Mixtures of endocrine disruptors: how similar must mechanisms be for concentration addition to apply?

Must molecular mechanisms be the same for concentration addition to predict the effect of a mixture or is it sufficient for compounds to affect similar pathways or common outcomes? Does concentration addition provide a closer approximation to observations than alternative models such as independent action? Suppose effects are mediated by hormone A through receptor R, synthesis of A is reduced by compound B, and that C is a competitive antagonist to A. Both B and C reduce levels of the receptor-ligand complex AR but via different specific mechanisms. Are combinations of B and C concentration additive? We used simple pharmacodynamic models, deriving mathematical models using equilibrium binding and mass balance. Assume A binds the receptor at one site of R with effects proportional to the concentration of AR. Let C act as a competitive antagonist via the Gaddum equation. Let B affect synthesis of A via a function g(B). We derive a model describing the joint response surface of B and C, and a function describing its isoboles. Under concentration addition, the isoboles must be negatively sloped straight lines. We show that linearity of the isoboles depends crucially on g(B). The mixture is concentration additive if g″(B), the second derivative of g(B) with respect to B, is always zero. Responses are greater than concentration additive if g″(B) is always positive and less than concentration additive if g″(B) is always negative. We describe functions g(B) that lead to all three cases as well as one that is greater than concentration additive in some regions and less than concentration additive in others. At least in this simple model, concentration addition cannot be assumed: mixtures of competitive antagonists and compounds that alter hormone synthesis can lead to results that are concentration additive, greater than concentration additive or less than concentration additive. Nevertheless, concentration addition appears to provide a closer approximation to the pharmacodynamic model examined here than independent action. Care needs to be taken in extrapolating to other situations, but analysis of simple pharmacodynamic models appears to be a useful strategy.