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In a Boolean network, each (target) gene is ‘predicted’ by several other genes by means of a Boolean function (predictor). Thus, after having inferred such a function from gene expression data, it could be concluded that if we observe the values of the predictive genes, we know, with full certainty, the value of the target gene. Conceptually, such an inherent determinism seems problematic as it assumes an environment with no uncertainty. However, the data that used for the inference exhibits uncertainty on several levels.
Another class model called Probabilistic Boolean Networks (PBNs) (Shmulevich et al., 2002) shares the appealing properties of Boolean networks, but is able to cope with uncertainty, both in the data and the model selection. A model incorporates only a partial description of a physical system. This means that a Boolean function giving the next state of a variable is likely to be only partially accurate.
The basic idea is to extend the Boolean network to accommodate more than one possible function for each node. Thus, to every node xi . , their corresponds a set Fi={ fj },j=1,..., l(i) , Where each fj is a possible function determining the value of gene xi and l(i) is the number of possible functions for gene xi . A realization of the PBN at a given instant of time is determined by a vector of Boolean functions, where the ith element of that vector contains the predictor selected at that instant for gene xi . In other words, the vector function fk:{0,1}^n mapps to {0,1}^n acts as a transition function (mapping) representing a possible realization of the entire PBN. Such functions are commonly referred to as multiple-output Boolean functions Each of the N possible realizations can be thought of as a standardBoolean network operates for one time step. In other words, at every state x(t) belongs to {0,1}^n , one of the N Boolean networks is chosen and used to make the transition to the next state x(t+1) belongs to {0,1}^n . The probability Pi that the ith (Boolean) network or realization is selected can be easily expressed in terms of the individual selection probabilities Cj see (Shmulevich et al., 2002). The dynamics of the PBN are essentially the same as for Boolean networks, but at any given point in time, the value of each node is determined by one of the possible predictors, chosen according to its corresponding probability.This can be interpreted by saying that at any point in time, we have one out of N possible networks. The basic building block of a PBN is shown in the Figure1.
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