Moiety vectors for DAS.

<p>Moiety vectors for DAS.</p

Characteristics of conserved moieties identified in the three metabolic networks treated here.

<p>The total number of instances of a moiety is plotted against the number of atoms per instance. Classification of moieties as transitive, internal, or integrative is described in, Methods, Section Classification of moieties.</p

Instantaneous iCore moieties.

<p>Carbon and phosphate containing moieties in an extreme pathway of the <i>E. coli</i> core network that corresponds to glycolysis. Four conserved moieties are distinguished by shape in the figure. The pathway also conserves one oxygen atom moiety and two hydrogen atom moieties that were omitted to simplify the figure. Metabolite abbreviations are, Glc: D-glucose (VMH [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.ref023" target="_blank">23</a>] ID: glc_D), PEP: phosphoenolpyruvate (VMH ID: pep), Pyr: pyruvate (VMH ID: pyr), F6P: D-fructose 6-phosphate (VMH ID: f6p), ATP: adenosine triphosphate (VMH ID: atp), ADP: adenosine diphosphate (VMH ID: adp), FDP: D-fructose 1,6-bisphosphate (VMH ID: fdp), DHAP: dihydroxyacetone phosphate (VMH ID: dhap), G3P: glyceraldehyde 3-phosphate (VHM ID: g3p), NAD: nicotinamide adenine dinucleotide (VMH ID: nad), P_<i>bluei</i>: orthophosphate (VMH ID: pi), NADH: reduced nicotinamide adenine dinucleotide (VMH ID: nadh), DPG: 1,3-bisphospho-D-glycerate (VMH ID: 13dpg), Lac: D-lactate (VMH ID: lac_D). The glucose moiety (circles) is transitive whereas the other three moieties are internal, including the phosphate moiety (squares) which was classified as integrative in the full iCore network.</p

A graphical representation of an atom transition network for the DOPA decarboxylase reaction.

<p>Nodes (open circles) represent atoms. Atoms can be matched to metabolite structures in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g002" target="_blank">Fig 2</a> on their metabolite identifiers, colours and numbers. Directed edges (arrows) represent atom transitions. All except one hydrogen atom are omitted to simplify the figure.</p

Moiety subnetworks of DAS.

<p>(a) Moiety vectors <i>l</i>₁, <i>l</i>₂, <i>l</i>₃, <i>l</i>₆, and <i>l</i>₇ (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.t004" target="_blank">Table 4</a>) were used to decompose the stoichiometric matrix for DAS (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.t003" target="_blank">Table 3</a>) into five subnetworks. Colours match the corresponding moieties in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g004" target="_blank">Fig 4</a>. Linestyles match the corresponding reactions in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g004" target="_blank">Fig 4</a>. The two hydrogen atom moiety subnetworks (<i>l</i>₄ and <i>l</i>₅) were omitted to simplify the figure. (b) A subnetwork derived from an extreme ray that did not represent moiety conservation. This subnetwork is not mass balanced as there is no mass transfer between Phe and BH₂, Tyr and BH₂, or BH₂ and CO₂ in the full metabolic network (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g004" target="_blank">Fig 4</a>).</p

Sets of conservation vectors for metabolic networks.

<p>The set of real-valued conservation vectors consists of all vectors in the left null space of a stoichiometric matrix. Real-valued basis vectors can be computed using efficient linear algebra algorithms but are difficult to interpret as they generally contain negative and noninteger elements. Nonnegative integer vectors are easier to interpret but more difficult to compute. Existing algorithms have exponential worst case time complexity. Algorithms exist to compute extreme rays, the set of all nondecomposable nonnegative integer vectors, and a maximal set of linearly independent nonnegative integer vectors. These vector sets intersect with the set of moiety vectors but are not equivalent to it. Moiety vectors represent conservation of an identifiable group of atoms in network metabolites. They are a property of the specific set of metabolites and reactions that constitute a metabolic network whereas other conservation vectors are a property of the network’s stoichiometric matrix. The method presented here computes moiety vectors in polynomial time.</p

The total stoichiometric matrix <i>S</i> = [<i>N</i>, <i>B</i>] for DAS.

<p>The total stoichiometric matrix <i>S</i> = [<i>N</i>, <i>B</i>] for DAS.</p

Identification of conserved carbon moieties in DAS.

<p>(a) The carbon atom transition network. Numbering of atoms and line styles of atom transitions refer to metabolite structures and reactions, respectively, in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g004" target="_blank">Fig 4</a>. The directed graph consists of 18 components, one for each of the nine carbon atoms in L-phenylalanine, and one for each of the nine carbon atoms in tetrahydrobiopterin. The single carbon atom (C1) in formate is in the same component as C9 in L-phenylalanine, since a path can be traced between the two atoms when directionalities of atom transitions are ignored. Isomorphic components have matching colours. A single instance of a conserved moiety consists of all equivalent atoms in a set of isomorphic components. (b) Moiety graphs for the three carbon moieties in DAS. Each graph was obtained by merging a set of isomorphic components in (a) into a single directed graph. Each node represents an instance of a conserved moiety. Each edge represents conservation of a moiety between two metabolites in a particular reaction in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g004" target="_blank">Fig 4</a> with matching line style. Colours match the background colours of the corresponding moieties in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g004" target="_blank">Fig 4</a>. Analysis of the full atom transition network for DAS yielded four additional conserved moieties (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.g006" target="_blank">Fig 6</a>).</p

Internal moieties in iCore.

<p>Internal moieties in iCore.</p

DAS: a small metabolic network consisting of reactions in the human dopamine synthesis pathway.

<p>Metabolite abbreviations are, Phe: L-phenylalanine (VMH [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004999#pcbi.1004999.ref023" target="_blank">23</a>] ID: phe_L), Tyr: L-tyrosine (VMH ID: tyr_L), L-DOPA: levodopa (VMH ID: 34dhphe), DA: dopamine (VMH ID: dopa), BH₄: tetrahydrobiopterin (VMH ID: thbpt), BH₂: dihydrobiopterin (VMH ID: dhbpt). Internal reactions are labelled R1–R4. R1 (dashed lines) is the phenylalanine hydroxylase reaction (VMH ID: r0399). R2 (dotted lines) is the tyrosine hydroxylase reaction (VMH ID: TYR3MO2 and THBPT4ACAMDASE). R3 (dash-dotted lines) is the DOPA decarboxylase reaction (VMH ID: 3HLYTCL). R4 (solid line) is a composite of the formate dehydrogenase reaction (VMH ID: FDH) and the dihydropteridine reductase reaction (VMH ID: DHPR). Exchange reactions are labelled E1–E6. The hydrogen ion (H⁺) exchange reaction E7 was omitted to simplify the figure. Atoms are numbered according to their order in each metabolite’s molfile. Atoms of different elements are numbered separately, in colours matching their elemental symbol. Atoms belonging to the same conserved moiety have identically coloured backgrounds.</p