Negative feedback and oscillations in a model for mRNA translation

The ribosome flow model (RFM) is a phenomenological model for the unidirectional flow of particles along a one-dimensional chain of [Formula: see text] sites. The RFM has been extensively used to study the dynamics of ribosome flow along a single-mRNA molecule during translation. In this case, the particles model ribosomes and each site corresponds to a consecutive group of codons. Networks of interconnected RFMs have been used to model and analyse large-scale translation in the cell and, in particular, the effects of competition for shared resources. Here, we analyse the RFM with a negative feedback connection from the protein production rate to the initiation rate. This model is based on, for example, the production of proteins that inhibit the translation of their own mRNA. The RFM with negative feedback is a 2-cooperative dynamic system, i.e. its flow maps the set of vectors with up to one-sign variation to itself. Using tools from the theory of 2-cooperative dynamical systems, we provide a simple condition guaranteeing that the closed-loop system admits at least one non-trivial periodic solution. When this condition holds, we also explicitly characterize a large set of initial conditions such that any solution emanating from this set converges to a non-trivial periodic solution. Such a solution corresponds to a periodic pattern of ribosome densities along the mRNA and to a periodic pattern of protein production.