Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops

Interlinking a positive feedback loop (PFL) with a negative feedback loop (NFL) constitutes a typical motif in genetic networks, performing various functions in cell signaling. How time delay in feedback regulation affects the dynamics of such systems still remains unclear. Here, we investigate three systems of interlinked PFL and NFL with time delays: a synthetic genetic oscillator, a three-node circuit, and a simplified single-node model. The stability of steady states and the routes to oscillation in the single-node model are analyzed in detail. The amplitude and period of oscillations vary with a pointwise periodicity over a range of time delay. Larger-amplitude oscillations can be induced when the PFL has an appropriately long delay, in comparison with the PFL with no delay or short delay; this conclusion holds true for all the three systems. We unravel the underlying mechanism for the above effects via analytical derivation under a limiting condition. We also develop a stochastic algorithm for simulating a single reaction with two delays and show that robust oscillations can be maintained by the PFL with a properly long delay in the single-node system. This work presents an effective method for constructing robust large-amplitude oscillators and interprets why similar circuit architectures are engaged in timekeeping systems such as circadian clocks.

Figure 1

Schematic of three models and the model dynamics. (a) The synthetic genetic circuit where the activator (araC), repressor (lacI), and reporter (yemGFP) are under the control of the same hybrid promoters ${\mathrm{p}}_{\mathrm{lac}/\mathrm{ara}-1}$ (in two plasmids). ${\tau }_{\mathrm{a}}$ and ${\tau }_{\mathrm{r}}$ separately denote the time delays in production of araC and lacI proteins. (b) Time courses of the concentration of $\mathrm{X}$ for ${\tau }_{\mathrm{r}}=0.3$ and ${\tau }_{\mathrm{a}}=0$ (blue), 0.3 (green), or 1.175 (red). (c) A three-node model. The transcription factor $\mathrm{X}$ induces the production of ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_2$. ${\mathrm{Y}}_1$ (${\mathrm{Y}}_2$) activates (represses) the production of $\mathrm{X}$. ${\tau }_{\mathrm{X}}, {\tau }_{{\mathrm{Y}}_1}$, and ${\tau }_{{\mathrm{Y}}_2}$ separately denote the delays in production of $\mathrm{X}, {\mathrm{Y}}_1$, and ${\mathrm{Y}}_2$. (d) Time courses of $x$ for ${\tau }_{{\mathrm{Y}}_2}=1$ and ${\tau }_{{\mathrm{Y}}_1}=0$ (blue), 10 (green), or 99 (red). (e) A simplified model. $\mathrm{X}$ regulates the production of itself via positive and negative feedback. ${\tau }_{\mathrm{p}}$ and ${\tau }_{\mathrm{n}}$ denote the corresponding delays. [(f)–(h)] Time courses of $x$. Two feedback loops are coupled multiplicatively (f), additively (g), or competitively (h). ${\tau }_{\mathrm{n}}=1.8$ and ${\tau }_{\mathrm{M}}$ denotes the minimal ${\tau }_{\mathrm{p}}$ at which the largest amplitude is induced; ${\tau }_{\mathrm{M}}=6.4$ (f), 5.025 (g), or 4.6 (h).

Figure 2

Effects of time delays on Hopf bifurcation and oscillation amplitude. [(a)–(c)] Dependence of the stability of steady states on ${\tau }_{\mathrm{p}}$ and ${\tau }_{\mathrm{n}}$. The red and black curves denote Hopf bifurcation curves determined by Eqs. (15) and (16), respectively. The curve marked by $(j,k,l)$ is composed of the points $({\tau }_n^{j,k,l},{\tau }_p^{j,k,l})$ with $i=0,1$ and $j,k=0,1,2,\cdots$. The white and blue regions denote stable and unstable steady states, respectively. The gray dashed vertical line separates stable from unstable steady states for the NFL-only system. The circles and arrows denote the pointwise periodicity in delays for Hopf bifurcation. The magenta dashed curves in (b) denote the boundary of oscillatory behavior. [(d)–(f)] Dependence of oscillation amplitude on ${\tau }_{\mathrm{n}}$ and ${\tau }_{\mathrm{p}}$. The amplitude is represented by the heat map. The circles and arrows refer to the delays for the same oscillation with period $T$. $m_{\mathrm{p}}=4, m_{\mathrm{n}}=2, {\alpha }_{\mathrm{n}}=0.01, K=2$ and ${\tau }_{\mathrm{n}}=3$; ${\alpha }_{\mathrm{p}}=1.35$ and $\eta =5.738$ for $G_{\mathrm{p}}+G_{\mathrm{n}}<-1$ [(a) and (d)], ${\alpha }_{\mathrm{p}}=1.57$ and $\eta =5.376$ for $G_{\mathrm{p}}+G_{\mathrm{n}}=-1$ [(b) and (e)], ${\alpha }_{\mathrm{p}}=10$ and $\eta =1.573$ for $-1<G_{\mathrm{p}}+G_{\mathrm{n}}<1$ [(c) and (f)], or ${\alpha }_{\mathrm{p}}=1$ and $\eta =6.427$ for the NFL-only system.

Figure 3

Effects of time delays on the amplitude and period of oscillation. (a) Dependence of the amplitude on ${\tau }_{\mathrm{n}}$. Black, red, and blue curves correspond to the NFL-only system and the coupled systems with ${\tau }_{\mathrm{p}}=0$ and with ${\tau }_{\mathrm{p}}={\tau }_{\mathrm{n}}$, respectively. (b) Dependence of the amplitude on ${\tau }_{\mathrm{p}}$ at ${\tau }_{\mathrm{n}}=3$. The dashed horizontal line denotes the amplitude for the NFL-only system. The dashed and dotted vertical lines separately correspond to ${\tau }_{\mathrm{p}}=7.22$ and 8.02. The other parameter values are the same as in Fig. 2. [(c) and (d)] Dependence of the period (c) and amplitude (d) on ${\tau }_{\mathrm{p}}$ in a wide range [the same parameters as in panel (b)]. The solid and dashed curves denote the stable and unstable limit cycles, respectively. Red and black dots denote the periodic solutions with the period of $T_0=8.02$ and $T_1=7.50$, respectively.

Figure 4

Large-amplitude oscillations induced by the PFL with a long delay. [(a) and (b)] Statistic distributions for the minimal ${\tau }_{\mathrm{p}}$ at which the maximal amplitude is acquired relative to ${\tau }_n$. 1054 and 1030 sets of parameter values are sampled for $K>1$ (a) and $K<1$ (b), respectively. [(c)–(f)] Effects of ${\tau }_{\mathrm{p}}$ on the oscillation amplitude in the simplified model with additive (c) or competitive (d) coupling, in the synthetic oscillator (e), and in the three-node model (f). The solid and dashed curves denote the stable and unstable limit cycles, respectively. The insets show the amplitude near ${\tau }_{\mathrm{a}}=0$ (e) or ${\tau }_{{\mathrm{Y}}_1}=0$ (f). See Sec. 2 for parameter values.

Figure 5

Dynamics of the simplified model with $m_{\mathrm{p}},m_{\mathrm{n}}\to +\infty$. (a) Dependence of the amplitude on ${\tau }_{\mathrm{p}}$. The blue and red curves denote the analytical and numerical results, respectively. The dashed horizontal line marks the amplitude for ${\tau }_{\mathrm{p}}=0$. The dot-dashed, dashed, and dotted vertical lines correspond to ${\tau }_{\mathrm{p}}=0.8$, 7.9, and 8.152, respectively. [(b)–(d)] Upper panels: The black, magenta, and blue curves denote $x\left(t\right), x_{t-{\tau }_{\mathrm{p}}}$, and $x_{t-{\tau }_{\mathrm{n}}}$, respectively. Lower panels: $f^{*}\equiv f\left(x_{t-{\tau }_{\mathrm{p}}}\right)$ and $g^{*}\equiv g\left(x_{t-{\tau }_{\mathrm{n}}}\right)$. Parameter values are as follows: $K=5$; (b) ${\tau }_{\mathrm{p}}=0$ and $T=T_0=8.152$; (c) ${\tau }_{\mathrm{p}}=0.8$ and $T=8.034$; (d) ${\tau }_{\mathrm{p}}=7.9$ and $T=8.203$. The other parameter values are the same as in Fig. 2.

Figure 6

Role for the strong PFL in modulating oscillatory dynamics. [(a) and (c)] Dependence of the oscillation amplitude and its CV on ${\tau }_{\mathrm{p}}$. The dashed horizontal line marks the amplitude for ${\tau }_{\mathrm{p}}=0$. [(b) and (d)] The oscillation period and its CV versus ${\tau }_{\mathrm{p}}$. The blue solid and dashed curves in [(a) and (b)] denote the stable and unstable limit cycles, respectively, in the deterministic case. Black dots at ${\tau }_{\mathrm{p}}=6.245$ and 7.019 denote the fold bifurcation of limit cycle. The circles and bars denote the mean and the standard deviation, respectively, in the stochastic case. Each stochastic simulation is run over [0,20000]. The amplitude is rescaled by $K_{\mathrm{n}}\mathrm{\Omega }=10$. $m_{\mathrm{p}}=4, m_{\mathrm{n}}=4, {\alpha }_{\mathrm{p}}=10, {\alpha }_{\mathrm{n}}=0.02, K=3, \eta =1.767$, and ${\tau }_{\mathrm{n}}=1.8$.

Figure 7

Stochastic simulation of the simplified model. Sample trajectories of $x$ for ${\tau }_{\mathrm{p}}=0$ (a), 0.5 (b), or 6 (c). The amplitude is rescaled by $K_{\mathrm{n}}\mathrm{\Omega }=10$. The other parameters are the same as in Fig. 6.