Structure theorems and the dynamics of nitrogen catabolite repression in yeast

Abstract

By using current biological understanding, a conceptually simple, but mathematically complex, model is proposed for the dynamics of the gene circuit responsible for regulating nitrogen catabolite repression (NCR) in yeast. A variety of mathematical "structure" theorems are described that allow one to determine the asymptotic dynamics of complicated systems under very weak hypotheses. It is shown that these theorems apply to several subcircuits of the full NCR circuit, most importantly to the URE2-GLN3 subcircuit that is independent of the other constituents but governs the switching behavior of the full NCR circuit under changes in nitrogen source. Under hypotheses that are fully consistent with biological data, it is proven that the dynamics of this subcircuit is simple periodic behavior in synchrony with the cell cycle. Although the current mathematical structure theorems do not apply to the full NCR circuit, extensive simulations suggest that the dynamics is constrained in much the same way as that of the URE2-GLN3 subcircuit. This finding leads to the proposal that mathematicians study genetic circuits to find new geometries for which structure theorems may exist.

Fig. 1.

The NCR circuit. Green arrows indicate up-regulation; blunted red arrows represent down-regulation at the level of transcription. Dashed lines indicate a weaker response. Blue lines indicate repression that is not at the transcriptional level. A subset of the connections are numbered, and these are referred to in the text. Arrow 10 indicates up-regulation of DEH1 by GAT1.

Fig. 2.

mRNA expression from simulation of System 8 and from experiment with yeast strain BY4743, after a switch from good to poor nitrogen. Blue is Dal80, black is GAT1, red is GLN3, green is URE2, and yellow is DEH1. The time axis is in minutes. These data compare favorably; the key features are the relatively constant GLN3 levels, fast rise in the DAL80, and the crossing of the DAL80 and GAT1 curves in the first few minutes after the switch.