Boolean network

Let us give a simple example of a Boolean network, taken from [8], shown in fig. 13.4. Part (a) of that figure shows the connectivity of the network part (b) gives the logical (Boolean) rules and part (c) the state transition table which results from the network structure and the logical rules. For example, consider the second line in the transition table, and we reason from the primed letters to the unprimed If A = 0, then B = 0 if B = 0, then A or C = 1 if C = 0, then A and B = 0. The task of reverse engineering is to find the network and rules that produce the observed transitions as displayed in fig. 13.4. 

We have already introduced that subject in chapter 9 (see eqs. (9.2)-(9.5)). There we chose a new measure of a correlation distance, the mutual information. A natural measure of such a distance between the probability distributions of two variables is the number of states jointly available to them (the size of the support set) compared to the number of states available to them individually. Thus, two probability distributions are close and the support set is small, if the knowledge of one predicts the most likely state of the other, even if there exists simultaneously a substantial number of other states. Hence from the gene expression distribution calculated in the transition table, and the measured expression distribution, a mutual information can be calculated and the search for an appropriate network proceeds until a minimum mutual information is obtained. 

Other methods are available (see [9]). In that paper, calculated illustrations are given on the number of calculations necessary for Boolean network identification. For example, for a connectivity of 2 genes to each gene only 50 input-output pairs in the transition table had to be calculated from random guesses to identify a gene network of 320 genes. 

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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. 

We might mention here in passing that while class-1, class-3 and class-4 (but not class-2) can all be obtained from one another with these two simple rules, class-2 behavior can only be obtained if the system is first quenched [vich86b]. That is, if the lattice is initially randomly populated with AND and XOR rules according to the prescribed value of p and is then frozen for all later times. Such quenched random rules harbor some interesting properties of their own in dimensions d > 1, and are the basis of much of Kauffman s findings on random Boolean networks (see section 8.6). 

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 ... 

Fig. 8.16 Typical random Boolean network with connectivity k == 1. Arrows indicate inputs to given sites and represent the Boolean functions at the sites toward which the arrows are pointing.

DDLab is an interactive graphics program for studying many different kinds of discrete dynamical systems. Arbitrary architectures can be defined, ranging from Id, 2d or 3d CA to random Boolean networks. 

The random Boolean network of a seeker cell showing internal and external connections... 

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... 

Shmulevich I, Dougherty ER, Zhang W. Gene perturbation and intervention in probabilistic Boolean networks. Bioinformatics. 2002 18 1319-1331. 

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. 

T. Akutsu, S. Miyano, and S. Kuhara, Algorithms for identifying Boolean networks and related biological networks based on matrix multiplication and fingerprint fnnction. / Comput Biol 7(3 ) 331-343 (2000). 

T. Akntsn, S. Miyano, and S. Knhara, Identification of genetic networks from a small nnmber of gene expression patterns nnder the Boolean network mode. Pac Symp Bio-comput, pp. 17-28 (1999). 

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 ... 

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. 

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