MOTIVATION: Dynamical modeling of gene regulation via network models constitutes a key problem for genomics. The long-run characteristics of a dynamical system are critical and their determination is a primary aspect of system analysis. In the other direction, system synthesis involves constructing a network possessing a given set of properties. This constitutes the inverse problem. Generally, the inverse problem is ill-posed, meaning there will be many networks, or perhaps none, possessing the desired properties. Relative to long-run behavior, we may wish to construct networks possessing a desirable steady-state distribution. This paper addresses the long-run inverse problem pertaining to Boolean networks (BNs). RESULTS: The long-run behavior of a BN is characterized by its attractors. The rest of the state transition diagram is partitioned into level sets, the j-th level set being composed of all states that transition to one of the attractor states in exactly j transitions. We present two algorithmsfor the attractor inverse problem. The attractors are specified, and the sizes of the predictor sets and the number of levels are constrained. Algorithm complexity and performance are analyzed. The algorithmic solutions have immediate application. Under the assumption that sampling is from the steady state, a basic criterion for checking the validity of a designed network is that there should be concordance between the attractor states of the model and the data states. This criterion can be used to test a design algorithm: randomly select a set of states to be used as data states; generate a BN possessing the selected states as attractors, perhaps with some added requirements such as constraints on the number of predictors and the level structure; apply the design algorithm; and check the concordance between the attractor states of the designed network and the data states. AVAILABILITY: The software and supplementary material is available at http://gsp.tamu.edu/Publications/BNs/bn.htm
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