Probabilistic Boolean networks (PBNs) have recently been introduced as a promising class of models of genetic regulatory networks. The dynamic behaviour of PBNs can be analysed in the context of Markov chains. A key goal is the determination of the steady‐state (long‐run) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the long‐term influence of a gene on another gene or determine the long‐term joint probabilistic behaviour of a few selected genes. Because matrix‐based methods quickly become prohibitive for large sizes of networks, we propose the use of Monte Carlo methods. However, the rate of convergence to the stationary distribution becomes a central issue. We discuss several approaches for determining the number of iterations necessary to achieve convergence of the Markov chain corresponding to a PBN. Using a recently introduced method based on the theory of two‐state Markov chains, we illustrate the approach on a sub‐network designed from human glioma gene expression data and determine the joint steady‐state probabilities for several groups of genes. Copyright © 2003 John Wiley & Sons, Ltd.

Motivation: Ranking gene feature sets is a key issue for both phenotype classification, for instance, tumor classification in a DNA microarray experiment, and prediction in the context of genetic regulatory networks. Two broad methods are available to estimate the error (misclassification rate) of a classifier. Resubstitution fits a single classifier to the data, and applies this classifier in turn to each data observation. Cross-validation (in leave-one-out form) removes each observation in turn, constructs the classifier, and then computes whether this leave-one-out classifier correctly classifies the deleted observation. Resubstitution typically underestimates classifier error, severely so in many cases. Cross-validation has the advantage of producing an effectively unbiased error estimate, but the estimate is highly variable. In many applications it is not the misclassification rate per se that is of interest, but rather the construction of gene sets that have the potential to classify or predict. Hence, one needs to rank feature sets based on their performance. Results: A model-based approach is used to compare the ranking performances of resubstitution and cross-validation for classification based on real-valued feature sets and for prediction in the context of probabilistic Boolean networks (PBNs). For classification, a Gaussian model is considered, along with classification via linear discriminant analysis and the 3-nearest-neighbor classification rule. Prediction is examined in the steady-distribution of a PBN. Three metrics are proposed to compare feature-set ranking based on error estimation with ranking based on the true error, which is known owing to the model-based approach. In all cases, resubstitution is competitive with cross-validation relative to ranking accuracy. This is in addition to the enormous savings in computation time afforded by resubstitution.

Motivation: A number of models have been proposed for genetic regulatory networks. In principle, a network may contain any number of genes, so long as data are available to make inferences about their relationships. Nevertheless, there are two important reasons why the size of a constructed network should be limited. Computationally and mathematically, it is more feasible to model and simulate a network with a small number of genes. In addition, it is more likely that a small set of genes maintains a specific core regulatory mechanism. Results: Subnetworks are constructed in the context of a directed graph by beginning with a seed consisting of one or more genes believed to participate in a viable subnetwork. Functionalities and regulatory relationships among seed genes may be partially known or they may simply be of interest. Given the seed, we iteratively adjoin new genes in a manner that enhances subnetwork autonomy. The algorithm is applied using both the coefficient of determination and the Boolean-function influence among genes, and it is illustrated using a glioma gene-expression dataset. Availability: Software for the seed-growing algorithm will be available at the website for Probabilistic Boolean Networks: http://www2.mdanderson.org/app/ilya/PBN/PBN.htm

Canalizing genes possess such broad regulatory power, and their action sweeps across a such a wide swath of processes that the full set of affected genes are not highly correlated under normal conditions. When not active, the controlling gene will not be predictable to any significant degree by its subject genes, either alone or in groups, since their behavior will be highly varied relative to the inactive controlling gene. When the controlling gene is active, its behavior is not well predicted by any one of its targets, but can be very well predicted by groups of genes under its control. To investigate this question, we introduce in this paper the concept of intrinsically multivariate predictive (IMP) genes, and present a mathematical study of IMP in the context of binary genes with respect to the coefficient of determination (CoD), which measures the predictive power of a set of genes with respect to a target gene. A set of predictor genes is said to be IMP for a target gene if all properly contained subsets of the predictor set are bad predictors of the target but the full predictor set predicts the target with great accuracy. We show that logic of prediction, predictive power, covariance between predictors, and the entropy of the joint probability distribution of the predictors jointly affect the appearance of IMP genes. In particular, we show that high-predictive power, small covariance among predictors, a large entropy of the joint probability distribution of predictors, and certain logics, such as XOR in the 2-predictor case, are factors that favor the appearance of IMP. The IMP concept is applied to characterize the behavior of the gene DUSP1, which exhibits control over a central, process-integrating signaling pathway, thereby providing preliminary evidence that IMP can be used as a criterion for discovery of canalizing genes.

Transfection of tumor suppressor miRNAs such as miR-34a, miR-449a, and miR-16 with DNA damage can regulate apoptosis and senescence in cancer cells. miR-16 has been shown to influence autophagy in cervical cancer. However, the function of miR-34a and miR-449a in autophagy remains unknown. The functional and persistent G1/S checkpoint signaling pathways in HeLa cells via these three miRNAs, either synergistically or separately, remain a mystery. As a result, we present a synthetic Boolean network of the functional G1/S checkpoint regulation, illustrating the regulatory effects of these three miRNAs. To our knowledge, this is the first synthetic Boolean network that demonstrates the advanced role of these miRNAs in cervical cancer signaling pathways reliant on or independent of p53, such as MAPK or AMPK. We compared our estimated probability to the experimental data and found reasonable agreement. Our findings indicate that miR-34a or miR-16 may control senescence, autophagy, apoptosis, and the functional G1/S checkpoint. Additionally, miR-449a can regulate just senescence and apoptosis on an individual basis. MiR-449a can coordinate autophagy in HeLa cells in a synergistic manner with miR-16 and/or miR-34a.

Long non-coding RNA (lncRNA) such as ANRIL and UFC1 have been verified as oncogenic genes in non-small cell lung cancer (NSCLC). It is well known that the tumor suppressor microRNA-34a (miR-34a) is downregulated in NSCLC. Furthermore, miR-34a induces senescence and apoptosis in breast, glioma, cervical cancer including NSCLC by targeting Myc. Recent evidence suggests that these two lncRNAs act as a miR-34a sponge in corresponding cancers. However, the biological functions between these two non-coding RNAs (ncRNAs) have not yet been studied in NSCLC. Therefore, we present a Boolean model to analyze the gene regulation between these two ncRNAs in NSCLC. We compared our model to several experimental studies involving gain- or loss-of-function genes in NSCLC cells and achieved an excellent agreement. Additionally, we predict three positive circuits involving miR-34a/E2F1/ANRIL, miR-34a/E2F1/UFC1, and miR-34a/Myc/ANRIL. Our circuit- perturbation analysis shows that these circuits are important for regulating cell-fate decisions such as senescence and apoptosis. Thus, our Boolean network permits an explicit cell-fate mechanism associated with NSCLC. Therefore, our results support that ANRIL and/or UFC1 is an attractive target for drug development in tumor growth and aggressive proliferation of NSCLC, and that a valuable outcome can be achieved through the miRNA-34a/Myc pathway.