).
The elucidation of a complex biochemical system like the metabolism of human erythrocytes is no trivial task(17) . It is not sufficient to have detailed kinetic information for the individual reactions considered in isolation, even if this information were to reflect accurately the actual conditions in vivo. One needs in addition a systematic framework for integrating this type of information into a model of the intact system and rigorous methods for extracting the systemic implications latent within such a model. Moreover, if this approach is to succeed, one must demonstrate that the model is of sufficient quality to warrant our confidence.
In developing an integrated model of metabolism in human
erythrocytes we have shown that the need for a systematic framework and
rigorous methodology can be met by the power-law formalism ()and biochemical systems theory (18, 19) .
In the process, a comprehensive model of metabolism in human
erythrocytes (12, 13, 14, 15) has
been taken through three stages of assessment and refinement in an
effort to develop a model that would justify a detailed systemic
analysis. We found that the original model did not exhibit an
appropriate steady state; this deficiency was identified with several
inconsistencies, and these were resolved to produce a new reference
model(18) . Next we found that the steady state was unstable; a
systematic search for interactions that could stabilize the steady
state revealed one for which there is independent experimental support,
and this was included in the model(18, 19) . Finally,
we found that the model lacked robustness; the pattern of parameter
sensitivities identified the pathways of nucleotide metabolism with
this problem, and the addition of previously overlooked mechanisms to
the model improved robustness(18, 19) . The result to
emerge from this process (Model III) now will be subject to systemic
analysis.
In this paper we shall first analyze the pattern of biochemical signal propagation that determines the steady-state behavior of metabolism in erythrocytes. This analysis reveals the importance of glutathione peroxidase, transaldolase, and the concentration of total glutathione in determining the systemic concentrations and fluxes, as well as the critical behavior of nucleotide metabolism. Second, we analyze the range of accurate representation by comparing the behavior of alternative representations. This analysis shows that the nominal steady state for this model exists only over a very narrow range of variation in some of the enzyme activities and suggests that the model is still in need of refinement. Taken together, the results suggest areas of the model where additional experimental effort might be focused.
MODEL OF METABOLISM IN HUMAN RED BLOOD CELLS
A schematic diagram of Model III representing metabolism in
human red blood cells is given in Fig. 1. The fundamental
equations that characterize any such metabolic network are
Kirchhoff's node equations, which also are referred to as mass
balance equations. These can be written in the following form ()
Figure 1:
Schematic representation of metabolism
in human red blood cells. The names of the metabolites corresponding to
the dependent variables in this model are given in Table 2. The
full names of enzymes or reactions are given in Table 3. The
concentration of reduced glutathione (GSH) X
is a constrained variable whose
value must equal the difference between the total concentration of
glutathione (G
) X
, which is an
aggregate variable that is also independent, and the concentration of
oxidized glutathione (GSSG) X
, which is a
dependent variable in this model. Other independent variables
corresponding to fixed, extracellular, or aggregate metabolites, some
of which are shown explicitly in this diagram, include the
concentrations of glucose (GLC) X
, adenine (ADE) X
, CO
X
, total
magnesium (MG
) X
, total nicotinamide
adenine dinucleotide (N
) X
, total
nicotinamide adenine dinucleotide phosphate (NP
) X
, inorganic phosphate (P
) X
, extracellular lactate (LAC
) X
, extracellular pyruvate (PYR
) X
, extracellular potassium (K
) X
, extracellular sodium (Na
) X
, and extracellular hypoxanthine
(HX
) X
. Arrows pointing in
one direction represent essentially irreversible reactions; arrows pointing in both directions represent reversible reactions. The
rate laws, which account for regulatory interactions not shown in this
diagram, are given in (19) .
where i = 1, 2, . . . n; an individual
metabolite concentration is represented by X, an individual flux from X
to X
is
represented by
, aggregate influx and
aggregate efflux for the X
pool are given
by V
and V
, and the numbers of
dependent and independent concentration variables are n and m(20) . Kirchhoff's node equations for the
specific system in Fig. 1are given in Table 1, using both
the above notation, which correlates the flux with its substrate and
product, and common mnemonic abbreviations.
The 33 dependent
concentration variables of this system are enumerated in the third
column of Table 1. The individual fluxes entering and leaving
each node are given in the fourth column. The numbering in this case
involves 58 independent concentration variables as well as the 33
dependent concentration variables. The grouping of individual fluxes
indicates an aggregation strategy in which the net flux through a
reaction is considered as a single flux. This is one of the most common
forms of aggregation in the biochemical literature(21) . Other
aggregation strategies will be discussed in subsequent sections. The
names of the dependent metabolites abbreviated in the first column of Table 1are given in Table 2. The subscripts of the s
in the second column are abbreviations for enzymes or reactions (see Table 3for full names and EC numbers). The rate laws for these
reactions are represented in most cases by conventional expressions
within the Michaelis-Menten formalism. The detailed rate laws are given
elsewhere(19) .
In addition to the 33 dependent variables of this model, there are 43 independent variables corresponding to the activities of the reactions numbered 34-76, another 15 independent variables corresponding to the concentration variables numbered 77-91 (see Table 4), and one constrained variable corresponding to the concentration of reduced glutathione (numbered 92), whose value is determined by the difference between the total concentration of glutathione, which is an independent variable, and oxidized glutathione, which is a dependent variable.
The steady-state values for the concentrations of this model are calculated by conventional methods (22) and are shown in Table 2along with the nominal steady state that is observed experimentally. The steady-state values calculated for aggregate fluxes through the reactions of this model are shown in Table 3.
We follow the customary procedure in systems analysis of distinguishing between dependent and independent variables and between steady state and transient behavior(23) . The values for the dependent concentrations and fluxes of a biochemical system are determined by the internal dynamics of the system as dictated by Kirchhoff's node laws. The values for the independent concentrations and fluxes are determined independently by agents outside of the system; these variables can be fixed by experimental means, and they characterize the environment of the system. For our purposes the steady-state behavior is that which persists in a constant environment. The transient behavior is that which is exhibited by the system as it progresses from one steady state to another.
where i = 1, 2, . . .n
and the subscript 0 indicates that the results are evaluated at
the nominal steady state. The rate constants of the aggregate rate laws
are and
; the kinetic orders of the
aggregate rate laws are given by g
and h
. These variables and parameters are all
nonnegative quantities, except for the kinetic orders, which have
negative values when they represent inhibitory interactions. The
explicit S-system representation for Model III is given in the
Appendix.
The steady-state solution of can be obtained explicitly (24) and expressed in matrix form as
where (in matrix form)
and
Subscripts d and id signify that the corresponding matrices (or
vectors) contain kinetic orders (or logarithms of concentrations) for
dependent and independent variables, respectively. Thus, the solution
vector for the logarithms of the dependent variables Y] is a linear function of the logarithms of
the independent variables Y]
; the constant
matrix [L] is the slope, and [M] b] is the intercept.
The steady-state fluxes can be determined from the appropriate rate law and the steady-state metabolite concentrations given in . Within the S-system representation, steady-state fluxes can be expressed in the following matrix notation.
Henceforth, we shall refer to the flux through the X pool in steady state simply as V
, since V
= V
= V
under
these conditions. Other fluxes can be defined by appropriate rate laws
within the power-law formalism, and their steady-state values can be
determined in exactly the same fashion from the steady-state values of
the concentration variables.
The explicit solution in and completely determines the steady-state values for the dependent concentrations and fluxes through pools in terms of the values for the independent metabolite concentrations and the parameters of the S-system representation.
The logarithmic gain in a dependent variable X with respect to change in an independent
variable X
is defined as follows
where i = 1, 2, . . . n; j = n+1, n+2, . . . n+m; and again the subscript 0 indicates values determined at the steady state. To obtain the theoretically expected value, one simply differentiates the explicit solution for the dependent variable () with respect to the independent variable. The full set of these logarithmic gain factors can be written in the form of an n by m matrix.
The elements of the M and A matrices contain all of the kinetic orders and only the kinetic orders. Thus, these logarithmic gains are clearly systemic properties that in general depend upon all of the molecular interactions within the system (25) .
Similarly, the logarithmic gain in a dependent
variable V with respect to change in an
independent variable X
is defined as follows
where i = 1, 2, . . . n and j = n+1, n+2, . . . n+m.
To obtain the theoretically expected value, one simply differentiates the explicit solution for the dependent variable () with respect to the independent variable. The full set of these logarithmic gain factors also can be written in the form of an n by m matrix
where, again, the subscripts d and id signify that the corresponding matrices contain kinetic orders for dependent and independent variables, respectively.
As noted above, other fluxes can be defined by appropriate rate laws within the power-law formalism, and their logarithmic gains can be determined in exactly the same fashion from the logarithmic gains in the concentration variables. For example, given that the forward and reverse processes for the enzyme transketolase 1 have rate laws (19)
The corresponding logarithmic gains in flux that result from a
change in the concentration of the enzyme glutathione peroxidase (X) are given by
where the logarithmic gains in concentration are obtained from the primary solution (). The logarithmic gain for the net flux in the forward direction can then be obtained by the appropriate average of the logarithmic gains for the fluxes in the forward and reverse directions.
In general there is no explicit steady-state solution for the Michaelis-Menten model and, hence, no explicit expression for the logarithmic gains. Nevertheless, for purposes of comparison one can approximate the logarithmic gains of this model by an empirical procedure. First, make a small finite change in an independent variable. Second, determine the resulting change in the steady-state value of the dependent variable. We use the Newton-Raphson method for nonlinear systems (20, 22) to calculate a numerical approximation to the steady-state concentrations and fluxes. Finally, take the appropriate ratios. The results can be expressed as follows
where the subscript 0 indicates values determined at the steady
state. The difference between X and X
is a small finite value
. As
approaches zero, the empirical values for these logarithmic gains
converge to the theoretically expected values obtained for the S-system
model(20) .
A logarithmic gain with a magnitude greater than 1 implies amplification of the original signal; a magnitude less than 1 indicates attenuation. A positive sign for the logarithmic gain indicates that the changes are in the same direction, both increase in value or both decrease. A negative sign indicates that the changes are in the opposite direction.
The influence of a given
independent variable on a particular dependent variable is given by the
magnitude of the corresponding logarithmic gain, e.g. L(V
,X
)
or
L(X
,X
)
.
The total influence over a particular dependent variable is
given by the sum of the individual influences by each independent
variable.
We shall use two alternative measurements of fidelity for a kinetic model. First is the error of representation, which is defined as the difference between the output signals generated by the model and the reference in response to a small change in their input signals. In operational terms, one makes a 2% change in an independent variable, determines the response of a dependent variable, and measures the percentage difference between the predicted response of the model and the actual response of a reference system. Second is the range of accurate representation, which is defined as the range of variation in input signals within which the output signals of the model and the reference system being represented differ by less than a specified tolerance. In this case one makes changes of increasing size in an independent variable, determines the corresponding responses of a dependent variable, measures the percentage difference between the predicted response of the model and the actual response of a reference system, and determines the range of variation in the independent variable for which the percentage difference is within 10%.
A schematic representation of metabolism in human
erythrocytes is given in Fig. 1. Although the regulatory
interactions are not shown in this schematic, they are incorporated
into the detailed rate laws, which are summarized in the Appendix. The
names of the 33 dependent metabolite concentrations are given in Table 2; those of the 58 independent variables are given in Table 3and Table 4. These variables are grouped into four
categories in an effort to facilitate the recognition of significant
patterns in the profiles of logarithmic gains and accuracy. Group I contains variables related to the pentose pathway. Dependent
variables are X to X
and V
to V
; independent are X
to X
.
Group II
contains variables related to glycolysis. Dependent variables are X to X
and V
to V
; independent are X
to X
.
Group III
contains variables related to nucleotide metabolism. Dependent
variables are X to X
and V
to V
; independent are X
to X
.
Group IV
contains variables related to aggregate and extracellular metabolites.
Independent variables are X to X
.
The logarithmic gains characterize the propagation of biochemical signals throughout the system, and they reflect the influence exerted by independent variables on the dependent variables of the intact system. These gains are obtained from the explicit solution for the steady state that is available within the framework of the S-system representation of biochemical systems theory (see under ``Methods''). The results are summarized in Fig. 2Fig. 3Fig. 4.
Figure 2:
Influence of independent variables on
metabolite concentrations as determined by the magnitudes of the
logarithmic gains. The three-dimensional plot displays the magnitudes
of the logarithmic gains as a function of the metabolite concentrations X and the independent variables X
. Logarithmic gains with magnitudes less
than 1/1000 are shown as black squares. The two-dimensional
projection on the right gives the magnitudes for a particular
metabolite concentration X
summed over
all the independent variables X
. The
two-dimensional projection on the left gives the magnitudes
for an independent variable X
summed over
all the metabolite concentrations X
. The solid bars in each projection represent the sum of the
positive gains, and the hatched bars represent the sum of the
negative gains.
Figure 3:
Influence of independent variables on
fluxes through metabolite pools as determined by the magnitudes of the
logarithmic gains. The three-dimensional plot displays the magnitudes
of the logarithmic gains as a function of the fluxes V and the independent variables X
. Logarithmic gains with magnitudes less
than 1/1000 are shown as black squares. The two-dimensional
projection on the right gives the magnitudes for a particular
flux V
summed over all the independent
variables X
. The two-dimensional
projection on the left gives the magnitudes for an independent
variable X
summed over all the fluxes V
. The solid bars in each
projection represent the sum of the positive gains, and the hatched
bars represent the sum of the negative
gains.
Figure 4:
Simplified schematic of metabolism in
human erythrocytes. Only key branch points and net fluxes are
represented. The three aggregate nodes, which also must satisfy
Kirchhoff's node law, include the proximal portion of the pentose
phosphate pathway, the distal portion of the pentose phosphate pathway
and most of glycolysis, and nucleotide metabolism. The net fluxes can
be identified with the following enzymes: F,
hexokinase; F
, glucose-6-phosphate dehydrogenase; F
, xylulose-5-phosphate isomerase; F
, adenine phosphoribosyl transferase; F
, hypoxanthine transport process; F
, inosine transport process; F
, phosphoglucoisomerase; F
,
ribose-5-phosphate isomerase; F
, transketolase I; F
, adenosine transport process; and F
, enolase. The numbers associated with the net
fluxes are their logarithmic gains in response to a change in the
concentration of glutathione peroxidase X
, except
in the cases of F
and F
. In
these two cases the net flux is zero at steady state, so the
logarithmic gains are undefined. Instead, we have shown with an asterisk (*) the logarithmic gain for the efflux, which is
indicative of the change in net flux because the influx is unchanged.
See Fig. 1for a more detailed metabolic map of this
system.
First, there are five metabolites whose concentrations are
unaffected by change in most independent variables (X, X
, X
, X
, and X
). This can be seen most clearly in the
two-dimensional bar chart projected on the right in Fig. 2, where the magnitudes for a given metabolite
concentration X
with respect to change in the
independent variables X
are summed over all j. The solid portion of each bar represents the sum
of the positive logarithmic gains, whereas the hatched portion
represents the sum of the negative logarithmic gains. The concentration
of gluconolactone 6-phosphate X
has a pattern of
influence that is indicative of a rate-limiting step in the pentose
phosphate pathway. Gluconolactone 6-phosphate is the substrate of an
essentially irreversible reaction, and, with the exception of changes
in the activity of the enzyme that catalyzes this reaction, its
concentration should vary directly with the flux through the reaction.
Aside from the exception, X
is influenced only by X
(1.00) and X
(1.00). This
indicates that the flux through the proximal portion of the pentose
phosphate pathway is limited by the activity of glutathione peroxidase X
and by the concentration of total glutathione X
. This is to be expected because the pool of
reduced glutathione (GSH) is at its maximum, essentially equal to the
concentration of total glutathione (G
) X
, and the rate law for glutathione peroxidase is
represented as a simple first-order process (the product of substrate
concentration and a rate constant proportional to enzyme activity). The
concentration of gluconolactone 6-phosphate X
varies inversely with changes in the level of the enzyme
6-phosphogluconolactonase X
(-1.00), which
is expected in this case because the flux is fixed. The concentration
of oxidized glutathione X
is affected only by
changes in X
(0.944), X
(0.997), and the level of glutathione reductase X
(-0.979). The explanation is the same as that given above; X
varies directly with the independent variables
that determine the flux and inversely with the level of the enzyme for
which it is a substrate. The concentration of lactate X
is influenced primarily by the level of the lactate transport
process X
(-0.120) and by the concentration
of extracellular lactate X
(0.879). The
concentration of NADH X
is influenced by the
concentrations of extracellular lactate X
(0.575)
and extracellular pyruvate X
(-0.662), and
by the concentration of total nicotinamide adenine dinucleotide X
(0.992). The concentration of pyruvate X
is unaffected by changes in almost all
independent variables. This is indicated by the row of black
squares for the dependent variable X
. This
invariance can be understood as follows. In steady state the
concentration of pyruvate within the cell must equal the concentration
of pyruvate outside the cell, which is a constant in this model,
otherwise there would be some net flux entering or leaving the
intracellular pool X
. If there were such a flux,
then the flux through glyceraldehyde phosphate dehydrogenase would not
equal the flux through lactate dehydrogenase, and there could be no
steady state for the pools of NAD and NADH. Thus, the only independent
variable that has an influence on the intracellular concentration of
pyruvate is the concentration of extracellular pyruvate X
(1.00).
Second, the seven metabolite
concentrations most influenced by change in the independent variables
are those for gluconate 6-phosphate X,
nicotinamide adenine dinucleotide phosphate X
,
sedoheptulose 7-phosphate X
, glucose 6-phosphate X
, fructose 6-phosphate X
,
5-phosphoribosyl-1-pyrophosphate X
, and inosine X
. This can be seen most clearly in the
two-dimensional bar chart projected on the right in Fig. 2. There are large negative logarithmic gains as well as
positive ones. The distribution of total influence exerted
over each of the metabolite concentrations varies from a low of
1.00-3.48 for concentrations associated with the first pattern (X
, X
, X
, X
, and X
) to a high of 22.7-35.2 for
concentrations associated with the second pattern (X
, X
, X
, X
, X
, X
, and X
). The large gradient in the changes between
sedoheptulose 7-phosphate X
(35.2) and erythrose
4-phosphate X
(4.97), and the large increases in
5-phosphoribosyl-1-pyrophosphate X
(22.7),
suggest that the flux through the distal portion of the pentose
phosphate pathway is restricted, and that into nucleotide metabolism is
augmented. Further support for this hypothesis is given below.
Third, the independent variables that have the greatest total influence on the metabolite concentrations are the activities of
the enzymes glutathione peroxidase X (42.1) and
transaldolase X
(40.4) and the concentration of
total glutathione X
(42.1). This can be seen in
the two-dimensional projection on the left, where the
magnitudes of the logarithmic gains for the dependent concentrations X
with respect to change in a given independent
variable X
are summed over all i. Again,
contributions from positive and negative logarithmic gains are shown
separately by solid and hatched bars. These three
variables determine the flux through the pentose phosphate pathway and
its diversion into nucleotide metabolism. Those variables with the next
greatest level of total influence are the concentrations of total
phosphate X
(31.4) and total magnesium X
(23.9) and the activities of the enzymes
hexokinase X
(27.9) and adenosine monophosphate
phosphohydrolase X
(26.4), all of which are
concerned with nucleotide metabolism.
Fourth, there are many
independent variables that have almost no effect on any of the
metabolite concentrations in the model. The distribution of total influence exerted by each of these independent variables ranges
from 0 to 2.08. As can be seen in the two-dimensional projection on the left in Fig. 2, these are associated with the pentose
phosphate pathway (X to X
, X
, and X
) glycolysis (X
, X
, X
, X
, X
, X
to X
, X
, X
, and X
), and nucleotide
metabolism (X
, X
to X
, X
, X
, X
, X
, X
, and X
). This list includes the activities for 13 of
the 16 fast reactions; it also includes the activities for four
reactions that follow a rate-limiting step and three reactions involved
in transport of intra- and extracellular metabolites that are near
equilibrium.
Fifth, if one sums the signed logarithmic
gains with respect to the enzyme concentrations (X through X
), one sees that the value is zero
for each metabolite concentration (data not shown). Although this is
not generally true(26) , it is to be expected for a model, like
the one analyzed in this paper, in which each of the rate laws is
assumed to be independent and proportional to the concentration of the
corresponding enzyme(26, 27, 28) .
First,
five of the fluxes are unaffected by change in most of the independent
variables. This can be clearly seen from the two-dimensional projection
on the right, where the magnitudes of the logarithmic gains
for a given flux V with respect to change in the
independent variables X
are summed over all j. The solid portion of each bar represents the sum
of the positive logarithmic gains, whereas the hatched portion
represents the sum of the negative logarithmic gains. The fluxes
through the pools of gluconolactone 6-phosphate V
and glutathione V
are affected only by two
independent variables, the activity of glutathione peroxidase X
(0.992) and the intracellular concentration of
total glutathione X
(0.992). This is because the
pool of reduced glutathione (GSH) is at its maximum, essentially equal
to the concentration of total glutathione (G
), and thus the
flux through the pentose phosphate pathway is limited by the activity
of glutathione peroxidase X
and the intracellular
concentration of total glutathione X
. The flux
through the pool of sodium V
is a function only
of the level of sodium leakage X
(1.00) and of
the concentration of external sodium X
(0.419)
and is therefore insensitive to change in the value of all other
independent variables. The flux through the pool of potassium V
is only affected by changes in the activities
of the pump X
(0.322) and the leakage mechanisms
for sodium X
(0.170) and potassium X
(0.494); all other influences that would be
mediated via changes in ATP have no effect because the ATP
concentration is far in excess of its K
for the
pump(18) . The flux through 2,3-diphosphoglycerate V
is largely a function of diphosphoglycerate
phosphatase levels X
(0.959) because this enzyme
is operating at nearly its maximum rate with substrate concentrations
far in excess of its K
.
Second, the fluxes that
are most influenced by change in the independent variables are those
through the pools of nucleotide metabolism (adenosine monophosphate V, adenosine diphosphate V
,
adenosine triphosphate V
, inosine V
, hypoxanthine V
, and
ribose 1-phosphate V
) followed by those through
glycolytic intermediates (glucose 6-phosphate V
and fructose 6-phosphate V
) and the pentose
phosphate pathway (ribose 5-phosphate V
). This can
be seen from the two-dimensional projection on the right,
which shows large negative logarithmic gains as well as positive ones.
The distribution of total influence exerted over each of the
fluxes varies from a low of 1.28-2.40 for fluxes associated with
the first pattern (V
, V
, V
, V
, V
) to a high of 19.5-29.1 for the fluxes
associated with the second pattern (V
, V
, V
, V
, V
, V
, V
, V
, V
).
Third, the
independent variables that have the greatest total influence
on the dependent fluxes are the activities of the enzymes glutathione
peroxidase X (38.1) and transaldolase X
(33.4) and the concentration of total
glutathione X
(38.1). This can be seen in the
two-dimensional bar chart projected on the left. Increases in X
and X
and decreases in X
lead to large increases in flux through the
lower portions of the pentose phosphate pathway and nucleotide
metabolism. This will be examined below from another perspective (see Fig. 4). The independent variables with the next greatest total
influence are total phosphate X
(27.1) and total
magnesium X
(20.2) and the activities of the
enzymes hexokinase X
(21.9) and adenosine
monophosphate phosphohydrolase X
(25.5). This is
the same qualitative pattern that was observed for the metabolite
concentrations (see Fig. 2).
Fourth, many independent
variables have almost no effect on any of the fluxes in the model (see
the two-dimensional projection on the left in Fig. 3).
Among these are metabolites and activities of enzymes associated with
the pentose phosphate pathway (X to X
, X
, and X
). This is to be expected since the flux through
this pathway is limited by the activity of glutathione peroxidase X
, as noted above. Also among these are
metabolites and activities associated with glycolysis (X
, X
, X
, X
, X
, X
, X
, X
, X
, and X
) and nucleotide
metabolism (X
, X
, X
, X
, X
, X
, X
, and X
). Many, though not
all, of these reactions are fast, and some fast reactions are not
included in this list. Again, the pattern is very similar to that for
the metabolite concentrations.
Fifth, if one sums the signed logarithmic gains with respect to the enzyme concentrations (X through X
), one observes
that the value is unity for each flux (data not shown). Again, this is
to be expected for the special case of a model that is assumed to be
homogeneous of degree one in the enzyme concentrations, although it is
of no general biochemical significance. There are negative as well as
positive values in each of these sums, and the negative values can be
quite large, which is frequently the case(26, 28, 29, 30, 31, 32, 33) and
contrary to what some investigators have claimed. This provides further
evidence against the notion that a sum of unity is equivalent to a
distribution function for influence, with some enzymes required to have
less if others have more. In general, these summation relationships are
neither necessary (26) nor sufficient (32) for a valid
steady-state analysis.
The results for the 11 net fluxes are shown
in Fig. 4. The response to an increase in level of glutathione
peroxidase shows that the increase in flux into the pentose phosphate
pathway F (1.00) is preferentially diverted from
glycolysis F
(0.897) to nucleotide metabolism F
(1.21). Moreover, this results in a net increase
in flux into nucleotide metabolism since the increase in F
(1.21) is greater than the increase in F
(0.897), which returns flux to glycolysis. The
same pattern can be seen in the response to an increase in the
concentration of total glutathione or a decrease in the level of
transaldolase. Hence, in each case an increased proportion of the flux
through ribose 5-phosphate X
is being diverted
into nucleotide metabolism. This result shows that the distribution of
flux at the branch points of the pentose phosphate pathway is a
critical feature of this model. The diversion of flux from glycolysis
to nucleotide metabolism is accompanied by a dissociation in the
nominal rates of ATP production and utilization by the glycolytic
kinases. The difference is rectified by appropriate changes in the
rates of adenosine kinase and adenosine triphosphate phosphohydrolase
and to a lesser extent by changes in the rates of Na/K-ATPase,
adenylate kinase, and phosphoribosyl pyrophosphate synthetase.
For
comparison, we also have determined the logarithmic gains empirically
within the framework of the MichaelisMenten representation. This
involves making a small change in an independent variable, iteratively
solving for the new steady state, and then calculating the logarithmic
gains by taking finite differences. The empirically determined results
are identical, within computational precision, to the theoretically
expected results in Fig. 2, Fig. 3, and Fig. 4when independent variables are changed by ±
10%. However, this method is computationally
inefficient; it takes 172 times longer to compute the logarithmic gains
for the Michaelis-Menten model than it does for the S-system model. (
)
We determined the error of representation as described under ``Methods'' by changing each independent variable by -2% and then calculating the change produced in each dependent variable for both the S-system model and the Michaelis-Menten model considered as a reference. Only enzyme levels are considered in this analysis because the Michaelis-Menten model failed to realize a steady state when independent variables other than enzyme levels were changed by as little as 2% (data not shown).
The three-dimensional plot in Fig. 5shows the magnitude of the error as a function of the
independent and dependent concentration variables. Differences with
magnitude less than 2.0 10
are shown as black squares. The changes produced by both models are in most
cases identical within computational precision (black
squares). The average magnitude of all the percentage differences
is 0.0104%. The largest differences are produced by changes in the
levels of glutathione peroxidase X
and
transaldolase X
. The metabolic pools that exhibit
the greatest differences are gluconate 6-phosphate X
, nicotinamide adenine dinucleotide phosphate X
, sedoheptulose 7-phosphate X
, glucose 6-phosphate X
,
fructose 6-phosphate X
, and ATP X
. A very similar pattern of results was obtained
when independent variables were changed by +2% (data not shown),
and the average magnitude of all the percentage differences is
0.00995%.
Figure 5:
Error of representation based on a
comparison of power-law and Michaelis-Menten formalisms. Independent
variables are changed by -2%, and the Michaelis-Menten
representation is considered as the reference system. The
three-dimensional plot displays the magnitudes of the differences as a
function of metabolite concentrations X and of independent variables X
. Differences with magnitude less than
2.0E-5 are shown as black squares. The two-dimensional
projection on the right gives the average of the magnitudes
for a particular metabolite concentrations X
summed over all the independent variables X
. The two-dimensional projection on the left gives the average of the magnitudes for an independent
variable X
summed over all the metabolite
concentrations X
.
To explore further the nature of the deviation between
these alternative representations we have examined the range of
accurate representation (see ``Methods'') in response to
changes in the level of glutathione peroxidase X.
The four most significant differences are plotted in Fig. 6. The
Michaelis-Menten model shows drastic changes when the level of
glutathione peroxidase is reduced by as little as 9%. Beyond this point
the Michaelis-Menten model does not exhibit a steady state. This
behavior may be the result of a positive feedback effect: A decrease in
glutathione peroxidase level lowers the influx to the pentose pathway,
which in turn leads to a diversion of flux away from nucleotide
metabolism. The resulting depletion of the ATP pool causes a reduction
in the rate of synthesis of glucose 6-phosphate. The pool of glucose
6-phosphate becomes depleted and limits both the activity of
phosphoglucoisomerase, which leads to the depletion of fructose
6-phosphate, and the activity of glucose-6-phosphate dehydrogenase,
which completes the cycle by further lowering the influx to the pentose
pathway and causing the accumulation of NADP.
Figure 6:
Range of accurate representation based on
a comparison of power-law and Michaelis-Menten formalisms. The level of
the independent variable glutathione peroxidase is considered 100% at
the nominal steady state. As this independent variable is being reduced
the Michaelis-Menten model undergoes drastic changes in the
concentration of several metabolites. After a reduction of only about
9% there is an explosive accumulation of NADP, while
the concentrations of ATP, glucose 6-phosphate, and fructose
6-phosphate plummet; beyond this point the Michaelis-Menten model fails
to exhibit a steady state. See ``Results'' for further
discussion.
We have previously taken a model of metabolism in human red blood cells through three stages of assessment and refinement (18, 19) to produce the model used in this paper (Model III). With this model we have addressed two issues. First, we have examined the propagation of biochemical signals within this metabolic network. This requires an integrated systems approach, and the relevant concepts in this context are logarithmic gains in concentrations and fluxes. Second, we have examined the accuracy of representation. Here we have used a comparative approach involving alternative kinetic models to identify areas of metabolism that might not be well represented. The results of these studies point to specific areas of erythrocyte metabolism that are in need of further experimental investigation.
A similar pattern of systemic responses was identified
among the fluxes of the system ( Fig. 3and Fig. 4). The
flux through the proximal portion of the pentose phosphate pathway (e.g. V) and the fluxes through the pools of
sodium V
, potassium V
, and
2,3-diphosphoglycerate V
have highly attenuated
responses to nearly all of the independent variables. These fluxes are
fixed by a rate-determining step or, in the case of potassium, are
isolated from other influences by saturation of the key enzyme through
which these influences would otherwise be communicated. On the other
hand, fluxes that exhibit highly amplified responses include those
through the pools of nucleotide metabolism (V
through V
, and V
through V
) the early portion of glycolysis (V
and V
), and the lower
portion of the pentose phosphate pathway (V
).
Again, this manifests the redirection of flux into or out of nucleotide
metabolism.
The analysis of signal propagation also identified
independent variables (input signals) whose influence on the metabolite
concentrations and fluxes of the system is either very little or very
great. One half of the independent variables have almost no influence
on the metabolite concentrations or fluxes of the system. These include
enzyme levels for fast reactions operating near equilibrium and for
reactions that follow a rate-limiting step in a pathway. On the other
hand, three independent variables are highly influential in affecting
the metabolite concentrations and fluxes of the system: the activities
of the enzymes glutathione peroxidase X and
transaldolase X
, and the concentration of total
glutathione X
(Fig. 2, Fig. 3, and Fig. 4). Each of these three variables has a major influence on
the distribution of flux between the pentose phosphate pathway and
nucleotide metabolism.
We have not ruled out a physiological role for this diversion of flux in the context of the whole cell. If this behavior were to be part of a normal response, then the maintenance of ATP levels in the cell would require a minimal flux through glutathione peroxidase. Attempts to maintain ATP levels during blood storage by eliminating the oxidative stress responsible for this flux (34) might then be counterproductive (18) . On the other hand, in animals with a dietary deficiency for selenium, large changes in the activity of glutathione peroxidase (a selenium-dependent enzyme) occur without a major disruption of erythrocyte metabolism(35) ; whereas in the model, small changes (9%) in the activity of glutathione peroxidase cause diversions leading to the loss of the nominal steady state, which suggests that there are still unsolved issues here.
We have used
a Michaelis-Menten model of metabolism in human erythrocytes as a
reference to test the predicted accuracy of the local power-law
representation in the context of a larger system. Our previous analysis
of parameter sensitivities led us to predict (19) that Model
III would accurately represent the Michaelis-Menten model only over a
limited range of variation in its concentration variables. In this
paper, the question of accuracy was examined directly by comparing
alternative representations involving the Michaelis-Menten and
power-law formalisms. We first determined the error in response to a
small change in the independent variables as a function of independent
and dependent concentration variables (Fig. 5). On average the
discrepancies between these alternative representations is small
(0.0104%), which suggests that much of the system is well represented.
However, the discrepancies that do exist are large. The greatest errors
occur in the pentose phosphate pathway and in nucleotide metabolism,
and these occur in response to changes in the activities of glutathione
peroxidase X and transaldolase X
. Furthermore, in determining the range of
accurate representation we discovered that the large discrepancies are
due to the behavior of the Michaelis-Menten model, which fails to
exhibit a nominal steady state when the activity of glutathione
peroxidase is decreased by as little as 9% (Fig. 6). Thus, the
narrow range of accurate representation that was predicted on the basis
of parameter sensitivities (19) has been confirmed here and
associated with the lack of robustness of the reference model.
The comparative approach has demonstrated the ability of the theory to predict areas of problematic representation and has contributed to the diagnosis of problems in the Michaelis-Menten model. However, because of the inadequacies in the reference model, this approach could not yield useful information about the actual range of variation over which the biological system might be accurately represented by the power-law formalism.
Another issue in need of further attention is
the kinetic characterization of the fast reactions that operate near
equilibrium. It is frequently assumed that fast reactions near
equilibrium can be described by simple mass-action kinetics and the
choice of large values for the rate constants. In principle, the
results should be independent of the choice if the values are
sufficiently large and their ratio is equal to the equilibrium constant
for the reaction. However, when these values are not made sufficiently
large, then an arbitrary choice of values can have significant effects
on the state of the system(18) . We tried to eliminate these
arbitrary effects by assigning the same large number 10 for
all forward rate constants. We could not use larger numbers because
then the Michaelis-Menten model failed to maintain a steady state.
After having made various refinements in the model we have gone back
and reexamined this issue. We assumed the same value for the forward
rate constants of all of these fast reactions, varied this value over
the range 10
to 10
, and found the steady state
to be largely unchanged. Nevertheless, this procedure remains
questionable. The choice of very large values for the rate constants
forces reactions closer to equilibrium than they may actually be in the
organism, and this introduces error into the model. On the other hand,
too low a value may not bring the reactions close enough to
equilibrium, and this also introduces error. Choosing just the right
set of values in an ad hoc fashion to make the model fit the
data is unacceptable. The only satisfactory solution to this problem is
better data and a more complete kinetic characterization for these fast
reactions.
The rate laws for Model III of metabolism in human erythrocytes using the S-system representation, given in the notation of , are the following.
pp. 379-383, Cambridge University Press, New York