# A circuit-preserving mapping from multilevel to Boolean dynamics

## TL;DR

This paper introduces a new method for converting multilevel biological network models into Boolean models that preserves dynamics and interactions without creating non-admissible states, improving analysis capabilities.

## Contribution

A novel multilevel to Boolean conversion method that avoids non-admissible states and preserves variable interactions, with an implementation and application to counter a known conjecture.

## Key findings

- The new method eliminates non-admissible states in Boolean models.
- It preserves variable interactions and dynamics more effectively.
- Applied to challenge a conjecture on oscillations in logical models.

## Abstract

Many discrete models of biological networks rely exclusively on Boolean variables and many tools and theorems are available for analysis of strictly Boolean models. However, multilevel variables are often required to account for threshold effects, in which knowledge of the Boolean case does not generalise straightforwardly. This motivated the development of conversion methods for multilevel to Boolean models. In particular, Van Ham's method has been shown to yield a one-to-one, neighbour and regulation preserving dynamics, making it the de facto standard approach to the problem. However, Van Ham's method has several drawbacks: most notably, it introduces vast regions of "non-admissible" states that have no counterpart in the multilevel, original model. This raises special difficulties for the analysis of interaction between variables and circuit functionality, which is believed to be central to the understanding of dynamic properties of logical models. Here, we propose a new multilevel to Boolean conversion method, with software implementation. Contrary to Van Ham's, our method doesn't yield a one-to-one transposition of multilevel trajectories, however, it maps each and every Boolean state to a specific multilevel state, thus getting rid of the non-admissible regions and, at the expense of (apparently) more complicated, "parallel" trajectories. One of the prominent features of our method is that it preserves dynamics and interaction of variables in a certain manner. As a demonstration of the usability of our method, we apply it to construct a new Boolean counter-example to the well-known conjecture that a local negative circuit is necessary to generate sustained oscillations. This result illustrates the general relevance of our method for the study of multilevel logical models.