Minimization and equivalence in multi-valued logical models of regulatory networks

Streck, Adam; Lorenz, Therese; Siebert, Heike

doi:10.1007/s11047-015-9525-2

Cited by 3 publications

(4 citation statements)

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“…While both suppress inter-genes regulations, the stepwise tends to add positive self-regulations, whereas the asymptotic tends to add negative self-regulations. This work contributes to a better understanding of the different ways to represent a multilevel system, for different ways can represent the same model [15], which causes ambiguities in the notation.…”

Section: Discussionmentioning

confidence: 99%

“…The correspondence between evolution functions and state transition graphs is not bijective. In fact, as discussed by Streck et al [15], for a given (multilevel) grid graph Γ, there are multiple evolution functions which have Γ as their state transition graphs. To have a bijective correspondence between the two representations of the system, we restrict ourselves to a certain class of evolution functions.…”

Section: Methodsmentioning

confidence: 99%

“…A function which is neither asymptotic nor stepwise has in general more non-trivial partial derivatives than the asymptotic and the stepwise functions sharing the same state transition graph given in Proposition 1. (See [15] for a detailed discussion. The stepwise function in our paper can be seen as a special case of the canonical function defined there.…”

Section: Propositionmentioning

confidence: 99%

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A circuit-preserving mapping from multilevel to Boolean dynamics

Fauré

Kaji

2018

Journal of Theoretical Biology

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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...

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Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

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A circuit-preserving mapping from multilevel to Boolean dynamics

Fauré

Kaji

2018

Journal of Theoretical Biology

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“…As shown above, a parametrization gives rise to a dynamical behaviour, which is different (in most cases [23]) from the behaviour of any other possible parametrization of the system. Our primary interest is then to find the parametrizations of a system that match our expectations, if there are any, which is achieved by the process of parameter filtering.…”

Section: Parametrization Spacementioning

confidence: 99%