Figure 1

A gene promoter driven by constant and oscillating signals, $s(t)$. We compare the variances of $X$ for the two input signals, ${\sigma }_{X,c}^2$ and ${\sigma }_{X,o}^2$, at the same mean level $⟨X⟩$.Reuse & Permissions

Figure 2

(a) Relative noise for oscillating and constant signals, ${log}_2({\sigma }_{X,o}^2/{\sigma }_{X,c}^2)$, as the protein degradation rate $\mu$ and promoter deactivation rate $\lambda$ are varied. In red (blue) regions the oscillatory (constant) signal leads to lower noise. Other parameters are: $\tau =25\min$, $T=100\min$, $\kappa =1{\min }^{-1}$, $\rho =50{\min }^{-1}$. The dotted line indicates $T^{-1}$. (b),(c) Typical time-series of $X(t)$ for the parameter combinations denoted in (a) by ⋄ (b: $\lambda =0.2{\min }^{-1}$, $\mu =0.05{\min }^{-1}$) and ○ (c:$\lambda =0.2{\min }^{-1}$ $\mu =0.00167{\min }^{-1}$). Unless otherwise noted, parameters as in (c) are used in panels (d)–(g). (d) The promoter power spectrum $S_P(\omega )$ and gain $g^2(\omega )/{\rho }^2=[{\omega }^2+{\mu }^2{]}^{-1}$. For both inputs there is a noise background due to randomness in promoter switching. Peaks appear in $S_{P,o}(\omega )$ at $\omega =n{\omega }_T$ due to the periodicity of the oscillating input signal. (e) Distribution of the fraction of time the promoter is active in an interval ${\tau }_X$, calculated from stochastic simulations. The oscillatory signal leads to more reproducible promoter activity on the time scale ${\tau }_X$. (f) Variance against mean output level as $T$ is varied with $\tau$ held constant. The oscillating signal can achieve a lower variance over nearly the full range of output levels. In (f) and (g), lines show exact analytic results, points show results of stochastic simulations. (g) Fano factor ${\sigma }_X^2/⟨X⟩$ as $\kappa$ is varied with constant $K=\lambda /\kappa =0.2$. For slow switching $\kappa \tau \lesssim 1$ the two signals become equivalent. For extremely fast switching the constant signal minimizes ${\sigma }_X^2$. At intermediate switching rates, $0.1{\min }^{-1}\lesssim \kappa \lesssim 18{\min }^{-1}$, the oscillating signal allows for smaller ${\sigma }_X^2$.Reuse & Permissions

Figure 3

Effects of noise in (a) oscillation amplitude, and (b) the duration of on and off periods. Solid lines show results for constant (blue) and deterministic oscillatory (red) stimuli. Even with significant variability in amplitude or timing, the oscillating signal leads to a smaller ${\sigma }_X^2$ than a constant signal. Parameters are as in Fig. 2f.Reuse & Permissions