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This paper uses a Kinetic Monte Carlo model that includes processes to integrate curvature due to the formation of five- and seven-member rings to simulate polycyclic aromatic hydrocarbons (PAHs) growing in lightly sooting ethylene and acetylene counterflow diffusion flames. The model includes new processes to form seven-member rings via hydrogen-abstraction-acetylene-addition and bay closure reactions on sites containing partially embedded five-member rings. The model additionally includes bay closure and HACA bay capping reactions for the integration of five-member rings. The mass spectra of PAHs predicted by the model are assessed against experimental data, and the distribution of embedded five-member rings and seven-member rings is studied as a function of spatial location, molecule size and frequency of events sampled in the simulation. The simulations show that the formation of seven-member rings and the embedding of five-member rings is a competitive process. Both types of rings are observed more frequently as the simulation proceeds from the fuel outlet towards the stagnation plane. Approximately 15% of the events that integrate curvature resulted in the formation of a seven-member ring coupled to an embedded five-member ring, and the remaining 85% of events embedded five-member rings via the formation of six-member rings. The proportion of PAHs containing embedded five-member rings and/or seven-member rings is observed to be a function of PAH size, passing through a maximum for PAHs containing 15–20 six-member rings. However, the proportion of PAHs containing both types of ring increases with PAH size, where upwards of 10% of PAHs containing at least one five-member ring and 15 or more six-member rings also contain a seven-member ring. [ABSTRACT FROM AUTHOR]