Realistic network-like systems are usually composed of multiple networks with interacting relations such as school-enterprise research and development collaboration networks. Here we study the percolation properties of a special kind of that R&D collaboration networks, namely institute-enterprise R&D collaboration networks. We introduce two actual IERDCNs to show their structural properties, and present a mathematical framework based on generating functions for analyzing an interacting network with any connection probability. Then we illustrate the percolation threshold and structural parameter arithmetic in the sub-critical and supercritical regimes. We compare the predictions of our mathematical framework and arithmetic to data for two real R&D collaboration networks and a number of simulations, and we find that they are in remarkable agreement with the data. We show applications of the framework to electronics R&D collaboration networks.