Inferring (Biological) Signal Transduction Networks via Transitive Reductions of Directed Graphs

Abstract In this paper we consider the p-ary transitive reduction (TRp) problem where p>0 is an integer; for p=2 this problem arises in inferring a sparsest possible (biological) signal transduction network consistent with a set of experimental observations with a goal to minimize false positive inferences even if risking false negatives. Special cases of TRp have been investigated before in different contexts; the best previous results are as follows: (1) The minimum equivalent digraph problem, that correspond to a special case of TR1 with no critical edges, is known to be MAX-SNP-hard, admits a polynomial time algorithm with an approximation ratio of 1.617+ε for any constant ε>0 (Chiu and Liu in Sci. Sin. 4:1396–1400, 1965) and can be solved in linear time for directed acyclic graphs (Aho et al. in SIAM J. Comput. 1(2):131–137, 1972). (2) A 2-approximation algorithm exists for TR1 (Frederickson and JàJà in SIAM J. Comput. 10(2):270–283, 1981; Khuller et al. in 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 937–938, 1999). In this paper, our contributions are as follows: • We observe that TRp, for any integer p>0, can be solved in linear time for directed acyclic graphs using the ideas in Aho et al. (SIAM J. Comput. 1(2):131–137, 1972). • We provide a 1.78-approximation for TR1 that improves the 2-approximation mentioned in (2) above. • We provide a 2+o(1)-approximation for TRp on general graphs for any fixed prime p>1.

[1]  S. L. Wong,et al.  A Map of the Interactome Network of the Metazoan C. elegans , 2004, Science.

[2]  Lan V. Zhang,et al.  Evidence for dynamically organized modularity in the yeast protein–protein interaction network , 2004, Nature.

[3]  Nicola J. Rinaldi,et al.  Transcriptional Regulatory Networks in Saccharomyces cerevisiae , 2002, Science.

[4]  Sarah M Assmann,et al.  The Arabidopsis Putative G Protein–Coupled Receptor GCR1 Interacts with the G Protein α Subunit GPA1 and Regulates Abscisic Acid Signaling , 2004, The Plant Cell Online.

[5]  E D Sontag,et al.  Some new directions in control theory inspired by systems biology. , 2004, Systems biology.

[6]  James R. Knight,et al.  A Protein Interaction Map of Drosophila melanogaster , 2003, Science.

[7]  R. Albert,et al.  Predicting Essential Components of Signal Transduction Networks: A Dynamic Model of Guard Cell Abscisic Acid Signaling , 2006, PLoS biology.

[8]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[9]  Joseph JáJá,et al.  Approximation Algorithms for Several Graph Augmentation Problems , 1981, SIAM J. Comput..

[10]  R. Heinrich,et al.  The Regulation of Cellular Systems , 1996, Springer US.

[11]  Robert E. Tarjan,et al.  Finding optimum branchings , 1977, Networks.

[12]  Alain Vavasseur,et al.  Arabidopsis OST1 Protein Kinase Mediates the Regulation of Stomatal Aperture by Abscisic Acid and Acts Upstream of Reactive Oxygen Species Production Article, publication date, and citation information can be found at www.plantcell.org/cgi/doi/10.1105/tpc.007906. , 2002, The Plant Cell Online.

[13]  Radhika Desikan,et al.  A new role for an old enzyme: Nitrate reductase-mediated nitric oxide generation is required for abscisic acid-induced stomatal closure in Arabidopsis thaliana , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Eduardo D. Sontag,et al.  Algorithmic and complexity results for decompositions of biological networks into monotone subsystems , 2007, Biosyst..

[15]  D. S. Johnson.  On Strongly Connected Digraphs with Bounded Cycle Length , 1996 .

[16]  Alfred V. Aho,et al.  The Transitive Reduction of a Directed Graph , 1972, SIAM J. Comput..

[17]  Samir Khuller,et al.  A uniform framework for approximating weighted connectivity problems , 1999, SODA '99.

[18]  J. Davies,et al.  Molecular Biology of the Cell , 1983, Bristol Medico-Chirurgical Journal.

[19]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[20]  Samir Khuller,et al.  Approximating the minimum equivalent digraph , 1994, SODA '94.