On the stability of mass action reactions in open systems

Linear analysis near a steady-state of biochemical networks: control analysis, correlation metrics and circuit theory

Background

Several approaches, including metabolic control analysis (MCA), flux balance analysis (FBA), correlation metric construction (CMC), and biochemical circuit theory (BCT), have been developed for the quantitative analysis of complex biochemical networks. Here, we present a comprehensive theory of linear analysis for nonequilibrium steady-state (NESS) biochemical reaction networks that unites these disparate approaches in a common mathematical framework and thermodynamic basis.

Results

In this theory a number of relationships between key matrices are introduced: the matrix A obtained in the standard, linear-dynamic-stability analysis of the steady-state can be decomposed as A = SR^T where R and S are directly related to the elasticity-coefficient matrix for the fluxes and chemical potentials in MCA, respectively; the control-coefficients for the fluxes and chemical potentials can be written in terms of R^TBS and S^TBS respectively where matrix B is the inverse of A; the matrix S is precisely the stoichiometric matrix in FBA; and the matrix e^At plays a central role in CMC.

Conclusion

One key finding that emerges from this analysis is that the well-known summation theorems in MCA take different forms depending on whether metabolic steady-state is maintained by flux injection or concentration clamping. We demonstrate that if rate-limiting steps exist in a biochemical pathway, they are the steps with smallest biochemical conductances and largest flux control-coefficients. We hypothesize that biochemical networks for cellular signaling have a different strategy for minimizing energy waste and being efficient than do biochemical networks for biosynthesis. We also discuss the intimate relationship between MCA and biochemical systems analysis (BSA).