Predictive simulation of systems comprised of numerous interconnected, tightly coupled com-ponents promises to help solve many problems of scientific and national interest. Howeverpredictive simulation of such systems is extremely challenging due to the coupling of adiverse 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 togain computational efficiency. The traditional layering of uncertainty quantification aroundnonlinear solution processes is inverted to allow for heterogeneous uncertainty quantificationmethods to be applied to each component in a coupled system. Moreover this approachallows stochastic dimension reduction techniques to be applied at each coupling interface.The mathematical feasibility of these ideas is investigated in this report, and mathematicalformulations for the resulting stochastically coupled nonlinear systems are developed.3