[Paper Review] RBA like problem with thermo-kinetics is non convex

Marc Dinh, Vincent Fromion|arXiv (Cornell University)|May 23, 2017

Advanced Control Systems Optimization4 references3 citations

TL;DR

This paper demonstrates that metabolic flux optimization problems incorporating stoichiometric, enzyme allocation, and thermo-kinetic constraints are inherently non convex. Using a minimal three-reaction network with fixed external metabolite levels, the authors identify at least two distinct local optima via numerical optimization, proving non convexity in the class of problems studied, which has implications for algorithmic tractability in systems biology modeling.

ABSTRACT

The aim of this short note is to show that the class of problem involving kinetic or thermo-kinetic constraints in addition to the usual stoechiometric one is non convex.

Motivation & Objective

• To determine whether non convexity in systems biology optimization problems arises from the problem structure or its formulation.
• To investigate the impact of including thermo-kinetic constraints alongside stoichiometric and enzyme allocation constraints on the convexity of the optimization problem.
• To demonstrate that the presence of kinetic and thermodynamic constraints leads to multiple local optima, indicating non convexity.
• To provide a minimal counterexample showing non convexity in a biologically plausible metabolic network.

Proposed method

• Formulated a minimal metabolic network with three reversible enzymatic reactions and a biomass reaction.
• Applied stoichiometric constraints to enforce mass balance at steady state.
• Incorporated thermo-kinetic flux equations using a standard Michaelis-Menten-like rate law with all parameters set to 1.
• Defined an enzyme budget constraint with relative weights on E1, ET, and E2.
• Used numerical optimization via fmincon with 1000 random starts to explore the solution space.
• Validated solutions using linear programming after fixing internal metabolite levels X1 and X2.

Experimental results

Research questions

• RQ1 Is the optimization problem involving stoichiometric, enzyme allocation, and thermo-kinetic constraints convex?
• RQ2 Can multiple local optima exist in such a problem under biologically plausible conditions?
• RQ3 Does the inclusion of kinetic and thermodynamic constraints introduce non convexity into the RBA-like optimization framework?
• RQ4 Are the observed local optima consistent with elementary flux modes, as suggested by prior theory?

Key findings

• The optimization problem exhibits at least two distinct local optima, confirming non convexity in the problem class.
• The first local optimum achieves a biomass flux νμ = 0.0574 with νT = -0.0574 and ν1 = 0.
• The second local optimum achieves νμ = 0.0540 with ν1 = 0.0540 and νT = 0, indicating different flux distributions.
• The two optima correspond to different elementary flux modes, supporting prior theoretical claims.
• The optimal enzyme distribution differs significantly between the two solutions: E1 = 0.3719 and ET = 0 in one, E1 = 0 and ET = 1.0659 in the other.
• The existence of multiple local optima was confirmed through both global numerical search and local LP verification after fixing X1 and X2.