Metabolic pathway analysis using a nash equilibrium approach

Abstract

A novel approach to metabolic network analysis using a Nash Equilibrium (NE) formulation is proposed in which enzymes are considered players in a multi-player game. Each player has its own payoff function with the objective of minimizing the Gibbs free energy associated with the biochemical reaction(s) it catalyzes subject to elemental mass balances while the network objective is to find the best solution to the sum of the player payoff functions. Consequently, any NE solution may not be best solution for all players. Key advantages of the NE approach include the ability to account for (1) aqueous electrolyte behavior, (2) the consumption/production of co-factors, and (3) charge balancing. However, the proposed Nash equilibrium formulation results in a set of nonlinear programming sub-problems that are more demanding to solve than conventional flux balance analysis (FBA) formulations which rely on linear programming. A direct substitution solution methodology for pathways with feedback is described. The Krebs cycle is used to demonstrate the efficacy of the NE approach while comparisons with both FBA and experimental data are used to show that it represents a paradigm shift in metabolic network analysis.

Appendix A: Metabolite chemistry for the Krebs cycle

Model 1 for the Krebs cycle consists of the following eight chemical reactions:

$$\begin{aligned}&C_4 H_2 O_5^{-2} +C_{23} H_{34} N_7 O_{17} P_3 S^{-4}+H_2 O \rightleftharpoons C_6 H_5 O_7^{-3} \nonumber \\&\quad +\,C_{21} H_{32} N_7 O_{16} P_3 S^{-4}+H^{+} \end{aligned}$$

(A1)

$$\begin{aligned}&\quad C_6 H_5 O_7^{-3} \rightleftharpoons i C_6 H_5 O_7^{-3} \end{aligned}$$

(A2)

$$\begin{aligned}&\quad iC_6 H_5 O_7^{-3} \rightleftharpoons C_5 H_4 O_5^{-2} +CO_2 +H^{+} \end{aligned}$$

(A3)

$$\begin{aligned}&\quad C_5 H_4 O_5^{-2} +C_{21} H_{37} N_7 O_{16} P_3 S^{+1} \rightleftharpoons C_{25} H_{39} N_7 O_{19} P_3 S^{-1}+CO_2 \end{aligned}$$

(A4)

$$\begin{aligned}&\quad C_{25} H_{39} N_7 O_{19} P_3 S^{-1}+H_2 O \rightleftharpoons C_4 H_4 O_4^{-2} +C_{21} H_{37} N_7 O_{16} P_3 S^{+1} \end{aligned}$$

(A5)

$$\begin{aligned}&\quad C_4 H_4 O_4^{-2} \rightleftharpoons C_4 H_2 O_4^{-2} \end{aligned}$$

(A6)

$$\begin{aligned}&\quad C_4 H_2 O_4^{-2} +H_2 O \rightleftharpoons C_4 H_4 O_5^{-2} \end{aligned}$$

(A7)

$$\begin{aligned}&\quad C_4 H_4 O_5^{-2} \rightleftharpoons C_4 H_2 O_5^{-2} +2H^{+} \end{aligned}$$

(A8)