Ubiquitous monotone and almost-monotone Boolean functions in systems biology

Systems Biology is quickly becoming a major focus of applied and computational mathematics research. While traditionally numerics and continuous modeling techniques have been used, in recent years it has become apparent that qualitative modeling, data quantization and data mining questions play a central role for the field. Monotone Boolean functions are ubiquitous in these models, and, surprisingly, often both the function and its dual simultaneously convey practically useful information: Consider the hypergraph defined by the metabolites interacting in a metabolic network, expressed as the stoichiometric matrix. Then the socalled elementary modes are nothing but the extreme rays of the rational polyhedral cone given by the matrix, and their support patterns form a hypergraph. Every transversal of this hypergraph is a so called cut set, and the set of all (minimal) cut sets is of central interest when analyzing failure modes of the system under consideration (Haus, Klamt, & Stephen 2008). Many other biological phenomena can also be modeled using hypergraphs, or would profit from the combinatorial richness that simple graph approximations cannot provide (Klamt, Haus, & Theis 2009). Monotone functions also arise from non-hypergraph models: In the context of ancestral genome reconstruction one question is whether a matrix whose columns are genomic markers and whose rows list, for various species, which genomic markers are present. The hypothesis is that in a hypothetical ancestor these markers were consecutive. Therefore one is interested in deciding whether the columns of the matrix can be reordered to make nonzero entries in all rows consecutive. This property, known as consecutive-ones-property (C1P) can be checked in polynomial time(Fulkerson & Gross 1965; Tucker 1972), and is (up-)monotone under taking submatrices. If the matrix fails to be C1P, one needs to find out which rows are the cause: The minimal conflicting sets of rows are determined by the dual monotone function (Chauve et al. 2009). Finally, some applications give rise to boolean func-