Detection of haplotype-dependent allele-specific DNA methylation in WGBS data.

In heterozygous genomes, allele-specific measurements can reveal biologically significant differences in DNA methylation between homologous alleles associated with local changes in genetic sequence. Current approaches for detecting such events from whole-genome bisulfite sequencing (WGBS) data perform statistically independent marginal analysis at individual cytosine-phosphate-guanine (CpG) sites, thus ignoring correlations in the methylation state, or carry-out a joint statistical analysis of methylation patterns at four CpG sites producing unreliable statistical evidence. Here, we employ the one-dimensional Ising model of statistical physics and develop a method for detecting allele-specific methylation (ASM) events within segments of DNA containing clusters of linked single-nucleotide polymorphisms (SNPs), called haplotypes. Comparisons with existing approaches using simulated and real WGBS data show that our method provides an improved fit to data, especially when considering large haplotypes. Importantly, the method employs robust hypothesis testing for detecting statistically significant imbalances in mean methylation level and methylation entropy, as well as for identifying haplotypes for which the genetic variant carries significant information about the methylation state. As such, our ASM analysis approach can potentially lead to biological discoveries with important implications for the genetics of complex human diseases.

Computational and experimental approaches for modeling gene regulatory networks.

To understand most cellular processes, one must understand how genetic information is processed. A formidable challenge is the dissection of gene regulatory networks to delineate how eukaryotic cells coordinate and govern patterns of gene expression that ultimately lead to a phenotype. In this paper, we review several approaches for modeling eukaryotic gene regulatory networks and for reverse engineering such networks from experimental observations. Since we are interested in elucidating the transcriptional regulatory mechanisms of colon cancer progression, we use this important biological problem to illustrate various aspects of modeling gene regulation. We discuss four important models: gene networks, transcriptional regulatory systems, Boolean networks, and dynamical Bayesian networks. We review state-of-the-art functional genomics techniques, such as gene expression profiling, cis-regulatory element identification, TF target gene identification, and gene silencing by RNA interference, which can be used to extract information about gene regulation. We can employ this information, in conjunction with appropriately designed reverse engineering algorithms, to construct a computational model of gene regulation that sufficiently predicts experimental observations. In the last part of this review, we focus on the problem of reverse engineering transcriptional regulatory networks by gene perturbations. We mathematically formulate this problem and discuss the role of experimental resolution in our ability to reconstruct accurate models of gene regulation. We conclude, by discussing a promising approach for inferring a transcriptional regulatory system from microarray data obtained by gene perturbations.

Nonlinear multiresolution signal decomposition schemes--part I: morphological pyramids.

Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. This paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis. The proposed theory is based on the following ingredients: 1) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases toward higher levels and 2) each step toward a higher level is implemented by an (information-reducing) analysis operator, whereas each step toward a lower level is implemented by an (information-preserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. Several examples of pyramid decomposition schemes are shown to be instances of the proposed theory: a particular class of linear pyramids, morphological skeleton decompositions, the morphological Haar pyramid, median pyramids, etc. Furthermore, the paper makes a distinction between single-scale and multiscale decomposition schemes, i.e., schemes without or with sample reduction. Finally, the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks.

Nonlinear multiresolution signal decomposition schemes--part II: morphological wavelets.

In its original form, the wavelet transform is a linear tool. However, it has been increasingly recognized that nonlinear extensions are possible. A major impulse to the development of nonlinear wavelet transforms has been given by the introduction of the lifting scheme by Sweldens (1995, 1996, 1998). The aim of this paper, which is a sequel to a previous paper devoted exclusively to the pyramid transform, is to present an axiomatic framework encompassing most existing linear and nonlinear wavelet decompositions. Furthermore, it introduces some, thus far unknown, wavelets based on mathematical morphology, such as the morphological Haar wavelet, both in one and two dimensions. A general and flexible approach for the construction of nonlinear (morphological) wavelets is provided by the lifting scheme. This paper briefly discusses one example, the max-lifting scheme, which has the intriguing property that preserves local maxima in a signal over a range of scales, depending on how local or global these maxima are.

Binary random fields, random closed sets, and morphological sampling.

We theoretically formulate the problem of processing continuous-space binary random fields by means of mathematical morphology. This may allow us to employ mathematical morphology to develop new statistical techniques for the analysis of binary random images. Since morphological transformations of continuous-space binary random fields are not measurable in general, we are naturally forced to employ intermediate steps that require generation of an equivalent random closed set. The relationship between continuous-space binary random fields and random closed sets is thoroughly investigated. As a byproduct of this investigation, a number of useful new results, regarding separability of random closed sets, are presented. Our plan, however, suffers from a few technical problems that are prominent in the continuous case. As an alternative, we suggest morphological discretization of binary random fields, random closed sets, and morphological operators, thereby effectively implementing our problem in the discrete domain.