The regulation of gene expression in eukaryotes: Bistability and oscillations in repressilator models Rui Dilão a,b,n a Nonlinear Dynamics Group, Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisbon, Portugal b Institut des Hautes Études Scientifiques, 35, route de Chartres, 91440 Bures-sur-Yvette, France HIGHLIGHTS  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. article info Article history: Received 26 March 2013 Received in revised form 6 September 2013 Accepted 10 September 2013 Available online 19 September 2013 Keywords: Regulation of gene expression in eukaryotes Delays m-repressilator Limit cycles Bistability abstract 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 ¼1, the m- repressilator is deducible from the mass action law and, in the limit n-1, it is a Boolean type model. With this embedding and with delays, if m is odd and n 41, 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 41, there is always a choice of parameters for which the m-repressilator model has bistability. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction In the cell, protein synthesis begins with the transcription of mRNA from its DNA template. Then, the mRNA is translated into protein in the ribosomes. This is the central dogma of molecular biology, as discussed in Crick (1967, 1970). This mechanism of protein synthesis is common to both prokaryote and eukaryote organisms. As prokaryote organisms lack a nuclear membrane, both DNA molecules and ribosomes are located in the cytoplasm. In eukar- yotes, the DNA molecules are in the nucleoplasm and the ribo- somes are in the cytoplasm. This different localisations of the DNA implies that, in eukaryotes, for protein translation to occur, the mRNA molecules have to migrate through the nucleoplasm, cross- ing the nuclear membrane. The transcription of mRNA is done by a catalytic enzymatic process involving the RNA polymerase enzyme that is continu- ously produced in the cell. In this process, the DNA is the catalyst. The RNA polymerase binds to the promoter regions of the DNA and translation begins. Some genes have control regions where tran- scription activators and repressors can bind, enabling the activa- tion or inhibition of the mRNA transcription, Alberts et al. (2008). At the molecular spatial scale and during short time scales, it is assumed that RNA polymerase molecules move randomly in the cytoplasm (prokaryotes) or in the nucleoplasm (eukaryotes). In models, it is implicitly assumed that the mRNA motion is intrinsically stochastic, Larson et al. (2009). A general mathematical framework to describe quantitatively the transcription, translation and protein synthesis in prokaryotes, based on the mass action law, has been proposed in Alves and Dilão (2005). This class of models is built with genes and proteins. The mass action law assumes that reaction kinetics result from the Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology 0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.09.010 n Correspondence address: Nonlinear Dynamics Group, Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisbon, Portugal. Tel.: þ351 218 417 617; fax: þ351 218 419 123. E-mail address: rui@sd.ist.utl.pt Journal of Theoretical Biology 340 (2014) 199–208