Model development is a key task in systems biology, and involves different steps, such as model calibration, experimental design and model refinement which usually take place in an iterative way (see reviews in [1–5]). The process of building a model of a biological system typically starts by generating an initial model candidate, or by taking one from the pre-existing knowledge, and then involves an iterative cycle of hypotheses-driven model modifications, new experimentation and subsequent model identification steps, finally leading to a satisfactory refined model [6, 7]. Thus, model selection, experimentation and model refinement can be considered the basic elements of systems biology .
A number of researches have proposed different iterative schemes for model development involving the steps of parameter estimation, identifiability analysis, and optimal experimental design [9–12]. The related topic of optimal experimental design for parameter estimation [3, 13] and for model discrimination [14–16] is receiving increased attention in recent years. Lillacci and Khammash  introduced a new method for parameter estimation based on Kalman filtering that can also be used to discriminate among alternate models of the same biological process.
Verheijen  presented an overview of model selection practices, highlighting the main criteria for choosing out of a large set of models: level of rigor, accuracy with respect to data, adequacy of the model, and its flexibility and computational complexity. He also identifies developments in optimization-based approaches [19, 20] as very promising, but recognizing its limitations due to numerical and algorithmic challenges. Although research along this line has continued [21, 22], it still remains as a challenging numerical problem.
Here, we present a method to simultaneously select a model and calibrate it in a single step. This contribution is based on the following four key ideas: (i) frequently, iterative model development cycles can be considered in a more compact way if sets of hypotheses can be grouped together and formulated as a parameterized set of models, from which the best alternative must be selected; (ii) we consider the problem of model selection formulating it as a simultaneous model selection and parameter identification problem; (iii) further, in order to make the selection decision in a systematic way, we formulate it as an optimization problem  acting on the parameterized set of models; (iv) the optimization problem, which belongs to the class of mixed-integer nonlinear programming (MINLP) problems, is solved by recently developed algorithms based on metaheuristics.
The paper is organized as follows: First, we describe the framework used for model selection and identification, based on the nested models paradigm. Then we state the corresponding optimization problem using a formulation based on mixed-integer non-linear programming subject to differential and algebraic constraints. In the following sections, we describe the application of this methodology to a case study considering a dynamic model of the KdpD/KdpE system of Escherichia coli. Finally, we provide a discussion of the results and summarize the main conclusions of the study.