The Predictor

The predictor is a method for inferring Boolean networks using the expression data given by the matrix E. We seek a Boolean function f_n independently for each node a_n. To this end, we first pick the input variables to f_n: we determine a minimum set s_n of nodes, whose levels must be input to f_n, in order for s_n to explain the observed data E. Then, we construct a truth table using these nodes as inputs. Specifically, the function for node a_n is determined according to the following procedure:

1.

Build sets S_ij of nodes with different values in rows i and j
Consider all pairs of rows (i, j) in E in which the expression level of a_n differs, excluding rows in which a_n was itself forced to a high or low value. For each such pair, find the set S_ij of all other nodes whose expression level also differs between the two rows (i, j). Because the network is self-contained, a change in at least one of these genes or stimuli must have caused the corresponding difference in a_n. Therefore, at least one node in this set must be included as a variable in f_n.

2.

Find a minimum cover set S_min of $\{S_{ij}\}$
Identify the smallest set of nodes S_min required to explain the observed differences over all pairs of rows (i, j), i.e., S_min is such that at least one of its nodes is present in each set S_ij. This task is a classic combinatorial problem called minimum set covering which can be solved by the branch and bound technique. More than one smallest set S_min may be found, in which case a distinct function f_n is inferred and reported for each such set.

3.

Determine truth table of a_n from S_min and E
Once S_min has been determined for the node a_n, a truth table is determined for f_n in terms of the levels of genes and/or stimuli in S_min by taking relevant levels directly from E. If all combinations of input levels are not present in E, the corresponding output level for gene a_n cannot be determined and is represented by the symbol "*" in the truth table.

If a node has more than one minimum cover set, several networks are inferred, each with a distinct function corresponding to each set. If several such nodes exist, a separate network hypothesis is returned for each combination of functions at each node. The minimum set cover ensures that only the most parsimonious networks will be returned.