Joint Realizability of Monotone Boolean functions

TL;DR

This paper investigates the conditions under which collections of monotone Boolean functions can be realized as state transition graphs of differential equation models, revealing how algebraic restrictions influence realizability.

Contribution

It establishes a theoretical framework linking ODE models to collections of MBFs and analyzes how algebraic complexity restrictions affect joint realizability.

Findings

The class of jointly realizable MBFs decreases with increased algebraic restrictions.

Explicit examples demonstrate the limits of realizability under different ODE classes.

The results have implications for understanding gene regulation network dynamics.

Abstract

The study of monotone Boolean functions (MBFs) has a long history. We explore a connection between MBFs and ordinary differential equation (ODE) models of gene regulation, and, in particular, a problem of the realization of an MBF as a function describing the state transition graph of an ODE. We formulate a problem of joint realizability of finite collections of MBFs by establishing a connection between the parameterized dynamics of a class of ODEs and a collection of MBFs. We pose a question of what collections of MBFs can be realized by ODEs that belong to nested classes defined by increased algebraic complexity of their right-hand sides. As we progressively restrict the algebraic form of the ODE, we show by a combination of theory and explicit examples that the class of jointly realizable functions strictly decreases. Our results impact the study of regulatory network dynamics, as well as the classical area of MBFs. We conclude with a series of potential extensions and conjectures.