Joint Realizability of Monotone Boolean Functions

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.