September 5, 2023

He outer optimization level seeks to minimize the number of reactions in the network (Equation (2)), while fulfilling the inner constraints. The inner problem definition is a modification of the standard flux balance analysis (FBA) [23] formulation (cf. Equation (4) and Additional file 1). However, instead of biomass production, the objective function maximizes the sum over weighted fluxes (Equation (3)). A further constraint on the fluxes is imposed by demanding that a fraction fmin of the maximum biomass production fmax of the complete network is achieved (Equation (5) and [24]). This requirement is based on optical density data from the same experimental setup, which indicate a growth rate of about 10 to 30 of the growth rate under ambient conditions (cf. Additional file 1). To account for D(+)-Galactosamine (hydrochloride) the medium (modified MOPS minimal medium) on which the cell cultures were grown, constraints on the exchange reactions are taken from [21] and only inorganic compounds and glucose are allowed to enter and exit the system. To reduce the computational complexity, we seek to reduce the number of integer variables. To this end, we distinguish between indispensable reactions, which make up most of the biomass production and dispensable reactions, which have a negligible contributions to growth. To define these two groups, we delete, one by one, every reaction and performed FBA on the perturbed network. If the resulting biomass production remains above a defined threshold (99 ), we consider the reaction dispensable for the organism’s viability under ambient conditions (for robustness of the findings, see the Results section). Reactions that are considered indispensable are not assigned a Boolean variable. Altogether, we obtain the following formulationNwhere Sij is the stoichiometric coefficient of metabolite i in reaction j, cj is the contribution of vj to the objective function and vmax is the upper boundary on vj , while j D and N denote the reactions that are dispensable and indispensable, respectively. Although we investigate time-series data, the program formulation employs a the quasi-steady-state assumption (Equation (4)). We assume a separation of the timeconstants at which transcriptional and metabolic regulations take place. This is justified by the evidence that changes taking place on the metabolic level are generally much faster (seconds) compared to those taking place on the transcriptional level (minutes) [25]. In other words, enzyme dynamics occur more quickly compared to changes in gene expression. To solve the min-max MILP, it is transformed from a bilevel to a single-level MILP. This procedure employs two steps: (1) finding the dual for the inner linear program [26] and (2) removing the occurring bi-linear terms [27] (cf. Additional file 1).Fractional appearance of reactions in EFMsminimizej=yjThe reduced size of the networks allows the computation of sets of EFMs for the time- and condition specific minimal networks. It has already been shown that the importance of a reaction for network functionality can be characterized by the number of EFMs in which it is involved [13]. Extending this concept to the time domain, we define the fractional appearance Xij of PubMed ID:https://www.ncbi.nlm.nih.gov/pubmed/17139194 a reaction i at time j as the ratio between the number of elementary modes involving reaction i and the total number of elementary modes at time j: X(i, j) = Nr. of EFMs including reaction i at time j . This definition allows to Nr. of all EFMs at time j characterize the temporal.

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