Dynamic estimation of specific fluxes in metabolic networks using non-linear dynamic optimization

Metabolic network models describing the biochemical reaction network and material fluxes inside microorganisms open interesting routes for the model-based optimization of bioprocesses. Dynamic metabolic flux analysis (dMFA) has lately been studied as an extension of regular metabolic flux analysis (MFA), rendering a dynamic view of the fluxes, also in non-stationary conditions. Recent dMFA implementations suffer from some drawbacks, though. More specifically, the fluxes are not estimated as specific fluxes, which are more biologically relevant. Also, the flux profiles are not smooth, and additional constraints like, e.g., irreversibility constraints on the fluxes, cannot be taken into account. Finally, in all previous methods, a basis for the null space of the stoichiometric matrix, i.e., which set of free fluxes is used, needs to be chosen. This choice is not trivial, and has a large influence on the resulting estimates.

In this work, a new methodology based on a B-spline parameterization of the fluxes is presented. Because of the high degree of non-linearity due to this parameterization, an incremental knot insertion strategy has been devised, resulting in a sequence of non-linear dynamic optimization problems. These are solved using state-of-the-art dynamic optimization methods and tools, i.e., orthogonal collocation, an interior-point optimizer and automatic differentiation. Also, a procedure to choose an optimal basis for the null space of the stoichiometric matrix is described, discarding the need to make a choice beforehand. The proposed methodology is validated on two simulated case studies: (i) a small-scale network with 7 fluxes, to illustrate the operation of the algorithm, and (ii) a medium-scale network with 68 fluxes, to show the algorithm's capabilities for a realistic network. The results show an accurate correspondence to the reference fluxes used to simulate the measurements, both in a theoretically ideal setting with no experimental noise, and in a realistic noise setting.

Because, apart from a metabolic reaction network and the measurements, no extra input needs to be given, the resulting algorithm is a systematic, integrated and accurate methodology for dynamic metabolic flux analysis that can be run online in real-time if necessary.