Tài liệu Báo cáo khoa học: Chromokinetics of metabolic pathways doc

Chromokinetics of metabolic pathways
Jo¨ rg W. Stucki
Department of Pharmacology, University of Bern, Bern, Switzerland
Some methods to study and intuitively understand steady-
state flows in complicated metabolic pathways are dis-
cussed. For this purpose, a suitable decomposition of
complex metabolic schemes into smaller subsystems is used.
These independent subsystems are then interpreted as basic
colors of a chromatic coloring scheme. The mixture of these
basic colors allows an intuitive picture of how a steady state
in a metabolic pathway can be understood. Furthermore,
actions of drugs can be more easily investigated on this
basis. An anaerobic variant of pyruvate metabolism in rat
liver mitochondria is presented as a simple example. This
experiment allows measurement of the percentage that each
basic color contributes to the steady states resulting from
different experimental conditions. Possible implementations
of existing algorithms and rational design of new drugs
are discussed. A
MATHEMATICA
program, based on a
new algorithm for finding all basic colors of stoichiometric
networks, is included.
Keywords: stoichiometric network analysis; extremal cur-
rents; elementary flux modes; steady-state flux analysis;
rational drug design.
When confronted with a realistic metabolic pathway, e.g.
glycolysis and fatty acid breakdown supplemented with the
citric acid cycle plus oxidative phosphorylation, most
investigators rapidly lose track because of the sheer com-
plexity of such schemes. Therefore, a way is sought to split

large schemes into smaller, understandable pieces. One
popular decomposition method is to simplify the scheme by
educated guesses and to ignore the rest, for example looking
only at the citric acid cycle or only at glycolysis, etc.
Alternatively, one might attempt decomposition into differ-
ent ÔchemistriesÕ, such as carbohydrates, lipids, amino acids,
etc. This is the format adopted by most textbooks of
Biochemistry. These reduced reaction schemes can then be
analyzed in detail by simulating the individual reactions with
a computer, which amounts to solving a set of differential
equations. This does not mean that that part of metabolism
is then understood, let alone the rest, but at least some
quantitative predictions can be made.
A radically different, and yet not so popular, method
consists of decomposing the metabolic network into invari-
ant components, starting with the stoichiometry matrix.
This procedure allows the description of steady states as
convex (non-negative) mixtures of extremal steady states [1].
A close analog of such a mixture is well known from
kindergarten experience. Suppose that one wants to color a
picture, then what one basically needs are three different
crayons, viz. red, yellow and blue. Different colors can then
be created by mixing these elementary basic colors. Artists
use this kind of mixing on canvases; color TV and
computers produce colors on screens by a similar process.
The basic colors of a metabolic pathway are its
independent steady-state currents, which can be calculated
from the stoichiometry matrix. A mixture of these colors,
akin to painting, can then represent every steady state of the
metabolic scheme. In order to distinguish this type of

quantitative analysis from a computer simulation of the
differential equations, and owing to the close analogy with
coloring methods, I call this analysis ÔchromokineticsÕ.Note,
however, that this analysis applies to steady states only and
cannot cope with transients. I would like to carry over the
intuitive picture of coloring to the reader without insisting
too much on the powerful mathematical machinery stand-
ing behind it. I suspect that with the help of this
chromokinetic idea, metabolic pathways can be understood
more intuitively and predictions can also be made. The aim
of this article is to illustrate this interpretation, step-by-step,
with a simple metabolic example.
The concept behind these basic colors is not new. It came
already to kinetic schemes in different guises: Clarke called
them Ôextremal currentsÕ [1,2]; later, Schuster called some-
thing similar Ôelementary modesÕ [3,4]. In the present report,
I stick to the original terminology introduced by Clarke [1].
The results and definitions published by Clarke are precise,
exhaustive and nonintuitive. Here I just add some color to
these abstract pictures.
A simple example
Anaerobic mitochondrial pyruvate metabolism
First, a simple example is needed. Isolating small autonomic
metabolic units from a whole organism can bring about
simplification. One example of this is mitochondria iso-
lated from rat liver. However, mitochondrial metabolism
still contains far too many reactions to allow coherent
illustration of the procedure. Hence, further simplification is
needed, and this can be achieved by external constraints
Correspondence to J. W. Stucki, Department of Pharmacology,

University of Bern, Friedbu
¨
hlstrasse 49, CH-3010 Bern, Switzerland.
Fax: + 41 31 632 49 92, Tel.: + 41 31 632 32 81,
E-mail:
Enzymes: oxoglutarate dehydrogenase (EC 1.2.4.2); fumarase (EC
4.2.1.2); succinate dehydrogenase (EC 1.3.99.1). See also Appendix 1.
Note: a website is available at />(Received 4 February 2004, revised 17 March 2004,
accepted 5 May 2004)
Eur. J. Biochem. 271, 2745–2754 (2004) Ó FEBS 2004 doi:10.1111/j.1432-1033.2004.04203.x
such as offering only certain chosen substrates to metabolize
in the incubation medium, adding inhibitors to suppress
certain reactions, etc. By doing so, we finally arrive at a
caricature of what mitochondria are supposed to do, yet it
will be helpful to illustrate the analysis. Our caricature is the
following: pyruvate added to the incubation medium is
metabolized by pyruvate dehydrogenase and pyruvate
carboxylase (present in liver and kidney). The products
are then further metabolized in the citric acid cycle or by
condensation to ketone bodies. To further simplify the
scheme, we completely block oxidation and phosphoryla-
tion and collapse transmembrane electrical potentials. These
ÔanaerobicÕ mitochondria are, of course, no longer produ-
cing ATP and therefore we have to add it to the incubation
medium. Figure 1 displays a minimal metabolic scheme
describing the operative pathways.
How can we now find the Ôbasic colorsÕ of this scheme?
What is its Ôspectral chromatic decompositionÕ? Mentally,
we can reticulate the overall scheme into partial diagrams,
each one having a steady state with exactly one degree of

freedom. Each of these can be conceived of as running
separately, like a little independent clockwork toy. This
requires that we have to take care of conserved moieties. For
example, NADH and NAD
+
cannot pass the inner
mitochondrial membrane and are present only in minute
amounts in the mitochondria. Therefore, we must make
sure that the NADH produced can readily be consumed by
an appropriate oxidizing reaction, thus reproducing
NAD
+
. If not, the clockwork toy would come to a rapid
stop and no measurable net flow would result. The same
applies for the conserved pool of metabolite moieties linked
with CoA plus free CoASH. Figure 2 shows the unique four
partial diagrams, which the scheme admits as decomposi-
tions of the grand view. The calculation of these partial
diagrams will be shown below.
These diagrams are the basic colors used for the mixture
of any steady state. By defining the percentage that each of
these colors contributes, or weights, we can unambiguously
characterize the resulting steady state. To fix the chromatic
idea, we assign, to each partial diagram, a color, viz.: j
1
,
blue; j
2
,red;j
3

, green; j
4
white. The reader should not be
frustrated by the limitation of the human visual system to
three basic colors. Some insects see many more, and for the
computer they are ÔseenÕ as a string of numbers, akin to a
telephone number. A proper mix thereof will unambigu-
ously define a steady-state situation. Such a mixture of non-
negative only contributions of colors is called a convex
combination. That means that every steady state is limited
to be within a geometric object spanned by the basic colors
interpreted as vertices, like in a color triangle, for example.
We will explain this later. A pendent to our color scheme
may be thought of as the description of, e.g. the center of
gravity of a solid body by so-called barycentric coordinates.
Yet another legerdemain, suitable for a colorblind person,
would perhaps consist of stacking transparencies, each one
having a gray level of intensity (j
i
). Illumination through all
of these transparencies with an overhead projector would
then also yield an overall picture of the steady state.
Before proceeding to more detailed mathematical
descriptions, I will, however, continue this exposition by
reporting an incubation of isolated rat liver mitochondria.
This experiment was actually designed to illustrate the
chromokinetic method and will represent the measured
results in the form of a chromatic object formed from basic
colors. The measurements of the incubation are summarized
in Table 1. We decided to illustrate the usefulness of the

chromatic scheme by studying the action of a drug. In the
present scheme, we used fluorocitrate for inhibiting aconi-
tase. This substance is not a particularly useful drug, but is,
in fact, a deadly poison. Recalling that our ischemic
Fig. 1. Simplified model of mitochondrial pyruvate metabolism. This reaction scheme is a simplified version of a previously published model of
mitochondrial pyruvate metabolism, where computer-simulated and measured fluxes were compared [14]. The transport of CO
2
,ATP,ADPandP
i
through the inner mitochondrial membrane was omitted in this scheme. In other words, these molecules were treated as (constant) external
reactants (shown in italics). Furthermore, only the entry of citrate into the Krebs cycle is included and further metabolic changes were neglected, as
supported by the experimental results in Table 1. The r
i
at the arrows is used for the numbering scheme in Scheme 1. The elementary reactions are
listedinAppendix1;seealsoÔConcluding remarksÕ.
2746 J. W. Stucki (Eur. J. Biochem. 271) Ó FEBS 2004
mitochondria are already in a state which can be considered
dead for all practical purposes, because oxidative phos-
phorylation is blocked, still further inhibition of vital
reactions will cause no significant harm, but will be very
useful for illustrating the analysis. From Fig. 2 we can
readily deduce the balance equations for the contribution
(or fixed weight owing to stoichiometry) of each basic color,
j
i
, to the overall consumption and production of the
metabolites in the incubation medium:
Pyruvate ¼ 4j
1
þ 3j

2
þ 3j
3
þ 4j
4
Malate ¼ 2j
1
þ j
2
þ j
3
þ 2j
4
Acetoacetate ¼ j
1
3-Hydroxybutyrate ¼ j
2
Citrate ¼ j
3
(Eqn 1)
Considering our system, we are left with a set of five
equations for the measured metabolites and with only four
unknowns: j
1
–j
4
. In such fortuitous cases, the j
i
values can
easily be calculated by using standard methods as a least-

square solution of an overdetermined system. The results,
j
i
, calculated from our experiment, are also included in
Table 1.
The result represented in this form is stunning indeed. To
appreciate this, suppose that one had to describe the
outcome of the incubation with standard practice, without
having access to chromatic decomposition methods. One
would then write, for example, that pyruvate consumption
declines monotonically with increasing fluorocitrate con-
centration, whereas production of 3-hydroxy-butyrate first
increases, then decreases, etc. The final statement would
then somehow say that further experiments, probably based
on radioactive tracer methods, are needed to better under-
stand what is going on.
By contrast, we claim that we can already explain what is
going on, on the basis of the meagre data record in Table 1,
Fig. 2. Independent currents in anaerobic pyruvate metabolism. These diagrams are graphical representations of the independent currents into which
thereactionschemeinFig.1canbedecomposed.Theycorrespondtothebasiccolorsj
1
–j
4
and were calculated as described in the text (Scheme 2).
The numbers on the arrows are the weights of the j
i
on the different metabolites involved. Thus, for example, the pyruvate metabolized in (A) is
4 · j
1
; malate produced 2 · j

1
, etc. This allows setting up the balance equations that were used to convert measured metabolite flows into j
i
values.
Table 1. Metabolite turnover in rat liver mitochondria. Isolation and
incubation of mitochondria was performed exactly as described pre-
viously [14], with the exception that nonradioactive pyruvate was used
and the incubation time was 20 min after the addition of 20 mg of
mitochondrial protein. Moreover, the incubation medium was sup-
plemented with 10 m
M
ATP, 5 lg of rotenone, antimycin A and
oligomycin, and 0.3 lg of valinomycin (final volume 3 mL). The val-
ues in the Table represent metabolite values measured in duplicate,
with incubations carried out at 37 °C. The flows j
1
–j
4
were calculated
from the balance equations, using a least-square standard method.
Control
Fluorocitrate
(0.33 m
M
)
Fluorocitrate
(0.67 m
M
)
Metabolite produced or used (l moles/20 min)

Pyruvate (used) 12.56 7.92 2.00
Acetoacetate
0.34 0.23 0.00
3-Hydroxybutyrate
1.31 1.50 0.73
Malate
5.57 2.45 0.66
Citrate
0.67 0.69 0.01
Color (l moles/20 min)
j
1
(blue)
0.34 (9.41%) 0.23 (9.27%) 0.00 (0%)
j
2
(red)
1.23 (34.07%) 1.53 (61.69%) 0.72 (100%)
j
3
(green)
0.59 (16.34%) 0.72 (29.03%) 0.00 (0%)
j
4
(white)
1.45 (40.16%) )0.0 (0.0%) )0.0 (0%)
Ó FEBS 2004 Chromokinetics of metabolic pathways (Eur. J. Biochem. 271) 2747
even without sophisticated recourse to radioactive tracer
techniques. The coloring scheme of steady-state kinetics
allows us to clearly see what has happened. At the low

concentration of the inhibitor fluorocitrate (0.33 m
M
), the
contribution of the white color (j
4
) is wiped out, as expected,
as citrate can no longer be metabolized in the citric acid
cycle (reaction r
8
). The color scheme has now collapsed to a
mixture of blue, green and red. A further increase of the
inhibitor (0.67 m
M
) leads to unexpected side-effects: blue
(j
1
)andgreen(j
3
) vanish altogether and we are left with a
monochromatic red (j
2
). Note that this steady state corres-
ponds exactly to one basic color; hence the term Ôextremal
currentÕ. One might speculate about factors causing this
effect. Consulting Fig. 2, we might conjecture that citrate
synthase, as well as the transport of acetoacetate out of the
mitochondria, are also affected by fluorocitrate and then
propose new experiments on these hypotheses. We will,
however, not pursue this line of investigations any further.
It is instructive to examine a geometrical representation

of these experimental results. Most fortunately our scheme
has only four basic colors, and thus 3D space just allows
translation of the chromatic scheme into a simple platonic
solid: a tetrahedron. In Clarke’s terminology this would be
called the convex current polytope resulting from cutting
the current cone with the hyper plane.
Figure 3A shows this current polytope in 3D space. If we
happened to have more than four colors, say five, we would
need four dimensions for a drawing, which could then no
longer be visualized. Of course, our example has been
tailored such that it can be represented in 3D. But it is
important to stress that we deal here with a constructed
special case and that, in general, the current polytopes
cannot be visualized (see below in ÔConcluding remarksÕ).
In our experiments, we have three different steady states:
that of the control and two with different concentrations of
fluorocitrate. Each one of these steady states must be
located at a defined point within the strict interior of the
current polytope. As three points define a plane, we can thus
construct such an Ôexperimental planeÕ and cut it with the
tetrahedron, and also calculate its corresponding color by
mixing the basic colors involved. This is illustrated in
Fig. 3B. Of course, this plane is an artificial construct, as by
selecting a more detailed range of fluorocitrate concentra-
tions, the steady states would probably no longer all lie
within that plane but rather follow a curve off the
Ôexperimental planeÕ. However, this simplification is useful
for illustrating the basic idea. In Fig. 3C, the Ôexperimental
planeÕ in 3D is then finally rotated into the 2D plane. It is
apparent that each of the three steady states has a color,

which is a mixture of the ground colors given by the
contributions, j
i
, in Table 1. The control is greenish-gray
color, whereas with fluorocitrate the resulting mixtures are
reddish brown and pure red. As fluorocitrate eliminates the
contributions of white (j
4
) altogether, the experimental
points are on the baseline of the triangle. This was already
evident from Fig. 3B, where the steady states in the presence
of fluorocitrate lie on the ground surface of the tetrahedron
owing to a collapse of the third coordinate. In summary, this
geometrical illustration should provide a particularly lucid
and intuitive picture of the mixing of basic colors to describe
what is called Ôconvex combinationÕ in mathematics.
The paint box and how it is acquired
The composition of the paint box is, in our parlance, the
decomposition of the whole system into elementary units,
each one describing an independent steady state, viz. a basic
color. I will delay the formal description of a steady state and
assume here that one intuitively understands what is meant
by steady state, namely as much material flows out of the
system as flows in, and vice versa. There is no net
accumulation of molecules within the system, akin to a
bathtub in a steady-state configuration, wherein there is no
spillover of water, but rather a steady water level. A proper
balance of taps and sinks is all that is needed. The major
problem comprises finding all of the basic colors and not just
a few of all that exist. It is this quest for completeness that

makes this decomposition a complex mathematical problem.
Fig. 3. Current polytope and experimental steady states are represented as colored geometrical objects. The current polytope of anaerobic mito-
chondrial pyruvate metabolism is represented as a tetrahedron. For the coloring of the faces in (A) a corresponding RGB value was associated with
the spatial coordinates. In (B) the Ôexperimental planeÕ was calculated using the percentage contributions of the three experimental conditions in
Table 1. This plane was then colored in place within the tetrahedron again by assigning a corresponding RGB value furnished by the spatial
coordinates. In (C), the same triangular plane was projected and colored into the 2D plane. The experimental points were added as yellow circles of
increasing size corresponding to the control, 0.33 m
M
and 0.66 m
M
fluorocitrate successively. This geometrical representation clearly demonstrates
how all of the three steady states are located within the accessible region of the current polytope (see Concluding remarks).
2748 J. W. Stucki (Eur. J. Biochem. 271) Ó FEBS 2004
Combinatorial method
Confronted with the chromatic decomposition problem
for the first time, the immediate reaction is to attempt
an exhaustive trial-and-error approach. Via a rigorous
enumeration, one might then try out all possible combi-
nations of arrows in the reaction scheme and sort out the
ones allowing a steady state. Clarke has published such a
brute force method in a key publication [1]. The algorithm
described therein only needs to be supplemented by a
small routine which generates all the combinations using a
backtrack algorithm [5]. I will not reproduce the pro-
cedures here in detail, as they will be of little use for
realistic metabolic pathways. The major problem with this
brute force approach is that it will rapidly explode. This
has to do with the fact that the number of combinations
grows faster than exponentially with growing network
size.

In order to exemplify Clarke’s algorithm, we first set up
the stoichiometry matrix for our caricature system in Fig. 1.
Labeling the columns by the 12 reaction velocities from r
1
to r
12
(left to right) and the rows with the metabolites
(downwards) in the order pyruvate, acetyl-CoA, acetoace-
tate, 3-hydroxy-butyrate, oxaloacetate, citrate, malate and
NADH, we obtain the stoichiometry matrix shown in
Scheme 1, as can easily be verified by inspecting the reaction
scheme in Fig. 1 and Appendix 1. Owing to the conservation
conditions, not all species in Fig. 1 are independent, as
mentioned. The computer can detect this effortlessly,
because summing the rows standing for NAD
+
and
NADH, for example, results in zeroes, which means a
singular matrix. Thus, one of these rows has to be omitted
and the choice of which one is deleted is arbitrary. I kept
NADH and discarded NAD
+
. Similarly, CoASH was
omitted and acetyl-CoA was retained.
Processing this matrix by the above-mentioned algo-
rithm, and looking at what the computer is doing, we note
the following: 220 combinations have been generated, as
expected. The number of combinations is the binomial
coefficient (r, n +1) where r ¼ 12 is the number of reactions
and n ¼ 8 the number of independent species. We identified

the following statistics: four basic colors (extremal currents
with positive components only), which were repeatedly
found 15 times; 27 singular submatrices; and 174 sub-
matrices that contained positive as well as negative com-
ponents. It is important to remain firm and to insist on
non-negative components only. Trying to interpret negative
components as backward reactions will rapidly lead to a
quagmire of sheer confusion. Thus, we strictly followed the
practice, decreed by Clarke, of formulating backward
reactions as separate reactions (which in our scheme we
actually did not need to do).
Summarizing the results of the calculations, we obtain
a matrix containing the j
i
values of the basic colors in
terms of the reactions, r
i
, as columns (Scheme 2). The
rows j
1
to j
4
of this color matrix correspond to the partial
diagrams in Fig. 2. The numbers above the arrows in the
partial diagrams in Fig. 2 are the weights given in the
colors matrix, whereas the experimental data in Table 1
yield the percentage contributions or barycentric coordi-
nates of each basic color, j
i
. Clarke calls this matrix ÔEÕ

and obtains the reaction velocities by v ¼ Ej. I prefer
normalization by Sj
i
¼ 1 to obtain dimensionless bary-
centric coordinates (or percentage contribution of each
basic color). With this, the color matrix describes a rigid
convex polytope into which all steady states of the
different experiments can be inscribed and compared
consistently (see Fig. 3). If the normalization factors are
stored together with the color matrix, then the original
reaction velocities can be reproduced.
At that stage, one might feel contended by the result
achieved thus far. Yet, a small back-of-the-envelope calcu-
lation announces insurmountable obstacles. Suppose we had
to decompose a more realistic metabolic pathway, instead of
our caricature, and which would embrace c. 100 reactions
and 80 metabolites. After writing down the stoichiometry
matrix ÔNÕ for this system, we would have to expect the
binomial coefficient (100,81) of possible combinations,
which is approximately 10
20
. Hitting the enter key on the
computer to launch the number crunching, would, by all
means, necessitate resetting and restarting the machine. An
optimistic estimate, considering the advanced state-of-the-
art computer technology, would predict a computing time of
about 4000 years to find the basic colors for this problem.
Therefore, in order to further advance the painting of
metabolic pathways, other approaches need consideration.
Synthetic method

The observation, that out of the many combinations only
an exceedingly small minority of solutions are useful basic
colors, inspires a bottom-up instead of a top-down
approach, as used above. The advantage of this has to do
with the basic organization of metabolic pathways, which
seem to resemble a collection of hubs rather than a
completely connected system.
I have decided to be brief on using mathematical
notations in this report. However, sometimes the use of
mathematical notations is unavoidable, lest the impression
of a lukewarm presentation is given. Thus, we express, for
the time being, changes of the concentrations as:
dc/dt ¼ f(v), (Eqn 2)
The reaction velocities, v
i
, may be nonlinear functions of the
concentrations (or of the chemical activities) of the species
involved. Delegating nonlinearity into the v
i
values, in this
manner, we note that the concentrations, c
i
, are linear
functions of these velocities and we put:
Ó FEBS 2004 Chromokinetics of metabolic pathways (Eur. J. Biochem. 271) 2749
dc/dt ¼ Nv, (Eqn 3)
where N is the stoichiometry matrix, as explained above.
The very definition of the steady state declares that the
concentration changes, dc/dt, must vanish as time proceeds
to infinity. This allows:

Nv ¼ 0; (Eqn 4)
which also means that the ÔvÕ vectors are orthogonal to the
rows of N. Therefore, the basic colors have to be within the
null space (or kernel) of N. There are different arbitrary
ways to write down the null space of N, because no general
formal rules exist. Yet, all contain the minimal number of
basis vectors. For the sake of convenience, we asked
MATHEMATICA
to produce a kernel of N. An undocumented
feature of
MATHEMATICA
’s NullSpace command is that
towards an integer input it reacts with an integer orthogonal
answer (real numbers would give an orthonormal answer in
real numbers). In our case we obtained, with
k ¼ NullSpace[N]; (Eqn 5)
the kernel shown in Scheme 3, whose vectors were also
produced, among many others, using the combinatorial
method above. This kernel already contains one basic color
(feasible solution), viz. j
1
¼ k[[1]], having non-negative
contributions only. Finding all others requires further deep
insights into the geometrical properties of current cones and
current polytopes.
A recent detailed analysis, including proofs, has been
carried out by Wagner [6] and will be published
elsewhere. For the present purpose, two major results
will be briefly summarized: first, each vertex of the
current polytope has to lie within an orthogonal plane of

the Euclidean space spanned by the reaction velocities;
and, second, only one cone vector can be located within a
plane. All other vectors are not basic colors, but linear
combinations thereof. On the basis of these geometrical
properties, Wagner found a new algorithm for the
calculation of the current polytope from the kernel of
the stoichiometry matrix [6].
The
MATHEMATICA
program given in Appendix 2 imple-
ments this algorithm. Steps (1) and (2) have been already
mentioned above. Step (3) copies the kernel ÔkÕ into an initial
tableau and makes a bitmap with entries True for zeroes in
the tableau and False elsewhere. Step (4) sets up the filter
functions containing the necessary tests for rejection of
unfeasible solutions. In (5) the tableau is processed column-
wise, such that zeroes are produced by all possible linear
combinations of suitable rows of the tableau. If the resulting
combinations are neither identical nor themselves a linear
combination of already existing rows of the tableau, they
are attached to the tableau sequentially. The necessary test
is based on the geometrical properties mentioned above.
Finally, step (6) displays all basic colors of the network.
They correspond to the ones already given in Eqn (3).
Note that the above example is trivial insofar as only
pairwise combinations were needed. Other models may
require much more rich orchestration, such as trios,
quartets, etc., of combinations. An instructive example is
the model of the Belusov–Zhabotinsky reaction, discussed
by Clarke [2], which needs up to quintets. In addition, still-

larger systems may need preprocessing of the stoichiometry
matrix, as well as of the kernel, in order to prevent
premature explosion of the tableau. A more detailed
discussion of also including reversible reactions has been
published previously [6].
It is instructive to compare the statistics of the present
calculation with the combinatorial method of Clarke: nine
possible linear combinations were calculated, three passed
the test and were added to the tableau. The final tableau had
four rows, of which none contained negative coefficients
yielding a color matrix with four vectors. Evidently, the
synthetic algorithm is much more efficient than the brute
force combinatorial method. It is, however, far from trivial
to predict an upper limit for the number of colors for a given
model. Linear algebra can only assert that:
f ! r À n; (Eqn 6)
of such basic colors (f is frames or edges of the current cone
in the positive orthant or number of vertices of the current
polytope in Clarke’s exposition and number of basic colors
here; r is number of reactions; and n is number of
independent species). The number of basic colors, f, can
thus only be determined after the problem is actually solved.
Although the number of calculations necessary is drastically
reduced in comparison to the combinatorial method, the
problem remains probably non-polynomial (NP) and big
systems may not yet be decomposed into basic colors within
reasonable time.
Linear programming methods
The same goes for yet another type of newer algorithms,
which has already been applied successfully to decompose

systems as complicated as the metabolism of Escherichia coli
bacteria. This algorithm is based on a general solution of a
linear programming problem. A simplification thereof was
worked out in detail by Schuster and collaborators [3,4]. By
contrast to the solution based on the nullspace of the
stoichiometry matrix, this algorithm constructs an exhaust-
ive series of tableaux via a Gaussian elimination procedure
along with selection rules akin to those described above.
Detailed numerical examples, illustrating the application of
this algorithm, are presented in various publications by
Schuster [3,4] and notably in the valuable book of Heinrich
& Schuster [7]. A computer implementation of this algo-
rithm is available in the C language [4]. This algorithm
has further been implemented with the
MATLAB
language, in
a package called
FLUXANALYZER
, by S. Klamt and which is
available on the Internet. Moreover,
METATOOLS
by S. Sch-
uster and collaborators is another program in C, which can
also be downloaded from the Internet.
In his publications, Schuster emphasizes the so-called
Ôelementary flux modesÕ. These should not be confused with
the basic colors or vertices of the current polytope. The
elementary flux modes are identical to the basic colors only
2750 J. W. Stucki (Eur. J. Biochem. 271) Ó FEBS 2004
if there are solely irreversible reactions occurring in the

scheme. By including reversible reactions, without expressing
them as separate reactions in the stoichiometry matrix,
however, the elementary flux modes are a subset of the basic
colors. Stated differently, the elementary flux modes of
Schuster are a projection of Clarke’s current polytope into a
lower dimensional space and a proper projection operator is
known [6]. Benchmarking tests, comparing Schuster’s algo-
rithm to ours, are in progress and will be published elsewhere.
Concluding remarks
A few points merit further discussion. First, one must realize
that all information about the steady states is already
contained in the stoichiometry matrix. By applying a series of
matrix transformations, one can calculate the kernel or the
basic colors or the current polytope, etc., but by such
procedures no new information is introduced. In this sense,
these different representations are all invariants of the
stoichiometry matrix. Nonetheless, these transformations
lead to radical changes in viewpoints. In the representation of
the reaction velocity space in Fig. 1, the system is seen from
the outside. It is an exterior representation. Switching to the
basic colors or, equivalently, to the current polytope, gives an
interior representation. All the steady states are located in the
strict interior of the polytope and are a mixing of the basic
colors or, in other words, a convex combination of the
vertices of the polytope. In this sense, the steady state is blind
to whathappens outside the polytope; it is opaque. Of course,
during transients, the system makes some excursions outside
these opaque borders to then settle down in the interior at
steady state, provided it is globally asymptotically stable.
A further important point is the bookkeeping of conser-

vation conditions. By using the Clarke algorithm, the
stoichiometry matrix needs the deletion of all dependent
species, otherwise the algorithm would attempt to find
solutions of a singular matrix. In simple cases, such as in the
conservation condition, NAD
+
+NADH¼ constant,
these are quite simple to detect. But already in more
complicated models, which at first sight may look trivial,
conservation conditions become a real problem. One
innocently looking example is the glycosome of Trypano-
somes [8]. There, the conservation of the organic phosphates
bound to different carbohydrates already requires some
analysis and could easily escape detection. One of the
biggest advantages of the calculation of the kernel of the
stoichiometry matrix is that all of these complications are
circumnavigated because conservation conditions can be
completely ignored. The basis vectors of the kernel have
automatically eliminated these redundancies and have
reduced the problem to full rank, i.e. containing only
independent species. The explanation for this is that the
NullSpace algorithm uses the row echelon form of the
matrix. Applied to the above-mentioned reaction scheme in
glycosomes, including reversible reactions, exactly three
basic colors are found [9]. When one is interested only in the
proper decomposition of the system into independent and
dependent species, one may use the program
GEPASI
by
P. Mendes that can be downloaded from the Internet.

A simplified, qualitative version of flux analysis is
currently much in vogue. From genomic data, researchers
try to construct plausible stoichiometry matrices for micro-
organisms. By a type of dead-or-alive analysis it can then be
predicted which Ôelementary modesÕ have to be knocked out
in order to kill the organism. In stark contrast to this, it can
also be estimated under which conditions organisms show
optimal growth or how they adapt to a changing environ-
ment [10–13]. Of course, such an analysis would be much
more powerful if information about the weight of the
different modes were available, which necessitates a meas-
urement of the metabolites flowing in and out of the system.
Knocking out only the most important contributions, e.g. by
genetic manipulations or with inhibitors, could then lead to
useful therapies. A big handicap for pharmacological
interventions, but a big advantage for life is, however, the
redundancy built into biological systems which becomes
immediately apparent in a flux analysis. As many combina-
tions of arrows lead to the same product in large systems, it is,
of course, not sufficient to just eliminate one single enzyme.
For this very reason, occasionally researchers are frustrated
to observe that their knockout mice have no phenotype.
The alert reader may have noticed that the experi-
mental data in Table 1 are not fully consistent. A carbon
balance can be calculated as an independent test, which does
not need a chromatic decomposition. Realizing that every
molecule of acetoacetate, 3-hydroxybutyrate and citrate
results from metabolizing two pyruvate molecules, whereas
malate is the product of one pyruvate, we can compare the
calculated with the measured pyruvate utilization. This

calculation shows an 81%, 92% and 107% carbon recovery
in the control, with 0.33 m
M
and 0.66 m
M
fluorocitrate,
respectively. It can be estimated that % 50% of the missing
pyruvate units are caused by the fumarase reaction and the
remaining 50% have disappeared in the further metabolism
of the Krebs cycle beyond citrate. In the presence of
fluorocitrate, only fumarate production could explain the
difference, because aconitase is inhibited. Therefore, in
order to arrive at an exact result, the fumarase, as well as the
oxoglutarate dehydrogenase, reactions should be included
in the scheme, and the production of fumarate, ketoglut-
arate and succinate should be measured in the incubation
medium. However, by doing this we would immediately
leave 3D space and the current polytope could no longer be
visualized. This can be exemplified as follows: considering
the full scheme, shown previously [14], under anaerobic
conditions reveals that the Krebs cycle is interrupted at the
succinate dehydrogenase step as no oxidation of FADH
2
is
then possible. In other words, the Krebs cycle is now cut
into two trees: one hanging down from oxaloacetate to
fumarate; and the other from citrate to succinate. A
calculation of the current polytope for this case results
in a 7D geometrical object with 13 vertices. A similar
complication arises when including the exchange carriers

operative in the inner mitochondrial membrane (citrate-
malate, malate-P
i
, etc.). This is why I formulated the
pyruvate influx and effluxes of the products as unidirec-
tional transports only (see Appendix 1). In conclusion, the
scheme in Fig. 1 is a simplification, which allows the
illustration of the chromokinetic interpretation, and it
should be borne in mind that it represents only an
approximation to the complete scheme.
Our example with the application of fluorocitrate shows
not only the effect of a drug to inhibit a crucial enzyme, but
simultaneously gives an idea about possible side-effects at
Ó FEBS 2004 Chromokinetics of metabolic pathways (Eur. J. Biochem. 271) 2751
higher concentrations which can be precisely expressed in
terms of changing contributions of basic colors. Hence,
chromokinetics seems promising for a rational design of new
effective drugs with a minimum of unwanted side-effects.
Again, it must be stressed that such information cannot be
predicted from the color matrix alone but necessitates the
measurement of the metabolites actually turned over in
incubation. The shift induced by fluorocitrate from a
ÔnormalÕ to a ketotic state could even inspire an application
of chromokinetics in diagnosis. Healthy and pathological
metabolic situations could then be expressed in terms of color
regions. In our example, gray is healthy (ignoring the
poisoned mitochondria for the time being) and red is ketotic.
A major unsolved problem is cellular signaling. Many
proteins interact and change metabolic flows. Such interac-
tions can unfortunately not be captured in a stoichiometry

matrix. It seems, however, possible to make a qualitative
analysis based on an adjacency matrix, which contains the
known protein interactions. Using a variant of the methods
described above, useful information about the regulatory
functions of protein interactions can then be obtained. Such
investigations are currently in progress in our laboratory.
Acknowledgements
This work has been supported by grants from the Swiss National
Science Foundation. I am indebted to Dr Clemens Wagner and Dr
Robert Urbanczik for inspiring discussions. The technical expertise of
Mrs Lilly Lehmann, in performing the experiments, is gratefully
acknowledged.
References
1. Clarke, B.L. (1980) Stability of complex reaction networks.
In Advances in Chemical Physics XLIII (Prigogine,I.&Rice,S.A.,
eds), pp. 1–215. John Wiley & Sons, New York.
2. Clarke, B.L. (1981) Complete set of steady states for the general
stoichiometric dynamical system. J. Chem. Phys. 75, 4970–4979.
3. Schuster, S. & Ho
¨
fer, T. (1991) Determining all extreme semi-
positive conservation relations in chemical reaction systems: a test
criterion for conservativity. J. Chem. Soc. Faraday Trans. 87,
2561–2566.
4. Schuster, R. & Schuster, S. (1993) Refined algorithm and com-
puter program for calculating all non-negative fluxes admissible in
steady states of biochemical reaction systems with or without some
flux rates fixed. CABIOS 9, 79–85.
5. Stucki, J.W. (1978) Stability analysis of biochemical systems.
Progr. Biophys. Mol. Biol. 33, 99–187.

6. Wagner, C. (2004) Nullspace approach to determine the ele-
mentary modes of chemical reaction systems. J. Phys. Chem. B
108, 2425–2431.
7. Heinrich, R. & Schuster, S. (1996) The Regulation of Cellular
Systems. Chapman & Hall, New York.
8. Bakker, B.M., Mensonides, F.I.C., Teusink, B., van Hoek, P.,
Michels, P.A.M. & Westerhoff, H.V. (2000) Compartmentation
protects trypanosomes from the dangerous design of glycolysis.
Proc. Natl Acad. Sci. USA 97, 2087–2092.
9. Wagner, C. (2004) Einsatz Systembiologie gegen Parasiten.
Bioworld 1, 2–5.
10. Stelling, J., Klamt, S., Bettenbrock, K., Schuster, S. & Gilles, E.D.
(2002) Metabolic network structure determines key aspects of
functionality and regulation. Nature 420, 190–193.
11. Edwards, J.S. & Palsson, B.O. (2000) The Escherichia coli
MG1655 in silico metabolic genotype: its definition, character-
istics, and capabilities. Proc. Natl Acad. Sci. USA 97, 5528–5533.
12. Ibarra, R., Edwards, J.S. & Palsson, B.O. (2002) Escherichia coli
K-12 undergoes adaptive evolution to achieve in silico predicted
optimal growth. Nature 420, 186–189.
13. Almaas, E., Kovacs, B., Vicsek, T., Oltvai, Z.N. & Barabasi, A L.
(2004) Global organization of metabolic fluxes in the bacterium
Escherichia coli. Nature 427, 839–843.
14. Stucki, J.W. & Walter, P. (1972) Pyruvate metabolism in mito-
chondria from rat liver. Measured and computer-simulated fluxes.
Eur. J. Biochem. 30, 60–72.
Appendix 1.
Elementary reactions used in the simplified scheme (Fig. 1). Thesymbol{}isusedfortheextramitochondrialmetabolitepoolandthesymbol[]for
further metabolism in the Krebs cycle. The reaction scheme in Fig. 1 is a minimal model of anaerobic pyruvate metabolism in mitochondria and is,
in this sense, an approximation. Thus, it ignores the reactions in the Krebs cycle beyond isocitrate dehydrogenase and the reactions beyond malate

dehydrogenase (see Concluding remarks). The transport reactions across the inner mitochondrial membrane were simplified to influx and efflux
without detailed formulations as symports or antiports. The efflux of oxaloacetate was omitted altogether because only neglible concentrations
appear in the incubation medium. In order to allow the electrogenic passage of ATP into the mitochondria through the adeninenucleotide
translocase, the membrane potential was collapsed with valinomycin. The same applies for other nonelectroneutral transports.
No. Reaction Enzyme(s) EC No.
R
1
{}fi Pyruvate Influx
R
2
Pyruvate + CoASH + NAD
+
fi AcetylCoA + NADH + H
+
+CO
2
Pyruvate dehydrogenase 1.2.4.1.
R
3
2 AcetylCoA fi Acetoacetate + 2 CoASH Acetyltransferase + 2.3.1.9.
acetoacetylCoA hydrolase 3.1.2.11.
R
4
Acetoacetate + NADH + H
+
fi 3-OH-Butyrate + NAD
+
3-Hydroxybutyrate dehydrogenase 1.1.1.30.
R
5

Pyruvate + ATP + CO
2
fi Oxaloacetate + ADP + P
i
Pyruvate carboxylase 6.4.1.1.
R
6
AcetylCoA + Oxaloacetate fi Citrate + CoASH Citrate synthase 2.3.3.1.
R
7
Oxaloacetate + NADH + H
+
fi Malate + NAD
+
Malate dehydrogenase 1.1.1.37.
R
8
Citrate + NAD
+
fi NADH + H
+
+ [ ] Aconitase + isocitrate dehydrogenase 4.2.1.3.
1.1.1.41.
R
9
Acetoacetate fi { } Efflux
R
10
3-OH-Butyrate fi { } Efflux
R

11
Citrate fi { } Efflux
R
12
Malate fi { } Efflux
2752 J. W. Stucki (Eur. J. Biochem. 271) Ó FEBS 2004
Appendix 2.
Ó FEBS 2004 Chromokinetics of metabolic pathways (Eur. J. Biochem. 271) 2753
A color finder written in
MATHEMATICA
. This short program was written in
MATHEMATICA
4.
MATHEMATICA
code entries into
the computer are displayed as boldface Courier font. On a Macintosh Cube computer (450 Mhz) the calculation of the basic
colors for anaerobic pyruvate metabolism in mitochondria took 0.02 s. A more involved example, the reaction scheme for the
Belusov–Zhabotinsky reaction [2], took 0.8 s and produced all 34 basic colors. No attempt was made for further speed
optimization, such as matrix preprocessing or jumping out of prematurely finished loops. The major motivation to publish
this program here for the first time is to give the reader an interactive tool with which he or she can explore their own
problems. To do this, the user first has to enter his own stoichiometry matrix into
MATHEMATICA
’s front end, as in step (1). It
is mandatory to include reverse (backward) reactions as separate additional columns to the stoichiometry matrix. Then, the
MATHEMATICA
code from steps (2) to (6) has to be copied exactly. Step (5) contains timing and printing facilities, which may
be omitted. For large systems, containing many hundreds of basic colors, only their number may be of interest. This is
accessible with the last statement of the program. It is good practice to start a new problem with a new
MATHEMATICA
kernel.

This fairly general program can successfully manage a variety of networks. A big advantage is that it makes full use of the
exact calculations possible with
MATHEMATICA
. Thus, by using rational numbers instead of floating point approximations
thereof, the program escapes the possibility of missing some basic colors owing to round off or truncation errors. If one
accepts the prize to be paid for this in terms of computer time, the present program may serve as a standard to find all basic
colors also in large metabolic pathways.
2754 J. W. Stucki (Eur. J. Biochem. 271) Ó FEBS 2004