In the particular case of a metabolic system with only irreversible reactions, the set of admissible reactions reads:

P = {v ∈ ^q : Nv = 0 and v ≥ 0}     (3)

Compared with (1) P is in this case a particular, namely a pointed polyhedral cone (an example is depicted in Figure 1). This geometry can be intuitively understood, noting that there are certainly 'enough' intersecting half-spaces (given by the inequalities v ≥ 0) to have this 'pointed' effect in 0: P contains no real line (otherwise there coexist x and -x not null in P, a contradiction with the constraint v ≥ 0). The figure even suggests that a pointed polyhedral cone can be either defined in an implicit way, by the set of constraints as we did until now, or in an explicit or generative way, by its 'edges', the so-called extreme rays (or generating vectors) that unambiguously define its boundaries. In the following, we show that elementary modes always correspond to extreme rays of a particular pointed cone as defined in (3) and that their computation therefore matches to the so-called extreme ray enumeration problem, i.e. the problem of enumerating all extreme rays of a pointed polyhedral cone defined by its constraints. An overview on general and current issues on extreme ray enumeration can be found in 25. For the sake of consistency, we use this reference as a main source and adopt the same mathematical notations.

The pointed polyhedral cone is the central mathematical object throughout this work; therefore we shall introduce more precise definitions and results surrounding it.

P is a pointed polyhedral cone of ^d if and only if P is defined by a full rank h × d matrix A (rank(A) = d) such that,

P = P(A) = {x ∈ ^d : Ax ≥ 0}     (4)

The h rows of the matrix A represent h linear inequalities, whereas the full rank mention imposes the "pointed" effect in 0. Note that a pointed polyhedral cone is, in general, not restricted to be located completely in the positive orthant as in (3). For example, the cone considered in extreme-pathway analysis may have negative parts (namely for exchange reactions), however, by using a particular configuration it is ensured that the spanned cone is pointed 8.

Now we must characterize the extreme rays. A vector r is said to be a ray of P(A) if r ≠ 0 and for all α > 0, α · r ∈ P(A). We identify two rays r and r' if there is some α > 0 such that r = α · r' and we denote r ≃ r', analogous as introduced above for flux vectors. For any vector x in P(A), the zero set or active set Z(x) is the set of inequality indices satisfied by x with equality. Noting A_i• the i^th row of A, Z(x) = {i : A_i•x = 0}. Zero sets can be used to characterize extreme rays. For simplicity, we adopt in this document the following characteristic (25 for example) as a working definition of extreme rays.

Definition 1: Extreme ray

Let r be a ray of the pointed polyhedral cone P(A). The following statements are equivalent:

(a) r is an extreme ray of P(A)

(b) if r' is a ray of P(A) with Z(r) ⊆ Z(r') then r' ≃ r

Since A is full rank, 0 is the unique vector that solves all constraints with equality. The extreme rays are those rays of P(A) that solve a maximum but not all constraints with equalities. This is expressed in (b) by requiring that no other ray in P(A) solves the same constraints plus additional ones with equalities. Note that in (b) Z(r) = Z(r') consequently holds.

An important property of the extreme rays is that they form a finite set of generating vectors of the pointed cone (25 for example): any vector of P(A) can be expressed as a non-negative linear combination of extreme rays, and the converse is true: all non-negative combinations of extreme rays lie in P(A). Moreover, the set of extreme rays is the unique minimal set of generating vectors of a pointed cone P(A) (up to positive scalings).

Lemma 1: EMs in networks of irreversible reactions

In a metabolic system where all reactions are irreversible, the EMs are exactly the extreme rays of P = {v ∈ ^q : Nv = 0 and v ≥ 0}.

Proof: P is the solution set of the linear inequalities defined by where I is the q × q identity matrix. Since it contains I, A is full rank and therefore P is a pointed polyhedral cone. All v ∈ P obey Nv = 0, thus the 2m first inequalities defined by A hold with equality for all vectors in P and the inclusion condition of Definition 1 can be restricted to the last q inequalities, i.e. the inequalities corresponding to the reactions. Inclusion over the zero set can be equivalently seen as containment over the set of non-zeros in v, i.e. R(v). Consequently, e ∈ P is an extreme ray of P if and only if: for all e' ∈ P : R(e') ⊆ R(e) ⇒ e' = 0 or e' ≃ e, i.e. if and only if e is elementary. Thus, all three conditions in (2) are fulfilled.

In the general case, some reactions of the metabolic system can be reversible. Consequently, A does not contain the identity matrix and P (as given in (1)) is not ensured to be a pointed polyhedral cone anymore 7. Because they contain a linear subspace, non-pointed polyhedral cones cannot be represented properly by a unique set of generating vectors composed of extreme rays, albeit a set of generating vectors exists, sometimes also called convex basis 7. One way to obtain a pointed polyhedral cone from (1) is to split up each reversible reaction into two opposite irreversible ones. Note that this operation is completely analogous to a transformation step used in linear programming to obtain a linear optimization problem in canonical form: free variables v are also split into two variables v⁺ and v^- with v = v⁺ - v^- and v⁺, v^- ≥ 0 26. It has been noticed earlier that this virtual split does not change essentially the outcome: the EMs in the reconfigured network are practically equivalent to the EMs from the original network 10. Here we prove and precisely characterize this result.

We first introduce some notations. We denote the original reaction network by S and the reconfigured network (with all reversible reactions split up) by S'. The reactions of S are indexed from 1 to q. Remember that Irrev denotes the set of irreversible reaction indices and Rev the reversible ones. An irreversible reaction indexed i gives rise to a reaction of S' indexed i. A reversible reaction indexed i gives rise to two opposite reactions of S' indexed by the pairs (i,+1) and (i,-1) for the forward and the backward respectively. The reconfiguration of a flux vector v ∈ ^q of S is a flux vector v' ∈ ^{Irrev ∪ Rev × {-1;+1}} of S' such that

Let N' be the stoichiometry matrix of S'. N' can be written as N' = [N - N_Rev] where N_Rev consists of all columns of N corresponding to reversible reactions. Note that if v is a flux vector of S and v' is its reconfiguration then Nv = N'v'.

If possible, i.e. if v' ∈ ^{Irrev ∪ Rev × {-1;+1}} is such that for any reversible reaction index i ∈ Rev at least one of the two coefficients v'_(i,+1) or v'_(i,-1) equals zero, then we define the reverse operation, called back-configuration that maps v' back to a flux vector v such that:

Theorem 1: EMs in original and in reconfigured networks

Let S be a metabolic system and S' its reconfiguration by splitting up reversible reactions. Then the set of EMs of S' is the union of

a) the set of reconfigured EMs of S

b) the set of two-cycles made of a forward and a backward reaction of S' derived from the same reversible reaction of S

Proof: see Methods.

Thus, the set of EMs of the original network is equivalent (up to the two-cycles) to the set of EMs in the reconfigured network and therefore can be seen as a reduced set of extreme rays of the pointed convex polyhedron as defined by:

P = {v' ∈ ^{q + |Rev|} : N'v' = 0 and v' ≥ 0}     (5)

Hence, EMs computation can be derived from any extreme ray enumeration algorithm applied to the reconfigured network and followed by vector back-configuration and the elimination of meaningless vectors, namely the two-cycles.

Note that exactly the same procedure – splitting reversible reactions into two irreversible ones – was carried out also in the original work of Clarke 27 on stability analyses in stoichiometric networks. Clarke called the extreme rays of the corresponding cone (5) extreme currents. Thus, extreme currents are identical to the EMs in the reconfigured network and, hence, also (up to the 2-cycles) equivalent to the EMs from the original network

In the following we present a simple yet efficient algorithm for extreme ray enumeration, the so-called Double Description Method 28. We show that it serves as a common framework to the most prominent EM computation methods. To reach this generality, we concentrate on mathematical operations regardless to the actual data-structures used in the implementation. Therefore we manipulate objects such as matrices, vectors or inequalities and leave their implementation into tableaus, arrays and so on to the next section.

A generating matrix R of a pointed polyhedral cone P(A) is a matrix such that P(A) = {x ∈ ^d : x = Rλ for some λ ≥ 0}. The pair (A,R) is called a Double Description pair, or DD pair. As mentioned above, the extreme rays form the unique set of minimal generating vectors of P(A) and thus, considered as set of d-vectors, the extreme rays of P(A) form the columns of a generating matrix R that is minimal in terms of number of columns. The pair (A,R) is then called a minimal DD pair.

The strategy of the Double Description Method is to iteratively build a minimal DD pair (A_k, R_k) from a minimal DD pair (A_{k - 1}, R_{k - 1}), where A_k is a submatrix of A made of k rows of A. At each step the columns of R_k are the extreme rays of P(A_k), the convex polyhedron defined by the linear inequalities A_k. The incremental step introduces a constraint of A that is not yet satisfied by all computed extreme rays. Some extreme rays are kept, some are discarded and new ones are generated. The generation of new extreme rays relies on the notion of adjacent extreme rays. Here again, for the sake of simplicity, we adopt a characteristic (25 for example) as a working definition of adjacent extreme rays.

Definition 2: Adjacent extreme rays

Let r and r' be distinct rays of the pointed polyhedral cone P(A). Then the following statements are equivalent:

(a) r and r' are adjacent extreme rays

(b) if r" is a ray of P(A) with Z(r) ∩ Z(r') ⊆ Z(r") then either r" ≃ r or r" ≃ r'

The Double Description Method together with Theorem 1 offers a framework for computing EMs. The only steps to include are a reconfiguration step that splits reversible reactions and builds the matrix A, and a post-processing step that gets rid of futile two-cycles and computes the back-configuration. The dimension of the space is given by the number of reactions in the reconfigured network: q' = q + |Rev|. This results in the general algorithmic scheme as given in Table 1 (from here, all variables for the reconfigured network are written without prime):

As mentioned in the introduction section, the two most efficient algorithms for computing EMs available are the recently introduced null-space approach 24 and the Schuster algorithm 6, that we call "canonical basis approach" (implemented, for example, in METATOOL 29 version 4.3 and FluxAnalyzer version 5.0 30). Both algorithms handle reversible reactions directly. A direct handling of reversible reactions, meaning without network reconfiguration, is feasible in each setting and has been described in the respective original articles. This requires adapted adjacency tests. However, it does not affect the overall strategy. For simplicity, we describe these algorithms with networks of irreversible reactions only (the issue of reversible reactions is discussed below). We are now able to see that the algorithms of Schuster and Wagner differ basically only in the chosen initialization for R.

It is common practice to reduce the problem of extreme ray enumeration by restricting the input set to the set of irredundant constraints 25. Although the general problem of extreme ray enumeration is non-polynomial, the reduction into irredundant constraints is equivalent to linear programming and therefore of polynomial complexity. To our best knowledge, this important pre-processing has never been spelled out explicitly in the context of EM computation. However some network simplification steps have been proposed earlier 430 that deeply relate to the notion of redundancy removal. These simplifications include three heuristics that reduce the size of the original stoichiometric matrix N and thus the input size of the problem: the detection of conservation relations, of strictly detailed balanced reactions and of enzyme subsets.

Conservation relations of metabolites are captured as linear dependencies between rows of the stoichiometry matrix N (thus, in the left null-space of N; 31). This implies that some of the equality constraints in Nr = 0 are linearly dependent. Satisfying a maximal linearly independent subset of these equations suffices to satisfy all equations. Therefore the problem can be reduced to , where is the reduced stoichiometry matrix. For example, in Figure 3a, metabolites B and C build up one conservation relation and thus one of these metabolites can be removed. Note that conservation relations need not to be considered explicitly in the null-space approach since their removal does not affect the computed null-space matrix.

Conservation relations only consider redundancies among the equalities. The general approach handles also inequality constraints. Strictly detailed balanced reactions 32 and enzyme subsets 29 are particular cases of such redundancies. Strictly detailed balanced reactions are reactions with null flux at any steady-state. Many of them can be identified as null row vectors of K, the kernel matrix of N, and can be eliminated from the system. A non-trivial example is shown in Figure 3b, where R1 is strictly detailed balanced and would be detected by using the kernel matrix. However, there may be further reactions with a fixed zero-flux in steady state that cannot be identified by K. Some of those can be found by a simple analysis of N. For example, all the uni-directional reactions pointing into an internal sink (or emanating from a source) are certainly not participating in any steady-state flux (Figure 3c).

An enzyme subsets is defined as a group of reactions with relative constant flux ratio at steady state. Many of them can be identified as row vectors of K differing only in a scalar factor α. Reactions R1, R2 and R5 in Figure 3d would represent one enzyme subset. Assume one works on the reconfigured network and reactions R1 and R2 are members of the same enzyme subset. Thus, at steady state, we have for the respective rates r₁ = α · r₂. If α > 0, the constraints r₁ ≥ 0 and r₂ ≥ 0 are redundant, r₁ ≥ 0 being sufficient. In that case the practice is to lump both reactions into one lowering the number of reactions (and often also of the metabolites). If α < 0, the constraints r₁ ≥ 0 and r₂ ≥ 0 imply r₁ = r₂ = 0, hence, a special case of strictly detailed balanced reactions. In this case we say that the reactions contradict each other. Both reactions are not used and can be eliminated from the system as reactions R1 and R4 in Figure 3e.

We identified another kind of redundancies. We call a metabolite M uniquely produced (respectively consumed) if only one single reaction, say i, can produce (respectively consume) M for several consuming (respectively producing) it (see Figure 3f). In that case, balancing metabolite M at steady-state implies that r_i is always non-zero whenever the other reactions connected to M are active. We can therefore lump each reaction consuming (respectively producing) M with reaction i and remove metabolite M, decreasing the dimension of the problem further (see also the example in Figure 5 which is discussed below). Note that some enzyme subsets and strictly detailed balanced reactions can be seen as special cases of this type of redundancy.

Elimination of redundancies and network compression should be done in a pre-processing step leading to a compressed network structure. Thereby, it is important to detect and remove such redundancies iteratively until no further redundancy can be found. A MATLAB function compressSMat which removes all redundancies discussed above in an iterative fashion can be obtained from the corresponding author. After the computation of EMs, lumped reactions can be expanded to their single components.

There is a general approach for identifying redundancies in a set of linear constraints that uses linear programming, for example with the software redund distributed together with the software lrs 33. This approach does not require any iterative process, but only identify redundant inequalities. Rows of A can be eliminated but no consequent column-wise reduction is done. Therefore, a simple redundancy removal is not as powerful as the accompanying network compressions presented above. The method however has the advantage to be systematic and might lead in the future to further network simplifications not yet identified.

Using the reconfigured network with only irreversible reactions we have shown that the most important algorithms for EM computation belong to the same general framework. However, the original algorithms from Schuster and Wagner operate directly on the original network without splitting reversible reactions. At a first look, this seems to be more efficient since the dimension (number of reactions) is lower, decreasing seemingly also the memory requirement and the costs for adjacency tests. However, using the reconfigured network S' offers great simplifications. First, as already mentioned in an earlier section, the adjacency tests are easier to handle. The most important advantage, however, is the following. For the CBA in S' it follows that all non-zero elements of a ray r^k will be retained if a new ray is obtained by combining r^k with another (adjacent) ray because only positive combinations of rays are performed. The same holds for the NSA with respect to the p already processed inequality (irreversibility) constraints. This is of great importance since the adjacency test requires the information on zero/non-zero places in the rays only.

We illustrate this idea for NSA because this approach turned out to be more efficient than CBA. We assume that N has full rank m, i.e. there is no conservation relation. In this section, all variables correspond again to the network with split reversible reactions.

As described above, for an initialization of R we use a kernel matrix K of N having form (8):

Note that we use here the transposed representation of the tableau compared to Wagner's original article 24. Since by eq. (9) only positive column combinations are performed during the algorithm, no negative number can show up in the upper part (consisting of q - m rows (reactions)) during the next iterations. The first row to be processed now is p = q - m + 1. Using the general algorithmic scheme provided above all rays with non-negative entries at row p are retained and all negative entries can be combined with positive ones that are adjacent to them to obtain a zero at position p.

Assuming that the procession of the p-th row leads to a collection of t rays, we have:

The upper part, R₁, contains the p processed rows which only contain non-negative values. Again, positive combinations of rays performed during the next iterations lead in the upper part to sums of non-negative numbers. Hence, it is easy to keep track of the zeroes in the upper part R₁ by the use of bit masks. After the procession of the p-th inequality constraint the p-th row (i.e. the first row of R₂) can be transformed to its binary representation and moved from R₂ to R₁. Using a binary representation for R₁ has many advantages:

(i) For the next row p + 1 to be processed we have to perform the adjacency test for pairs of vectors , . This test only requires the first p elements of these rays (see (10)), hence, exactly the columns of R₁. Test (10) can then be written as a simple (and fast) bit operation. Two distinct vectors , are adjacent if and only if for all vectors r^k distinct from and , it holds:

( taken from R₁; r_1...p denotes the first p elements of r). Of course, the identical terms in the parentheses are computed only once.

(ii) Combination step of two adjacent rays (eq. (9)) reduces for the part in R₁ to a simple OR operation, which is already computed for (13). The other (real number) components of the two rays (contained in R₂) are combined as usual by eq. (9).

(iii) Bit operations as applied in R₁ are not only fast, they are numerically exact in contrast to operations on real numbers.

(iv) The binary representation requires much less memory. Taking a typical 64-bit floating-point variable, storing R₁ binary takes only 1.6 % of the memory needed for real numbers. Taking into account that in the worst case (all reactions reversible) the number of reactions in the reconfigured network is twice of that from the original one we still have a reduction in memory requirements of more than 96%. Note that R₂ is empty at the end of the algorithm, hence, all EMs are then stored binary.

Bitmap representations of EMs have already been used in earlier implementations for accelerating the adjacency (elementarity) tests. However, binary tableaus had then been stored and updated in parallel to the full (real number) tableau of EMs which is not necessary here.

After the whole processing, EMs (extreme rays) are obtained for the reconfigured network S' as binary vectors. Binary patterns of EMs are completely sufficient for many applications of EMs (see discussion). However, a well-known lemma (25 for example) ensures that this information is also sufficient to retrieve the real values up to a positive scalar:

Lemma 2

In a d-dimensional Euclidean space, let r be a ray of the pointed polyhedral cone P(A). The following statements are equivalent:

(a) r is an extreme ray of P(A)

(b) rank(A_Z(r)) = d - 1

Each obtained binary vector provides the zero set Z_q(e) and its complement the reaction set R(e) of an EM e in the reconfigured network S'. Lemma 2 says that the equation and therefore

N_R(e)e_R(e) = 0     (14)

admit a one-dimensional solution space, i.e. the dimension of the null space of N_R(e) is 1. N_R(e) denotes the m × |R(e)| sub-matrix of N containing all those reactions (columns) of N which are involved in e. Solving the homogeneous linear system (14) gives a vector that can be normalized and properly oriented for example by dividing it by the value on its first participating reaction (see the example below). The reconstruction process reflects the fact that an EM is – up to a scalar – determined by its participating reactions.

In a second post-processing step, we transform the (real number) EMs of S' back into their representation in the original network S by using the rules given before Theorem 1. Note that it is also possible to transform first the binary EMs from S' into the binary EMs of S and then to reconstruct the real numbers (by using eq. (14) for the stoichiometric matrix of the non-reconfigured network S; see pseudo-code). In both cases, if the original network had been compressed during pre-processing, the EMs can finally be expanded to their corresponding modes in the uncompressed network.

Using the results of the previous sections we are now able to give a pseudo-code of the binary (null-space) approach (Figure 4). The code follows MATLAB style, which provides a convenient and comprehensible notation for operations on vectors and matrices. We use several native MATLAB routines (written in bold). For concision, we also make the use of some other routines (indicated in italic). The code of the latter routines is not given here explicitly but their names and accompanying comments should allow the reader to implement them. For readers not familiar with MATLAB notation we give in the Methods section some basic explanations which should suffice for understanding the pseudo-code.

Note that the pseudo-code in Figure 4 is not given in its computationally most efficient form. It should just present the basic structure of the algorithm. There are two important issues in the algorithm we still have to discuss.

This section is devoted to illustrate our binary approach for computing elementary modes. Figure 5(a) shows a simple example network consisting of four metabolites (A,B,C,D) and 7 reactions (R1...R7), whereof R5 is reversible. The stoichiometric matrix N of this network reads accordingly:

Using our rules for removing redundancies, this network can be compressed as depicted in Figure 5(b). Metabolite A is uniquely produced, hence, R1 and R2 can be combined to R1c and reactions R1 and R3 are lumped into R2c. R3c and R4c correspond to the original reactions R4 and R5, respectively. Finally, R6 and R7 are enzyme subsets and are combined to R5c. Metabolites A and D can be removed, since they do not occur in any reaction anymore. Thus, the network dimension could be reduced by two metabolites and two reactions. The stoichiometric matrix N_C of the compressed system reads:

From this compressed network, we can compute a null space matrix having structure (8), here even without permuting rows (reactions):

K_C would be the starting tableau in the original null-space approach. Applying our binary approach we have now to split the (only) reversible reaction R4c in the compressed network (Figure 5(c)). This results in the stoichiometric matrix N_C', where R4c_b denotes the additionally introduced column of the backward direction of R4c:

Now we need to determine a null space matrix K_C' of N_C', if possible in the sparse form as in eq. (M1) (Methods section). K_C – as given in (18) – contains only irreversible reactions in the identity sub-matrix. Therefore, without further rearrangements, we can already use it to construct K_C' as described in the Methods section. We introduce an additional row in the identity sub-matrix of K_C (corresponding to R4c_b) and an additional column representing the two-cycle from the split reversible reaction R4c:

K_C' is now a proper initialization for the R tableau according to (11). The first four rows (in the identity sub-matrix) can be seen as already completed, we therefore denote the starting tableau as R⁴. According to (12) we can divide R⁴ into a binary (a non-zero entry is demarked by "×") and unprocessed real number part:

We proceed now with the 5-th row (R4c). All columns with non-negative entries in R4c are retained (columns 1 and 4). Columns 2 and 3 have a negative entry at position R4c and are therefore combined with 1 and 4 to obtain a zero at position R4c. In the binary sub-tableau, the combination step is a simple OR operation. Thereby, using the obtained binary patterns, the adjacency test (13) must be performed for each pair of combined columns. Here, all 4 possible pairs are adjacent. Accordingly, after completing row 5, tableau R⁵ has 6 columns and reads:

Now we have already reached the last iteration step where R5c – the last row in real number format – is processed. Columns 1–5 are retained and column 6 is combined with columns 1,3 and 5. However, the column pairs (1,6) and (3,6) are not pairs of adjacent rays. This can be detected in two alternative ways. The usual way is that both column pairs violate condition (13) because of column 4. The second and quicker way is to observe that the minimal number of zeros in this network is 3 (q'-m-1 = 6-2-1) and that their respective combinations would give columns with only 2 zeros. These combinations are therefore not included in the tableau. We obtain:

Tableau R⁶ is the binary representation of the EMs (extreme rays) from the split compressed network. Now, the post-processing begins. First, we remove the spurious 2-cycle (second column in R⁶) raised by splitting R4c. Then, rows R4c and R4c_b are combined by an OR operation and row R4c_b is dropped. Note, if a completely reversible elementary mode exists in the non-split network, it would lead to two EMs – one for each direction – in the split network. In such a case, either both are kept or only one, then marked as reversible EM. We have now obtained the 5 EMs of the compressed network as binary vectors:

Here, it is easy to reconstruct the real numbers of the EMs from their binary patterns. For illustrating the general case, we reconstruct the first mode e¹ using eq. (14):

The dimension of the null space of , hence of the solution space of eq. (25) is 1 (as it is for all EMs). A scalable solution vector is (2,1,1)^T, normalizing to the first component yields the unique solution (1,0.5,0.5)^T. Thus, the first EM in the compressed network is e¹ = (1,0,0,0.5,0.5)^T. Reminding that we lumped the original reactions R1 and R2 into R1c and R6 and R7 into R5c, we can finally reconstruct the original elementary mode from the uncompressed network, that is R1 + R2 + 0.5 × R5 + 0.5 × R6 + 0.5 × R7.

We implemented the binary null-space approach (binary NSA) in MATLAB (Mathworks Inc.) and incorporated it into the FluxAnalyzer 3034. The function includes a pre-processing step where the network is compressed as described. Some sub-routines of the algorithm are performed by compiled C-code (via MATLAB MEX interface), since this proved to accelerate the implementation drastically. In order to check the capabilities of our algorithm we computed the elementary modes in realistic and large metabolic networks. The three networks (S1-S3) considered here are variants from a model of the central metabolism of Escherichia coli investigated originally in 1123. For considering networks with different complexities we inserted an increasing number of substrate uptake or/and product excretion (pseudo) reactions, which increase the number of EMs much faster than the insertion of internal reactions. For a (rough) comparison with the original NSA we used the program coverN (developed by Clemens Wagner and co-workers; available upon request from clemens.wagner@pki.unibe.ch, which is also implemented in MATLAB and uses external C-files for some sub-routines. The original as well as the binary CBA algorithm proved to be slower than both methods of NSA (not shown).

Table 2 summarizes the computations. As a first result, it can be noted that redundancy removal and network compression during pre-processing results in much smaller networks. Note that the dimensions of the compressed networks of S1 and S2 are even lower than given in 23 due to the additional removal of uniquely produced/consumed metabolites. A lower number of reactions reduces the dimension of the null-space (hence, the number of iterations) and, in particular, the effort for adjacency tests. Generally, the proportion of the pre-processing on the overall computation time is negligible.