Noise characteristics of the generalized repressilator (14). (a) Detailed diagram of the reactions in the standard repressilator with three genes involving six chemical species, as implemented with our stochastic algorithm. In the simplified cartoon, each circle represents a gene repressing the subsequent gene in a cycle. The generalized repressilator studied here considers cycles with odd number of genes n = 3, 5, 7, 9. (b) The top panel shows time series of one of the proteins for the deterministic model of the repressilator with n = 3 (filled) and n = 9 (dashed) genes with parameters k
= 25, d
= 3, θ = 3, k
= 4 and α = 2. The lower panel shows the corresponding time series of the SSA started from stationarity, guaranteed by the DCFTP-SSA. For the top panel, the y-axis has units of protein concentration, whereas for the lower panel the y-axis has unitos of number of proteins. (c) The top panel shows the distribution of the period for the repressilator with n = 3 genes, while the bottom panel shows the same distribution for the generalized repressilator with n = 9 genes. Note that the distribution for n = 3 is skewed with a long right tail, while that of n = 9 is more symmetric, but has fatter tails than would be expected for a Gaussian distribution. The histograms were obtained from time-series with 104 periods. (d) The top two panels show the dependence of the mean (∘) and variance (□) of the period distribution with n. The lines indicate a linear fit for the means and a quadratic fit for the variances. The inset in the top right panel, shows that, for this set of parameters, the relative noise of the period, as measured by the coefficient of variation (*), is minimal for a length of n = 7 genes in the loop. The two lower panels show the skewness (∇) and kurtosis (◇) for the period distribution. The skewness decreases to zero as n grows, in accordance with the observed decrease of the asymmetry of the distribution. The kurtosis does not disappear as n grows indicating the presence of long-tails. Note that the kurtosis also reaches an apparent minimum at n = 7.