A Nash Equilibrium Approach to Metabolic Network Analysis

Abstract

A novel approach to metabolic network analysis using a Nash Equilibrium formulation is proposed. Enzymes are considered to be players in a multi-player game in which each player attempts to minimize the dimensionless Gibbs free energy associated with the biochemical reaction(s) it catalyzes subject to elemental mass balances. Mathematical formulation of the metabolic network as a set of nonlinear programming (NLP) sub-problems and appropriate solution methodologies are described. A small example representing part of the production cycle for acetyl-CoA is used to demonstrate the efficacy of the proposed Nash Equilibrium framework and show that it represents a paradigm shift in metabolic network analysis.

Appendix

1.1 Biochemical Reactions for a Simplified Metabolic Network for Acetyl CoA Production

The biochemical reactions involved at each node in the metabolic network are as follows:

• 1. Acetyl CoA production from pyruvate and acetate

  $$\begin{aligned} C_3 H_3 O_3 \; + \; C_2 H_3 O_2 \; + \; 2 C_{21} H_{32} N_7 O_{16} P_3 S \; \rightleftharpoons \; 2 C_{23} H_{34} N_7 O_{17} P_3 S \; + \; C O_2 \; + \; H_2 O \end{aligned}$$

  (20)

  Pyruvate $\rightarrow$ acetyl CoA: pyruvate dehydrogenase (genes: lpd, aceE, and aceF) [26]

  Acetate $\rightarrow$ acetyl CoA: acetyl CoA synthetase (genes: acs) [26]

• 2. Acetyl phosphate production from acetyl CoA

  $$\begin{aligned} C_{23} H_{34} N_7 O_{17} P_3 S \; + \; H O_4 P \; \rightleftharpoons \; C_2 H_3 O_5 P \; + \; C_{21} H_{32} N_7 O_{16} P_3 S \end{aligned}$$

  (21)

  Acetyl CoA $\leftrightarrow$ acetyl phosphate: phosphotransacetylase (genes: pta or eutD) [26]

• 3. Acetate production from acetyl phosphate

  $$\begin{aligned} C_2 H_3 O_5 P \; \rightleftharpoons \; C_2 H_3 O_2^{-} \; + \; P O_3^{3-} \end{aligned}$$

  (22)

  Acetyl phosphate $\leftrightarrow$ acetate: acetate kinase (genes: ackA, tdcD, or purT) [26]

Table 4. Unknown fluxes, components and Gibbs energy of formation.