Modeling the Hes1 oscillator.

Somitogenesis describes the segmentation of vertebrate embryonic bodies, which is thought to be induced by ultradian clocks (i.e., clocks with relatively short cycles compared to circadian clocks). One candidate for such a clock is the bHLH factor Hes1, forming dimers which repress the transcription of its own encoding gene. Most models for such small autoregulative networks are based on delay equations where a Hill function represents the regulation of transcription. The aim of the present paper is to estimate the Hill coefficient in the switch of an Hes1 oscillator and to suggest a more detailed model of the autoregulative network. The promoter of Hes1 consists of three to four binding sites for Hes1 dimers. Using the sparse data from literature, we find, in contrast to other statements in literature, that there is not much evidence for synergistic binding in the regulatory region of Hes1, and that the Hill coefficient is about three. As a model for the negative feedback loop, we use a Goodwin system and find sustained oscillations for systems with a large enough number of linear differential equations. By a suitable variation of the number of equations, we provide a rational lower bound for the Hill coefficient for such a system. Our results suggest that there exist additional nonlinear processes outside of the regulatory region of Hes1.