The question of how nanoparticles tethered with single stranded DNA self-assemble in solution is studied in a coarse-grained model which incorporates the molecular recognition properties of nucleic acids. It is a lattice model, on a high-coordination cubic lattice, and enhanced sampling is obtained through Monte Carlo simulations with configurational biased moves combined with parallel tempering and multicanonical methods. The energetic and structural mechanisms of self-assembling, particularly those driving the formation of multiple bridges of double-stranded DNA, are investigated by calculating the potential of mean force between two tethered nanoparticles. This direct interaction potential, and its dependence on interparticle distance, is systematically studied as function of a number of system parameters and assembling architectures. The temperature dependence of this potential is shown to be directly correlated to bulk self-assembling transition temperatures. In addition, the relationship of these results and those from experiment is also established.