Modelling Oscillatory Patterns in the Bovine Estrous Cycle with Boolean Delay Equations

Susanna Röblitz,Heike Siebert,Julia Plöntzke,Mascha Berg

doi:10.1007/s11538-021-00942-z

Abstract

Boolean delay equations (BDEs), with their relatively simple and intuitive mode of modelling, have been used in many research areas including, for example, climate dynamics and earthquake propagation. Their application to biological systems has been scarce and limited to the molecular level. Here, we derive and present two BDE models. One is directly derived from a previously published ordinary differential equation (ODE) model for the bovine estrous cycle, whereas the second model includes a modification of a particular biological mechanism. We not only compare the simulation results from the BDE models with the trajectories of the ODE model, but also validate the BDE models with two additional numerical experiments. One experiment induces a switch in the oscillatory pattern upon changes in the model parameters, and the other simulates the administration of a hormone that is known to shift the estrous cycle in time. The models presented here are the first BDE models for hormonal oscillators, and the first BDE models for drug administration. Even though automatic parameter estimation still remains challenging, our results support the role of BDEs as a framework for the systematic modelling of complex biological oscillators.

Highlights

There exist different approaches to create mathematical models of biological systems
We present the two different Boolean delay equations (BDEs) models as explained in the previous section
This is not surprising, since the BDE model was derived from the ordinary differential equation (ODE) trajectories and equations

Summary

Introduction

There exist different approaches to create mathematical models of biological systems. Two commonly used modelling types are ordinary differential equations (ODEs), with continuous time and continuous values for the model components, i.e. 121 Page 2 of 25 space”, and Boolean models, with discrete time and discrete space Both types of models have their advantages and disadvantages. There exist various modelling frameworks with discrete time and continuous space as, for instance, difference equations, or with continuous time and discrete space. One approach for the latter, presented by Stoll et al (2012), uses continuous-time Markov chains. Another example are Boolean delay equations (BDEs). The delays enable the modelling of different time scales in processes such as signalling or gene transcription

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