A circuit-preserving mapping from multilevel to Boolean dynamics

Abstract

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. BackgroundBoolean models have proved very useful in the analysis of various networks in biology. However, it is often convenient to introduce multilevel variables to account for multiple threshold effects. We are often faced with choices between using Boolean variables or multilevel variables. This can be crucial since theoretical results are sometimes proved only for Boolean or multilevel networks. A particular example of this situation is in René Thomas' conjecture that a local negative circuit is necessary to produce sustained (asynchronous) oscillations. This paper stems from the simple idea that a Boolean counter-example to that conjecture could be found by transposing a multilevel counter-example found earlier by Richard and Comet. However, we believe the method developed in this paper, together with a handy script which implements it, is widely applicable to other theoretical studies which involves discrete networks. We also find the notion of asymptotic evolution function defined in this paper sheds light on the understanding of relation between the state transition graph and the interaction graph.1.1. Introduction. Introduced in the 1960s-70s to model biological regulatory networks, the logical (discrete) formalism has...

On the conversion of multivalued to Boolean dynamics

“…In the conversion, two variables are introduced to represent the second component. The interaction graphs G F (1, 1, 0 The interaction graph at (1, 1, 0, 0) consists therefore of a positive cycle that involves all four variables, and corresponds to a non-elementary cycle found at G f (1, 1, 0) as the composition of the two negative cycles in (3). Two negative cycles corresponding to the two cycles in G f (1, 1, 0) appear instead in G F (1, 0, 1, 0).…”

“…In this section we present a Boolean conversion of this map which demonstrates that the absence of local negative cycles does not imply the existence of a fixed point or the absence of cyclic attractors, for Boolean networks with n ≥ 6. An alternative conversion of the same counterexample can be found in [3]; in Section 7 we compare the two approaches. First, to create a Boolean conversion of Richard's example with a unique cyclic attractor, we can take the stepwise version f , which is given in Figure 2a.…”

“…For comparison, the conversion introduced in Section 4 writes as F = b • f • ψ * . Properties of the binarisation are studied in [3] for maps that are asymptotic. The asymptotic versionf : X → X of f is defined as follows:f…”

“…Here we apply the conversion method from multivalued to Boolean to the multivalued counterexample of Richard, and exhibit a simple example of map on 6 Boolean components having no fixed points and no local negative cycles (Section 5). An alternative Boolean conversion for the same map was recently presented by Fauré and Kaji [3]; a comparison to their conversion method is given in Section 7. As a second application of the conversion, we give a multivalued version of the result of Ruet [9] on the existence of local cycles in presence of mirror pairs (Theorem 4).…”

Local Negative Circuits and Cyclic Attractors in Boolean Networks with at most Five Components

“…Ruet [12] exhibited a procedure to create counterexamples in the Boolean case, for every n ≥ 7, n being the number of variables; these are maps admitting an antipodal attractive cycle and no local negative circuits in the regulatory graph. Tonello [17] and Fauré and Kaji [3] identified different Boolean versions of Richard's discrete example, that provide counterexamples to Question 1 for n = 6. A map with an antipodal attractive cycle and no local regulatory circuits also exists for n = 6 (we present such a map in appendix A).…”