Maximum Number of Fixed Points in Regulatory Boolean Networks

Departamento Ingeniería Matemática, Universidad de Concepción, Av. Esteban Iturra s/n, Casilla 160-C, Concepción, Chile.
Bulletin of Mathematical Biology (Impact Factor: 1.39). 08/2008; 70(5):1398-409. DOI: 10.1007/s11538-008-9304-7
Source: PubMed


Boolean networks (BNs) have been extensively used as mathematical models of genetic regulatory networks. The number of fixed points of a BN is a key feature of its dynamical behavior. Here, we study the maximum number of fixed points in a particular class of BNs called regulatory Boolean networks, where each interaction between the elements of the network is either an activation or an inhibition. We find relationships between the positive and negative cycles of the interaction graph and the number of fixed points of the network. As our main result, we exhibit an upper bound for the number of fixed points in terms of minimum cardinality of a set of vertices meeting all positive cycles of the network, which can be applied in the design of genetic regulatory networks.

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    • "In Information Theory, the study of fix(G, F) is strongly related to the network coding solvability problem [17] [5], and here again upper-bound are of special interest. It is then not so surprising that the following fundamental bound has been established independently in both contexts [3] [2] [17]: For every digraph G, "
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    ABSTRACT: For a graph $G$, let $\mathcal{C}$ be the set of conjunctive networks with interaction graph $G$, and let $\mathcal{H}$ be the set of graphs obtained from $G$ by contracting some edges. Let $\mathrm{fix}(f)$ be the number of fixed points in a network $f\in \mathcal{C}$, and let $\mathrm{mis}(H)$ be the number of maximal independent sets in $H\in\mathcal{H}$. Our main result is \[\mathrm{mis}(G)~\leq~\max_{H\in\mathcal{H}}\mathrm{mis}(H)~\leq~ \max_{f\in\mathcal{C}}\mathrm{fix}(f)~\leq~ \left(\frac{3}{2}\right)^{m(G)}\mathrm{mis}(G) \] where $m(G)$ is the maximum size of a matching $M$ of $G$ such that every edge of $M$ is contained in an induced copy of $C_4$ that contains no other edge of $M$. Thus if $G$ has no induced $C_4$ then $\max_{H\in\mathcal{H}}\mathrm{mis}(H)=\max_{f\in\mathcal{C}}\mathrm{fix}(f)=\mathrm{mis}(G)$, and this contrasts with following complexity result: It is coNP-hard to decide if $\max_{f\in\mathcal{C}}\mathrm{fix}(f)=\mathrm{mis}(G)$ or if $\max_{H\in\mathcal{H}}\mathrm{mis}(H)=\mathrm{mis}(G)$, even if $G$ has a unique induced copy of $C_4$.
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    • "positive) cycles, then every boolean G-function has no fixed points (resp. at least two fixed points) and is thus not nilpotent [1]. These observations lead us to study the unsigned version of the question, which is more tractable. "
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    ABSTRACT: Does the interaction graph of a finite dynamical system can force this system to have a "complex" dynamics ? In other words, given a finite interval of integers $A$, which are the signed digraphs $G$ such that every finite dynamical system $f:A^n\to A^n$ with $G$ as interaction graph has a "complex" dynamics ? If $|A|\geq 3$ we prove that no such signed digraph exists. More precisely, we prove that for every signed digraph $G$ there exists a system $f:A^n\to A^n$ with $G$ as interaction graph such that $f^{\lfloor\log_2 n\rfloor+2}$ is a constant. The boolean case $|A|=2$ is more difficult, and we provide partial answers instead. We exhibit large classes of unsigned digraphs which admit boolean dynamical systems which converge in linear time. We also prove that any symmetric digraph, and any graph with a loop on each vertex admits a boolean dynamical system which converges in constant time.
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    • "For instance, the min-net introduced in Section II-C is a non-decreasing coding function. Non-decreasing coding functions have been widely studied (see [25], [29], [28]); they are usually represented by an interaction graph with positive signs on all arcs (see [29] and the references therein for a survey of the work on signed interaction graphs). "
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    ABSTRACT: Linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations towards their destinations. The solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding. The guessing number approach converts this problem to determining the number of fixed points of coding functions $f:A^n\to A^n$ over a finite alphabet $A$ (usually referred to as Boolean networks if $A = \{0,1\}$) with a given interaction graph, that describes which local functions depend on which variables. In this paper, we generalise the so-called reduction of coding functions in order to eliminate variables. We then determine the maximum number of fixed points of a fully reduced coding function, whose interaction graph has a loop on every vertex. Since the reduction preserves the number of fixed points, we then apply these ideas and results to obtain four main results on the linear network coding solvability problem. First, we prove that non-decreasing coding functions cannot solve any more instances than routing already does. Second, we show that triangle-free undirected graphs are linearly solvable if and only if they are solvable by routing. This is the first classification result for the linear network coding solvability problem. Third, we exhibit a new class of non-linearly solvable graphs. Fourth, we determine large classes of strictly linearly solvable graphs.
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