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: 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) 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 . Chromokinetics of metabolic pathways Jo¨ rg W. Stucki Department of Pharmacology, University of Bern, Bern, Switzerland Some. 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