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Boolean networks

The Boolean network model, introduced by Kauffman (Kauffman, 1969, 1974; Kauffman and Glass, 1973)and recently developed by Shmulevich(Shmulevich, 2002), has received the most attention, not only from the biology community, but also in physics. In this model, gene expression is quantized to only two levels: ON and OFF. The expression level (state) of each gene is functionally related to the expression states of some other genes, using logical rules. A Boolean network G(V,F) is defined by a set of nodes corresponding to genes V = {x1, . . . , xn} and a list of Boolean functions F = (f1, . . . , fn) . The state of a node (gene) is completely determined by the values of other nodes at time t by means of underlying logical Boolean functions. The model is represented in the form of directed graph.Each xi represents the state (expression) of gene i, where xi=1 represents the fact that gene i is expressed and xi=0 means it is not expressed. The list of Boolean functions F represents the rules of regulatory interactions between genes. That is, any given gene transforms its inputs (regulatory factors that bind to it) into an output, which is the state or expression of the gene itself. The maximum connectivity of a Boolean network is defined by K= maxi (ki) . All genes are assumed to update synchronously in accordance with the functions assigned to them and this process is then repeated. The artificial synchrony simplifies computation while preserving the qualitative, generic properties of global network dynamics (Kauffman, 1993; Huang, 1999; Wuensche, 1998).

Below the example is presented. Consider a Boolean network consisting of 5 genes {x1, . . . , x5} with the corresponding Boolean functions given by the truth tables shown in Figure1. The maximum connectivity is K=3, although we allow some input variables to duplicate, essentially reducing the connectivity.The dynamics of this Boolean network are shown in Figure2. Since there are 5 genes, there are 2^5 = 32 possible states that the network can be in. Each state is represented by a circle and the arrows between states show the transitions of the network according to the functions in Table 1., Figure1. . It is easy to see that because of the inherent deterministic directionality in Boolean networks as well as only a finite number of possible states.

Truth tables of the functions in a Boolean network with 5 genes. The indices j1, j2, and j3 indicate the input connections for each of the functions.

The state-transition diagram for the Boolean network defined in table 1. (Figure1).

In the context of Boolean networks as models of genetic regulatory networks, there is no doubt that the binary approximation of gene expression is an oversimplification (Huang, 1999). However, even though most biological phenomena manifest themselves in the continuous domain, they are often described in a binary logical language such as‘on and off,’‘upregulated and downregulated’, and‘responsive and nonresponsive.’There is a several examples showing that a Boolean formalism is meaningful in biology, in (Shmulevich and Zhang, 2002), one reasoned that if the genes, when quantized to only two levels (1 or 0), would not be informative in separating known sub-classes of tumors, then there would be little hope for Boolean modeling of realistic genetic networks based on gene expression data.

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Source:  OpenStax, Introduction to bioinformatics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10240/1.3
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