Figure 9. Non-random timing of off-phases increases the cyclicity of transcriptional cycles.

Histograms showing the distribution of (A) on-times, (B) off-times, (E) off-times greater than 3 h with the refractory period, and (H) off-times greater than 3 h without the refractory period, estimated from the Markov Chain Monte Carlo algorithm (Text S1 Section 3.4). The superimposed black lines show the fit of an exponential distribution with the same mean value as the data. (C, F, and I) The off-phases in the system are not memoryless. The probability of having to wait for t hours in the off state given that the off-time has already lasted for s hours is plotted for a range of values of t for the distributions in (B, E, and H), respectively. The dashed lines represent the exponential probabilities, and the solid lines are the sample probability estimates. The uppermost lines are calculated when t = 0, the lines beneath that are calculated for t = 0.5, and so on in increments of 0.5. In a memoryless system such as that described by the telegraph process this is independent of s (hence s is constant for a given value of t), but for our system this probability decreases significantly with s during the refractory period (F). The decrease of this probability for higher values of s is due to the finite length of our time-series. (D, G, and J) Autocorrelation functions for a number of mRNA time-series simulated using on and off durations selected at random from the distributions above (B, E, and H, respectively). The variance of the time of the first peak (which estimates period) is given in each plot. In (E) the refractory period is indicated by RF.