Edward R. Dougherty

Gene-expression profiles in hereditary breast cancer.

BACKGROUND Many cases of hereditary breast cancer are due to mutations in either the BRCA1 or the BRCA2 gene. The histopathological changes in these cancers are often characteristic of the mutant gene. We hypothesized that the genes expressed by these two types of tumors are also distinctive, perhaps allowing us to identify cases of hereditary breast cancer on the basis of gene-expression profiles. METHODS RNA from samples of primary tumor from seven carriers of the BRCA1 mutation, seven carriers of the BRCA2 mutation, and seven patients with sporadic cases of breast cancer was compared with a microarray of 6512 complementary DNA clones of 5361 genes. Statistical analyses were used to identify a set of genes that could distinguish the BRCA1 genotype from the BRCA2 genotype. RESULTS Permutation analysis of multivariate classification functions established that the gene-expression profiles of tumors with BRCA1 mutations, tumors with BRCA2 mutations, and sporadic tumors differed significantly from each other. An analysis of variance between the levels of gene expression and the genotype of the samples identified 176 genes that were differentially expressed in tumors with BRCA1 mutations and tumors with BRCA2 mutations. Given the known properties of some of the genes in this panel, our findings indicate that there are functional differences between breast tumors with BRCA1 mutations and those with BRCA2 mutations. CONCLUSIONS Significantly different groups of genes are expressed by breast cancers with BRCA1 mutations and breast cancers with BRCA2 mutations. Our results suggest that a heritable mutation influences the gene-expression profile of the cancer.

Probabilistic Boolean Networks: a rule-based uncertainty model for gene regulatory networks.

MOTIVATION Our goal is to construct a model for genetic regulatory networks such that the model class: (i) incorporates rule-based dependencies between genes; (ii) allows the systematic study of global network dynamics; (iii) is able to cope with uncertainty, both in the data and the model selection; and (iv) permits the quantification of the relative influence and sensitivity of genes in their interactions with other genes. RESULTS We introduce Probabilistic Boolean Networks (PBN) that share the appealing rule-based properties of Boolean networks, but are robust in the face of uncertainty. We show how the dynamics of these networks can be studied in the probabilistic context of Markov chains, with standard Boolean networks being special cases. Then, we discuss the relationship between PBNs and Bayesian networks--a family of graphical models that explicitly represent probabilistic relationships between variables. We show how probabilistic dependencies between a gene and its parent genes, constituting the basic building blocks of Bayesian networks, can be obtained from PBNs. Finally, we present methods for quantifying the influence of genes on other genes, within the context of PBNs. Examples illustrating the above concepts are presented throughout the paper.

Ratio-based decisions and the quantitative analysis of cDNA micro-array images

Gene expression can be quantitatively analyzed by hybridizing fluor-tagged mRNA to targets on a cDNA micro-array. Comparison of gene expression levels arising from co-hybridized samples is achieved by taking ratios of average expression levels for individual genes. In an image-processing phase, a method of image segmentation identifies cDNA target sites in a cDNA micro-array image. The resulting cDNA target sites are analyzed based on a hypothesis test and confidence interval to quantify the significance of observed differences in expression ratios. In particular, the probability density of the ratio and the maximum-likelihood estimator for the distribution are derived, and an iterative procedure for signal calibration is developed.

From Boolean to probabilistic Boolean networks as models of genetic regulatory networks

Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles governing biological systems in an integrative and holistic manner. It also paves the way toward the development of systematic approaches for effective therapeutic intervention in disease. The central theme in this paper is the Boolean formalism as a building block for modeling complex, large-scale, and dynamical networks of genetic interactions. We discuss the goals of modeling genetic networks as well as the data requirements. The Boolean formalism is justified from several points of view. We then introduce Boolean networks and discuss their relationships to nonlinear digital filters. The role of Boolean networks in understanding cell differentiation and cellular functional states is discussed. The inference of Boolean networks from real gene expression data is considered from the viewpoints of computational learning theory and nonlinear signal processing, touching on computational complexity of learning and robustness. Then, a discussion of the need to handle uncertainty in a probabilistic framework is presented, leading to an introduction of probabilistic Boolean networks and their relationships to Markov chains. Methods for quantifying the influence of genes on other genes are presented. The general question of the potential effect of individual genes on the global dynamical network behavior is considered using stochastic perturbation analysis. This discussion then leads into the problem of target identification for therapeutic intervention via the development of several computational tools based on first-passage times in Markov chains. Examples from biology are presented throughout the paper.

Gene selection: a Bayesian variable selection approach

UNLABELLED Selection of significant genes via expression patterns is an important problem in microarray experiments. Owing to small sample size and the large number of variables (genes), the selection process can be unstable. This paper proposes a hierarchical Bayesian model for gene (variable) selection. We employ latent variables to specialize the model to a regression setting and uses a Bayesian mixture prior to perform the variable selection. We control the size of the model by assigning a prior distribution over the dimension (number of significant genes) of the model. The posterior distributions of the parameters are not in explicit form and we need to use a combination of truncated sampling and Markov Chain Monte Carlo (MCMC) based computation techniques to simulate the parameters from the posteriors. The Bayesian model is flexible enough to identify significant genes as well as to perform future predictions. The method is applied to cancer classification via cDNA microarrays where the genes BRCA1 and BRCA2 are associated with a hereditary disposition to breast cancer, and the method is used to identify a set of significant genes. The method is also applied successfully to the leukemia data. SUPPLEMENTARY INFORMATION http://stat.tamu.edu/people/faculty/bmallick.html.

External Control in Markovian Genetic Regulatory Networks

Probabilistic Boolean Networks (PBNs) have been recently introduced as a rule-based paradigm for modeling gene regulatory networks. Such networks, which form a subclass of Markovian Genetic Regulatory Networks, provide a convenient tool for studying interactions between different genes while allowing for uncertainty in the knowledge of these relationships. This paper deals with the issue of control in probabilistic Boolean networks. More precisely, given a general Markovian Genetic Regulatory Network whose state transition probabilities depend on an external (control) variable, the paper develops a procedure by which one can choose the sequence of control actions that minimize a given performance index over a finite number of steps. The procedure is based on the theory of controlled Markov chains and makes use of the classical technique of Dynamic Programming. The choice of the finite horizon performance index is motivated by cancer treatment applications where one would ideally like to intervene only over a finite time horizon, then suspend treatment and observe the effects over some additional time before deciding if further intervention is necessary. The undiscounted finite horizon cost minimization problem considered here is the simplest one to formulate and solve, and is selected mainly for clarity of exposition, although more complicated costs could be used, provided appropriate technical conditions are satisfied.

Performance of feature-selection methods in the classification of high-dimension data

Contemporary biological technologies produce extremely high-dimensional data sets from which to design classifiers, with 20,000 or more potential features being common place. In addition, sample sizes tend to be small. In such settings, feature selection is an inevitable part of classifier design. Heretofore, there have been a number of comparative studies for feature selection, but they have either considered settings with much smaller dimensionality than those occurring in current bioinformatics applications or constrained their study to a few real data sets. This study compares some basic feature-selection methods in settings involving thousands of features, using both model-based synthetic data and real data. It defines distribution models involving different numbers of markers (useful features) versus non-markers (useless features) and different kinds of relations among the features. Under this framework, it evaluates the performances of feature-selection algorithms for different distribution models and classifiers. Both classification error and the number of discovered markers are computed. Although the results clearly show that none of the considered feature-selection methods performs best across all scenarios, there are some general trends relative to sample size and relations among the features. For instance, the classifier-independent univariate filter methods have similar trends. Filter methods such as the t-test have better or similar performance with wrapper methods for harder problems. This improved performance is usually accompanied with significant peaking. Wrapper methods have better performance when the sample size is sufficiently large. ReliefF, the classifier-independent multivariate filter method, has worse performance than univariate filter methods in most cases; however, ReliefF-based wrapper methods show performance similar to their t-test-based counterparts.

Optimal infinite-horizon control for probabilistic Boolean networks

External control of a genetic regulatory network is used for the purpose of avoiding undesirable states, such as those associated with disease. Heretofore, intervention has focused on finite-horizon control, i.e., control over a small number of stages. This paper considers the design of optimal infinite-horizon control for context-sensitive probabilistic Boolean networks (PBNs). It can also be applied to instantaneously random PBNs. The stationary policy obtained is independent of time and dependent on the current state. This paper concentrates on discounted problems with bounded cost per stage and on average-cost-per-stage problems. These formulations are used to generate stationary policies for a PBN constructed from melanoma gene-expression data. The results show that the stationary policies obtained by the two different formulations are capable of shifting the probability mass of the stationary distribution from undesirable states to desirable ones.

Fuzzification of set inclusion: theory and applications

Abstract Fuzzification of set inclusion for fuzzy sets is developed in terms of an indicator for set inclusion, the indicator giving the degree to which a fuzzy set is a subset of another fuzzy set. To date, such indicators have been called ‘inclusion grades’; however, in contrast to most existing indicators, it is proposed in the present paper that the indicator must be two-valued for crisp sets. The approach, herein, is to begin by postulating desired properties of indicators for fuzzified set inclusion, to then assume a specific mathematical form for such indicators, and then derive the necessary and sufficient conditions under which the specified formula gives rise to indicators possessing the desired properties. The investigation results in a very general class of indicators based on the bold union operation, and, most importantly, in a complete measure-theoretic characterization of this class. The characterization takes the form of a constrained representation providing explicit formulation for all indicators in the class of interest. The paper closes with applications of fuzzified set inclusion to shape recognition via image processing, in particular, mathematical morphology, and to the measurement of fuzziness in fuzzy sets by means of entropy.

Coefficient of determination in nonlinear signal processing

Abstract For statistical design of an optimal filter, it is probabilistically advantageous to employ a large number of observation random variables; however, estimation error increases with the number of variables, so that variables not contributing to the determination of the target variable can have a detrimental effect. In linear filtering, determination involves the correlation coefficients among the input and target variables. This paper discusses use of the more general coefficient of determination in nonlinear filtering. The determination coefficient is defined in accordance with the degree to which a filter estimates a target variable beyond the degree to which the target variable is estimated by its mean. Filter constraint decreases the coefficient, but it also decreases estimation error in filter design. Because situations in which the sample is relatively small in comparison with the number of observation variables are of salient interest, estimation of the determination coefficient is considered in detail. One may be unable to obtain a good estimate of an optimal filter, but can nonetheless use rough estimates of the coefficient to find useful sets of observation variables. Since minimal-error estimation underlies determination, this material is at the interface of signal processing, computational learning, and pattern recognition. Several signal-processing factors impact application: the signal model, morphological operator representation, and desirable operator properties. In particular, the paper addresses the VC dimension of increasing operators in terms of their morphological kernel/basis representations. Two applications are considered: window size for restoring degraded binary images; finding sets of genes that have significant predictive capability relative to target genes in genomic regulation.