  One of the most critical problems we face in the study of biological systems
is building accurate statistical descriptions of them. This problem has been
particularly challenging because biological systems typically contain large
numbers of interacting elements, which precludes the use of standard brute
force approaches. Recently, though, several groups have reported that there may
be an alternate strategy. The reports show that reliable statistical models can
be built without knowledge of all the interactions in a system; instead,
pairwise interactions can suffice. These findings, however, are based on the
analysis of small subsystems. Here we ask whether the observations will
generalize to systems of realistic size, that is, whether pairwise models will
provide reliable descriptions of true biological systems. Our results show
that, in most cases, they will not. The reason is that there is a crossover in
the predictive power of pairwise models: If the size of the subsystem is below
the crossover point, then the results have no predictive power for large
systems. If the size is above the crossover point, the results do have
predictive power. This work thus provides a general framework for determining
the extent to which pairwise models can be used to predict the behavior of
whole biological systems. Applied to neural data, the size of most systems
studied so far is below the crossover point.
