#### Abstract

Metabolic modeling has proved to be a very powerful tool to get a better insight into the metabolism of an organism. For example, metabolic modeling has clarified production of triacylglycerols from microalgae and carbohydrates from cyanobacteria (Baroukh et al., 2015). Both compounds can then be turned into biofuel (biodiesel and bioethanol, respectively) with expected reduced environmental impacts (Lardon et al., 2009).

Metabolisms of microalgae and cyanobacteria are driven by the solar flux which supports fixation of CO2. Periodic fluctuation of light induces instationarity of their metabolisms, with accumulation of metabolites (especially lipids and carbohydrates). Therefore, such metabolisms are never at steady state.

However, most of the approaches dedicated to metabolism analysis assume balanced growth (i.e. systems at steady state). Furthermore, metabolic models are of high dimension, which makes their mathematical analysis and parameter identification complex. Identifying conditions to maximize productivity by a rigorous mathematical analysis is generally not possible.

Here we propose a method to reduce the dimension of a dynamical metabolic system, which is appropriate to derive model based control strategies. Contrary to nearly all existing works, the idea is to keep some dynamical components of the model, that are necessary especially when dealing with microalgae and cyanobacteria.

A first attempt in this direction was carried out with the DRUM method (Baroukh et al., 2014). DRUM approach has provided sound results, with very efficient representation of accumulation of lipids and carbohydrates in microalgae submitted to light/dark cycles. However, as almost all methods developed for metabolic analysis, it relies on a series of assumptions whose mathematical bases have not been rigorously established.

The main objective of our work is to provide mathematical foundations for the reduction of metabolic networks to low dimensional dynamical models. For reducing metabolic systems accurately, we propose a dynamical approach that relies on time scale separation and the QSSA. Additionally, an algorithm to numerically estimate the parameters of the reduced system is proposed.