  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.
