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Biased Eukaryotic Gene Regulation Rules Suggest Genome Behavior Is Near Edge Of Chaos
, 1997
"... Introduction The present article reports an analysis of data pertaining to certain biases in the observed patterns of transcription regulation of eukaryotic genes. In the Boolean idealization, a small subset of possible switching rules, the canalizing functions, are highly utilized. To draw inferenc ..."
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Introduction The present article reports an analysis of data pertaining to certain biases in the observed patterns of transcription regulation of eukaryotic genes. In the Boolean idealization, a small subset of possible switching rules, the canalizing functions, are highly utilized. To draw inferences about the implications of the observed biases, a statistical ensemble approach was used. Representative networks constructed within the ensemble of networks that satisfy the biases were studied numerically. The consequences indicate that modeled genomic regulatory systems are in a dynamical "ordered" regime, measurably close to a transition to a "chaotic" regime. A number of testable consequences are derived. Section 2 Transcription State Spaces, Trajectories, Attractors, and Boolean Net Models A state space is a mathematical abstraction used to describe a dynamical system consisting of a number of interacting variables. The human genomic regulatory
Large attractors in cooperative bi-quadratic Boolean networks
, 2007
"... Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960’s as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular impor ..."
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Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960’s as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers. An important question is which conditions on the network topology can rule out exponentially long periodic orbits in the system. In this paper we consider cooperative systems, i.e. systems with positive feedback interconnections among all variables, which in a continuous setting guarantees a very stable dynamics. In Part I of this paper we presented a construction that shows that for an arbitrary constant 0 < c < 2 and sufficiently large n there exist n-dimensional Boolean cooperative networks in which both the indegree and outdegree of each for each variable is bounded by two (bi-quadratic networks) and which nevertheless contain periodic orbits of length at least c n. In this part, we prove an inverse result showing that for sufficiently large n and for 0 < c < 2 sufficiently close to 2, any n-dimensional cooperative, bi-quadratic Boolean network with a cycle of length at least c n must have a large proportion of variables with indegree 1. Such systems therefore share a structural similarity to the systems constructed in Part I.
Extremely Chaotic Boolean Networks
, 811
"... It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic regime, even under the simultaneous assumption of several conditio ..."
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It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic regime, even under the simultaneous assumption of several conditions which in randomized studies have been separately shown to correlate with ordered behavior. These properties include using at most two inputs for every variable, using biased and canalyzing regulatory functions, and restricting the number of negative feedback loops. We also prove for n-dimensional Boolean networks that if in addition the number of outputs for each variable is bounded and there exist periodic orbits of length c n for c sufficiently close to 2, any network with these properties must have a large proportion of variables that simply copy previous values of other variables. Such systems share a structural similarity to a relatively small Turing machine acting on one or several tapes. The concept of a Boolean network was originally proposed in the late 1960’s by Stuart Kauffman to model gene regulatory behavior at the cell level [18, 19]. This type of modeling
AVOIDING BIAS IN BOOLEAN NETWORK STATISTICAL STUDIES
, 2000
"... Stuart Kauffman and others have proposed Random Boolean networks (RBN) as models of genetic regulation in living cells. Common practice in statistical studies of RBN ensembles has been to sample Boolean rules without explicit concern for their effective connectance. We show that this practice may le ..."
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Stuart Kauffman and others have proposed Random Boolean networks (RBN) as models of genetic regulation in living cells. Common practice in statistical studies of RBN ensembles has been to sample Boolean rules without explicit concern for their effective connectance. We show that this practice may lead to substantial bias in estimating important dynamical features of the sparselyconnected nets which have been of particular interest. We present exact formulas for (1) the number of distinct network state transition maps as a function of the numbers of elements (N) and nominal inputs (Kn); and (2) the number of rules by effective connectance (Ke), given Kn. We introduce the concept of multiplicity, relating the numbers of nominal and effective mappings, and show how to partition an ensemble into equal-multiplicity sub-ensembles. The necessary (exact) formulas are provided. This approach enables fine-grained assessment of dynamical features and construction of unbiased estimates with respect to the overall ensemble. We illustrate how the formulas apply to ensembles with Kn = 1 and Kn = 2. Our results may be relevant to any study that employs samples of sparsely-connected Boolean nets 1.
Recent Results on Ordering Parameters in Boolean Networks
"... The dynamics of discrete mathematical systems can be related to the val-ues of certain ordering parameters. Experiments with the total set of possible structures of a boolean network with six elements and extended sets of two-valued boolean functions show that three simple parameters describing the ..."
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The dynamics of discrete mathematical systems can be related to the val-ues of certain ordering parameters. Experiments with the total set of possible structures of a boolean network with six elements and extended sets of two-valued boolean functions show that three simple parameters describing the topology, the bias of boolean functions, and their cana-lyzing potential yield necessary conditions for complex dynamics. The crucial factor determining dynamics turns out to be the topology. The impact on simulations of real systems is discussed. 1.
