Identifying appropriate modulation of gene expressions through optimization is a rational approach to design engineered organisms for biotechnological purposes. This strategy would greatly benefit from the development of detailed kinetic models of the target metabolism. Such models should incorporate regulatory influences, for example through a set of ordinary differential equations accounting for the system's dynamics. Models based on systematic approximated kinetic representations, such as power-laws, Saturating and Cooperative or convenience kinetics, provide uniform forms that are guaranteed to be accurate over a wide range of conditions.
Unfortunately, non-convexities included in such models may lead to the existence of multiple local optima in which standard optimization algorithms may get trapped during the search. Several stochastic and deterministic global optimization methods have been proposed to overcome this limitation, but only the second group can rigorously guarantee global optimality. Deterministic global optimization algorithms rely on the use of convex underestimators to formulate lower-bounding convex problems that yield lower bounds on the global optimum. Most of these methods are general purpose and originally devised to exploit special structures in the continuous terms of the formulation. Because of this, they can encounter numerical difficulties in some instances, particularly when a large number of non-convexities of different nature are found, as is the case in metabolic engineering problems.
In previous works [1-3], the authors developed highly efficient optimization algorithms and a set of related strategies for understanding the evolution of adaptive responses in cellular metabolisms described by the GMA canonical representation. GMA models provide a more realistic representation than stoichiometric linear models. The Saturable and Cooperative (SC) formalisms  are more accurate representations valid for a wider range of values. Unfortunately, the strategies proposed for GMA models are not directly applicable to the SC formalisms. In order to side-step this shortcoming and extend the global optimization methods developed for power-law models, we explore herein the use of systematic techniques for recasting SC models into GMA ones. Recasting permits the exact transformation of a continuous non-linear model of arbitrary form into a canonical GMA model [5-6]. This transformation is performed at the expense of increasing the number of variables of the original model. Through this technique, GMA models become a canonical form that can be used for simulation and optimization purposes, which opens the door for effectively extending the optimization and feasibility analysis originally devised for GMA models to other detailed kinetic models.
In this work, we shall show the practical utility of recasting SC models into GMA models for optimization purposes. This technique is similar to the symbolic reformulation algorithm proposed by Smith and Pantelides . In contradistinction with this method, we focus on obtaining a power-law representation rather than bringing the model to a standard form containing linear constraints and a set of nonlinearities corresponding to bilinear product, linear fractional, simple exponentiation and univariate function terms. Our results show that recasting non-linear kinetic models into GMA models is indeed an appropriate strategy that helps overcoming some of the numerical difficulties that arise during the global optimization task.
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- Guilléen-Gosálbez G, Pozo C, Jiménez L, Sorribas A: A global optimization strategy to identify quantitative design principles for gene expression in yeast adaptation to heat shock. Computer Aided Chemical Engineering 2009, 26:1045-1050.
- Pozo C, Guillén-Gosálbez G, Sorribas A, Jiménez L: A Spatial Branch-and-Bound Framework for the Global Optimization of Kinetic Models of Metabolic Networks. Industrial and Engineering Chemistry Research 2010.
- Sorribas A, Hernandez-Bermejo B, Vilaprinyo E, Alves R: Cooperativity and saturation in biochemical networks: a saturable formalism using Taylor series approximations. Biotechnology and bioengineering 2007, 97(5):1259-1277.
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