The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

Given a matrix of values in which the rows correspond to objects and the columns correspond to features of the objects, rearrangement clustering is the problem of rearranging the rows of the matrix such that the sum of the similarities between adjacent rows is maximized. Referred to by various names and reinvented several times, this clustering technique has been extensively used in many fields over the last three decades. In this paper, we point out two critical pitfalls that have been previously overlooked. The first pitfall is deleterious when rearrangement clustering is applied to objects that form natural clusters. The second concerns a similarity metric that is commonly used. We present an algorithm that overcomes these pitfalls. This algorithm is based on a variation of the Traveling Salesman Problem. It offers an extra benefit as it automatically determines cluster boundaries. Using this algorithm, we optimally solve four benchmark problems and a 2,467-gene expression data clustering problem. As expected, our new algorithm identifies better clusters than those found by previous approaches in all five cases. Overall, our results demonstrate the benefits of rectifying the pitfalls and exemplify the usefulness of this clustering technique. Our code is available at our websites.

Abstract. In order to efficiently analyze the complicated regulatory systems often encountered in biological settings, abstraction is essential. This paper presents an automated abstraction methodology that systematically reduces the small-scale complexity found in genetic regulatory network models, while broadly preserving the large-scale system behavior. Our method first reduces the number of reactions by using rapid equilibrium and quasi-steady-state approximations as well as a number of other stoichiometry-simplifying techniques, which together result in substantially shortened simulation time. To further reduce analysis time, our method can represent the molecular state of the system by a set of scaled Boolean (or n-ary) discrete levels. This results in a chemical master equation that is approximated by a Markov chain with a much smaller state space providing significant analysis time acceleration and computability gains. 1

Microarrays are often used to identify target genes that trigger specific diseases, to elucidate the mechanisms of drug effects, and to check SNPs. However, data from microarray experiments are well known to contain biases resulting from the experimental protocols. Therefore, in order to elucidate biological knowledge from the data, systematic biases arising from their protocols must be removed prior to any data analysis. To remove these biases, many normalization methods are used by researchers. However, not all biases are eliminated from the microarray data because not all types of errors from experimental protocols are known. In this paper, we report an effective way of removing various types of biases by treating each microarray dataset independently to detect biases present in the dataset. After the biases contained in each dataset were identified, a combination of normalization methods specifically made for each dataset was applied to remove biases one at a time. Key words: cDNA microarray; normalization; print-order bias 1.

matrix decomposition methods. Abstract. High throughput technologies produce large biological datasets that may lead to greater understanding of the biological mechanisms behind diseases such as cancer. However, progress has been slow in extracting meaningful information from these datasets. We describe a method of clustering lists of genes mined from a microarray dataset using functional information from the Gene Ontology. The method uses relationships between terms in the ontology both to build clusters and to extract meaningful cluster descriptions. The approach is general and may be applied to assist explanation other datasets associated with ontologies. 1

Abstract. We explore the problem of designing oligonucleotides that help locate organisms along a known phylogenetic tree. We develop a suffix-tree based algorithm to find such short sequences efficiently. Our algorithm requires O(Nm) time and O(N) space in the worst case where m is the number of the genomes classified by the phylogeny and N is their total length. We implemented our algorithm and used it to find these discriminating sequences in both small and large phylogenies. We believe our algorithm will have wide applications including: high-throughput classification and identification, oligo array design optimally differentiating genes in gene families, and markers for closely related strains and populations. It will also have scientific significance as a new way to assess the confidence in a given classification. 1

understand the processes that underlie the regulatory networks and pathways controlling inter-cellular and intra-cellular activities. In recent times microarray datasets are extensively used for this purpose. The scope of such analysis has broadened in recent times towards reconstruction of gene networks and other holistic approaches of Systems Biology. Evolutionary methods are proving to be successful in such problems and a number of such methods have been proposed. However all these methods are based on processing of genotypic information. Towards this end, there is a need to develop evolutionary methods that address phenotypic interactions together with genotypic interactions. We present a novel evolutionary approach, called Phenomic algorithm, wherein the focus is on phenotypic interaction. We use the expression profiles of genes to model the interactions between them at the phenotypic level. We apply this algorithm to the yeast sporulation dataset and show that the algorithm can identify gene networks with relative ease. Keywords—Evolutionary computing, Gene expression analysis,

In the framework of gene expression data analysis, the selection of biologically relevant sets of genes and the discovery of new subclasses of diseases at bio-molecular level represent two significant problems. Unfortunately, in both cases the correct solution is usually unknown and the evaluation of the performance of gene selection and clustering methods is difficult and in many cases unfeasible. A natural approach to this complex issue consists in developing an artificial model for the generation of biologically plausible gene expression data, thus allowing to know in advance the set of relevant genes and the functional classes involved in the problem. In this work we propose a mathematical model, based on positive Boolean functions, for the generation of synthetic gene expression data. Despite its simplicity, this model is sufficiently rich to take account of the specific peculiarities of gene expression, including the biological variability, viewed as a sort of random source. As an applicative example, we also provide some data simulations and numerical experiments for the analysis of the performances of gene selection methods. Key words: Gene expression modeling, gene selection, gene expression data clustering, positive Boolean functions, DNA microarrays. 1

A key role for ontologies in bioinformatics is their use as a standardised, structured terminology, particularly to annotate the genes in a genome with functional and other properties. Since the output of many genome-scale experiments results in gene sets it is natural to ask if they share common function. A standard approach is to apply a statistical test for overrepresentation of ontological annotation, often within the Gene Ontology. In this paper we propose an alternative to the standard approach that avoids problems in over-representation analysis due to statistical dependencies between ontology categories. We use a feature construction approach to pre-process Gene Ontology annotation of gene sets and incorporate these features as input to a standard supervised machine learning algorithm. Our approach is shown to allow the straightforward use of an ontology in the context of data sourced from multiple experiments to learn a classifier predicting gene function as part of cellular response to an environmental stress. 1

understand the processes that underlie the regulatory networks and pathways controlling inter-cellular and intra-cellular activities. In recent times microarray datasets are extensively used for this purpose. The scope of such analysis has broadened in recent times towards reconstruction of gene networks and other holistic approaches of Systems Biology. Evolutionary methods are proving to be successful in such problems and a number of such methods have been proposed. However all these methods are based on processing of genotypic information. Towards this end, there is a need to develop evolutionary methods that address phenotypic interactions together with genotypic interactions. We present a novel evolutionary approach, called Phenomic algorithm, wherein the focus is on phenotypic interaction. We use the expression profiles of genes to model the interactions between them at the phenotypic level. We apply this algorithm to the yeast sporulation dataset and show that the algorithm can identify gene networks with relative ease. Keywords—Evolutionary computing, Gene expression analysis, Gene networks, Microarray data analysis, Phenomic algorithms. I.