Motivation: We have hypothesized that the construction of transcriptional regulatory networks using a method that optimizes connectivity would lead to regulation consistent with biological expectations. A key expectation is that the hypothetical networks should produce a few, very strong attractors, highly similar to the original observations, mimicking biological state stability and determinism. Another central expectation is that, since it is expected that the biological control is distributed and mutually reinforcing, interpretation of the observations should lead to a very small number of connection schemes. Results: We propose a fully Bayesian approach to constructing probabilistic gene regulatory networks (PGRNs) that emphasizes network topology. The method computes the possible parent sets of each gene, the corresponding predictors and the associated probabilities based on a nonlinear perceptron model, using a reversible jump Markov chain Monte Carlo (MCMC) technique, and an MCMC method is employed to search the network configurations to find those with the highest Bayesian scores to construct the PGRN. The Bayesian method has been used to construct a PGRN based on the observed behavior of a set of genes whose expression patterns vary across a set of melanoma samples exhibiting two very different phenotypes with respect to cell motility and invasiveness. Key biological features have been faithfully reflected in the model. Its steady-state distribution contains attractors that are either identical or very similar to the states observed in the data, and many of the attractors are singletons, which mimics the biological propensity to stably occupy a given state. Most interestingly, the connectivity rules for the most optimal generated networks constituting the PGRN are remarkably similar, as would be expected for a network operating on a distributed basis, with strong interactions between the components.