Avoiding Bias in Boolean Network Statistical Studies

Stuart Kauffman and others have proposed Random Boolean networks (RBN) as models of genetic regulation in living cells. Common practice in statistical studies of RBN ensembles has been to sample Boolean rules without explicit concern for their effective connectance . We show that this practice may lead to substantial bias in estimating important dynamical features of the sparselyconnected nets which have been of particular interest. We present exact formulas for (1) the number of distinct network state transition maps as a function of the numbers of elements (N) and nominal inputs (Kn); and (2) the number of rules by effective connectance (Ke), given Kn. We introduce the concept of multiplicity, relating the numbers of nominal and effective mappings, and show how to partition an ensemble into equal-multiplicity sub-ensembles. The necessary (exact) formulas are provided. This approach enables fine-grained assessment of dynamical features and construction of unbiased estimates with respect to the overall ensemble. We illustrate how the formulas apply to ensembles with Kn = 1 and Kn = 2. Our results may be relevant to any study that employs samples of sparsely-connected Boolean nets