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# 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.
(Impact Factor: 1.29). 08/2008; 70(5):1398-409. DOI: 10.1007/s11538-008-9304-7
Source: PubMed

ABSTRACT 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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• "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. "
##### Article: Simple dynamics on graphs
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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). "
##### Article: Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability
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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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• "The first result in this area, proposed by Thomas [38], is that networks whose interaction graphs do not have a positive cycle (i.e. a cycle with an even number of negative arcs) have at most one fixed point. This was then generalised into an upper bound on the number of fixed points of Boolean networks: a network has at most 2 k + fixed points, where k + is the minimum size of a positive feedback vertex set of its interaction graph [5] [3]. This upper bound was then refined via the use of local graphs [36] [28] [29] [30]. "
##### Article: Fixed points of Boolean networks, guessing graphs, and coding theory
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ABSTRACT: In this paper, we are interested in the number of fixed points of functions $f:A^n\to A^n$ over a finite alphabet $A$ defined on a given signed digraph $D$. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on $D$. We then discover relationships between the number of fixed points of $f$ and problems in coding theory, especially the design of codes for the asymmetric channel. Using these relationships, we derive upper and lower bounds on the number of fixed points, which significantly improve those given in the literature. We also unveil some interesting behaviour of the number of fixed points of functions with a given signed digraph when the alphabet varies. We finally prove that signed digraphs with more (disjoint) positive cycles actually do not necessarily have functions with more fixed points.