Figure 1
Dependence of Protein Noise on k d . Dependence of fano factor (variance of number of protein molecules per cell divided by mean number of protein molecules per cell) and coefficient of variation (standard deviation divided by mean) on k d of the DNA binding site, for physiological values of k d ranging between 0.01 nM and 100 nM . In all panels, k p = 0.1s-1, γ m = 5 × 10-3s-1 and γ p = 2 × 10-4s-1. In the top two panels (a) and (b), k d is varied, and the model is controlled by holding all other parameters fixed. In the bottom two panels (c) and (d), k d is varied, and the model is controlled to keep a constant protein abundance of 100 molecules per cell by also varying k m . In the left-hand panels (a) and (c), the fano factor is plotted as a noise measure; in the right-hand panels (b) and (d), the coefficient of variation is plotted. In (a) and (b) k m = 0.1s-1. In (c) and (d), k m is varied along with k d so that mean protein abundance is held constant. Each of the data points is the measure of noise from a stochastic simulation of the model. The black lines show the respective noise measure as derived by our mathematical analysis; the blue lines, where distinguishable from the black lines, show the noise measure as derived by Thattai and van Oudenaarden; the red lines show the noise measure for the equivalent unregulated model. In all panels it can be seen that (i) our noise measures are very close to the expression derived by Thattai and Van Oudenaarden; (ii) the simulations match the noise level for weak values of k d greater than 1 nM ; (iii) for strong values of k d less than 1 nM, the level of noise increases with repressor strength, and is very much greater than predicted by the linearized QSS model. (a), (c) and (d) all show that the noise level is predicted to be lower in the regulated system than the equivalent unregulated system. However, the stochastic simulations demonstrate that for strong values of k d , the noise of the regulated system can be greater than that of the unregulated system. In (b), the red line would appear to indicate that the unregulated system is consistently less noisy than the regulated system. However, in this panel, because all parameters are held constant, the protein abundance is much higher than in the unregulated system than the regulated system, and because of the Poisson-like nature of the noise (variance proportional to mean), the coefficient of variation is necessarily lower. In (d) it can be seen that when the unregulated system is adjusted so that protein levels are the same, a consistent pattern of behaviour is observed.