Efficient Reconstruction of Metabolic Pathways by Bidirectional Chemical Search
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Abstract
One of the main challenges in systems biology is the establishment of the metabolome: a catalogue of the metabolites and biochemical reactions present in a specific organism. Current knowledge of biochemical pathways as stored in public databases such as KEGG, is based on carefully curated genomic evidence for the presence of specific metabolites and enzymes that activate particular biochemical reactions. In this paper, we present an efficient method to build a substantial portion of the artificial chemistry defined by the metabolites and biochemical reactions in a given metabolic pathway, which is based on bidirectional chemical search. Computational results on the pathways stored in KEGG reveal novel biochemical pathways.
Keywords
Artificial chemistry Biochemical reaction Metabolic pathway1 1. Introduction
Metabolism can be regarded as a network of chemical reactions activated by enzymes and connected via their substrates and products, and a metabolic pathway can be regarded as a coordinated sequence of biochemical reactions (Deville et al., 2003). The definition of a metabolic pathway is not exact, and most pathways constitute indeed highly intertwined cyclic networks. In a cell, the substrates of a pathway are usually the products of another pathway, and there are junctions where pathways meet or cross (Karp and Mavrovouniotis, 1994).
The analysis of metabolic pathways is motivated by the rapidly increasing quantity of available information on metabolic pathways for different organisms. One of the most comprehensive sources of metabolic pathway data is the Roche Applied Science Biochemical Pathways chart (Michal, 1999). There are also several databases on metabolic pathways, such as aMAZE (Lemer et al., 2004), BRENDA (Schomburg et al., 2002), MetaCyc (Caspi et al., 2006), KEGG (Kanehisa and Goto, 2000), and WIT (Overbeek et al., 2000). These databases contain hundreds of metabolic pathways and thousands of biochemical reactions, and even the metabolic pathway for a small organism constitutes a large network. For instance, the proposed metabolic pathway for the bacterium E. coli consists of 436 compounds (substrates, products, and intermediate compounds) linked by 720 reactions (Edwards and Palsson, 2000).
An artificial chemistry (Dittrich et al., 2001), on the other hand, is a computational model of a chemical system that consists of a set of objects (molecules), a set of reaction rules (that allow for the production of new molecules from already existing molecules), and a definition of the dynamics of the system (that is, application conditions for the reaction rules), aimed at answering qualitative questions about the chemical system. Thus, artificial chemistries model real chemistries, in which molecules represent chemical compounds and reaction rules represent chemical reactions and, in particular, artificial chemistries model organic chemistries (Benkö et al., 2003a, 2003b, 2004).
The chemical description of molecules in an artificial chemistry can be made at different levels of resolution, from simple molecular descriptors to structural formulas. One of these representations are chemical graphs, with nodes corresponding to the atoms of the molecules and edges indicating the bonds between them. Chemists have used chemical graphs to distinguish isomers since the second half of the nineteenth century, and in first course organic chemistry classes, chemical reactions are explained in terms of constitutional formulas and a handful of reaction mechanisms, which are nothing but chemical graphs and rules to modify them by means of breaking, forming, and changing the type of bonds. This leads in a natural way to artificial chemistries based on labeled graphs as molecules and graph transformation rules as reactions. Several such artificial chemistries have been proposed so far: see, for instance, (Benkö et al., 2003a, 2003b, 2004; McCaskill and Niemann, 2001; Rosselló and Valiente, 2005a).
Artificial chemistries can also be used to model biochemical systems such as metabolic pathways, in which molecules represent metabolites and reaction rules represent biochemical reactions (Rosselló and Valiente, 2005b), and they allow for answering qualitative questions about metabolism. In this paper, we present an efficient method to build a substantial portion of the artificial chemistry defined by the metabolites and biochemical reactions in a given metabolic pathway. Our method is based on bidirectional chemical search, and its implementation uses chemical graphs to represent sets of molecules. We report also on the results of some experiments applying this method to pathways stored in KEGG, which reveal novel biochemical pathways.
2 2. Modeling biochemical reactions as chemical graph transformations
Following (Rosselló and Valiente, 2005a), by a chemical graph, we understand a complete labeled weighted graph (V,E, ℓ,μ), with (V, E) an undirected graph (without multiple edges or selfloops), ℓ a labeling mapping that labels every node v ∈ V with a chemical element ℓ(υ), and μ : E → ℕ an edge weight function. We shall denote the weight of the edge joining nodes υ and w by μ(υ, w); notice that μ(υ, w) = μ(w, υ) because the graph is undirected. A weight of 0 stands for a nonexisting bond, a weight of 1 for a single bond, a weight of 2 for a double bond, etc. The valence of a node in a chemical graph is the total weight of the edges incident to it.

\({\ell _1}\left( {{\upsilon _1}} \right) = {\ell _2}\left( {M\left( {{\upsilon _1}} \right)} \right)\).

\(\sum\nolimits_{{w_1} \in {V_1}} {{\mu _1}\left( {{\upsilon _1},{w_1}} \right) = } \sum\nolimits_{{w_1} \in {V_1}} {{\mu _2}\left( {M\left( {{\upsilon _1}} \right),M\left( {{w_1}} \right)} \right)} \).
When there exists an atom mapping between two chemical graphs G _{1} and G _{2}, these chemical graphs (and the multimolecules they represent) are said to be compatible: this means that they have the same number of nodes for each possible pair (label, valence). Notice that there is no stereochemical information in this simplified representation, and thus stereoisomers are represented by the same chemical graph. There is no electrical charge information either, and anions and cations are also represented by the same chemical graph.
A chemical reaction graph is a structure R = (G _{1},G _{2},M), where G _{1} = (V _{1}, E _{1}, ℓ _{1}, μ _{1}) and G _{2} = (V _{2},E _{2}, ℓ _{2},μ _{2}) are compatible chemical graphs, called the substrate and the product chemical graphs, respectively, and M : V _{1} → V _{2} is an atom mapping between them.
The application of a chemical reaction graph to a given chemical graph, consists of breaking, forming, and changing bonds in a subgraph of the chemical graph which is isomorphic to the substrate of the chemical reaction graph. Reversible chemical reaction graphs can also be applied in the opposite direction, by breaking, forming, and changing bonds in a subgraph of the chemical graph which is isomorphic to the product of the chemical reaction graph.
Given two compatible chemical graphs G _{1} = (V _{1},E _{1}, ℓ _{1},μ _{1}) and G _{2} = (V _{2},E _{2}, ℓ _{2} μ _{2}), an optimal atom mapping between them is an atom mapping of minimal size, which always exists (but it needs not be unique). An optimal atom mapping models the classical principle of minimum structure change, by which a chemical reaction normally occurs through the redistribution of the minimum number of valence electrons, that is, the formation and breaking of the least number of covalent bonds (Temkin et al., 1996).
The size of a chemical reaction graph R = (G _{1},G _{2},M) is simply the size of the corresponding atom mapping M.
3 3. Reconstructing metabolic pathways by bidirectional chemical search
Artificial chemistries (Dittrich et al., 2001) are computational models of chemical systems and, in particular, of biochemical systems such as metabolic pathways. An artificial chemistry consists of a set of molecules, a set of reaction rules that produce new molecules from already existing molecules, and the definition of the dynamics of the system, which specifies the application conditions of the rules, the preference in their application, etc. (Rosselló and Valiente, 2005b).
A metabolic pathway can be regarded as a coordinated sequence of biochemical reactions and is often described in symbolic terms, as a succession of transformations of one set of substrate molecules into another set of product molecules (Rosselló and Valiente, 2004). Substrate and product must be compatible chemical graphs for a pathway between them to exist (Rosselló and Valiente, 2004, 2005a, 2005b).
Metabolic pathways are often represented as directed hypergraphs, with substrate and product molecules as nodes and biochemical reactions as hyperarcs. Since a chemical graph can represent the disjoint union of a set of molecules, though, the equivalent representation of artificial chemistries and, in particular, metabolic pathways as directed graphs becomes more natural. An artificial chemistry defined by a set of chemical reaction graphs, is thus represented as a directed secondorder graph with the chemical graphs that represent the sets of substrate and product molecules as vertices and applications of the chemical reaction graphs, including information on atom mapping, as arcs.
Unfortunately, the size of the artificial chemistry defined by a setM of chemical graphs and a set R of chemical reaction graphs is often exponential in the size of M and R, and thus artificial chemistries are known for very small instances only, involving a few dozens of molecules and biochemical reactions. Therefore, we consider in this paper the problem of obtaining a substantial portion of the artificial chemistry defined by a set of biochemical reactions while avoiding the complexity of reconstructing the whole artificial chemistry.
 (1)
The initial chemical graphs represent all sets of at most m metabolites among those involved in the set R of reactions, for some fixed, but arbitrary, m (in examples and applications in this paper we shall always take m = 2).
 (2)
The reconstruction process is restricted to a fixed, but arbitrary, number k of derivation steps.
 (3)
The initial and final sets of metabolites of every metabolic pathway belong to the set of initial chemical graphs.
While the first two constraints (on the size of the initial chemical graphs and the lengths of the metabolic pathways under inspection) are motivated by complexity considerations alone, the third constraint allows for directing the search of new metabolic pathways inside the artificial chemistry. That is, instead of building the artificial chemistry by applying the biochemical reactions in every possible way to each of the initial chemical graphs, we perform a bidirectional search by constructing forward metabolic pathways of length at most k starting in initial chemical graphs and backward metabolic pathways of length at most k ending in initial chemical graphs, and then gluing them to obtain all metabolic pathways of length at most 2k starting and ending in initial chemical graphs.

First, we extract the set M of all chemical graphs representing sets of at most m any metabolites appearing in substrates and products of the reactions in R. We call the elements of M the initial chemical graphs.

Next, we identify all compatibility classes in M (maximal subsets of compatible initial chemical graphs). Biochemical reactions transform chemical graphs into compatible chemical graphs and, therefore, the origin and the end of a metabolic pathway will be compatible sets of metabolites. Thus, since we restrict ourselves to metabolic pathways starting and ending in initial chemical graphs, we can restrict ourselves to search for metabolic pathways starting and ending in each compatibility class of initial chemical graphs.

Then each compatibility class C in M is considered as a set of potential substrates C _{ F } ^{(0)} and a set of potential products C _{ R } ^{(0)} for the reactions in R.

For every i = 1, …, k, the forward application of the reactions in R to the elements of C _{ F } ^{(i−1)} produces a set of multimolecules C _{ F } ^{(i)} , while the reverse application of these reactions to the molecules in C _{ R } ^{(i−1)} produces a set of multimolecules C _{ R } ^{(i)} .
 Any nonempty intersection of a set obtained by forward application and a set obtained by reverse application of reactions yields a new pathway between elements of C. To avoid repetitions, it is enough to check whether each C _{ F } ^{(i)} intersects C _{ R } ^{(i)} and C _{ R } ^{(i−1)} . More specifically:

For i = 1, the forward application of the reactions in R to the molecules in C _{ F } ^{(0)} produces a set C _{ F } ^{(1)} of new molecules, and the reverse application of the reactions in R to the molecules in C _{ R } ^{(0)} produces a set C _{ R } ^{(1)} of new molecules. Open image in new window
Then

Every member of C _{ F } ^{(1)} ∩ C _{ R } ^{(0)} yields a new pathway C _{ F } ^{(0)} → C _{ F } ^{(1)} ∩ C _{ R } ^{(0)} of length 1.

Every member of C _{ F } ^{(1)} ∩ C _{ R } ^{(1)} yields a new pathway C _{ F } ^{(0)} → C _{ F } ^{(1)} ∩ C _{ R } ^{(1)} → C _{ R } ^{(0)} of length 2.

For i = 2, the forward application of the reactions in R to the molecules in C _{ F } ^{(1)} produces a set C _{ F } ^{(2)} of new molecules, and the reverse application of the reactions in R to the molecules in C _{ R } ^{(1)} produces a set C _{ R } ^{(2)} of new molecules. Open image in new window
Then
 Every member of C _{ F } ^{(2)} ∩ C _{ R } ^{(1)} yields a new pathway of length 3.$$C_F^{\left( 0 \right)} \to C_F^{\left( 1 \right)} \to C_F^{\left( 2 \right)} \cap C_R^{\left( 1 \right)} \to C_R^{\left( 0 \right)}$$
 Every member of C _{ F } ^{(2)} ∩ C _{ R } ^{(2)} yields a new pathway of length 4.$$C_F^{\left( 0 \right)} \to C_F^{\left( 1 \right)} \to C_F^{\left( 2 \right)} \cap C_R^{\left( 2 \right)} \to C_R^{\left( 1 \right)} \to C_R^{\left( 0 \right)}$$

And, recursively, the forward application of the reactions in R to the molecules in I _{ F } = C _{ F } ^{(i−1)} produces a set C _{ F } = C _{ F } ^{(i)} of new molecules, and the reverse application of the reactions in R to the molecules in I _{ R } = C _{ R } ^{(i−1)} produces a set C _{ R } = C _{ R } ^{(i)} of new molecules. Open image in new window
Then
 Every member of C _{ F } ∩ I _{ R } yields a new pathway of length 2i − 1.$$C_F^{\left( 0 \right)} \to ... \to {I_F} \to {C_F} \cap {I_R} \to ... \to C_R^{\left( 0 \right)}$$
 Every member of C _{ F } ∩ C _{ R } yields a new pathway of length 2i.$$C_F^{\left( 0 \right)} \to ... \to {I_F} \to {C_F} \cap {C_R} \to {I_R} \to ... \to C_R^{\left( 0 \right)}$$
The following result shows that in this way we obtain all metabolic pathways of length at most 2k under constraints (1) and (3) above.
Lemma 1. For every i = 1, …, k, all metabolic pathways of length 2i−1 and 2i starting and ending in initial chemical graphs are obtained in the ith iterative step of the procedure explained above.
Proof: Ifis a pathway with m _{0} and m _{2i−1} initial chemical graphs, then m _{ j } ∈ C _{ F } ^{(j)} for every j = 0, …, i and m _{2i−1−l } ∈ C _{ R } ^{(l)} for every l = 0, …, i − 1, and hence in particular, m _{ i } ∈ C _{ F } ^{(i)} ∩ C _{ R } ^{(i−1)} . Therefore, this path is obtained in the ith iterative step of the procedure explained above$${m_0} \to {m_1} \to ... \to {m_i} \to ... \to {m_{2i  1}}$$On the other hand, ifis a pathway with m _{0} and m _{2i } initial chemical graphs, then m _{ j } ∈ C _{ F } ^{(j)} for every j = 0, …, i and m _{2i−l } ∈ C _{ R } ^{(l)} for every l = 0, …, i, and hence, in particular, m _{ i } ∈ C _{ F } ^{(i)} ∩ C _{ R } ^{(i)} . Therefore, this path is also obtained in the ith iterative step of that procedure.$${m_0} \to {m_1} \to ... \to {m_i} \to ... \to {m_{2i}}$$ 
C _{ F } ^{(0)}  →  C _{ F } ^{(1)}  C _{ R } ^{(1)}  →  C _{ R } ^{(0)} 

c  →  (ab, bbe)  (def , ddf)  →  af 
ab  →  (c, bde)  (dde, dee)  →  ae 
ad  →  dde  (dde, ddd)  →  ad 
ae  →  dee  (c, bde, bdd)  →  ab 
af  →  (def, bee)  ab  →  c 
Notice that some elements of C _{ F } ^{(1)} and C _{ R } ^{(1)} do no longer belong to M, as we warned
C _{ F } ^{(0)}  →  C _{ F } ^{(1)}  →  C _{ F } ^{(2)}  C _{ R } ^{(2)}  →  C _{ R } ^{(1)}  →  C _{ R } ^{(0)} 

c  →  (ab, bbe)  →  ((c, bde), (bbd, def))  ((af , bbe), bbd)  →  (def , ddf)  →  af 
ab  →  (c, bde)  →  ((ab, bbe), (ab, bdd, bee))  (ad, ae)  →  (dde, dee)  →  ae 
ad  →  dde  →  (ad, ae)  (ad, Ø)  →  (dde, ddd)  →  ad 
ae  →  dee  →  ae  (ab, (ab, bdd, bee), bde)  →  (c, bde, bdd)  →  ab 
af  →  (def , bee)  →  ((af , bbe), bde)  (c, bde, bdd)  →  ab  →  c 
c → ab → c → ab,  c → ab → bde → ab, 
ab → c → ab → c,  ab → bde → bdd → ab, 
af → bee → bde → ab,  c → bbe → def → af, 
ab → bde → ab → c,  c → ab → c → ab → c, 
c → bbe → bbd → ddf → af,  c → ab → bde → ab → c, 
c → ab → bde → bdd → ab,  ab → c → ab → c → ab, 
ab → c → ab → bde → ab,  ab → c → bbe → def → af, 
ab → bde → ab → c → ab,  ab → bde → ab → bde → ab, 
ab → bde → bdd → ab → c,  ab → bde → bdd → bde → ab, 
ab → bde → bee → bde → ab,  ad → dde → ad → dde → ad, 
ad → dde → ad → dde → ae,  ad → dde → ae → dde → ae, 
ae → dee → ae → dde → ae,  af → def → af → def → af, 
af → def → bbe → def → af,  af → bee → bde → ab → c. 
af → bee → bde → bdd → ab, 
 (a)
to produce all metabolic pathways of length up to 2k
 (b)
to produce all shortest metabolic pathways of length up to 2k
 (c)
to produce all minimal acyclic metabolic pathways of length up to 2k in all cases under restrictions (1) to (3) made explicit above.
We give our reconstruction algorithms in full pseudocode next. Algorithm 1 one formalizes the procedure explained above.

It receives the sets I _{ F } = C _{ F } ^{(i−1)} and I _{ R } = C _{ R } ^{(i−1)} of the results of all direct and reverse applications, respectively, of i − 1 consecutive rules in R to multimolecules in C (when i = 1, C _{ F } ^{(0)} = C and C _{ R } ^{(0)} = C) and it produces the sets N _{ F } = C _{ F } ^{(i)} and N _{ R } = C _{ R } ^{(i)} of the results of all direct and reverse applications, respectively, of rules in R to multimolecules in I _{ F } and I _{ R }, respectively. That is, the sets of the results of all direct and reverse applications, respectively, of i consecutive rules in R to multimolecules in C.
 The lines starting with output call a procedure that outputs the list of all metabolic pathways of lengths 2i − 1 and 2i obtained so far. When i = 1:

the first output line gives all length 1 pathways m → m _{ f } ^{(1)} , with m ∈ C,

the second output line gives all length 2 pathways m → m _{ r } ^{(1)} → m′ with m,m′ ∈ C.
And when i > 1:
Algorithm 1. Given a set R of biochemical reactions and a number k of derivation steps, obtain the set of all metabolic pathways of length up to 2k using the metabolites and reactions in R starting and ending in sets of at most m metabolites among those involved in the reactions in R. Open image in new window
 Thefirst output line gives all length 2i − 1 pathwayswith m,m′ ∈ C.$$m \to m_f^{\left( 1 \right)} \to ... \to m_f^{\left( {i  1} \right)} \to m_f^{\left( i \right)} = m_r^{\left( {i  1} \right)} \to m_r^{\left( {i  2} \right)} \to ... \to m_r^{\left( 1 \right)} \to m'$$
 The second output line gives all length 2i pathwayswith m,m′ ∈ C.$$m \to m_f^{\left( 1 \right)} \to ... \to m_f^{\left( {i  1} \right)} \to m_f^{\left( i \right)} = m_r^{\left( i \right)} \to m_r^{\left( {i  1} \right)} \to ... \to m_r^{\left( 1 \right)} \to m'$$

Algorithm 2 produces a metabolic network (X, Y) containing all metabolic pathways up to a given length, where the vertex set X contains the initial and final metabolite sets together with all those new metabolite sets produced by the forward and reverse application of the given biochemical reactions, and the arc set Y consists of all direct derivations thus obtained.
Now, upon the metabolic network (X, Y ) obtained with the previous algorithm, the set of all shortest metabolic pathways of length up to 2k, using the metabolites and reactions in R starting and ending in sets of at most m metabolites among those involved in the reactions in R, can be obtained by using an allpairs shortest path algorithm (Dijkstra, 1959; Floyd, 1962; Johnson, 1977; Takaoka, 1998) upon each element of C as source vertex and each element of C as target vertex in turn.
Algorithm 2. Given a set R of biochemical reactions and a number k of derivation steps, obtain the metabolic network (X, Y) containing all metabolic pathways of length up to 2k, using the metabolites and reactions in R starting and ending in sets of at most m metabolites among those involved in the reactions in R. Open image in new window
Algorithm 3 extracts the set of all minimal acyclic metabolic pathways of length up to 2k, using the metabolites and reactions in R starting and ending in sets of at most m metabolites among those involved in the reactions in R, from the metabolic network (X, Y) produced by Algorithm 2.
In this algorithm, each path of the form u → …→ υ is extended in all possible ways by arcs in Y of the form υ → w until reaching an element w ∈ C, where the test w ∉ p ensures the resulting paths are acyclic.
Algorithm 3. Given a metabolic network (X, Y) and a set C of initial and final metabolite sets, enumerate all minimal acyclic metabolic pathways contained in (X, Y) which start and end in metabolite sets from C. Open image in new window where acyclic(C,E, υ, p) is defined as follows: Open image in new window
Remark 1. Notice that the shortest path derivation ab → c → bbe → def → af is not minimal, and the minimal acyclic derivation c → bbe → bbd → ddf → af is not shortest.
4 4. Results and discussion
 (1)
Obtaining all pathways of length up to 2k by bidirectional search,
 (2)
Storing them in a compact representation, and
 (3)
Extracting shortest pathways and minimal acyclic pathways from the compact representation, where m and k are the only parameters of the reconstruction algorithms.
The metabolic reconstruction algorithm was implemented as a Perl script, using the Chemistry::Reaction module from the PerlMol collection of Perl modules for computational chemistry (TubertBrohman, 2004). The core of the methodology is embodied in the Chemistry::Artificial Perl module, which is available from the authors and will also be available from the PerlMol collection of Perl modules for computational chemistry (TubertBrohman, 2004). This module can be used to reconstruct the artificial chemistry defined by a given set of reaction equations written in reaction SMILES format (Weininger, 1988). For instance, the following Perl script first stores the artificial chemistry containing all derivations of length up to 2k = 4 starting and ending in sets of at most m = 2 metabolites using the reaction equations in file rctn.smi (Algorithm 2) and then, extracts all shortest derivations and all minimal acyclic derivations (Algorithm 3). Open image in new window
 (1)
Obtain reference pathway maps from the KEGG (Kanehisa et al., 2006) database. We have used KEGG release 42.0 in all our experiments.
 (2)
Solve the optimal atom mapping problem for all of the reactions in the reference pathways, using the optimal atom mapping by chemical substructure search algorithm and tool support (Félix and Valiente, 2007).
 (3)
Reconstruct metabolic pathways of length up to 8 for each reference pathway.
 (4)
Orient the reactions, according to the study of irreversibility of reactions in KEGG carried out in (Ma and Zeng, 2003).
 (5)
Filter out those metabolic pathways that involve irreversible reactions applied in the reverse direction.
 (6)
Identify the new metabolites thus obtained, by chemical structure search in CheBi (Brooksbank et al., 2005), MetaCyc (Caspi et al., 2006), KEGG (Kanehisa et al., 2006), and SciFinder Scholar (Wagner, 2006).
 (7)
Analyze the new metabolic pathways for coexistence of metabolites and enzymes in each particular organism.
Number of vertices (n) and arcs (m) of the metabolic network containing all metabolic pathways of length up to 2k found by bidirectional chemical search upon the metabolites and reactions stored in KEGG for several reference maps (map), for k = 1, 2, 3, 4
map  k = 0  k = 1  k = 2  k = 3  k = 4  

n  n  m  n  m  n  m  n  m  
00010  529  870  690  931  854  931  854  931  854 
00020  82  253  350  458  818  737  1712  785  1876 
00030  314  1148  1678  2284  4788  2988  6770  3021  6836 
00031  23  33  20  33  20  33  20  33  20 
00040  330  707  756  870  1178  888  1214  915  1268 
00051  702  913  422  943  488  943  488  943  488 
00053  201  660  1108  1285  2982  1819  4618  2276  6046 
00061  53  102  118  102  118  102  118  102  118 
00062  290  2359  4188  5042  10884  5706  12212  6012  12824 
00071  372  2550  4418  4977  10322  5314  10996  5314  10996 
00072  8  8  0  8  0  8  0  8  0 
00100  229  229  0  229  0  229  0  229  0 
00120  292  1901  3254  3442  7680  3442  7680  3442  7680 
00130  267  289  44  296  58  296  58  296  58 
00150  290  290  0  290  0  290  0  290  0 
00190  14  14  0  14  0  14  0  14  0 
00220  238  399  326  437  422  439  426  439  426 
00231  18  45  54  45  54  45  54  45  54 
00251  24  44  44  52  60  52  60  52  60 
00252  146  235  186  260  2742  274  270  280  282 
00260  604  841  482  915  632  929  676  929  676 
00271  386  633  502  788  850  943  850  943  850 
00272  95  110  36  111  38  111  38  111  38 
00280  320  1206  1778  2595  5200  3129  6286  3134  6298 
00290  161  350  390  350  390  350  390  350  390 
00300  152  287  276  287  276  287  276  287  276 
00310  188  380  394  381  396  381  396  381  396 
00311  14  27  26  27  26  27  26  27  26 
00330  289  376  180  383  194  383  194  383  194 
00340  129  1293  390  385  536  385  536  385  536 
00360  157  244  178  246  182  246  182  246  182 
00400  37  54  34  54  34  54  34  54  34 
00410  106  264  320  293  382  316  428  316  428 
00471  13  30  34  37  54  37  54  37  54 
00590  870  3128  4672  5501  10278  7052  14456  7189  14824 
00906  594  1181  1250  1345  1780  1357  1818  1357  1818 
Number of shortest pathways (short) and the number of minimal acyclic pathways (min) of length up to 2k found by bidirectional chemical search upon the metabolites and reactions stored in KEGG for several reference maps (map), for k = 1, 2, 3, 4
map  k = 1  k = 2  k = 3  k = 4  

short  min  short  min  short  min  short  min  
00010  8  8  8  8  8  8  8  8 
00020  8  8  8  8  8  8  8  8 
00030  6  10  6  44  6  326  6  1714 
00040  2  2  2  2  2  2  2  2 
00053  50  194  52  672  52  3250  52  17412 
00061  20  20  20  20  20  20  20  20 
00062  50  50  50  50  50  50  50  50 
00071  62  62  62  62  62  62  62  62 
00120  24  36  30  192  30  984  30  4716 
00220  4  4  4  4  4  4  4  4 
00251  2  4  2  4  2  4  2  4 
00252  6  8  8  12  8  12  8  12 
00260  8  8  8  8  8  8  8  8 
00271  8  8  8  12  8  24  8  36 
00272  6  6  6  6  6  6  6  6 
00280  6  6  6  6  6  6  6  6 
00290  12  12  12  12  12  12  12  12 
00300  4  6  4  6  4  6  4  6 
00310  10  10  10  10  10  10  10  10 
00330  6  6  6  6  6  6  6  6 
00340  2  2  2  2  2  2  2  2 
00360  4  4  4  4  4  4  4  4 
00410  4  4  4  6  4  6  4  6 
00590  156  156  156  180  156  228  156  228 
00906  76  76  76  78  76  78  76  78 
The biological significance of these results can be assessed by examining the actual pathways found by bidirectional search, using the metabolites and reactions stored in KEGG for a particular reference pathway map. Besides obtaining again some of these reactions, an intermediate step is added in some metabolic pathways to one of the reactions stored in KEGG. For instance, using the metabolites and reactions stored in KEGG for glycine, serine, and threonine metabolism (reference pathway map 00260), we have obtained the following pathway: While the methylation of LSerine to 2Methylserine and demethylation of Pyruvate to Glyoxylate followed by the methylation of Glyoxylate to LAlanine and demethylation of 2Methylserine to Hydroxypyruvate is chemically feasible, the Serine pyruvate aminotransferase enzyme (2.6.1.51) allows for the oxidative deamination of LSerine into LAlanine, as stated in KEGG reaction R00585:
Among the novel metabolic pathways found by bidirectional search, using the metabolites and reactions stored in KEGG for carotenoid biosynthesis (reference pathway map 00906), we have obtained the following metabolic pathway: Open image in new window
A KEGG pathway reference map contains information for several organisms. Thus, it is important to find evidence that all four metabolites appearing in this pathway are present in a same organism, and also that the enzyme activating the reverse biochemical reaction R06961 (carotene 7,8desaturase, 1.14.99.30) is indeed expressed in that particular organism.
Carotenoid biosynthesis spans several related pathways: spheroidene, normalspirilloxanthin, unusualspirilloxanthin, abscisic acid biosynthesis, and astaxanthin biosynthesis. However, there are organisms whose metabolism does not include both carotenoid biosynthesis and abscisic acid biosynthesis. In fact, Arabidopsis thaliana (thale cress) is the only organism for which the four metabolites are annotated in KEGG to carotenoid biosynthesis, and the gene coding for carotene 7,8desaturase, AT3G04870, is indeed expressed in A. thaliana (Bartley et al., 1999; Scolnik and Bartley, 1995).
Number of potential biochemical reactions between sets of at most m metabolites among those involved in the reactions stored in KEGG for several reference maps. For each value of m, the first column gives the number of classes with two or more molecules (which indicates the possibility of a biochemical reaction among them) and the second column gives the total number of classes
map  m = 1  m = 2  m = 3  m = 4  m = 5  

00010  6  45  293  920  5552  11199  60502  94731  446942  601910 
00020  1  37  97  665  2250  7068  24833  51429  170858  279264 
00030  7  41  263  714  3696  6828  28860  42957  151233  198172 
00031  2  17  40  160  365  955  2121  4254  9072  15268 
00040  9  41  346  768  4986  7977  40053  53668  213052  258300 
00051  11  34  301  482  2985  3794  17651  20210  75027  81554 
00053  7  32  199  439  2086  3207  11631  14982  43952  51606 
00061  0  45  113  814  2791  8237  28979  55956  183880  281947 
00062  0  33  91  362  1118  2118  5855  8397  20396  25711 
00071  1  60  271  1415  7218  18353  88066  156445  664502  965103 
00072  0  14  7  112  99  560  642  2072  2675  6137 
00100  14  70  825  2262  20756  40791  285566  454527  2439893  3381759 
00120  10  50  396  10  6685  12885  67353  106346  471120  654410 
00130  5  46  246  1038  5395  13902  64431  120826  467334  719792 
00150  9  41  294  744  4201  7645  35515  52648  205587  269069 
00190  0  14  8  1221  106  561  712  2146  3148  6706 
00220  0  63  244  1771  8846  26869  125700  242959  921233  1387232 
00231  0  22  102  263  248  2022  2564  11662  17577  53983 
00251  3  48  209  1122  5512  15603  74831  143221  604301  911037 
00252  5  531  301  1239  7186  17277  89453  154299  664930  941960 
00260  5  84  624  3100  23630  63164  96407  790950  4252609  64074 
00271  2  69  336  3622  12816  41942  236574  518317  2567554  4365113 
00272  2  44  138  944  3409  12221  45450  109489  389767  724507 
00280  8  42  312  762  4409  7540  34765  49006  185510  233275 
00290  8  37  274  645  3895  6601  33131  46469  196273  245176 
00300  2  54  251  1318  6750  18193  85928  163830  47724  1047724 
00310  4  69  393  2210  13161  40178  220249  464308  2165271  30821 
00311  1  38  62  749  1656  9189  23656  78677  209181  492812 
00330  6  68  490  2088  14025  35627  203409  377970  1741156  2671199 
00340  1  59  255  2551  7897  22521  104759  1153  763662  1153907 
00360  7  51  364  1105  6230  12342  55100  85952  309536  418892 
00400  2  53  191  1379  6345  20855  95031  198504  781143  1261828 
00410  3  53  234  1329  6273  18727  81546  165437  610070  967461 
00471  2  22  55  259  670  1939  5061  10810  27605  48249 
00590  10  29  213  404  2178  3349  14417  19596  70620  88693 
00906  18  67  786  1608  13100  20264  02026412  159842  719545  869908 
Notes
Acknowledgement
The research described in this paper has been partially supported by the Spanish CICYT, project TIN 200407925C0301 GRAMMARS and project MTM200607773 COMGRIO, and by EU project INTAS IT 04777178. A preliminary version of this paper has appeared in (Félix et al., 2007).
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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