Predictive simulation of systems comprised of numerous interconnected, tightly coupled components promises to help solve many problems of scientific and national interest. However predictive simulation of such systems is extremely challenging due to the coupling of a diverse set of physical and biological length and time scales. This report investigates un-certainty quantification methods for such systems that attempt to exploit their structure to gain computational efficiency. The traditional layering of uncertainty quantification around nonlinear solution processes is inverted to allow for heterogeneous uncertainty quantification methods to be applied to each component in a coupled system. Moreover this approach allows stochastic dimension reduction techniques to be applied at each coupling interface. The mathematical feasibility of these ideas is investigated in this report, and mathematical formulations for the resulting stochastically coupled nonlinear systems are developed.