Article

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: 1.29). 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.

1 Follower
 · 
156 Views
  • [Show abstract] [Hide abstract]
    ABSTRACT: Some articles of this volume will be reviewed individually.
  • Source
    [Show abstract] [Hide abstract]
    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.
  • Source
    [Show abstract] [Hide abstract]
    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.

Preview

Download
9 Downloads
Available from