Constructing a Knowledge Base for Gene Regulatory Dynamics by Formal Concept Analysis Methods

Constructing a Knowledge Base for Gene Regulatory Dynamics by Formal Concept Analysis Methods Johannes Wollbold12, Reinhard Guthke2, and Bernhard Ganter1 1 University of Technology, Institute of Algebra, Dresden, Germany http://www.math.tu-dresden.de/alg/algebra.html jwollbold@gmx.de 2 Leibniz Institute for Natural Product Research and Infection Biology - Hans-Kn¨oll-Institute (HKI) Jena, Germany Abstract. Our aim is to build a set of rules, such that reasoning over temporal dependencies within gene regulatory networks is possible. The underlying transitions may be obtained by discretizing observed time se- ries, or they are generated based on existing knowledge, e.g. by Boolean networks or their nondeterministic generalization. We use the mathemat- ical discipline of formal concept analysis (FCA), which has been applied successfully in domains as knowledge representation, data mining or soft- ware engineering. By the attribute exploration algorithm, an expert or a supporting computer program is enabled to decide about the validity of a minimal set of implications and thus to construct a sound and com- plete knowledge base. From this all valid implications are derivable that relate to the selected properties of a set of genes. We present results of our method for the initiation of sporulation in Bacillus subtilis. However the formal structures are exhibited in a most general manner. Therefore the approach may be adapted to signal transduction or metabolic net- works, as well as to discrete temporal transitions in many biological and nonbiological areas. Keywords: complete lattices, reasoning, temporal logic, gene expression 1 Introduction As the mathematical methodology of formal concept analysis (FCA) is little known within systems biology, we give a short overview of its history and pur- poses. During the early years 1980, FCA emerged within the community of set and order theorists, algebraists and discrete mathematicians. Its first aim was to find a new, concrete and meaningful approach to the understanding of complete lattices (ordered sets such that for every subset the supremum and the infimum exist). The following discovery showed to be very fruitful: Every complete lat- tice is representable as a hierarchy of concepts, which were conceived as sets of objects sharing a maximal set of attributes. This paved the way for using the developed field of lattice theory for a transparent and complete representation of very different types of knowledge. FCA was inspired by the pedagogue Hartmut arXiv:0807.3287v1 [q-bio.MN] 21 Jul 2008 2 von Hentig [7] and his program of restructuring sciences, with a view to interdis- ciplinary collaboration and democratic control. The philosophical background goes back to Charles S. Peirce (1839 - 1914), who condensed some of his main ideas to the pragmatic maxim: Consider what effects, that might conceivably have practical bear- ings, we conceive the objects of our conception to have. Then, our conception of these effects is the whole of our conception of the object. [14, 5.402] In that tradition, FCA aims at unfolding the observable, elementary proper- ties defining the objects subsumed by scientific concepts. If applied to temporal transitions, effects of homogeneous classes of states can be modeled and pre- dicted in a clear and concise manner. Thus FCA seems to be appropriate to describe causality - and the limits of its understanding. At present, FCA is a richly developed mathematical theory, and there are practical applications in various fields as data and text mining, knowledge man- agement, semantic web, software engineering or economics [3]. FCA has been used for the analysis of gene expression data in [2] and [13], but this is the first approach of applying it to model (gene) regulatory networks. The math- ematical framework of FCA is very general and open, such that multifarious refinements are possible, according to current approaches of modeling dynamics within systems biology. On the other hand, we developed a formal structure for general discrete temporal transitions. They occur in a variety of domains: control of engineering processes, development of the values of variables or objects in a computer program, change of interactions in social networks, a piece of music, etc. In this paper, however, the examples are uniquely biological. The purpose is to construct a knowledge base for reasoning about temporal dependencies within gene regulatory or signal transduction networks, by the attribute exploration algorithm: For a given set of interesting properties, it builds a sound, complete and nonredundant knowledge base. This minimal set of rules has to be checked by an expert or a computer program, e.g. by comparison of knowledge based predictions with data. Since there exist relatively fixed thresholds of activation for many genes, it is a common abstraction to consider only two expression levels offand on. The classical approach of Boolean networks [8] is able to capture ess

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