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.
Highlights
When modeling biological phenomena, different levels of abstraction can be used to capture the mechanisms of the underlying system [1]
We present a systematic transition scheme for a large class of ordinary differential equations (ODEs) into Boolean networks
We present our procedure for deriving a discrete model from an ODE model
Summary
Different levels of abstraction can be used to capture the mechanisms of the underlying system [1]. A global attractor analysis as often infeasible for ODE models, e.g., allows to understand capabilities of the system in terms of stable behavior and to uncover decision processes determining long-term dynamics In application, such insights are of particular interest when analyzing differentiation or multistable response systems [8, 9]. The validation step ensures that the derived Boolean model satisfies the given requirements in terms of property conservation It can be conducted utilizing Boolean-like biological data, aiming at a model capturing all important biological observations. The proposed validation method demonstrates how to check the conservation of oscillatory behavior and is suitable for both transient trajectories and attractors It nicely illustrates the potential of our more pragmatic approach in application
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