State estimation in constraint based models of metabolic-genetic networks

Common sensors for bioreactors can only measure extracellular variables and total biomass, but not the intracellular biomass distribution and biological reaction fluxes. Reconstructing the intracellular variables requires the application of estimation methods and observers, which are commonly based on ordinary differential equation models for the biological dynamics. However, due to a lack of knowledge on kinetic mechanisms, many metabolic models are formulated as constraint based stoichiometric models, where an optimization problem is solved to determine intracellular reaction fluxes. The recently proposed deFBA model is a dynamic constraint based model that also takes into account the intracellular biomass distribution, and thus appears suitable to estimate intracellular states in bioreactors. This paper introduces the state estimation problem for the deFBA model. Since the model already involves an optimization problem, doing state estimation with it yields a bilevel optimization problem. The bilevel problem is transformed to a mixed integer quadratic program to achieve an efficient numerical solution. The approach is illustrated with an example using artificial data.