Enumerating constrained elementary flux vectors of metabolic networks

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1 Enumerating constrained elementary flux vectors of metabolic networks Robert Urbanczik Institute of Physiology, University of Bern Bühlplatz 5, 3012-Bern, Switzerland Abstract I generalize the concept of elementary vector to the case where the steady state space of the metabolic network is not a flux cone but is a general polyhedron due to further inhomogeneous constraints on the flows through some of the reactions. On one hand, this allows to selectively enumerate elementary modes which satisfy certain optimality criteria and this can yield a large computational gain compared to full enumeration. On the other hand, in contrast to the single optimum found by executing a linear program, this enables a comprehensive description of the set of alternate optima often encountered in flux balance analysis. The concepts are illustrated on a metabolic network model of human cardiac mitochondria. 1 Introduction Constrained based modeling has proven to be a very useful approach for analyzing complex metabolic networks [1]. The first kind of constraints considered in such modeling derives from a steady state assumption on the operation of the network and requires that there be not net accumulation or depletion of metabolites which are internal to the system. The second kind of constraints reflects that some of the reactions in the network will run almost exclusively in one direction and requires the flow through these so called irreversible reactions to be nonnegative. Sometimes, as a third kind of constraints, absolute bounds are placed on the flow through some or all of reactions in the system. In geometrical terms, the fluxes satisfying the first two kinds of constraints form a polyhedral cone which is called the flux cone and one technique used in constrained based modeling has been to obtain an exhaustive description of the cone by enumerating its elementary vectors. Each elementary vector represents a minimal stoichiometrically viable pathway through the network and is thus often called an elementary mode [2]. Despite recent algorithmic advances [3, 4, 5] however, due to a combinatorial explosion of the number of elementary vectors, this approach is not useful for studying the large networks encountered in genome scale reconstructions of microorganisms such as E. coli or S. cerevisiae. Recently, 1

2 it has been suggested to consider instead of the flux cone as a simpler object the conversion cone [6]. This amounts to projecting the flux cone on to the exchange reactions of the system and only provides a description of the overall reactions which can be effected by the network. Thanks to the reduction of dimensionality achieved by the projection, enumerating the elementary vectors of the conversion cone can be feasible even for genome scale networks. While getting an overview of the metabolisms functionality in this manner is certainly useful, the description is by design incomplete since it abstracts away the internal workings of the system. If one incorporates the third kind of constraints into the analysis, the set of admissible fluxes is no longer a polyhedral cone but a general polyhedron. The most popular approach for studying this polyhedron has been flux balance analysis [7] where Linear Programming is used to search for a point satisfying an optimality criterion. While executing a single Linear Program for a network with say 1000 reactions is not a challenge for current computers, great care must often be taken in interpreting the result. The reason is that the optimum is usually not achieved at just a single point but is degenerate and the Linear Program returns an arbitrarily chosen point from the often high dimensional sub-polyhedron of all optimal flux vectors. Hence, for a more thorough analysis of the solution space, one is once again faced with the task of adequately describing a high dimensional polyhedron. The purpose of this paper is to extend the definition and the algorithms for elementary vectors from the special case of polyhedral cones to that of general polyhedra. In the context of metabolic network analysis two things are achieved by this. Considering the full flux cone, we obtain a procedure for selectively enumerating flux vectors which satisfy a given optimality criterion. Secondly, in the context of flux balance analysis, this provides a method for systematically studying the set of alternate optima. In Section 3, these concepts are illustrated on an existing model of human cardiac mitochondria [8]. The next section is devoted to the mathematical basis of the approach. 2 Describing flux polyhedra In formal terms the three kinds of constraints can be expressed as: a) Nν = 0. b) ν i 0 for i Irrev. (1) c) α i ν i β i for i = 1,..., n. Here N is the m by n stoichiometry matrix of the metabolic network, ν an n- dimensional flux vector and Irrev {1,..., n} the set of irreversible reactions. We shall allow α i = and β i = in condition (1c) to cover the cases where there is no lower or upper bound on some components of the flux vector. The set of solutions to Eqs. (1a) and (1b) is commonly referred to as the flux cone. In analogy, I shall call the solution set a flux polyhedron, when, in addition to these two constraints, inhomogeneous inequalities such as (1c) are imposed. 2

3 The above system of constraints can be written in more compact form as Aν b, (2) with a suitable d by n matrix A and an d-dimensional vector b. For instance, the equality constraint in (1a) can be replaced by the inequality constraints Nν 0 and N ν 0. Hence, the homogeneous constraints (1a,1b) could be represented in the above manner by choosing b = 0 and a matrix A given in block form as N A = N D. (3) Here D is a diagonal n by n matrix with D ii = 1 if reaction i is irreversible (otherwise D ij = 0). When one additionally incorporates the inhomogeneous constraints (1c) into such a transformation, a non-zero value of b arises. While computationally it may not be the best strategy to reduce to such a normal form, for presenting the main mathematical concepts Eq. (1) is both too specialized and too cumbersome. Hence I shall first consider Eq. (2). 2.1 Minimal generating sets A notion one can entertain about what solving Eq. (2) means is that in entails finding two finite sets of generating vectors {v (l) } p l=1 and {u(k) } q k=1 such that a vector satisfying (2) can be written as ν = p λ l v (l) + l=1 q µ k u (k) where k=1 λ l, µ k 0 and (4) p l=1 λ l = 1. Indeed, a classical result in Polyhedral Geometry, the Minkowski-Weyl theorem [9, 10], states that one can always find {v (l) } and {u (k) } such that a vector ν is a solution of Eq. (2) if and only if it can be expressed in the form (4) by choosing appropriate weights λ l and µ k. Note that the weights for the v (l) are bounded whereas the ones for the u (k) may be arbitrarily large. I shall hence call the v (l) bounded and the u (k) unbounded generators. Further, the bounded generators satisfy Eq. (2) whereas, in general, the unbounded generators may not. An important exception is the homogenous case b = 0. The polyhedron is then a cone and the unbounded generators lie in the cone. Even in this case, however, we need one trivial bounded generator, v (1) = 0, just to be able to satisfy the constraint p l=1 λ l = 1. When it is clear that we are dealing with a cone, it is customary to forget about this technicality, and use phrases such as generators of the cone to refer just to the unbounded generators. Sometimes one will be interested in parsimonious descriptions of the polyhedron and look for a minimal set of generators, i.e for {v (l) } and {u (k) } with the 3

4 additional property that Eq. (4) ceases to generate the whole polyhedron if any one of the v (l) or u (k) is removed. While useful, describing a polyhedron in this manner is not ideal in the context of metabolic networks. The first reason is, that the sets {v (l) } and {u (k) } may not be unique even if one requires minimality. It is then difficult to attribute biological significance to such a non unique solution. The second reason is that one cannot easily study the effect of deletions, i.e. the removal of one or more reactions from the network. In particular, it is often of interest whether additionally requiring ν i = 0 for the flow through reaction i means that Eq. (2) ceases to have a solution. But, if there are reversible reactions, this need not be the case even if the i-th component is nonzero in each of the generators v (l) and u (k). After all, positive and negative flows through a reversible reaction can cancel when combining the generators via Eq. (4). While, for single deletions, it is still straightforward to take this into account, this becomes increasingly complicated when one wants to consider the effect of double (ν i = ν j = 0) or even higher order deletions. This is why, in studying the full flux cone, elementary modes have received much more attention than minimal generating sets. 2.2 Elementary vectors of flux polyhedra In order to extend the concept of elementary vector from flux cones to flux polyhedra, it is useful to first have a definition of elementary vectors which, in the case of cones, works not just for flux cones but for general cones. In the context of analyzing the conversion cone [6], I have recently provided such a definition which applies to generic cones obtained as the solution set of an inequality of the form Aν 0, (5) i.e. of the homogenous version of Eq. (2). A nonzero vector e is called an elementary vector of the cone if it lies on an edge of the pointed cone obtained by intersecting the cone of solutions of (5) with one of the 2 n orthants of R n. For flux cones, i.e. if A is of the special form given by Eq. (3), this definition is entirely equivalent to the original definition based on maximal zero-sets [11]. But even for a general choice of the matrix A, many desirable properties still hold. On the one hand the set of elementary vectors is unique and on the other hand it is useful for studying deletions. In particular any nonzero solution ν of (5) can be decomposed as a sum of elementary vectors, ν = s l=1 e(l), in such a way that the following holds: Each row vector r of the matrix (e (1), e (2),..., e (s) ) satisfies r 0 or r 0, i.e no cancellation occurs when adding up the vectors in the decomposition. For deletions, this means that a nonzero vector ν can satisfy the constraints ν i = 0 and Aν 0 only if there is an elementary vector e of (5) with e i = 0. Based on this extended definition, the concept of elementary vector can be generalized from cones to polyhedra by mapping the inhomogeneous system (2) to a homogenous one, as illustrated in Fig. 1. This is achieved quite simply by introducing an additional scalar variable λ and considering the following homoge- 4

5 a 1 b 1 Λ Ν 1 Ν 2 Ν 1 Ν 2 c 1 d 1 Λ Ν 1 Ν 2 Ν 1 u Ν 2 Figure 1: a: Example for a bounded polyhedron. The endpoints of the vectors of the minimal generating set are shown by green circles. Since the polyhedron is bounded, the set is unique and coincides with the vertices of the polyhedron. Further, the vertices are also elementary vectors (marked red) but there are two additional elementary vectors with first component ν 1 = 0 resulting from intersecting the polygon with the ν 2 -axis. b: The cone associated to the polygon of Fig. 1a via Eq. (6). For this bounded case, the cone is geometrically obtained by lifting the polygon to the level λ = 1 and taking all rays emanating from the origin of the coordinate system which intersect the polygon at λ = 1. The edges of the cone correspond to the vertices of the polyhedron. (For clarity, the elementary vectors of the cone are not shown). c: Example of a polyhedron, unbounded in the directions indicated by the arrows. A minimal generating set has two unbounded generators (green circles connected to the origin by a line) and two bounded ones ( green circles). For the bounded generators any two points lying on opposing edges of the polyhedron could have been used. The elementary vectors are shown as red circles and the two unbounded elementary vectors are connected to the origin by a line. Two of the four bounded elementary vectors have ν 1 = 0 while the other two have ν 2 = 0. d: The cone associated to the polyhedron of Fig 1c. In contrast to all other rays in the cone, the edge of the cone (marked by u) does not intersect the polyhedron at λ = 1. This gives rise to the unbounded generators of the polyhedron. 5

6 nous inequalities in n + 1 variables (λ, ν): Aν λb 0, (6) λ 0. Clearly, if a vector ν satisfies condition (2), then (1, ν) satisfies (6) and, conversely, a solution (λ, ν) of (6) yields the solution ν/λ of (2), unless λ = 0. This allows to convert a generating set of the cone to one of the polyhedron by setting φ(λ, ν) = { ν if λ = 0 ν/λ else. (7) If u (k) = (λ (k), ν (k) ), with k = 1,..., q, is a set of unbounded generators of the cone (6), then {φ(λ (k), ν (k) )} q k=1 is a generating set of the polyhedron (2) and φ(λ (k), ν (k) ) is a bounded generator if λ (k) 0 and unbounded otherwise. We now define two kinds of elementary vector for the polyhedron (2). We shall say that φ(λ, e) is a bounded elementary vector of the polyhedron if (λ, e) is an elementary vector of the cone (6) with λ > 0. In case that (λ, e) is an elementary vector of the cone, but if λ = 0, we say that φ(0, e) is an unbounded elementary vector of the polyhedron. The next step is to investigate how decomposing into elementary vector works for polyhedra. Let ν be a solution of (2) and decompose (1, ν) into elementary vectors of the cone (6) as: (1, ν) = s u l=1 (λ (l), e (l) ) + s l=s u+1 (0, e (l) ). (8) For convenience, we have assumed that the decomposing vectors are ordered such that the s u vectors with λ (l) > 0 come first. This system of equations can be rewritten as 1 = ν = s u l=1 s u l=1 λ (l) (9) λ (l) φ(λ (l), e (l) ) + s l=s u+1 φ(0, e (l) ), where the last line yields the desired decomposition of ν into bounded and unbounded elementary vectors of the polyhedron. Note that the constraint on the λ (l) means that at least one bounded elementary vector must occur in the decomposition. As mentioned above, when decomposing (1, ν) the elementary vectors in Eq. (8) can be chosen such that each row of the matrix (e (1),..., e (s) ) is semipositive or semi-negative. Then we also have that r 0 or r 0 holds for each row r of the matrix ( ) φ(λ (1), e (1) ),..., φ(λ (su), e (su) ), φ(0, e (su+1) ),..., φ(0, e (s) ). (10) Since we can always decompose so that no cancellation occurs, as in the case of a cone, the elementary vectors of a polyhedron can be used to study the effect of 6

7 deletions. In particular, there is no vector ν in the polyhedron with ν i = 0 if the i-th component of every elementary vector of the polyhedron is nonzero. In fact, since a least one bounded elementary vector must occur when decomposing ν, we have an even simpler criterion: If the i-th component of every bounded elementary vector is nonzero, then every vector in the polyhedron has ν i 0. Based on the above analysis, it is straightforward to generalize the computational procedures for elementary vectors from flux cones to flux polyhedra. I have used the open-source package SNA (Stoichiometric Network Analysis [12]), which provides the functions ZH2gset and ZH2elvs, each taking two matrices Z and H as arguments and dealing with the cone given by the mixed system of equalities (Zν = 0) and inequalities (Hν 0). The routine ZH2gset computes a minimal generating set for the cone, whereas ZH2elvs enumerates the elementary vectors. Since the routines deal directly with the mixed system, it is possible to map the computation for the flux polyhedron (1) to that of a suitable cone, without first having to transform to the normal form (2). 3 Example Applications To test the above concepts, I shall consider a metabolic model [8] of human cardiac mitochondria. Besides reactions located inside the mitochondria, the model also comprises the cytosolic support apparatus of the mitochondria and the transport reactions between the mitochondria and the cytosol as well as between cytosol and the extracellular environment. In total there are 191 reactions, not counting exchange reactions, linking 232 compartmentalized metabolites, of which 27 are external. The above reaction count contains one reaction for the synthesis of protoheme and one for the production of phospholipid. Protoheme production is an important function of mitochondria, since heme plays an essential part in assembling many apoproteins. On the other hand, phospholipid synthesis is of interest due to it importance for the assembly and maintenance of the mitochondria itself. 3.1 The conversion cone of the network In analyzing the network, it is convenient to get an overview of the possible overall reactions between the 27 external metabolites by analyzing the conversion cone using SNA. This yields that the conversion cone has 1249 elementary vectors, of which 779 produce protoheme and 1154 produce phospholipid. Most of the protoheme conversions produce phospholipid as well, and only 60 of 779 elementary conversions yielding protoheme, do not produce phospholipid. While counting the number of conversions indicates considerable overlap between the production of the two compounds, a quite different picture emerges when one looks at minimal media. By inspecting the inputs of the elementary conversions, one finds that only fe2, gly and o2 are essential for protoheme synthesis, while for phospolipid synthesis the following are essential: 12dgr m, 7

8 Number of elementary conversions Number of elementary conversions protoheme produced per oxygen consumed phospholipid produced per ocdcea consumed Figure 2: Histogram of the yield ratios w.r.t. oxygen for the 779 elementary conversions producing protoheme (left) and w.r.t. ocdcea for the 1154 elementary conversions producing phospholipid (right). c204, c226, cdpchol, ocdca, ocdcea, ocdcya, ps m 1. For both products, the essential metabolites are by themselves not sufficient to enable the synthesis. For protoheme, the three essential compounds can be supplemented by glu-l or glc-d to obtain minimal media for the synthesis. One way of obtaining a minimal medium for phospholipid production is to supplement the essential inputs by glc-d, glyc3p and hdca. The examples show that, while the minimal media for the two products can overlap (if glc-d is used for protoheme), they do not have to overlap (if glu-l is used for protoheme). Hence there need not be any competition between protoheme and phospholipid synthesis. In case that a metabolite is an essential input for some product, it is interesting to ask what the yield is, i.e. the amount of product per consumed input metabolite. As an example, the possible yields of protoheme with respect to oxygen as well as the yield of phospholipid with respect to ocdcea are shown in Fig. 2. While only a small fraction of the elementary conversions producing protoheme utilize oxygen optimally (with yield 50 ), the majority of the conversions producing phospholipid 519 have the maximal yield of 125. So the criterion of optimal yield is much more 9 selective for oxygen consumption during protoheme synthesis than for ocdcea use when producing phospholipid. 3.2 Optimal elementary flux vectors We can now combine the theory in Section 2 with the findings from the analysis of the conversion cone, to selectively enumerate elementary vectors of the flux cone with maximal yield. For this, we add two inhomogeneous constraints of the form (1c) to the flux cone conditions (1a) and (1b). The first one requires the uptake of the essential input to be smaller or equal to 1, whereas the second one requires 1 The following abbreviations are used for the metabolites: 12dgr m: 1,2-Diacylglycerol, c204: nc20:4, c226: nc22:6, co2: CO 2, coa: Coenzyme A, cys-l: L-Cysteine, fe2: Fe2+, glc-d: D-Glucose, glu-l: L-Glutamate, gly: Glycine, glyc: Glycerol, glyc3p: Glycerol 3- phosphate, h: H+, h2o: H 2 O, hdca: Hexadecanoate (n-c16:0), lac-l: L-Lactate, o2: O 2, ocdca: Octadecanoate (n-c18:0), ocdcea: Octadecenoate (n-c18:1), ocdcya: Octadecynoate (n-c18:2), pi: Phosphate, ps m: Phosphatidylserine, urea: Urea. 8

9 optimal protoheme optimal phospholipid jointly optimal elementary vectors average essential bounded unbounded reactions reactions Table 1: Selective enumeration of the elementary vectors of the flux cone for maximal yield of protoheme w.r.t. consumed oxygen and of phospholipid w.r.t. ocdcea. The column average reactions gives the average number of reactions used in the bounded elementary fluxes (the unbounded elementary vectors do not yield the product and do not consume the substrate). The column essential reactions gives the number of reactions with nonzero flow in each bounded elementary vector. Removing one of these reactions, would impede optimal yield. the output of the product to be larger or equal to the maximal yield found in the conversions cone analysis. Table 1 list the results for studying the optimal use of oxygen in producing protoheme in this way, as well as for optimal use of ocdcea in producing phospholipid. As one might expect from the conversion cone analysis, there are far fewer optimal elementary vectors in the protoheme case than for phospholipid. It is interesting to study the optimal production of protoheme in more detail. Of the 69 optimal elementary fluxes, 56 also produce phospholipid. The synthesis path in the remaining 13 fluxes is much simpler, with only an average of 64 reactions used. However, the 56 elementary fluxes producing phospholipid have the rather intriguing property, that all of them show the maximal yield ( 125 ) of phospholipid relative to consumed ocdcea (also shown in Tale 1). So they simultane- 9 ously optimize protoheme production w.r.t. consumed oxygen and phospholipid w.r.t. ocdcea. Instead of enumerating just the elementary flux vectors with optimal yield, we could also have obtained the above results by first computing all elementary flux vectors and then picking the ones with the desired properties. So it is important to compare the computational cost of the two approaches. Considering the full flux cone, one finds that a minimal generating set has just 2543 elements which are computed in 5 seconds on a standard PC. However, when one goes on to calculate all elementary flux vectors, are found, and SNA needs about 6 minutes for the enumeration. In contrast to this, enumerating the elementary fluxes for optimal protoheme yield just takes 4 seconds, mainly spent in preprocessing. However, for maximal phospholipid yield, due to the large number of optimal elementary fluxes, the computation time is only reduced by 13% compared to full enumeration. While selective enumeration of optimal elementary vectors is computationally highly useful in cases where optimality is a stringent condition, in the other cases 9

10 full enumeration seems preferable. An assessment of how stringent a condition optimal yield is can, as we have seen, be obtained by analyzing the conversion cone. 3.3 The state space of flux balance analysis Above, we have used inhomogeneous constraints sparingly, in essence picking a face of the flux cone. In contrast to this, in flux balance analysis a much richer set of inhomogeneous constraints is typically used, substantially changing the geometry of the state space. In particular, the analysis of human cardiac mitochondria in [8] was based on the following constraints. A uniform upper bound of 1 was imposed on the uptake rate of the following 9 external metabolites used as input to the network: c204, c226, gly, glyc, glyc3p, hdca, ocdca, ocdcea, ocdcya. Further, the uptake rate was bounded by for oxygen, by 4.9 for glc-d, and the flow through the reversible exchange glu-l 0 was constrained to lie between 94 and 39. To make contact with experimental findings, the unit of measurement for the above flows is µmol/min/g proteins. Finally, constraints were placed on the flow for 15 internal reactions, based on the experimental literature on (not necessarily human cardiac) mitochondria. Details and references can be found in [8]. However, it turns out that, with one exception, all of the constraints on the internal flows are automatically satisfied if the constraints on the exchange reactions are fulfilled. The one exception is the upper bound of 32.6 on the ADP/ATP transporter between cytosol and mitochondria: adp c +atp m adp m +atp c. Hence, in addition to the conditions on the exchange flows, I shall only consider this internal constraint here. I first computed a minimal generating set for the flux polyhedron satisfying the above constraints. This yields bounded and 0 unbounded generating vectors. So the polyhedron is bounded and the bounded generating vectors are its vertices. By inspecting the vertices, the maximal production rate of protoheme is found to be 1 and vertices display this optimal rate. For 8 phospholipid, vertices have the maximal production rate of 100. As before, 27 it is possible to simultaneously optimize protoheme and phospolipid, with vertices being jointly optimal. To further investigate this jointly optimal sub-polyhedron, I computed its elementary fluxes. This yields elementary vectors, with an average of reactions per elementary vector having nonzero flow. Checking which reactions are always used, shows that 80 reactions are essential for jointly optimal production of protoheme and phospholipid given the above set of additional constraints. Since the essential reactions are by themselves not sufficient for jointly optimal synthesis, it also of interest to find the elementary vectors using the smallest number of reactions. Inspecting the whole set yields that in each elementary vector at least 113 reactions have nonzero flow and that there are only 13 elementary vectors which use just 113 reactions. These 13 elementary vectors provide minimal models of the jointly optimal synthesis pathway. 10

11 4 Discussion The main objective of this paper has been to extend the definition and computational procedures for elementary vectors from cones to general polyhedra. One application of the generalized approach has been to selectively enumerate elementary vectors of the flux cone which satisfy an optimality criterion. We have seen, that large computational savings result compared to a full enumeration, provided there are relatively few optimal elementary vectors. As a second application, we have studied the set of alternate optima in a flux balance analysis problem. With regard to computational performance in this application, it should be mentioned that, for the above model of human cardiac mitochondria, enumerating the vertices jointly optimal w.r.t to protoheme and phospholipid was found computationally infeasible in [8]. Here, we have not only been able to find these vertices but also the elementary vectors of the jointly optimal polyhedron. This may be indicative of the computational advantages of the present approach compared to the technique used in [8] which was based on lrs [13]. When comparing the flux cone to the polyhedron considered in flux balance analysis, one is impressed by the increase in complexity resulting from the large number of additional inhomogeneous constraints imposed in Section 3.3. While a minimal generating set of the cone has just 2543 vectors, the corresponding figure for the polyhedron is Obviously, including experimentally well documented bounds on the flows through certain reactions is an important way of enhancing the biological relevance of the model. On the other hand, additional conditions can also increase complexity. One advantage of the present approach may be that it works equally for bounded and unbounded polyhedra. So there is no need to add inhomogeneous constraints to the model just to ensure that the polyhedron is bounded. While one arrives at a comprehensive description of the optimal flux polyhedron by enumerating the elementary vectors, many of of the biological insights thus obtained can also be found by executing a large number of linear programs, especially if one is prepared to tackle the NP-complete task of mixed integer linear programming. The practical difficulties of arriving at comprehensive descriptions of the flux polyhedron in this manner are highlighted by the abundance of suggested methods [14, 15, 16, 17, 18], each one representing a different way of striking a compromise between detailedness of description and computational tractability. All of the results found by these approaches can also be obtained by enumerating elementary flux vectors, but the converse is often not true. For instance, the minimal models determined in Section 3.3 are outside of the scope of flux variability [14] or flux coupling analysis [15]. A quite different way of studying the optimal flux polyhedron is the random sampling of its vectors [19]. While this has been used to simply obtain the statistical analogues to flux variability and coupling analysis, a usage for optimal experimental design has also been suggested [20]. The idea is to determine the most informative single reaction measurement by considering the possible reduction of the state space resulting from measuring the flow through a particular reaction. 11

12 Based on a random sample, one can, for a single reaction measurement, estimate the probability of each possible outcome (assuming a uniform probability density on the flux polyhedron). Then it makes sense to choose to measure a reaction for which the information entropy of the probability distribution thus obtained is high since this optimizes the expected information gain. It is interesting to speculate what would happen if one replaced the random sample in this design strategy with the set of all elementary vectors. Of course, the latter are not all random but highly biased towards flux vectors where many reactions have zero flow. So, while basing the strategy on a random sample, is expected to lead to large reduction of uncertainty in terms of the admissible state space volume, basing it on elementary vector, should lead to large reduction of uncertainty with regard to which reactions are actually being used by the metabolism. References [1] N. Price, J. Reed, and B. Palsson. Genome-scale models of microbial cells: Evaluating the consequences of constraints. Nat. Rev. Microbiol., 2: , [2] S. Schuster, D.A. Fell, and T. Dandekar. A general definition of metabolic pathways useful for systematic organisation and analysis of complex metabolic pathways. Nature Biotech., 18: , [3] R. Urbanczik and C. Wagner. An improved algorithm for stoichiometric network analysis: theory and applications. Bioinformatics, 21: , [4] A. von Kamp and S. Schuster. Metatool 5.0: fast and flexible elementary modes analysis. Bioinformatics, 22: , [5] S. Klamt, J. Gagneur, and A. von Kamp. Algorithmic approaches for computing elementary modes in large biochemical reaction networks. IEE Proc.- Syst. Biol., 152: , [6] R. Urbanczik and C. Wagner. Functional stoichiometric analysis of metabolic networks. Bioinformatics, 21: , [7] R.U. Ibarra, J.S. Edwards, and B.O. Palsson. Escherichia coli k-12 undergoes adaptive evolution to achieve in silico predicted optimal growth. Nature, 420: , [8] T. Vo, H. Greenberg, and B. Palsson. Reconstruction and functional characterization of the human mitrochondrial metabolic network based on proteomic and biochemical data. J. Biol. Chem., 279: , [9] R.T. Rockafellar. Convex Analysis. Princeton University Press, Princeton,

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