Robustness of synthetic oscillators in growing and dividing cells

Synthetic biology sets out to implement new functions in cells, and to develop a deeper understanding of biological design principles. Elowitz and Leibler [Nature (London) 403, 335 (2000)] showed that by rational design of the reaction network, and using existing biological components, they could create a network that exhibits periodic gene expression, dubbed the repressilator. More recently, Stricker et al. [Nature (London) 456, 516 (2008)] presented another synthetic oscillator, called the dual-feedback oscillator, which is more stable. Detailed studies have been carried out to determine how the stability of these oscillators is affected by the intrinsic noise of the interactions between the components and the stochastic expression of their genes. However, as all biological oscillators reside in growing and dividing cells, an important question is how these oscillators are perturbed by the cell cycle. In previous work we showed that the periodic doubling of the gene copy numbers due to DNA replication can couple not only natural, circadian oscillators to the cell cycle [Paijmans et al., Proc. Natl. Acad. Sci. (USA) 113, 4063 (2016)], but also these synthetic oscillators. Here we expand this study. We find that the strength of the locking between oscillators depends not only on the positions of the genes on the chromosome, but also on the noise in the timing of gene replication: noise tends to weaken the coupling. Yet, even in the limit of high levels of noise in the replication times of the genes, both synthetic oscillators show clear signatures of locking to the cell cycle. This work enhances our understanding of the design of robust biological oscillators inside growing and diving cells.

Figure 1

Models for the synthetic oscillators and cell cycle. (a) Network architecture of the repressilator [1]: $P_1$ represses the production of $P_2$, $P_2$ represses $P_3$, and $P_3$ represses $P_1$ again. (b) Illustration of a circular chromosome, with the origin (Ori) and termination (Ter) of replication. When the three genes $p_1$, $p_2$, and $p_3$ are placed at a distance on the chromosome, there are temporal delays $\mathrm{\Delta }{t}_{1,2}^{}$ and $\mathrm{\Delta }{t}_{2,3}^{}$ between when the genes are replicated. (c) Gene copy numbers (top) and gene densities (bottom) of the genes $p_1$ (red, solid), $p_2$ (blue, dashed), and $p_3$ (orange, dotted), respectively. They are replicated at times $t_1$, $t_2$, and $t_3$, respectively, as indicated by the vertical lines. The thick gray vertical lines indicate cell divisions. For the gene copy number, lines are shifted vertically for clarity. (Bottom) Gene densities for each gene, normalized by their average. (d) Network architecture of the dual-feedback oscillator [13]: The activator ($A$) autoactivates its production and enhances the production of the repressor ($R$). The repressor autorepresses its production and suppresses the production of the activator. (e) Schematic of the circular chromosome. The genes for the activator ($a$) and repressor ($r$) are placed at different positions on the DNA, such that there is a temporal delay, $\mathrm{\Delta }t{}_{a,r}$, between their respective replication times. (f) Gene copy numbers (top) and gene densities (bottom) of the genes $a$ (green, solid) and $r$ (red, dashed), respectively. Genes $a$ and $r$ are replicated at times $t_a$ and $t_r$, respectively, indicated by the vertical lines. The thick gray vertical lines indicate cell divisions. For the gene copy number, lines are shifted vertically for clarity.

Figure 2

Models to determine the gene replication times. (a) Time trace of cell volume, which is a deterministic function of time in all models, where cell division occurs with a period $T_d$, indicated by the vertical solid gray lines. The vertical dashed gray lines indicate the times at which DNA replication is initiated when the timing of the initiation of replication is deterministic. The horizontal dashed lines show the volume (a) or the concentration of the origin of replication (b)–(d), at which DNA replication, on average, initiates. (b)–(d) Time traces of the density of the origin of replication of the chromosome (red solid line), a gene precisely half-way between the origin and terminus of replication (green dotted line) and the terminus of replication (blue dashed line). Arrows below the $x$ axis indicate the replication times of these sites. Note that the gene densities show no discontinuity at cell division. All gene densities are normalized by the critical density for replication initiation. (b) Fully deterministic model. Initiation of replication and the replication of the two genes occur at fixed times each cell cycle. (c) When there is stochasticity in DNA replication progression, the timing between initiation of replication and the replication of genes further along the DNA becomes stochastic. (d) When the initiation of replication is stochastic, but the replication rate is constant, all replication events move in concert, and the time between initiation and replication of the genes is fixed. (e) Probability density of gene replication times $P\left(\delta {\tau }_{i,j}\right)$. A tentative replication time $\delta {\tau }_{i,j}^{\prime }$ drawn from a Gaussian distribution (gray thick line) could lie outside the domain $[0,2\mathrm{\Delta }{t}_{i,j}]$ (dotted vertical lines). In this case, $\delta {\tau }_{i,j}^{\prime }$ is mapped back onto this domain by mirroring the value across the nearest domain boundary (black dashed lines). After the mapping, the replication times follow a flatter distribution (red solid line).

Figure 3

The repressilator [1] can strongly lock to the cell cycle, and the strength of locking depends sensitively on how the genes are positioned on the DNA. (a) Average (solid line) and standard deviation (shaded region) of the peak-to-peak time $T_{\mathrm{PtP}}$ as a function of the division time, where the time between replicating genes is $\mathrm{\Delta }{t}_{1,2}=\mathrm{\Delta }{t}_{2,3}=0$. The repressilator has an intrinsic period of $T_{\mathrm{int}}=125$. The locking regions around $T_{\mathrm{int}}$ and $2T_{\mathrm{int}}$ are almost absent. (b) and (c) Representative time traces of the concentrations of the three repressilator proteins, $p_1\left(t\right)$ (red, solid), $p_2\left(t\right)$ (blue, dashed), and $p_3\left(t\right)$ (orange, dotted), for the cell-division times indicated by the arrows in panel (a). (b) When $T_d=T_{\mathrm{int}}$, the oscillations are very regular (almost no variance in the PtP-times), but each protein concentration has a different amplitude. (c) At $T_d=2T_{\mathrm{int}}$, all three protein concentrations switch between a small and a large amplitude in successive oscillation cycles. Panels (d) ($\mathrm{\Delta }{t}_{1,2}>0$) and (e) ($\mathrm{\Delta }{t}_{1,2}<0$) show the effect of varying the timing of replication of the three genes, assuming $\mathrm{\Delta }{t}_{2,3}=\mathrm{\Delta }{t}_{1,2}$. We show results for four values of $\mathrm{\Delta }{t}_{1,2}$, given in the legend, and expressed as a fraction of the mean DNA replication time $T_{\mathrm{rep}}$. In panels (d) and (e), from top to bottom, the value of $\mathrm{\Delta }{t}_{1,2}$ decreases. For clarity, we only show the average peak-to-peak time as a function of $T_d$, not the standard deviation. Remarkably, for all $\mathrm{\Delta }{t}_{1,2}\ne 0$, there is significant locking. Clearly, the timing of gene replication can markedly affect locking, which means that the spatial distribution of the genes over the chromosome can be of critical importance in the interaction between the clock and the cell cycle. Parameters used in all simulations of the repressilator (from [1]): ${\alpha }_0=2.16\times {10}^{-3},\alpha =2.16,\beta =0.2,\gamma =20$, and $n=2$. (Figure adapted from [26].)

Figure 4

In the repressilator, locking persists in the presence of physiological levels of noise in the gene replication times. In all panels the solid lines show the peak-to-peak time in the activator concentration for different periods of the cell division time $T_d$. Standard deviation in $T_{\mathrm{PtP}}$ omitted for clarity, but it is similar in all panels. The amplitudes of the functions decrease as the time interval $\mathrm{\Delta }{t}_{1,2}$ decreases, as given in the legend. We used the physiologically motivated values for the standard deviations in the timing of the initiation, ${\sigma }_{\mathrm{init}}^{}=0.2T_d$, and the progression, ${\sigma }_{\mathrm{rep}}^{}=0.35T_{\mathrm{rep}}$, of DNA replication. The two panels show a different order of the genes $p_1$, $p_2$, and $p_3$ with $\mathrm{\Delta }{t}_{1,2}>0$ (a) and $\mathrm{\Delta }{t}_{1,2}<0$ (b). Clearly, at these noise levels, locking is strongly reduced compared to the deterministic case [see Figs. 3 and 3], but it is still clearly observable around $T_d=T_{\mathrm{int}}$.

Figure 5

The dual-feedback oscillator [13] can strongly lock to the cell cycle, and the strength of locking depends on the temporal order in which the genes are replicated during the cell cycle. The intrinsic period of the oscillator $T_{\mathrm{int}}=73$ min. (a) Average (solid line) and standard deviation (shaded region) of the peak-to-peak time $T_{\mathrm{PtP}}$ as a function of the division time $T_d$  when both genes are replicated simultaneously, $\mathrm{\Delta }{t}_{a,r}=0$. There is a wide region of cell division times (around $T_d=T_{\mathrm{int}}$) where the oscillator has a $T_{\mathrm{PtP}}$ equal to the cell cycle (left dashed line). (b) and (c) Representative time traces for the division times indicated by the arrows in panel (a). Shown are the activator and repressor concentrations $a\left(t\right)$ (green, dashed) and $r\left(t\right)$ (red, solid), respectively. At a cell-division time of $T_d=98$ min (b), just outside the region where the oscillator is locked to the cell cycle, the time traces show very irregular behavior resulting in a large variance in the PtP times. At $T_d=2T_{\mathrm{int}}$ (c), the oscillations switch between a small and a large amplitude in successive oscillation cycles, a signature of the periodic gene replications. (d) and (e) The effect of the order of gene replication during the cell cycle. For clarity, only the average peak-to-peak time as a function of $T_d$ is shown, not the standard deviation. Values of $\mathrm{\Delta }{t}_{\mathrm{a},\mathrm{r}}$ are given in the legend, and they are written as a fraction of the DNA replication time $T_{\mathrm{rep}}$. The amplitudes of the functions increase as the time interval $\mathrm{\Delta }{t}_{a,r}$ decreases. (d) Positive $\mathrm{\Delta }{t}_{a,r}$; the repressor gene is replicated after the activator gene. (e) Negative $\mathrm{\Delta }{t}_{a,r}$; the repressor gene is replicated before the activator gene. Remarkably, contrary to the behavior of the repressilator, locking decreases with increasing time delay between replicating genes $\mathrm{\Delta }{t}_{a,r}$. This illustrates that the influence of the cell cycle on the clock depends in a nontrivial way on the architecture of the clock and on the nature of the driving signal. Parameters used in all simulations of the dual-feedback oscillator (from [13]): $b_a=b_r=0.36{\min }^{-1}$, $\alpha =20$, $k_{-a}=k_{-r}=1.8{\min }^{-1}$, $t_a=t_r=90{\min }^{-1}$, $d_a=d_r=0.54{\min }^{-1}$, $k_{fa}=k_{fr}=0.9{\min }^{-1}$, $k_{da}=k_{dr}=k_t=0.018{\min }^{-1}{\mathrm{mol}}^{-1}$, $k_{-da}=k_{-dr}=k_{-t}=0.00018{\min }^{-1}$, $k_l=0.36{\min }^{-1}$, $k_{ul}=0.18{\min }^{-1}$, $\gamma =1080\mathrm{mol}/\min$, $c_e=0.1\mathrm{mol}$, $\gamma =2.5$, $\epsilon =0.2$, $k_a=0.059{\min }^{-1}$, and $k_r=0.018{\min }^{-1}$. For the full set of equations describing the model, see [33]. The rate constants $k_a$ and $k_r$ set the rate at which activators and repressors bind to the promoter, respectively. In the original experiment, these rates can be tuned by the concentration of the inducers arabinose and IPTG. For the value of the rate constants given, we used $\left[\text{IPTG}\right]=2\text{nM}$ and $\left[\text{ara}\right]=0.7\%$. (Figure adapted from [26].)

Figure 6

In the dual-feedback oscillator, locking persists in the presence of physiological levels of noise in the gene replication times. In all panels, solid lines show the peak-to-peak time in the activator concentration for different periods of the cell division time $T_d$. Standard deviation in $T_{\mathrm{PtP}}$ omitted for clarity, but it is similar in all panels. The amplitudes of the functions are similar for different time intervals $\mathrm{\Delta }{t}_{a,r}$. Legends are defined in Fig. 5. We used the physiologically motivated values for the standard deviations in the timing of the initiation, ${\sigma }_{\mathrm{init}}^{}=0.2T_d$, and the progression, ${\sigma }_{\mathrm{rep}}^{}=0.35T_{\mathrm{rep}}$, of DNA replication. The two panels show a different order of the activator and repressor gene with $\mathrm{\Delta }{t}_{a,r}>0$ (a) and $\mathrm{\Delta }{t}_{a,r}<0$ (b). As observed for the repressilator, locking is strongly reduced compared to the deterministic case [see Figs. 5 and 5], but it is still clearly observable around $T_d=T_{\mathrm{int}}$.

Figure 7

In the repressilator, stochasticity in the initiation of DNA replication plays the dominant role in attenuating the effects of gene replications. The top row, panels (a) and (b), shows results with only noise in the progression of DNA replication, ${\sigma }_{\mathrm{rep}}^{}=0.35T_{\mathrm{rep}}$, and the bottom row, panels (c) and (d), corresponds to the situation in which there is only noise in replication initiation, ${\sigma }_{\mathrm{init}}^{}=0.2T_d$ (see Fig. 2). In both panels, solid lines show the peak-to-peak time $T_{\mathrm{PtP}}$ in the oscillations of $P_1$ as a function of the cell division time $T_d$. Standard deviation in $T_{\mathrm{PtP}}$ omitted for clarity, but it is similar in all panels. Legends are defined in Fig. 3. The amplitudes of the functions decrease as the time interval $\mathrm{\Delta }{t}_{1,2}$ decreases. (a),(b) When there is noise in the time intervals between the gene replication events, but the initiation of DNA replication is fixed, locking seems little affected compared to the deterministic case [see Figs. 3 and 3]. (c),(d) When there is noise in the initiation of DNA replication, but the time between replications is fixed, the effects of the cell cycle almost disappear for division times $T_d>T_{\mathrm{int}}$, in both ways of ordering the genes. However, strong locking persists at the 1:1 locking region and for $T_d<T_{\mathrm{int}}$. Comparing with panels (a) and (b), noise in the initiation of DNA replication seems to be more effective in protecting the clock against the cell cycle.

Figure 8

In the dual-feedback oscillator, stochasticity in the initiation of DNA replication plays the dominant role in attenuating the effects of gene replications. In both panels, solid lines show the peak-to-peak time in the activator concentration for different periods of the cell division time $T_d$. Standard deviation in $T_{\mathrm{PtP}}$ omitted for clarity, but it is similar in all panels. Legends are defined in Fig. 5. The amplitudes of the functions decrease as the time interval $\mathrm{\Delta }{t}_{1,2}$ increases. We compare a scenario with only noise in DNA replication progression, with a standard deviation ${\sigma }_{\mathrm{rep}}^{}=0.35T_{\mathrm{rep}}$, panels (a) and (b), to a scenario with only noise in the initiation of DNA replication, with a standard deviation ${\sigma }_{\mathrm{init}}^{}=0.2T_d$, panels (c) and (d). (a),(b) As observed for the repressilator, with noise in replication progression but not in replication initiation, locking is little affected, compared to the deterministic case [see Figs. 5 and 5]. (c),(d) In the opposite scenario, with noise in the initiation of DNA replication but not in the progression of replication, most signatures of locking disappear both when the activator or repressor gene is replicated first. Only around $T_d=T_{\mathrm{int}}$ does locking persist. Clearly, comparing with panels (a) and (b), noise in the initiation of DNA replication has a stronger attenuating effect on locking.