In this work, we combined automated measurements of bacterial growth with theoretical tools from statistical physics to explore how drug interactions accumulate as the number of drugs, N, in a combination increases. First, we developed a statistical model that associates drug interactions with correlations between random variables. By construction, then, purely N-drug interactions correspond to correlation functions, allowing us to exploit formal statistical methods for measuring the contributions of all K-body interactions (K ≤ N) to a given N- drug effect. Using this framework, we then experimentally showed that the bacterial responses to drug pairs are sufficient to predict the effects of larger drug combinations in both gram-negative bacteria (E.coli) and gram-positive bacteria (S. aureus). Furthermore, we have extended this approach to include multiple types of human cancer cells. Our findings have enormous practical applications, as they circumvent the exponential combinatorial explosion associated with empirically characterizing large drug combinations. Equally interesting, these results raise new theoretical questions. Cells respond to single or two-drug treatments in a manner that depends on the specific biological and biochemical intricacies of the cells and drugs in question. Remarkably, however, the way that pairwise interactions accumulate as additional drugs are added appears to obey a simple and universal quantitative relationship, raising the question of whether emergent statistical properties ultimately dominate in this regime.
Growth in response to multiple drugs can be predicted from the growth in response to those drugs singly and in pairs using maximum entropy
(A) Schematic axes showing that the growth responses of bacteria to pairs of drugs (g12, g23, g13) are used to predict the growth response to all three drugs (g123). We use the three-drug case as an example; but growth in response to any number (N) of drugs can be predicted, as long as we know all pairwise responses. (B) We estimate growth in the presence of drugs using nonlinear least squares fitting to optical density time series. For each drug i, we define a random variable Xi whose expectation value is equal to the growth gi. (C) We made predictions by first estimating the maximum entropy distribution, P, using growth rate data from cells exposed to single drugs and drug pairs. The distribution takes an exponential form parameterized by resilience coefficients (hi, blue circles) and drug-drug coupling coefficients (Jij, pink boxes) that characterize the single drug response and the response to pairs of drugs, respectively. The resilience and coupling coefficients are chosen to ensure the moments, <Xi> and <XiXj>, of Ppair match the two-drug growth rate data at each drug dosage. After determining the maximum entropy distribution, the N-drug growth response can be predicted by calculating the expectation values of the product X1X2...XN. We find that these expectation values are related to the moments <Xi> and <XiXj> by simple algebraic expressions.