The regulation of gene expression in eukaryotes: Bistability and oscillations in repressilator models

• We use delays with the mass action law to model protein synthesis in eukaryotes.
• We have derived the properties of m-repressilator models, with and without delays.
• If m is even, the m-repressilator model shows bistability.
• If m is odd, the m-repressilator model has limit cycle solutions.
• The 1-repressilator is the simplest genetic mechanisms with stable oscillations.

To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n=1, the m  -repressilator is deducible from the mass action law and, in the limit n→∞n→∞, it is a Boolean type model. With this embedding and with delays, if m   is odd and n>1n>1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m   is even and n>1n>1, there is always a choice of parameters for which the m-repressilator model has bistability.