The convex basis of the left null space of the stoichiometric matrix leads to the definition of metabolically meaningful pools

Abstract

The stoichiometric matrix, S, represents a mapping of reaction rate vectors into a space of concentration time derivatives. The left null space of the stoichiometric matrix contains the dynamic invariants: a combination of concentration variables, referred to as metabolic pools, whose total concentration does not change over time. By analogy to the traditional reaction map formed by S, a compound map can be derived from -S(T). The analogy to flux analysis of the (right) null space of S enables us to classify the metabolic pools into three categories: Type A that contains chemical elements and their combinations in the form of certain moieties, Type B that contains such moieties in addition to cofactors carrying such moieties that are internal to the network, and Type C that contains only the cofactors. A convex formulation of the basis for the left null space allows us to directly classify the metabolic pools into these three categories. Type B metabolic pools include conservation pools that form conjugates of moiety-occupied and moiety-vacant concentration states of metabolites and cofactors. Type B metabolic pools thus describe the various states of moiety exchange between the primary substrates and the cofactors that capture properties like energy and redox potential. The convex basis gives clear insight into this exchange for glycolytic pathway in human red blood cell, including the identification of high and low energy pools that form conjugates. Examples suggest that pool maps may be more appropriate for signaling pathways than flux maps. The analysis of the left null space of the stoichiometric matrix allows us to define the achievable states of the cell and their physiological relevance.

FIGURE 1

The stoichiometric matrix, S, as a linear transformation between the reaction rate vectors, v, and time derivative of metabolite concentrations, dx/dt or . Each two subspaces in the domain (i.e., the null space and row space) and codomain (i.e., the left null space and column space) form orthogonal pairs with one another. The components that reside on each subspace are therefore independent from the others. (dyn, dynamic; ss, steady state; consv, conserved).

FIGURE 2

Stoichiometric matrix and the system's boundary. (A) A system with internal reactions, v_i, and metabolites, x_i, is closed to the environment and represented by S_int. (B) When the internal metabolites are allowed to be exchanged with the environment, the stoichiometric matrix includes exchange reactions, b_i, and the system becomes “open”. (C) Once the external metabolites (x_oi) are included, the system becomes “closed” and the stoichiometric matrix contains all the reactions and metabolites, S_tot. (int, internal; exch, exchang; tot, total).

FIGURE 3

Reaction maps versus compound maps. Reaction maps (left) show metabolites as nodes and reactions as directed edges. The reaction map includes both the internal and exchange fluxes, if present. In contrast, compound maps of the same systems (right) show the reactions as nodes and metabolites as directed edges. A system boundary that allows for the exchange of the internal nodes is open on a map. The compound map of an open reaction map is closed and vice versa, as it is shown by changing the network from A to B and to C.

FIGURE 4

Conserved pools and extreme pathway classifications. (A) Type I, II, and III extreme pathways correspond to through pathways, futile cycles coupled to cofactor utilization, and internal loops that are thermodynamically infeasible (Beard et al., 2002; Price et al., 2002), respectively. (B) Type A, B, and C metabolic pools correspond to the conservation of biochemical elements, metabolic moieties common to the primary and secondary metabolites, and cofactor conservation, respectively. Systems shown are presented as schematic examples of pools and flux pathways.

FIGURE 5

Metabolic pool classification schema and structure of L. Partitioning the metabolites into primary and secondary allows for classification of metabolic pools. In the absence of the secondary and primary metabolite participation in conservation pools, the vectors are classified as Type A and C, respectively. The remaining vectors are grouped as Type B pools with both metabolite types present.

FIGURE 6

Convex conservation pool representation for schematic biochemical systems. Seven example systems (I–VII) are graphically shown as reaction maps, compound maps, convex left null basis, and pool maps. Dash lines shown on the L matrix delineate metabolite and pool types.

FIGURE 7

Graphical depiction of the null space for the cofactor-coupled reaction. The cofactor-coupled reaction contains four convex conservation relationships as shown in Fig. 6 III. The relative distance between the carbon and cofactor metabolites remains constant and is determined by the magnitude of the conservation quantities so C + CP = a₁, C + A = a₂, CP + AP = a₃, and A + AP = a₄. The points with the same shading (black, gray, white) correspond to an identical state depicted in two distinct two-dimensional spaces. For example, the concentration state in which C = 2, CP = 1, A = 2, and AP = 3 is depicted by the gray point and satisfies a₁ = 3, a₂ = 4, a₃ = 4, and a₄ = 5. The states represented by the white and black points also satisfy these pool sizes. The concentration solution space is the solid line shown in the two spaces. The symbol ≈ indicates that the origin is not common between the two two-dimensional spaces.

FIGURE 8

Pool analysis of the human red blood cell glycolysis and Rapoport-Leubering shunt. (A) The reaction map of glycolysis and the Rapoport-Leubering shunt in human red blood cell contains 13 reactions and 20 metabolites. (B) The compound map of the reaction set shown in part A shows glycolytic reactions as the nodes and metabolites as the edges. (C) Pool maps of all 10 convex basis vectors in the human red blood cell glycolysis. ℓ₃ and ℓ₄ form high and low energy conjugates and ℓ₅, ℓ₆, and ℓ₇ comprise the redox conjugate set in the network. The conservation values are shown on the pool maps.

FIGURE 9

Schematic representation of signaling pathway (e.g., a tyrosine kinase-type signaling pathway) and its corresponding conservation pools. (A) Reaction map of the sample tyrosine kinase pathway. (B) Compound map of the tyrosine kinase pathway. (C) Metabolite pool maps of conservation pools. (L, ligand; R, receptor; LR, ligand-receptor complex; LR-P, phosphorylated ligand-receptor complex; S, signal transducers and activators of transcription (STAT); SP, phosphorylated STAT; T, transcriptional factor; TP, phosphorylated transcriptional factor; DNA-TP, active DNA-transcriptional factor complex; P, inorganic phosphate; act, active; inact, inactive).