Pairwise Maximum Entropy Models for Studying Large Biological Systems: When They Can Work and When They Can't

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, then the results may 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 large biological systems. Applied to neural data, the size of most systems studied so far is below the crossover point. Author Summary: Biological systems are exceedingly complicated: They consist of a large number of elements, those elements interact in nonlinear and highly unpredictable ways, and collective interactions typically play a critical role. It would seem surprising, then, that one could build a quantitative description of biological systems based only on knowledge of how pairs of elements interact. Yet, that is what a number of studies have found. Those studies, however, focused on relatively small systems. Here, we ask the question: Do their conclusions extend to large systems? We show that the answer depends on the size of the system relative to a crossover point: Below the crossover point the results on the small system have no predictive power for large systems; above the crossover point the results on the small system may have predictive power. Moreover, the crossover point can be computed analytically. This work thus provides a general framework for determining the extent to which pairwise models can be used to predict the behavior of large biological systems. It also provides a useful heuristic for designing experiments: If one is interested in understanding truly large systems via pairwise interactions, then make sure that the system one studies is above the crossover point.