Abstract
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.
Highlights
Many fundamental questions in biology are naturally treated in a probabilistic setting
Support for the efficacy of pairwise models has, necessarily, come from relatively small subsystems—small enough that the true probability distribution could be measured experimentally [7,8,9,11]. While these studies have provided a key first step, a critical question remains: will the results from the analysis of these small subsystems extrapolate to large ones? That is, if a pairwise model predicts the probability distribution for a subset of the elements in a system, will it predict the probability distribution for the whole system? Here we find that, for a biologically relevant class of systems, this question can be answered quantitatively and, importantly, generically—independent of many of the details of the biological system under consideration
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
Summary
Many fundamental questions in biology are naturally treated in a probabilistic setting. Deciphering the neural code requires knowledge of the probability of observing patterns of activity in response to stimuli [1]; determining which features of a protein are important for correct folding requires knowledge of the probability that a particular sequence of amino acids folds naturally [2,3]; and determining the patterns of foraging of animals and their social and individual behavior requires knowledge of the distribution of food and species over both space and time [4,5,6]. Are we faced with the problem of estimating probability distributions in high dimensional spaces, we must do this based on a small fraction of the neurons in the network
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