Ribosome self-assembly leads to overlapping reproduction cycles and increases growth rate

In permissive environments, E. coli can double its dry mass every 21 minutes. During this time, ribosomes, RNA polymerases, and the proteome are all doubled. Yet, the question of how to relate bacterial doubling time to other biologically relevant time scales in the growth process remains illusive, due to the complex temporal nesting pattern of these processes. In particular, the relation between the cell's doubling time and the ribosome assembly time is not known. Here we develop a model that connects growth rate to ribosome assembly time and show that the existence of a self-assembly step increases the overall growth rate, because during ribosome self-assembly existing ribosomes can start a new round of reproduction, by making a new batch of ribosomal proteins prior to the completion of the previous round. This overlapping of ribosome reproduction cycles increases growth rate beyond the serial-limit that is typically assumed to hold. Using recent data from ribosome profiling and well known measurements of the average translation rate, rigid bounds on the in-vivo ribosome self-assembly time are set, which are robust to the assumptions regarding the biological noises involved. At 21 minutes doubling time, the ribosome assembly time is found to be approximately 6 minutes --- three fold larger than the common estimate. We further use our model to explain the detrimental effect of a recently discovered ribosome assembly inhibitor drug, and predict the effect of limiting the expression of ribosome assembly chaperons on the overall growth rate.