The underlying pathway structure of biochemical reaction networks

Abstract

Bioinformatics is yielding extensive, and in some cases complete, genetic and biochemical information about individual cell types and cellular processes, providing the composition of living cells and the molecular structure of its components. These components together perform integrated cellular functions that now need to be analyzed. In particular, the functional definition of biochemical pathways and their role in the context of the whole cell is lacking. In this study, we show how the mass balance constraints that govern the function of biochemical reaction networks lead to the translation of this problem into the realm of linear algebra. The functional capabilities of biochemical reaction networks, and thus the choices that cells can make, are reflected in the null space of their stoichiometric matrix. The null space is spanned by a finite number of basis vectors. We present an algorithm for the synthesis of a set of basis vectors for spanning the null space of the stoichiometric matrix, in which these basis vectors represent the underlying biochemical pathways that are fundamental to the corresponding biochemical reaction network. In other words, all possible flux distributions achievable by a defined set of biochemical reactions are represented by a linear combination of these basis pathways. These basis pathways thus represent the underlying pathway structure of the defined biochemical reaction network. This development is significant from a fundamental and conceptual standpoint because it yields a holistic definition of biochemical pathways in contrast to definitions that have arisen from the historical development of our knowledge about biochemical processes. Additionally, this new conceptual framework will be important in defining, characterizing, and studying biochemical pathways from the rapidly growing information on cellular function.

Figure 1

(A) Chemical reaction scheme consisting of five metabolites and 11 fluxes. (B) Conversion of the reaction scheme using conventions described within the text. A system boundary is drawn with internal fluxes (v₁–v₇) and external fluxes (b₁–b₄) labeled.

Figure 2

Representation of reaction balance equations in matrix notation. From the five balance equations, a 5 × 11 stoichiometric matrix is generated.

Figure 3

Transformation of the theoretically feasible basis ℬ (spanning the null space of S) to a set of basis vectors 𝒫 (also spanning the null space of S), which are both theoretically and biochemically feasible. Note that all internal fluxes (above the dashed line) are positive in the basis 𝒫.

Figure 4

Fundamental pathways traced out by each of the six linearly independent basis vectors in 𝒫. Exchange flux activity that enters the system (i.e., negative) is indicated by the gray filled arrows, whereas activity that exits the system (i.e., positive) is indicated by the black filled arrows. Active internal fluxes are indicated by the black arrows.

Figure 5

Metabolic reaction network analyzed from the human erythrocyte containing 29 metabolites and the related 51 fluxes (37 internal and 14 exchange fluxes). Not shown in the figure is the involvment of ATP, ADP, NAD, NADH, NADP, NADPH, CO₂, H⁺, and P_i in associated reactions.