Diatoms often dominate planktonic communities in the ocean and phototrophic biofilms in streams and rivers, greatly contributing to global biogeochemical fluxes. In pelagic ecosystems, these microscopic algae can form chain-like microcolonies, which seem advantageous for nutrient uptake and protect against grazing, and at the same time reduce sinking. Despite the capability of many diatoms to form chains, their contribution to the architecture of phototrophic biofilms remains elusive. Here we propose a computational model to simulate the growth and behaviour of Diatoma chains in contrasting flow environments. This mass-spring mechanical model captures the natural behaviour of Diatoma chains well, emphasising the relevance of chain growth and entanglement for biofilm morphogenesis. The model qualitatively describes formation of intricate dome-shaped structures and of dreadlock-type streamers as observed in nature in multidirectional and unidirectional flow, respectively. The proposed model is a useful tool to study the effect of fluid dynamics on biofilm morphogenesis.