The basic colors of a metabolic pathway are its independent steady-state currents, which can be calculated from the stoichiometry matrix.. These diagrams are the basic colors used for th
Trang 1colors 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
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: joerg.stucki@pki.unibe.ch
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 http://www.cx.unibe.ch/pki/index.html
(Received 4 February 2004, revised 17 March 2004,
accepted 5 May 2004)
Trang 2such 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 ATPand 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.: j1, blue; j2, red; j3, green; j4 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 (ji) 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 , ADP and P 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 listed in Appendix 1; see also ÔConcluding remarksÕ.
Trang 3mitochondria 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,
ji, to the overall consumption and production of the metabolites in the incubation medium:
Pyruvate¼ 4j1þ 3j2þ 3j3þ 4j4 Malate¼ 2j1þ j2þ j3þ 2j4 Acetoacetate¼ j1
3-Hydroxybutyrate¼ j2
Citrate¼ j3
(Eqn 1)
Considering our system, we are left with a set of five equations for the measured metabolites and with only four unknowns: j1–j4 In such fortuitous cases, the jivalues can easily be calculated by using standard methods as a least-square solution of an overdetermined system The results,
ji, 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 the reaction scheme in Fig 1 can be decomposed They correspond to the basic colors j 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 valval-ues 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)
Color (l moles/20 min)
Trang 4even 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 mM), the
contribution of the white color (j4) is wiped out, as expected,
as citrate can no longer be metabolized in the citric acid
cycle (reaction r8) The color scheme has now collapsed to a
mixture of blue, green and red A further increase of the
inhibitor (0.67 mM) leads to unexpected side-effects: blue
(j1) and green (j3) vanish altogether and we are left with a
monochromatic red (j2) 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, ji, 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 (j4) 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).
Trang 5brute 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 r1
to r12 (left to right) and the rows with the metabolites
(downwards) in the order pyruvate, acetyl-CoA,
acetoace-tate, 3-hydroxy-butyrate, oxaloaceacetoace-tate, 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 r¼ 12 is the number of reactions
and n¼ 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)
i and obtains the reaction velocities by v¼ Ej I prefer normalization by Sji¼ 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 1020 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:
The reaction velocities, vi, may be nonlinear functions of the concentrations (or of the chemical activities) of the species involved Delegating nonlinearity into the vivalues, in this manner, we note that the concentrations, ci, are linear functions of these velocities and we put:
Trang 6dc/dt¼ 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:
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
MATHEMATICAto 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
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 j1¼ 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]
TheMATHEMATICAprogram 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:
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 algoalgo-rithm has further been implemented with theMATLABlanguage, in
a package calledFLUXANALYZER, by S Klamt and which is available on the Internet Moreover,METATOOLSby 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
Trang 7contained 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 what happens 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-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 mM and 0.66 mM 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 FADH2is 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-Pi, 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
Trang 8higher 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 (P rigogine, 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) The symbol { } is used for the extra mitochondrial metabolite pool and the symbol [ ] 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 ATPinto the mitochondria through the adeninenucleotide translocase, the membrane potential was collapsed with valinomycin The same applies for other nonelectroneutral transports.
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.
Trang 10A color finder written inMATHEMATICA This short program was written inMATHEMATICA4.MATHEMATICAcode 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 intoMATHEMATICA’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 MATHEMATICAcode 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 newMATHEMATICAkernel 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 withMATHEMATICA 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