Boolean network modeling

Many algorithms have been proposed for the inference of BNs including two widely used approaches (1) the consistency problem (Akutsu et al., 1998) to determine the existence of a BN that is consistent with the observed data and (2) the best-fit extension problem (Boros et al., 1998). 

Then the error size of the Boolean function/is defined as 

In the best-fit extension problem, we are looking for Boolean functions from a certain class of functions, denoted as C, which make as few misclassifications as possible. If w(x) = 1 for all X G F U F, then the error size is just the number of miselassifieations. The goal is to find subsets T and F such that F fl F = 0 and F U F = F U F for which pdBf(F, F ) has an extension where s(/) is the minimum in the previously chosen class of functions, C. Consequently, any extension/G C of pdBf(F, F ) has minimum error size. The consistency problem can be viewed as a special case of the best-fit extension problem when s(/) = 0. We may have measured identieal true and/or false vectors several times, each of them possibly associated with a different positive weight. 

3 Computation Times for Large-Scale Boolean Networks (Advanced Readers) 

More formally, a PBN G(V, F) is defined by a set of binary-valued nodes (genes) V = Xi. X and a list of function sets F = (Fi. F ), where each function set F,- consists of /(/) Boolean functions, that is, F,- = //, . / (. The value of each node X, is updated by a Boolean function taken from flie corresponding set F,. A realization of the PBN at a given time instant is defined by a vector of Boolean functions. If there are N possible realizations for the PBN, then there are N vector functions fi. f v, where each f, = jjl. .. I j N, 1 ji l(i), and each G Fi. Here, f, can take all possible realizations of PBN assuming independence of the random variables in f N = l(i) is the number of possible PBN realizations. 

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Elaborate models have been developed to account for the behavior of cellular biochemical networks. Boolean network models use a set of logical rules to illustrate the progress of the network reactions [10]. These models do not take into explicit account the participation of specific biochemical reactions. Models that account for the details of biochemical reactions have been proposed [11,12]. The behavior of these models depends on the rate constants of the chemical reactions and the concentrations of the reactants. Measurements like those described below of reaction fluxes and reactant concentrations will be able to test such network models. In the following sections, we will use simple examples to illustrate the characteristic steady-state behavior and propose an approach to measure fluxes and concentrations. 

Silvescu A. and Honavar V. (2001). Temporal Boolean network models of genetic networks and their inference from gene expression time series. Complex Systems. 13, pp 54-70. 

Lahdesmaki, H., Shmtrlevich, L, and YU-Haqa, O. (2003). On Learning gene regulatory networks under the boolean network model. Machine Learning, 52 147. 

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

Boolean Network with connectivity k- or N, )-net - generalizes the basic binary k = 2) CA model by evolving each site variable Xi 0,1 of according to a randomly selected Boolean function of k inputs ... 

In this section, two models of development were presented, a complex model consisting of a multioperon genome and a cytoplasm, and a simple model based on random Boolean networks. The simpler model was explained in more detail, as it is the basis for the extended example described here. This model utilizes both development and evolution to get to a cell that can develop into a multicellular organism able to seek a chemical trace. 

Price ND, Shmulevich I. Biochemical and statistical network models for systems biology. Curr. Opin. Biotechnol. 2007. Shmulevich I, et al. Probabilistic Boolean Networks a rule-based uncertainty model for gene regulatory networks. Bioinformatics. 2002 18 261-274... 

It is envisaged that most complex naturally occurring networks are driven by small world networks. Thus, the model developed and implemented in this work is inspired from Boolean networks (Thomas and Kauffman, 2001a 2001b Silvescu and Honavar, 2001 Albert, 2004), weight matrices (Weaver et al, 1999 and Ando and Iba, 2001) and small world phenomenon (Milgram, 1967 Watts and Strogatz, 1998 ... 

It is difficult to cover the wide range of so many different network modeling approaches, but we will attempt to briefly introduce three widely used network modeling techniques in this chapter Boolean network (BN), Bayesian belief network, and metabolic network modeling methods. 

Shmulevich, I., Dougherty, E. R., Kim, S., and Zhang, W. (2002). ProbabiUstic Boolean networks A mle-based uncertainty model for gene regulatory networks. Bioinformatics, 18 261—274. 

Abstract. In this paper we show that a well-known model of genetic regulatory networks, namely that of Random Boolean Networks (RBNs), allows one to study in depth the relationship between two important properties of complex systems, i.e. dynamical criticality and power-law distributions. The study is based upon an analysis of the response of a RBN to permanent perturbations, that may lead to avalanches of changes in activation levels, whose statistical properties are determined by the same parameter that characterizes the dynamical state of the network (ordered, critical or disordered). Under suitable approximations, in the case of large sparse random networks an analytical expression for the probability density of avalanches of different sizes is proposed, and it is shown that for not-too-smaU avalanches of critical systems it may be approximated by a power law. In the case of small networks the above-mentioned formula does not maintain its validity, because of the phenomenon of self-interference of avalanches, which is also explored by numerical simulations. 

It is also often assumed that the presence of power-law distributions is the hallmark of criticality. Indeed, slightly different (although overlapping) notions of criticality have been used [7]. In this paper we show that a well-known model of genetic regulatory networks, introduced hy one of us several years ago [8], i.e. that of Random Boolean Networks (RBNs), can he used to study the relationships between power-law distributions and criticality issues. 

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