(582as) Multi-Level Optimization Framework for the Identification of Multi-Objective Design Principles in Metabolic Networks

While in optimization scientists establish the objective and search the way to accomplish it, in natural systems the emergence of new designs results from the evolution driven by natural selection. While one can argue that natural systems are optimized by natural selection, it is unclear which is the objective function and constraints (i.e., optimization problem) being solved. Determining these design principles is a central topic in systems biology.

The identification of design principles can be posed as a reverse optimization problem, that is, we know the actual solution (the actual system) and we aim to determine which criteria (if any) are being optimized. Previous attempts to solve this problem have relied on single-objective models that assume that there is a single universal objective function that drives an organism’s evolution. In a recent work, however, Sauer and co-workers have shown that there might be more than one fitness function driving the metabolic machinery (Schuetz et al., 2012), and that the optimization of a given criterion might vary according to the environmental conditions.

Bearing this in mind, in this work we propose a new optimization tool for indentifying the set of objective functions that describes in the best manner the operation of metabolic networks. Our approach is inspired by the bi-level linear programming framework proposed by Burgard and Maranas (2002), which is adequately extended in order to deal with several objective functions simultaneously. Particularly, given a set of experimental observations, the goal of the analysis is to infer the form of the multi-objective optimization problem that shapes the adaption of microorganisms to the environment. To this end, we have developed a multi-level linear programming formulation based on flux balance analysis that includes an outer problem and a set of inner models. The outer problem seeks to optimize the least square difference between the experimental observations and the optimal solution predicted in silico, while the inner problems, which are defined for every experimental condition, optimize a linear combination of objectives subject to the weights imposed by the outer problem. The overall multi-level model seeks those combinations of weights that make each experimental condition optimal. A nonzero weight implies that the objective is biologically meaningful, while a zero weight implies the converse. Binary variables are added in order to limit the number of plausible fitness functions, making it possible to evaluate the effect of considering an increasing number of objectives. 

We reformulate this multi-level model into a single mixed-integer linear program (MILP) by replacing the inner problems by the corresponding Karush–Kuhn–Tucker conditions. This requires definition of auxiliary binary variables for each inequality present in the inner problems. Such binaries take a value of one if the constraint is active and zero otherwise. The reformulated MILP, which can be solved by standard branch and cut methods, provides, for each experimental condition being analyzed, the set of meaningful objective functions and associated weights. Hence, our approach allows testing a large number of objectives functions from which the best ones according to the calculated error are finally identified. 

The capabilities of our methodology are illustrated through its application to the central metabolism of Escherichia coli using flux data derived from isotopomer analysis. Particularly, we consider as surrogates for cellular fitness functions a set of reaction fluxes involving production/consumption of ATP and redox potential, as well as biomass growth rate, to predict 13C-determined in vivo fluxes. One of the main advantages of our approach is that it allows studying several objectives under various conditions, as oppose to the approach by Burgard and Maranas (2002), which was developed under the assumption that there is a single universal objective function that is valid under all possible environmental conditions. Numerical results show that experimental observations can be well explained by a reduced number of objectives (i.e., 3), and that there are different combinations of objectives leading to similar errors. Furthermore, biomass formation is always among the objectives found, which indicates its importance as a fitness function in evolution.   

References

• Schuetz, R., Zamboni, N., Zampieri, M., Heinemann, M., Sauer, U. (2012). Multidimensional optimality of microbial metabolism, Science, 601:336.
• Burgard, C. Maranas. (2002). Optimization-Based Framework for Inferring and Testing Hypothesized Metabolic Objective Functions.  Biotechology and bioengineering, 82(6): 670-677.