**1. Introduction**

Mathematical models in systems biology are mostly represented by ordinary differential equations (ODEs). They provide a representation of the information obtained from experimental observations about the structure and function of a particular biological network [1,2]. The integral component of ODEs is parameters that correspond to the kinetic characteristics of a reaction catalyzed by a specific enzyme in particular conditions. Typically, the parameters are identified by fitting the model to experimental data or are measured for individual reactions separately. Once parameter values are determined, dynamic models could be used to confirm hypotheses, draw predictions and find such (time-varying) stimulation conditions that result in a particular desired behavior of a system [2–4]. However, the problem of fitting a dynamical model to experimental data is non-differentiable, thus, derivative-free optimization methods should be used instead of gradient-based or higher-order optimizers [5,6].

Here, we present a framework that implements a set of population-based optimization methods to identify parameters in a dynamic model of a biological network of interest, from limited available experimental data. In other words, the presented framework allows finding parameter values of a dynamical model while only selected quantities are observed. This could drastically decrease the time of fitting separate reactions to data and improve

**Citation:** Weglarz-Tomczak, E.; Tomczak, J.M.; Eiben, A.E.; Brul, S. Population-Based Parameter Identification for Dynamical Models of Biological Networks with an Application to *Saccharomyces cerevisiae*.*Processes* **2021**, *9*, 98. https://doi.org/ 10.3390/pr9010098

Received: 14 December 2020 Accepted: 30 December 2020 Published: 5 January 2021

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estimation quality because all reactions are considered as a whole, thus, it takes into account interaction among reactions. The implementation of the approach is a stand-alone Python program. It utilizes PySCeS (Python Simulator for Cellular System) [7], a modeling tool for formulating dynamical models of biological networks and running simulations by solving ODEs numerically. Our framework loads a model developed using PySCeS or from the JWS database [8] together with experimental data, and outputs parameter values for which a difference between the experimental data and the simulation is smallest. Moreover, the framework allows adding new optimizers to a single file, without the necessity of changing any other parts of the program. Please see *Supplementary Data* for details. We refer to this framework as POPI4SB, see its schematic representation in Figure 1.

In this study, we chose *glycolysis* that is a crucial metabolic pathway and its upregulation is correlated with diseases like cancer [9,10]. Nearly all living organisms carry out glycolysis as a part of cellular metabolism. One of the most intensively studied organisms in the context of, among others, glycolysis is *Saccharomyces cerevisiae* species, also known as baker's yeas<sup>t</sup> [11–15]. We applied our optimization framework to a model of glycolysis in yeas<sup>t</sup> proposed in [16]. This model contains lumped reactions of the glycolytic pathway and includes production of glycerol, fermentation to ethanol and exchange of acetaldehyde between the cells, and trapping of acetaldehyde by cyanide.

**Figure 1.** A schematic representation of our framework. A dynamic model in the PySCeS format and experimental data are inputs to the program. The core component is the parameter identification with population-based optimization methods. Eventually, parameters values are returned, for with the lowest error (i.e., the difference between simulated data and experimental data) was achieved.

> The contribution of the paper is threefold:


#### **2. Materials and Methods**
