Statistical Properties and Robustness of Biological Controller-Target Networks

et al. (2012) Statistical Properties and Robustness of Biological Controller-Target Networks. PLoS ONE 7(1): e29374. doi:10.1371/journal.pone.0029374 Statistical Properties and Robustness of Biological Controller-Target Networks Jacob D. Feala 0 Jorge Cortes 0 Phillip M. Duxbury 0 Andrew D. McCulloch 0 Carlo Piermarocchi 0 Giovanni Paternostro 0 Vladimir Brezina, Mount Sinai School of Medicine, United States of America 0 1 Sanford-Burnham Medical Research Institute, La Jolla, California, United States of America, 2 Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, California, United States of America, 3 Department of Physics and Astronomy, Michigan State University , East Lansing, Michigan , United States of America, 4 Department of Bioengineering, University of California San Diego , La Jolla, California , United States of America Cells are regulated by networks of controllers having many targets, and targets affected by many controllers, in a ''many-tomany'' control structure. Here we study several of these bipartite (two-layer) networks. We analyze both naturally occurring biological networks (composed of transcription factors controlling genes, microRNAs controlling mRNA transcripts, and protein kinases controlling protein substrates) and a drug-target network composed of kinase inhibitors and of their kinase targets. Certain statistical properties of these biological bipartite structures seem universal across systems and species, suggesting the existence of common control strategies in biology. The number of controllers is ,8% of targets and the density of links is 2.5%61.2%. Links per node are predominantly exponentially distributed. We explain the conservation of the mean number of incoming links per target using a mathematical model of control networks, which also indicates that the ''many-to-many'' structure of biological control has properties of efficient robustness. The drug-target network has many statistical properties similar to the biological networks and we show that drug-target networks with biomimetic features can be obtained. These findings suggest a completely new approach to pharmacological control of biological systems. Molecular tools, such as kinase inhibitors, are now available to test if therapeutic combinations may benefit from being designed with biomimetic properties, such as ''many-to-many'' targeting, very wide coverage of the target set, and redundancy of incoming links per target. - Funding: This work was supported by National Science Foundation grant 0829891 and National Institutes of Health (NIH) grant R21AG030685. Partial support was also received from NIH grants P41-RR08605 and P01-HL098053. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. Control of cellular function depends on bipartite (two-layer) networks, in which one class of nodes (the controller) acts on the other class (the target) to regulate its function. Examples of cellular control networks include transcription factors, microRNAs, and protein kinases, which control genes, mRNA transcripts, and protein substrates, respectively. In these networks, the control layer interacts with the target layer in a combinatorial, many-to-many fashion (see Figure 1). In other words, each controller has many targets, the targets themselves are under the influence of many controlling molecules, and the target sets of different controllers overlap. Moreover, the number of controllers is usually significantly lower than the number of targets. This many-to-many structure is well recognized in biological systems [1], not only in intracellular control but also in many other types of complex control in biology, including the nervous system (see Text S1, section S1.1). The idea of a many-to-many bipartite control structure is similar to the concept of dense overlapping regulon (DOR) [2] in bacterial gene networks, which indicates a motif (i.e. a pattern that recurs within a network), in which transcription factors and genes are connected through many overlapping interactions. Here we extend this concept to different biological structures and describe the many-to-many property as a feature of entire control networks, for different types of control molecules, contrasting it with the other possible bipartite structures, such as one-to-one and one-tomany, described in Figure 1. One important question concerns the statistical properties of these control structures with strong overlap and redundancy. It was shown [2] that dense overlapping regulons deviate substantially from random networks. Here we explicitly characterize the global statistical properties of several bipartite control structures, and we show that the degree distribution of the two types of nodes is well approximated by exponentials. A key issue related to network topology is robustness. What are the advantages of the many-to-many structure in terms of robustness, and why, as we show here, do some parameters of the networks seem to be universal across different control structures and species? In order to explore the link between the network properties and robustness we introduce a simplified Boolean signaling model. Boolean network models of biological regulation were first pioneered by Kaufmann [3] [4], and have been used to model specific interactions in small, well-characterized biological pathways [5,6,7]. The control problem i.e. calculating the specific input sequence required to achieve a desired output has also been explored within these systems [8,9]. None of these Figure 1. Possible combinatorial control strategies. There are several qualitatively different structures for control networks of M controllers (x1,x2,xM) and N targets (y1,y2,yN). In the one-to-one case (left panel), M = N. doi:10.1371/journal.pone.0029374.g001 models explicitly considered bipartite structures, i.e. networks with two classes of nodes in which there are no links between nodes of the same class. While there have been many genome-wide network analyses [10,11,12,13,14,15], and one recent work on coregulation of transcription and phosphorylation networks [16], here we focus exclusively on universal features of bipartite networks, neglecting the fact that some of the targets might also act in turn as controllers on other downstream biological entities or on other targets. This simplified approach captures some peculiar and universal properties of control in biology that may help design biomimetic drug-target control strategies. Naturally occurring biological control networks share statistical properties We examine quantitative characteristics of three biological control systems in three different species (human, yeast, and E. coli), from the perspective of bipartite combinatorial control. First we consider the numbers of nodes. Table 1 (...truncated)