## Abstract

### Background

Flux Balance Analysis (FBA) based mathematical modeling enables *in silico* prediction of systems behavior for genome-scale metabolic networks. Computational methods have been derived in the FBA framework to solve bi-level optimization for deriving “optimal” mutant microbial strains with targeted biochemical overproduction. The common inherent assumption of these methods is that the surviving mutants will always cooperate with the engineering objective by overproducing the maximum desired biochemicals. However, it has been shown that this optimistic assumption may not be valid in practice.

### Methods

We study the validity and robustness of existing bi-level methods for strain optimization under uncertainty and non-cooperative environment. More importantly, we propose new pessimistic optimization formulations: P-ROOM and P-OptKnock, aiming to derive robust mutants with the desired overproduction under two different mutant cell survival models: (1) ROOM assuming mutants have the minimum changes in reaction fluxes from wild-type flux values, and (2) the one considered by OptKnock maximizing the biomass production yield. When optimizing for desired overproduction, our pessimistic formulations derive more robust mutant strains by considering the uncertainty of the cell survival models at the inner level and the cooperation between the outer- and inner-level decision makers. For both P-ROOM and P-OptKnock, by converting multi-level formulations into single-level Mixed Integer Programming (MIP) problems based on the strong duality theorem, we can derive exact optimal solutions that are highly scalable with large networks.

### Results

Our robust formulations P-ROOM and P-OptKnock are tested with a small *E. coli* core metabolic network and a large-scale *E. coli* iAF1260 network. We demonstrate that the original bi-level formulations (ROOM and OptKnock) derive mutants that may not achieve the predicted overproduction under uncertainty and non-cooperative environment. The knockouts obtained by the proposed pessimistic formulations yield higher chemical production rates than those by the optimistic formulations. Moreover, with higher uncertainty levels, both cellular models under pessimistic approaches produce the same mutant strains.

### Conclusions

In this paper, we propose a new pessimistic optimization framework for mutant strain design. Our pessimistic strain optimization methods produce more robust solutions regardless of the inner-level mutant survival models, which is desired as the models for cell survival are often approximate to real-world systems. Such robust and reliable knockout strategies obtained by the pessimistic formulations would provide confidence for in-vivo experimental design of microbial mutants of interest.