Periodic enzyme synthesis and oscillatory repression: why is the period of oscillation close to the cell cycle time?

During exponential growth of a cell culture, some enzymes are synthesized periodically. In a synchronous culture, in which all cells undergo DNA synthesis and division more-or-less synchronously, the burst of enzyme synthesis also occurs synchronously in each cell once per division cycle. However, there are a number of interesting cases in which periodic enzyme synthesis continues in the absence of synchronous DNA replication or cell division. In all cases of periodic enzyme synthesis in asynchronous cultures, the time between bursts of enzyme synthesis, though no longer identical to the cell cycle time, is still close to the interdivision time of the growing, replicating cells. The theory of oscillatory repression looks for an explanation of this phenomenon in the periodic repression of gene transcription caused by periodic fluctuations in the concentration of the endproduct of the metabolic pathway of which the enzyme is a part. A major difficulty with this theory is that there is no obvious relationship between the periodicity of the negative feedback loop, which is determined by the kinetics of synthesis and degradation of the individual components of the feedback loop, and the periodicity of the cell cycle, which is determined by overall net synthetic rates of cellular macromolecules. Why should the period of oscillation of a repressible gene transcription system be close to the interdivision time of a population of growing cells? In this paper, I show that the relationship may be coincidental: the two fundamental periods are close to each other because they are both close to the mass-doubling time of the cell culture. That the mean interdivision time must be close to the mass-doubling time is a consequence of "balanced" growth: there is a stable size distribution of cells in a growing culture. That the period of oscillation of the negative feedback loop is also close to the mass-doubling time is shown to be a consequence of the large, nearly constant demand for endproduct and the assumed stability of the enzyme. The period of oscillation is largely attributable to the slow dilution of the stable enzyme by cell growth. For reasonable values of the parameters describing the gene-control system, I show that the enzyme must be diluted by a factor of two (approximately), that is, by the growth accomplished by one mass-doubling (nearly).