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What is a good (gene) network?

We are fashion victims. Networking is fashionable so we decided last year that it was time to start to work on networks. In the last years, biology has evolved to understand how the relationships between a large number of elements (genes, proteins, ...) can influence the way a living organism functions. This question is well modelled by the use of biological networks that are a huge topic of interest in recent literature. As an example, the Leipzig WCGALP included several articles about biological networks (e.g. Tesson et al.; Jager et al.; Kadarmideen et al.; Reverter et al.). One of those was our work (Liaubet et al.) which, at the end of the talk, gave rise to THE question: ‘What is a good network?’ This innocent and apparently simple question has haunted us ever since. The answer is complex, much too complex for a short editorial, but we can draw some evidence from our experience. When working with biological networks, we are dealing with many different underlying questions: gene networks, protein networks or when speaking about the kind of relationships that they model, transcriptomic networks, regulation networks and interaction networks. We may consider the particular case of a gene co-expression network based on high throughput transcriptomic data. Usually, in this field, very limited prior biological knowledge is available as well as a frustrating annotation level (usually, in that kind of experiment in livestock species, about half of the genes have no functional or ontological annotation). With that restricted background, the use of a network model can help to improve the knowledge about the way genes interact and to emphasize key genes implicated in a given process. In Leipzig, we admired Trudy Mackay’s talk because the Drosophila model is so powerful. However, network inference with that kind of data has to be handled with care. For example, our first attempt was to build co-expression networks of differential genes (genes whose expression varies according to a phenotype of interest) for a developmental trait, between species. The results were very disappointing because networks were unstable, had similar structure, whatever the kind of genes considered (differential or chosen at random), and no pertinent biological conclusion could be drawn. This was the perfect example of a bad network! In that first experiment, the main problem was the too low number (about five) of observations available for each species. However, it was the starting point to understand the key features needed to obtain a good network. Now note that the question ‘What is a good network?’ can be divided into two sub-questions: What is a good network for biologists? What is a good network for statisticians? Statisticians like robustness. The number of observations used to define the network is never large enough. A simple simulation study can illustrate the fact that at least 20–30 observations are needed to accurately estimate a correlation coefficient, and even more to infer a medium size network (Schäfer and Strimmer, 2005a, Bioinformatics 21, 754–764). Furthermore, methods designed to deal with ‘small’ sample sizes and a large number of variables are required: When using the common Graphical Gaussian Models, this can consist of regularization (Dobra et al., 2004, J. Multiv. Anal. 90, 196–212; Mainshausen and Bühlmann, 2006, Ann. Stat. 34, 3), shrinkage (Schäfer and Strimmer, 2005b, Stat. Appl. Genet. Mol. Biol. 4, 32), bootstrap (Schäfer and Strimmer, 2005a), etc. But once the strength of the relations between two genes has been inferred by the chosen method, the subsequent question is: What are the important interactions? No other processing could be performed (e.g., keeping the complete network with edges weighted by the partial correlations), but use of ad hoc or significant thresholds can be beneficial for the readability and further interpretability of the network. Biologists like simple outputs to understand the data and recognize biological processes and molecular pathways. Moreover, they are on a quest for the ‘Grail of causality’, with the help of statistical models (Schadt et al., 2005, Nat. Genet. 37, 710–717; the Leipzig WCGALP papers by Rosa et al.; Tesson et al.; Mackay), or based on the properties of a gene network. In a good network, one should be able to emphasize the key genes in the biological process: For instance, hubs [nodes (genes) that are connected with a large number of genes] are straightforward candidates for being interesting genes but they can sometimes be disappointing because they are much J. Anim. Breed. Genet. ISSN 0931-2668