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The Convex Basis of the Left Null Space of the Stoichiometric Matrix Leads to the Definition of Metabolically Meaningful Pools

Department of Bioengineering, University of California San Diego, La Jolla, California 92093-0412 USA

Correspondence: Address reprint requests to Bernhard O. Palsson, E-mail: palsson{at}ucsd.edu.

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.

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