220 DETERMINATION OF COMPLEX REACTION MECHANISMS

is a marginal conditional probability, which averages the probability of the data over all possible parameters. Applied perturbations, such as mutations, deletions, and overexpression, require special considerations.

The calculations proceed as described, one stipulated Bayesian network after another, until an optimized score is attained. The optimized network, or the set of equivalent such networks, is then investigated for direct interactions of two, or more, species at a time to find so-called Markov neighbors; for separators, that is a species Z that separates species X and Y, and for activators and inhibitors. Such features are sought in all networks and a posterior probability of each feature, given the data, is established.

In [13] the authors chose microarray experiments on 565 genes from Saccharomyces cerevisiae to analyze by these methods. A great deal of useful information is obtained, of which we show two subnetworks inferred in some detail (fig. 13.9). The analysis can discover intracluster structure among correlated genes and intercluster connections between weakly correlated genes. The gene connections shown are the goal of such studies and are extremely useful for making predictions, for suggesting experiments, and ultimately for the treatment of genetic diseases. Are the subnetworks in fig. 13.9 sparse? That depends on an operational definition of sparse, yet to be formulated.

The Bayesian network approach is promising but there are limitations; we have mentioned already the restriction to acyclic graphs (no loops in the network) and the issue of Markovian independence. There are others: the presence of causal connectivity, as discussed in chapters 5 and 6, can be implied, can be indicated, but can not be determined. Finally, there is no rational basis, as yet, for connecting a Bayesian network with a chemical, biological, or genetic reaction mechanism: the equivalents of the concepts of temporal dynamics of reaction mechanisms, of rate coefficients, and of reversibility of elementary reactions are missing from Bayesian networks.

Following the unraveling of the connectivity of a genetic network, the next step is the experimental and theoretical study of the organization of the genome into regulatory modules [14]. A regulatory model is a given set of genes regulated together by a shared regulation program. It is assumed that the regulators are also transcriptionally regulated, and that their expression levels indicate their activity. The approach in this work is again based in part on Bayesian graphical methods, but goes further in ascertaining the presence and structure of modules; the computational requirements are substantial. We give here only an indication of the success of this approach by reproducing one of the figures from [14], fig. 13.10, with the sufficiently descriptive caption.

13.7Some Other Illustrative Approaches

In [15] there is a “systematic determination of genetic architecture” based on a combination of clustering and statistical algorithms.

The responses to systematic perturbations of a metabolic network (galactose utilization) have been measured with microarrays and methods of quantitative proteomics. Nearly a thousand mRNA were identified in twenty systematic perturbations, which led to the identification of the physical interactions governing the cellular response to each perturbation [16].

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An investigation of transcriptional regulatory networks in Sacch. cerevisiae is reported in [17]. The authors have determined the association of most of the transcriptional regulators with genes across the genome. They use statistical analysis of the measurements to show the presence of network motifs, and then demonstrate that an automated process can use these motifs to assemble a transcriptional regulatory network. For three review articles in this field, see [18–20].

We have encountered a number of difficult conceptual and practical problems in this chapter and throughout the entire book, and yet there are additional complications not treated. In a biological cell the concentrations of the various species may not be large, but many of the molecules (enzymes) may be large in size. Therefore molecular “crowding” [21] may occur; because of this effect, mass action kinetics does not apply and a form of fractal kinetics applies [22]. These problems have not yet been considered in the interpretation of most experimental investigations.

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