Ronaldo Fumio Hashimoto

Steady‐state analysis of genetic regulatory networks modelled by probabilistic Boolean networks

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 steadystate probabilities for several groups of genes.

Growing genetic regulatory networks from seed genes

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

Microarray gridding by mathematical morphology

DNA chips (i.e., microarrays) biotechnology is a hybridization (i.e., DNA matching) based process that makes it possible to quantify the relative abundance of mRNA from two distinct samples by analysing their fluorescence signals. This technique requires robotic placement (i.e., spotting) of thousands of cDNAs (i.e., complementary DNA) in an array format on glass microscope slides which provide gene-specific hybridization targets. The two different samples of mRNA, usually labeled with Cy3 and Cy5 fluorochromes, are cohybridized onto each spotted gene and two digital images, one for each fluorochrome, are acquired after hybridization. Before estimating the signal and background of each spot, it is necessary to locate the region of the spot in order to map the gene information with the corresponding spot. Therefore, these images must be segmented for analysis, that is, the spotting geometric structure must be found. That implies segmenting the subarrays (i.e., the set of grouped spots), and then the positions of the spots in each subarray. The authors introduce a new technique using morphological operators that performs automatic gridding procedures (i.e., subarrays and spot segmentation). This technique has been implemented and tested in a variety of microarray images with success.

Segmentation of microarray images by mathematical morphology

DNA chips (i.e., microarrays) biotechnology is a hybridization (i.e., matching of pairs of DNA)-based process that makes possible to quantify the relative abundance of mRNA of two distinct samples by analyzing their fluorescence signals. This technique requires robotic placement (i.e., spotting) of thousands of cDNAs (i.e., complementary DNA) in an array format on glass microscope slides. The spotted cDNAs are the hybridization targets for the mRNA samples. The two different samples of mRNA, usually labeled with Cy3 and Cy5 fluorochromes, are cohybridized onto each spotted gene. After hybridization, one digital image is acquired for each fluorochrome wavelength. Then, it is necessary to recognize each gene by its position in the array and to estimate its signal (i.e., hybridization information). For that, it is necessary to segment the image in three classes of objects: subarrays (i.e., set of grouped spots), spot box (i.e., the rectangular neighborhood that contains a spot) and spot (i.e., region of the image where there exists signal). In this paper, we present a technique based on mathematical morphology that performs this segmentation. In the website http://www.vision.ime.usp.br/demos/ microarray/detailed experimental results are presented.

Intrinsically Multivariate Predictive Genes

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.

Feature selection algorithms to find strong genes

The cDNA microarray technology allows us to estimate the expression of thousands of genes of a given tissue. It is natural then to use such information to classify different cell states, like healthy or diseased, or one particular type of cancer or another. However, usually the number of microarray samples is very small and leads to a classification problem with only tens of samples and thousands of features. Recently, Kim et al. proposed to use a parameterized distribution based on the original sample set as a way to attenuate such difficulty. Genes that contribute to good classifiers in such setting are called strong. In this paper, we investigate how to use feature selection techniques to speed up the quest for strong genes. The idea is to use a feature selection algorithm to filter the gene set considered before the original strong feature technique, that is based on a combinatorial search. The filtering helps us to find very good strong gene sets, without resorting to super computers. We have tested several filter options and compared the strong genes obtained with the ones got by the original full combinatorial search.

Efficient selection of feature sets possessing high coefficients of determination based on incremental determinations

Feature selection is problematic when the number of potential features is very large. Absent distribution knowledge, to select a best feature set of a certain size requires that all feature sets of that size be examined. This paper considers the question in the context of variable selection for prediction based on the coefficient of determination (CoD). The CoD varies between 0 and 1, and measures the degree to which prediction is improved by using the features relative to prediction in the absence of the features. It examines the following heuristic: if we wish to find feature sets of size m with CoD exceeding δ, what is the effect of only considering a feature set if it contains a subset with CoD exceeding λ δ | max{θ1,θ2,...,θv} < λ), where θ is the CoD of the feature set and θ1,θ2,...,θv are the CoDs of the subsets. Such probabilities allow a rigorous analysis of the following decision procedure: the feature set is examined if max{θ1,θ2,...,θv} ≥ λ. Computational saving increases as λ increases, but the probability of missing desirable feature sets increases as the increment δ - λ decreases; conversely, computational saving goes down as λ decreases, but the probability of missing desirable feature sets decreases as δ - λ increases. The paper considers various loss measures pertaining to omitting feature sets based on the criteria. After specializing the matter to binary features, it considers a simulation model, and then applies the theory in the context of microarray-based genomic CoD analysis. It also provides optimal computational algorithms.

A Combinatorial Optimization Technique for the Sequential Decomposition of Erosions and Dilations

This paper presents a general algorithm for the automatic proof that an erosion (respectively, dilation) has a sequential decomposition or not. If the decomposition exists, an optimum decomposition is presented. The algorithm is based on a branch and bound search, with pruning strategies and bounds based on algebraic and geometrical properties deduced formally. This technique generalizes classical results as Zhuang and Haralick, Xu, and Park and Chin, with equivalent or improved performance. Finally, theoretical analysis of the proposed algorithm and experimental results are presented.

A note on Park and Chin's algorithm [structuring element decomposition]

A finite subset of Z/sup 2/ is called a structuring element. A decomposition of a structuring element A is a sequence of subsets of the elementary square (i.e., the 3/spl times/3 square centered at the origin) such that the Minkowski addition of them is equal to A. H. Park and R.T. Chin (see ibid., vol.17, no.1, p.2-15, 1995) developed an algorithm for finding the optimal decomposition of simply connected structuring elements (i.e., 8-connected structuring elements that contain no holes), imposing the restriction that all subsets in this decomposition are also simply connected. The authors show that there exist infinite families of simply connected structuring elements that have decompositions but are not decomposable according to Park and Chins definition.

Entropic Biological Score: a cell cycle investigation for GRNs inference

Inference of gene regulatory networks (GRNs) is one of the most challenging research problems of Systems Biology. In this investigation, a new GRNs inference methodology, called Entropic Biological Score (EBS), which linearly combines the mean conditional entropy (MCE) from expression levels and a Biological Score (BS), obtained by integrating different biological data sources, is proposed. The EBS is validated with the Cell Cycle related functional annotation information, available from Munich Information Center for Protein Sequences (MIPS), and compared with some existing methods like MRNET, ARACNE, CLR and MCE for GRNs inference. For real networks, the performance of EBS, which uses the concept of integrating different data sources, is found to be superior to the aforementioned inference methods. The best results for EBS are obtained by considering the weights w1=0.2 and w2=0.8 for MCE and BS values, respectively, where approximately 40% of the inferred connections are found to be correct and significantly better than related methods. The results also indicate that expression profile is able to recover some true connections, that are not present in biological annotations, thus leading to the possibility of discovering new relations between its genes.