Complementing ODE-Based System Analysis Using Boolean Networks Derived from an Euler-Like Transformation

Abstract

In this paper, we present a systematic transition scheme for a large class of ordinary differential equations (ODEs) into Boolean networks. Our transition scheme can be applied to any system of ODEs whose right hand sides can be written as sums and products of monotone functions. It performs an Euler-like step which uses the signs of the right hand sides to obtain the Boolean update functions for every variable of the corresponding discrete model. The discrete model can, on one hand, be considered as another representation of the biological system or, alternatively, it can be used to further the analysis of the original ODE model. Since the generic transformation method does not guarantee any property conservation, a subsequent validation step is required. Depending on the purpose of the model this step can be based on experimental data or ODE simulations and characteristics. Analysis of the resulting Boolean model, both on its own and in comparison with the ODE model, then allows to investigate system properties not accessible in a purely continuous setting. The method is exemplarily applied to a previously published model of the bovine estrous cycle, which leads to new insights regarding the regulation among the components, and also indicates strongly that the system is tailored to generate stable oscillations.

Fig 1. ODE simulation from the model BovCycle and the generated discrete time series.

This time series is used to create a sequence of binary states for the validation of the discrete model. (a) Simulation output for GnRH, FSH, and LH. (b) Simulation output for Fol, CL, and P4. (c) Simulation output for E2, Inh, PGF, and IOF.

Fig 2. Dependency graph of the Boolean model for the bovine estrous cycle.

This graph directly follows from the discrete updates calculated in Step 1, which are derived from the right hand sides of the ODE model. Implemented in GINsim [23].