Discrete gene replication events drive coupling between the cell cycle and circadian clocks

Significance

Huygens famously showed that two mechanically connected clocks tend to tick in synchrony. We uncovered a generic mechanism that can similarly phase-lock two rhythmic systems present in many living cells: the cell cycle and the circadian clock. DNA replication during the cell cycle causes protein synthesis rates to show sharp, periodic jumps that can entrain the clock. To faithfully keep time in the face of these disturbances, circadian clocks must incorporate specific insulating mechanisms. We argue that, in cyanobacteria, the presence of multiple, identical chromosome copies and the clock’s core protein-modification oscillator together play this role. Our results shed new light on the complex factors that constrain the design of biological clocks.

Abstract

Many organisms possess both a cell cycle to control DNA replication and a circadian clock to anticipate changes between day and night. In some cases, these two rhythmic systems are known to be coupled by specific, cross-regulatory interactions. Here, we use mathematical modeling to show that, additionally, the cell cycle generically influences circadian clocks in a nonspecific fashion: The regular, discrete jumps in gene-copy number arising from DNA replication during the cell cycle cause a periodic driving of the circadian clock, which can dramatically alter its behavior and impair its function. A clock built on negative transcriptional feedback either phase-locks to the cell cycle, so that the clock period tracks the cell division time, or exhibits erratic behavior. We argue that the cyanobacterium Synechococcus elongatus has evolved two features that protect its clock from such disturbances, both of which are needed to fully insulate it from the cell cycle and give it its observed robustness: a phosphorylation-based protein modification oscillator, together with its accompanying push–pull read-out circuit that responds primarily to the ratios of different phosphoform concentrations, makes the clock less susceptible to perturbations in protein synthesis; the presence of multiple, asynchronously replicating copies of the same chromosome diminishes the effect of replicating any single copy of a gene.

Circadian clocks—autonomous oscillators with a roughly 24-h period that can be entrained to daily cycles of light and dark—are thought to confer important advantages on living cells by allowing them to anticipate diurnal environmental changes. Recent decades have seen considerable progress in elucidating both the architecture and the function of these biological timekeepers. Circadian clocks, however, are not the only oscillatory systems present in living cells. Most notably, cell growth and division are governed by a cell cycle, which can in many contexts be viewed as an autonomous oscillator. Much recent attention has been directed toward the connections between these two rhythmic systems, which are relevant for processes ranging from plants’ response to shade (1) to cancer susceptibility (2, 3). In particular, it is now clear that circadian clocks can exert specific regulatory influences on the cell cycle, and a number of experimental and modeling studies have sought to tease out the implications of this regulation (4–11). Here, we argue that, in addition to direct, specific regulation of one oscillator by the other, there must also be more generic connections between the circadian clock and the cell cycle (2, 10–12). In particular, we focus on the consequences of the discrete gene replication events that accompany DNA replication. We show that, as a result of the regular jumps in gene copy number caused by these events, the cell cycle must, very generally, contribute a periodic forcing to the circadian clock. This forcing can markedly change clock behavior and degrade clock function. We propose that cyanobacterial clocks have evolved specific features that can mitigate this effect. More broadly, this generically strong coupling to the cell cycle implies important constraints on the design of biological timekeepers if they are to remain accurate in dividing cells.

It is widely accepted that protein levels depend on a cell’s gene dosage. Typically, a doubling of the number of chromosomal copies of a gene should lead to an approximate doubling of its mRNA synthesis rate and thus to a corresponding increase in its protein levels. Most often, however, such effects are considered in the context of a change in the number of autosomal gene copies that persists throughout an organism’s lifetime (13), as, for example, in the haploinsufficiency of certain genes (14). It is less often acknowledged that the number of copies of all genes varies over each cell cycle, despite evidence that these variations have measurable consequences (15–18). Because of the well-known phenomenon of phase-locking of oscillators (19), regular, periodic changes in gene dose are likely to be especially relevant to cellular oscillators that depend on gene expression. A circadian clock that became slaved to the cell cycle, for example, would lose its identity as an autonomous timekeeper, and thus much of its ability to perform its biological function. Here, we show that negative transcriptional feedback oscillators (NTFOs)—a common motif in both prokaryotic and eukaryotic clocks—are indeed very strongly affected by driving from periodic gene replication events. This immediately raises the question of how real biological clocks are able to function in growing, dividing cells. To address this, we study the circadian clock of the cyanobacterium Synechococcus elongatus, which is known to exhibit stable rhythms over a wide range of growth rates (20, 21), but whose clock seems not to regulate DNA replication (4), suggesting exactly the sort of unidirectional forcing of the clock by the cell cycle that might have been expected to impair clock function.

The S. elongatus clock combines an NTFO (the transcription–translation cycle, or TTC) with a core phosphorylation-based posttranslational oscillator (the protein phosphorylation cycle, or PPC). Remarkably, the PPC can be reconstituted in vitro with the purified cyanobacterial clock proteins KaiA, KaiB, and KaiC (22), allowing detailed study of the mechanisms behind its oscillation. A number of studies have begun to converge on the view that the PPC works by synchronizing the intrinsic phosphorylation cycles of individual KaiC hexamers, primarily through phosphorylation-dependent sequestration of KaiA by KaiC (23–29). Although many details of the TTC remain murkier, it seems clear that the protein RpaA plays a central role, regulating the expression of clock components in a manner that ultimately reflects the KaiC phosphorylation state (30–34). Depending on light and nutrient levels, S. elongatus can have doubling times ranging from 6 to 72 h (21); the cell-cycle period is thus of the same order as the clock period of roughly 24 h, opening the way for interactions between the two. Indeed, the circadian clock is known to gate mitosis, prohibiting cell division during certain clock phases (4, 7, 8), although in constant light this gating leaves both DNA replication and cell growth essentially unchanged (4). Conversely, Mori and Johnson (20) argued that cell growth and division do not affect the S. elongatus circadian clock. We use mathematical modeling to study the unidirectional forcing of the clock by the cell cycle. We identify specific features of the S. elongatus clock that tend to insulate it from entrainment by regular gene replication events. Nonetheless, we argue that, under certain conditions, it should be possible to observe signatures of periodic forcing of the clock by the cell cycle. We further suggest how some of the clock’s protective mechanisms might be weakened experimentally, leading to much stronger signatures of its coupling to the cell cycle.

Below, we first model the effects of cell growth and division on a constitutively expressed protein. We show that gene replication, not cell division, is the essential cell-cycle event that influences protein concentrations and that, as long as the constitutively expressed protein is not subject to rapid, active degradation, its concentration varies little over the cell cycle. In contrast, gene replication can dramatically affect the behavior of an NTFO: The NTFO locks to the cell cycle over a range of cell-division times of many hours and shows erratic behavior outside this regime (12). We next ask how the real cyanobacterial clock can be so apparently undisturbed by the cell cycle. We find that incorporating both a PPC and a TTC into the clock significantly weakens coupling to the cell cycle, especially when the clock is read out by a push–pull network that is more sensitive to ratios of concentrations of different phosphorylation states than to their absolute values. The presence of multiple chromosome copies has a still more striking effect: If the cell has four copies after division (rather than only one), as can often be the case in S. elongatus, and if these are replicated one after the other (35), then the dose of the clock genes changes much more gradually, and cell-cycle effects are almost completely lost. Thus, S. elongatus may have evolved to carry multiple, identical chromosome copies in part to insulate its circadian clock from its DNA replication cycles.

Models and Results

The Cell Cycle’s Effect on a Constitutively Expressed Gene Is Weak.

Before turning to the more complex case of a circadian clock, we first investigate how the concentration of a single, constitutively expressed protein varies over a cell cycle. To this end, we add regular, rhythmic DNA replication and mitosis to a simple model of protein production and dilution.

The key quantities in our description are the number of copies $g\left(t\right)$ of the gene of interest and the cell volume $V\left(t\right)$. These vary periodically in time as sketched in Fig. 1 A and B, with a period given by the cell division time $T_{\text{d}}$. We assume for now that there is only one gene copy present immediately after cell division. This copy is replicated at some time before the next division, at which point $g\left(t\right)$ jumps from 1 to 2. When the cell divides, the chromosomes are split between the daughter cells, and $g\left(t\right)$ returns to 1. The cell volume grows exponentially: $V\left(t\right)=V_0\hspace{0.17em}\exp \left({\mu }_{\text{d}}t\right)$, with ${\mu }_{\text{d}}=\log \left(2\right)/T_{\text{d}}$. When t reaches $T_{\text{d}}$, division occurs, and $V\left(t\right)$ drops back from $2V_0$ to $V_0$.

Fig. 1.

DNA replication, but not cell division, affects average expression levels; for a protein that is constitutively expressed and decays by dilution only, the effect is small. Schematic time courses of the gene copy number $g\left(t\right)$ (A), the cell volume $V\left(t\right)$ (B), the gene density, $G\left(t\right)=g\left(t\right)/V\left(t\right)$ (C), and the concentration $C\left(t\right)$ of a constitutively expressed protein that decays only by dilution (D). Time in units of the cell division time $T_{\text{d}}$; vertical axes, arbitrary units. The gene density (C) has a discontinuity when the gene is replicated (vertical dotted lines) but not at cell division (vertical solid lines), when both $g\left(t\right)$ and $V\left(t\right)$ are halved. Even though the protein synthesis rate doubles when the gene is replicated, the maximum deviation of $C\left(t\right)$ from its time average is less than 4% (D). (E) The NTFO model: A protein with concentration $C\left(t\right)$ represses its own transcription with a delay $\mathrm{\Delta }$. (F) Zwicker et al. (36) model for coupled phosphorylation (PPC, purple background) and transcription–translation (TTC, blue background) cycles. KaiC hexamers switch between an active conformational state (circles) in which their phosphorylation level tends to rise and an inactive state (squares) in which it tends to fall. Active KaiC activates RpaA and inactive KaiC inactivates RpaA; active RpaA (red) activates kaiBC expression, leading (after a delay) to the injection of fully phosphorylated KaiC (pink) into the PPC.

The variables $g\left(t\right)$ and $V\left(t\right)$ define the gene density $G\left(t\right)\equiv g\left(t\right)/V\left(t\right)$. As long as noise and spatial variations are neglected, the behavior of a biochemical network depends only on protein concentrations, not separately on protein numbers and cell volume. As a result, the system responds to the protein synthesis rate per unit volume, proportional to $G\left(t\right)$, but not to $g\left(t\right)$ and $V\left(t\right)$ individually (Eq. 1). Fig. 1C shows that $G\left(t\right)$ has only a single discontinuity during the cell cycle, corresponding to the doubling of $g\left(t\right)$ when the gene is copied; at cell division, both $g\left(t\right)$ and $V\left(t\right)$ are halved, so their ratio is unchanged. Importantly, then, the mean-field, deterministic dynamics of a biochemical network is sensitive to the timing of DNA replication but not of cell division. This dynamics is likewise unaffected by any gating of cell division by the circadian clock, provided, as is the case in S. elongatus (4, 8), that this gating does not affect DNA replication or cell growth. Similarly, regardless of when during the division cycle the gene is copied, the time dependence of $G\left(t\right)$ is always the same: It doubles, decays exponentially for a time $T_{\text{d}}$, then doubles again, and so on. The exact moment of gene replication affects only the average value of $G\left(t\right)$, which can be absorbed, for modeling purposes, into the parameter β (Eq. 1). For simplicity, we thus always assume that the gene is replicated exactly at $t=T_{\text{d}}/2$.

Given the gene density $G\left(t\right)$, the concentration $C\left(t\right)$ of a constitutively expressed protein evolves as

$$\frac{dC\left(t\right)}{dt}=\beta G\left(t\right)-{\mu }_{\text{d}}C\left(t\right).$$ [1]

Here, proteins are expressed at a rate β per gene copy and diluted by cell growth at a rate ${\mu }_{\text{d}}=\log \left(2\right)/T$. We thus assume that, as is true for many bacterial proteins, the protein is not subject to active degradation (37). Fig. 1D shows how $C\left(t\right)$ varies over the cell cycle. Remarkably, even though the protein production rate doubles each time the gene is replicated, the protein concentration varies by no more than a few percent: The discrete jumps in protein production are smoothed out by the slow protein dilution. Thus, a protein that is constitutively expressed and not actively degraded is little affected by the cell cycle.

The Cell Cycle Strongly Perturbs Both the Period and the Amplitude of an NTFO.

Although the concentration of a protein that is constitutively expressed does not vary much over the cell cycle, oscillators are known to be far more sensitive to periodic driving than nonoscillatory systems (19). We thus next consider a simple model for a clock built on delayed, negative transcriptional feedback (Fig. 1E). The model consists of a single variable, $C\left(t\right)$, describing the concentration of proteins that inhibit their own production:

$$\frac{dC\left(t\right)}{dt}=\beta \hspace{0.17em}\tilde{G}\left(t\right)\hspace{0.17em}\frac{K_{\text{c}}^n}{K_{\text{c}}^n+C{\left(t-\mathrm{\Delta }\right)}^n}-{\mu }_{\mathrm{tot}}C\left(t\right).$$ [2]

We impose a fixed delay $\mathrm{\Delta }$ between the initiation of transcription and the appearance of functional proteins. Therefore, protein production at time t is proportional to the gene copy number $g\left(t-\mathrm{\Delta }\right)$ at time $t-\mathrm{\Delta }$. These proteins “arrive” in the cell volume $V\left(t\right)$ at time t. The protein synthesis rate per unit volume at time t is thus proportional to the protein production density $\tilde{G}\left(t\right)\equiv g\left(t-\mathrm{\Delta }\right)/V\left(t\right)$. $\tilde{G}\left(t\right)$ is a generalization of the gene density $G\left(t\right)$ of the preceding section to the case with a delay $\mathrm{\Delta }$ and parameterizes the periodic forcing of the NTFO by gene replication. Proteins disappear with a total rate ${\mu }_{\mathrm{tot}}={\mu }_{\text{d}}+{\mu }_{\mathrm{act}}$, where as before ${\mu }_{\text{d}}$ describes dilution due to cell growth and ${\mu }_{\mathrm{act}}$ describes possible active degradation. Including both terms allows us to vary the doubling time $T_{\text{d}}$ while holding ${\mu }_{\mathrm{tot}}$ constant and hence, in our simulations, to distinguish the trivial influence of the cell cycle on the clock through the dilution rate ${\mu }_{\text{d}}$ from other effects.

We next define the peak-to-peak time $T_{\mathrm{PtP}}$ as the time between successive peaks in $C\left(t\right)$ (Fig. 2 and Supporting Information); $T_{\mathrm{PtP}}$ reduces to the period of the circadian clock when oscillations are regular but remains defined when the cell cycle induces more erratic behavior. In Fig. 2A we plot the average peak-to-peak time $\langle T_{\mathrm{PtP}}\rangle$ for a range of division times $T_{\text{d}}$ at fixed ${\mu }_{\mathrm{tot}}$.

Fig. 2.

Periodic gene replication dramatically affects an NTFO. (A) The average peak-to-peak time $\langle T_{\mathrm{PtP}}\rangle$ (solid curve) versus the cell division time $T_{\text{d}}$ at fixed ${\mu }_{\mathrm{tot}}$ and β. The shaded region shows the SD of the peak-to-peak times (Supporting Information). Dashed lines indicate regions where the clock locks to the cell cycle with periods in a 1:1 (left) or 2:1 (right) ratio. (Smaller locking regions around $T_{\text{d}}=\mathrm{6,12},\mathrm{and}\hspace{0.17em}36$ h are not marked.) (B–D) Protein concentration $C\left(t\right)$ (blue solid line) and the protein production density $\tilde{G}\left(t\right)=g\left(t-\mathrm{\Delta }\right)/V\left(t\right)$ (red dashed line) for the values of $T_{\text{d}}$ indicated by the arrows in A; horizontal brackets in B–D illustrate the definition of the peak-to-peak time $T_{\mathrm{PtP}}$. At $T_{\text{d}}=24$ h (B), the clock locks firmly to the cell cycle. For $T_{\text{d}}=27$ h (C), the cell-cycle period is just too large for locking; as a result, the cell cycle dramatically disrupts the clock, leading to a large SD of $T_{\mathrm{PtP}}$ (see A). At $T_{\text{d}}=48$ h (D), two oscillation cycles of the NTFO fit exactly in one division time. The larger-amplitude oscillation cycle corresponds to cell-cycle phases where $\tilde{G}\left(t\right)$ is higher and the smaller amplitude to phases where $\tilde{G}\left(t\right)$ is lower. Similar results are obtained upon varying $T_{\text{d}}$ at constant ${\mu }_{\mathrm{act}}$ (Fig. S8).

As expected from the general theory of driven oscillators (19), the curve shows two striking features. First, around division times that are fractions or multiples of the clock’s intrinsic period of 24 h, the cell cycle determines the period of the clock. Especially around $T_{\text{d}}=24$ and 48 h, the average peak-to-peak time is directly proportional to $T_{\text{d}}$. At $T_{\text{d}}=24$ h (1:1 locking), $\langle T_{\mathrm{PtP}}\rangle =T_{\text{d}}$, and the amplitude of each clock oscillation cycle is the same (Fig. 2B). At $T_{\text{d}}=48$ h (2:1 locking), however, $\langle T_{\mathrm{PtP}}\rangle =T_{\text{d}}/2$, and two full clock cycles are required to make up a single division time. Because these two cycles occur at different gene densities, successive peaks in the trace of $C\left(t\right)$ have alternately large and small amplitudes.

Second, the standard deviation (SD) of $T_{\mathrm{PtP}}$ becomes very large just outside the locking regions. Fig. 2C shows that this variability in the phase of $C\left(t\right)$ is accompanied by substantial fluctuations in the amplitude for $T_{\text{d}}=27$ h. Because the difference between $T_{\text{d}}$ and the intrinsic clock period is just too large to allow stable locking, the clock constantly tries to lock to the cell cycle, but slips from time to time. As a result, the cell cycle dramatically disrupts the clock. In Supporting Information we show that both of these effects survive the introduction of intrinsic noise in chemical reactions and of stochasticity in the timing of DNA replication (Figs. S1 and S2; see also Fig. S3). Fig. 3 qualitatively explains how locking arises in the NTFO.

Fig. S1.

The effect of intrinsic noise on the locking of the NTFO to the cell cycle. (A) Average (solid line) and SD (shaded region) of the peak-to-peak time $T_{\text{PtP}}$ as a function of the division time $T_{\text{d}}$ for an NTFO with intrinsic noise and initial gene copy number $N=1$. The region of 1:1 locking with the cell cycle (left dashed line) has widened considerably compared with the deterministic case (Fig. 2A of the main text), and the SD in $T_{\text{PtP}}$ outside the locking region has increased. In contrast, the region of 2:1 locking (right dashed line) has shrunk almost to nothing. (B–D) Representative time traces for the division times indicated by the arrows in A. Shown are the protein concentration $C\left(t\right)=N_C\left(t\right)/V\left(t\right)$ of the NTFO (blue solid line) and the protein production density $\tilde{G}\left(t\right)$ (red dashed line), both normalized by their time average values. At a division time of $T_{\text{d}}=24$ h (B), the NTFO is locked to the cell cycle. Because of the intrinsic noise, the amplitude varies slightly from one oscillation cycle to the next. At $T_{\text{d}}=27$ h (C), just outside the locking region, the oscillator exhibits irregular behavior. At $T_{\text{d}}=48$ h (D), the NTFO oscillations switch between a small and a large amplitude in successive oscillation cycles, just as in the deterministic case.

Fig. S2.

The effect of stochasticity in the timing of gene replication on locking to the cell cycle. (A) The average and variance of $T_{\text{PtP}}$ for an NTFO in which the timing of gene replication is deterministic (red) or is drawn from a Gaussian distribution with a width σ that is 30% of the cell-division time $T_{\text{d}}$(blue). The stochasticity in the replication times decreases the width of the locking regions but increases the variance in the peak-to-peak times. (B) Same as A, but for $T_{\text{PtP}}$ of the phosphorylation fraction $p\left(t\right)$ of the TTC–(PPC_Zwicker) model (36) (see the supporting information of ref. 36). Again, stochasticity in the timing of replication reduces locking, but in this case the increase in the variance of $T_{\text{PtP}}$ outside the locking region is much less marked. We attribute this to the ability of the PPC to insulate the clock from variability in gene expression levels. (C) Representative time traces of the production density $\tilde{G}\left(t\right)$ (normalized to its time average), the phosphorylation fraction $p\left(t\right)$, and the total KaiC concentration $C_{\mathrm{tot}}\left(t\right)$ for the Zwicker model for $T_{\text{d}}$= 48 h. As in the deterministic limit, the amplitude of the $C_{\mathrm{tot}}\left(t\right)$ oscillations tends to alternate between a high and a low value, due to gene replication occurring every 48 h, on average; in contrast, the amplitude of $p\left(t\right)$ is relatively constant. The effect of periodic gene replication on $C_{\mathrm{tot}}\left(t\right)$ should thus be observable experimentally.

Fig. S3.

Heat plots of the width of the 1:1 locking region (A, E, and I) and the average SD of the peak-to-peak time (C, G, and K) as a function of the initial gene copy number N and the SD in the gene replication time ${\sigma }_{\text{rep}}/T_{\text{d}}$, for the NTFO model (A–D), the TTC–(PPC_Zwicker) model (36) (E–H), and the TTC–(PPC_Rust) model (26, 36) (I–L). B, D, F, H, J, and L show the same data as in the heat plots immediately above them, but as a function of ${\sigma }_{\text{rep}}/T_{\text{d}}$, for different values of N; note the difference in scale of the y axes in these panels. The major results of B, D, F, and H are also summarized in Fig. 5 C and D of the main text. The average SD in the peak-to-peak time is the SD in the peak-to-peak time averaged over $6<T_{\text{d}}<52$; it is a measure for the erratic behavior of the clock outside the locking regions. It is seen that in all models the width of the locking region rapidly decreases with both N and ${\sigma }_{\text{rep}}/T_{\text{d}}$. However, the average SD in the peak-to-peak time decreases with N but increases with ${\sigma }_{\text{rep}}/T_{\text{d}}$. Clearly, although having multiple chromosome copies is a powerful strategy for preventing locking, increasing the stochasticity in the timing of gene replication is not—decreasing locking at the expense of much greater variation in the length of the periodic is unlikely to be functionally advantageous. Comparing the two models with a PPC to the NTFO model shows that adding a PPC to a TTC also decreases both the width of the locking region and the average SD in the peak-to-peak time. Combining both features—a PPC and multiple chromosome copies—gives the strongest reduction in the coupling of the clock to the cell cycle.

Fig. 3.

Locking mechanism for the NTFO. Shown are time courses of the production density $\tilde{G}\left(t\right)=g\left(t-\mathrm{\Delta }\right)/V\left(t\right)$ (dashed red lines) and the protein concentration $C\left(t\right)$ (solid blue lines). For clarity, we consider the limit $n\to \infty$, in which the Hill function describing autoregulation (Eq. 2) reduces to a step function with repression threshold $K_{\text{c}}$, denoted by the dotted horizontal line. Shaded regions indicate times when $C\left(t\right)$ is rising. The panels correspond to two different initial phase differences between the NTFO and the cell cycle. In each case, when $C\left(t\right)$ drops below $K_{\text{c}}$ at time $t\ast -\mathrm{\Delta }$, protein production starts, but because of the delay $\mathrm{\Delta }$, new molecules are injected into the system only at time $t\ast$. (A) The gene has replicated just before $t\ast -\mathrm{\Delta }$, and $\tilde{G}\left(t\ast \right)$ is hence large, yielding a large amplitude for the next NTFO cycle. Because the rate of protein decay is independent of $\tilde{G}\left(t\right)$, the period of the NTFO cycle is correspondingly long. The subsequent NTFO cycle thus begins at smaller $\tilde{G}\left(t\ast \right)$, causing it to have a smaller amplitude and a shorter period. (B) The gene has not yet replicated at time $t\ast -\mathrm{\Delta }$, and $\tilde{G}\left(t\ast \right)$ is therefore low; consequently, the amplitude and period of the next NTFO cycle are small. The beginning of the subsequent cycle is then shifted toward higher $\tilde{G}\left(t\ast \right)$, increasing its period. In both cases, the result is that, after a few cell cycles, the period of the NTFO oscillation approaches that of the cell cycle, yielding stable 1:1 locking where the two oscillators have a well-defined phase relation. The largest amplitude and thus longest possible clock period arise when the protein synthesis phase (gray bar) coincides with the maximal $\tilde{G}\left(t\ast \right)$; if $T_{\text{d}}$ increases beyond this maximal period, locking cannot occur. An analogous loss of locking occurs if $T_{\text{d}}$ decreases below the minimal possible clock period. In either case, the clock shows erratic behavior until $T_{\text{d}}$ approaches values where 1:2 or 2:1 locking is possible.

A Phosphorylation Cycle Makes the Clock More Robust Against a Time-Varying Gene Density.

To study how a more realistic clock can become resilient to variability in the gene density, we turn to the S. elongatus circadian clock, and more specifically to the model of Zwicker et al. (25, 36) (Fig. 1F). This model provides a detailed description of the clock, including the synchronization of the phosphorylation state of different KaiC hexamers via KaiA sequestration and the coupling of the PPC oscillator to the TTC via RpaA. It represents KaiC as a hexamer but does not explicitly take into account that each KaiC monomer has two distinct phosphorylation sites (26, 38). In Supporting Information we show that a model based on that of Rust et al. (26), which describes KaiC at the level of monomers with two phosphorylation sites, gives similar results (Fig. S4). We thus expect that still more elaborate models of the PPC, which include hexameric KaiC with two phosphorylation sites per monomer (29), will lead to similar results. To include gene replication, we modify the model of ref. 36 so that the delayed negative feedback on KaiC production is modulated by a regularly oscillating protein production density $\tilde{G}\left(t\right)$ (Supporting Information). We follow both the total KaiC concentration $C_{\mathrm{tot}}\left(t\right)$ and the KaiC phosphorylation fraction $p\left(t\right)={\displaystyle {\sum }_{n=1}^{6}n{C}_n\left(t\right)/\left(6\hspace{0.17em}{C}_{\mathrm{tot}}\left(t\right)\right)}$, where $C_n$ is the concentration of n-fold phosphorylated KaiC hexamers.

Fig. S4.

The TTC–(PPC_Rust) model, which combines the TTC of Zwicker et al. (36) with the PPC of Rust et al. (26), is susceptible to periodic gene replication. The figure shows the average and SD of the peak-to-peak time $T_{\text{PtP}}$ of the phosphorylation fraction $p\left(t\right)$ for the TTC–(PPC_Rust) model and the TTC–(PPC_Zwicker) model (36). Clearly, the two models are similarly affected by the presence of the cell cycle.

Fig. 4A shows that a model with a PPC coupled to a TTC has a smaller locking window than an NTFO and lacks the large deviations in $T_{\mathrm{PtP}}$ just outside the locking region. The S. elongatus clock is hence more robust to gene replication than one based only on negative transcriptional feedback.

Fig. 4.

A clock with interlocked phosphorylation and transcriptional cycles is more robust against perturbations from periodic gene replication. (A) The average peak-to-peak times $\langle T_{\mathrm{PtP}}\rangle$ of the phosphorylation level $p\left(t\right)$ of the coupled TTC–PPC model of the Kai system (36) (red solid curve) and of $C\left(t\right)$ of the NTFO (solid blue curve, same as Fig. 2A), as a function of the cell division time $T_{\text{d}}$. The shaded regions show the SD of $T_{\mathrm{PtP}}$. Both the widths of the locking regions and the SDs of the peak-to-peak time outside the locking regions are smaller for $p\left(t\right)$ of the Kai system than for $C\left(t\right)$ of the NTFO. Arrows indicate division times for which we show time traces in B and C. (B) The total KaiC concentration $C_{\mathrm{tot}}\left(t\right)$ (dashed line) and $p\left(t\right)$ (solid line) at $T_{\text{d}}=26$ h. Although the amplitude of $C_{\mathrm{tot}}\left(t\right)$ is strongly affected by gene replication, the amplitude of $p\left(t\right)$ is nearly constant. (C) Plots of $p\left(t\right)$ and $C_{\mathrm{tot}}\left(t\right)$ at $T_{\text{d}}=48$ h, where the amplitude of $C_{\mathrm{tot}}\left(t\right)$ alternates between a low and a high value depending on the gene copy number in the cell. In contrast, $p\left(t\right)$ is almost unaffected by gene replication.

Clock Readout Through an RpaA-Based Push–Pull Network Filters Out Cell-Cycle-Dependent Variations in Protein Concentrations.

Although the variance of $T_{\mathrm{PtP}}$ outside of the locking region is relatively small for the combined TTC–PPC model, Fig. 4B shows that $C_{\mathrm{tot}}\left(t\right)$ exhibits strong amplitude fluctuations, mirroring those observed for the NTFO (Fig. 2). The phosphorylation fraction $p\left(t\right)$, in contrast, is far more resilient, suggesting that the clock encodes temporal information more reliably in $p\left(t\right)$ than in $C_{\mathrm{tot}}\left(t\right)$. Intriguingly, the RpaA-centered push–pull network that transmits this timing signal to downstream genes (30–34, 39) in fact responds primarily to $p\left(t\right)$: Because the rates of RpaA phosphorylation and dephosphorylation are indirectly controlled by different KaiC phosphoforms, variations in $C_{\mathrm{tot}}$ at fixed p change both rates together, leaving the fraction of phosphorylated RpaA largely unaffected. In contrast, changes in p shift the balance between the two opposing reactions and so modify the RpaA phosphorylation fraction (Fig. S5 and Supporting Information). Thus, not only is the basic PPC-based timekeeping mechanism insulated from variations in protein synthesis, but the readout mechanism selectively follows this more robust signal.

Fig. S5.

A push–pull network can read out the phosphorylation fraction $p\left(t\right)$ while remaining insensitive to the total concentration $C_{\mathrm{tot}}\left(t\right)$ of KaiC. (A) Steady-state output of the push–pull network, the fraction of phosphorylated substrate $\left[{\mathrm{S}}_{\text{p}}\right]/\left[{\text{S}}_{\mathrm{tot}}\right]$, plotted against the ratio of the active kinase concentration, $\left[\mathrm{K}\ast \right]$, to the active phosphatase concentration $\left[\mathrm{P}\ast \right]$; here, $\left[{\mathrm{S}}_{\text{p}}\right]/\left[{\text{S}}_{\mathrm{tot}}\right]$ mimics the phosphorylation fraction of RpaA. In steady state (but not necessarily in the general, time-varying case; see B) $\left[\mathrm{K}\ast \right]$ directly reports the concentration of KaiC in the phosphorylation phase of the clock and $\left[\mathrm{P}\ast \right]$ the concentration of KaiC in the clock’s dephosphorylation phase. For the solid red line, we change $\left[\mathrm{K}\ast \right]/\left[\mathrm{P}\ast \right]$ from 0.1 to 10, while keeping $\left[\mathrm{P}\ast \right]$ equal to $\left[{\text{S}}_{\mathrm{tot}}\right]$. The dashed and dotted lines show the result when both the kinase and phosphatase concentrations are halved or doubled, respectively. Because of the push–pull architecture, a change in the total concentration $\left[\mathrm{K}\ast \right]+\left[\mathrm{P}\ast \right]$ at fixed $\left[\mathrm{K}\ast \right]/\left[\mathrm{P}\ast \right]$ has only a small effect on the steady-state level of phosphorylated substrate $\left[{\mathrm{S}}_{\text{p}}\right]/\left[{\text{S}}_{\mathrm{tot}}\right]$; the network predominantly responds to the ratio $\left[\mathrm{K}\ast \right]/\left[\mathrm{P}\ast \right]$. (B) Schematic of our model of a simple push–pull network. The amount of active kinase, $\left[\mathrm{K}\ast \right]$, is controlled by the time-dependent rate $k_{\text{K}}\left(t\right)$ of conversion from K to $\mathrm{K}\ast$, and similarly for $\left[\mathrm{P}\ast \right]$ and $k_{\text{P}}\left(t\right)$; we imagine that these rates are proportional to the amount of KaiC in the phosphorylation and dephosphorylation phases of the clock, respectively. The two enzymes return to their inactive states at constant rates d. The interconversion between S and ${\mathrm{S}}_{\text{p}}$ follows the standard Michaelis–Menten reaction scheme. (C) Reading out time-varying rates $k_{\text{K}}$ and $k_{\text{P}}$. (Top) We let $k_{\text{K}}\left(t\right)$ oscillate with a peak-to-peak time of 24 h, with the amplitude of each consecutive oscillation cycle changing by a factor of two to mimic the variability in the total amount of KaiC when $T_{\text{d}}=48$ h (Fig. 2 of the main text). $k_{\text{P}}\left(t\right)$ has the same behavior as $k_{\text{K}}\left(t\right)$, but phase shifted by 12 h. (Bottom) With these time-varying inputs, the active enzyme concentrations $\left[\text{K}\ast \right]$ and $\left[\text{P}\ast \right]$ track the conversion rates $k_{\text{K}}\left(t\right)$ and $k_{\text{P}}\left(t\right)$, but $\left[{\mathrm{S}}_{\text{p}}\right]$ shows an essentially constant amplitude from one cycle to the next. Thus, even with time-varying inputs, the activity of RpaA is sensitive primarily to the ratio $k_{\text{K}}\left(t\right)/k_{\text{P}}\left(t\right)$, which plays the role of the phosphoryation ratio $p\left(t\right)$, not to each rate individually, or by extention to the absolute concentrations of KaiC phosphoforms. In all calculations, $K_M=\left[{\text{S}}_{\mathrm{tot}}\right]$.

Multiple Chromosome Copies Weaken the Cell Cycle’s Influence on the Clock.

Although the PPC reduces gene replication’s effect on the clock, it does not eliminate it entirely (Fig. 4). What other mechanisms might explain the observed resistance of the S. elongatus clock to the cell cycle? It is known that S. elongatus has multiple, identical copies of its chromosome (35, 40–42). These are not duplicated simultaneously, but rather one at a time, so that DNA replication occurs at a roughly constant rate throughout the cell cycle; furthermore, the timing of chromosome duplication seems to be independent of the phase of the clock (4, 35, 40, 42, 43). Motivated by this observation, we consider a cell that starts with N chromosomes after division and let $g\left(t\right)$ rise to $2N$ in N evenly spaced steps (Fig. 5 A and B). For larger N, the gene-copy number $g\left(t\right)$ increases more gradually, and hence the discrete jumps in the gene density $G\left(t\right)$ are considerably smaller. The effect on the clock is dramatic: In both the NTFO (solid line) and the TTC–PPC (dashed line), not only do the locked regions almost disappear for $N=4$ (Fig. 5C), but the variance in the peak-to-peak time $T_{\mathrm{PtP}}$ becomes very small (Fig. 5D). This latter effect persists even when significant stochastic variability in the rhythm of gene replication (parameterized by the SD ${\sigma }_{\mathrm{rep}}$ in the replication times; Supporting Information) is introduced. In fact, whereas S. elongatus can have as many as $N\approx 4$ chromosome copies at the beginning of the cell cycle (35, 40–42), these changes are already apparent when N is increased from 1 to 2 (Fig. S3).

Fig. 5.

Multiple chromosome copies strongly reduce the cell cycle’s effect on the circadian clock. (A) Gene copy number $g\left(t\right)$ for initial gene copy numbers $N=4$ (thick curve, left axis) and $N=1$ (thin curve, right axis) versus time (in units of cell-cycle time $T_{\text{d}}$). The increase in $g\left(t\right)$ is more gradual for $N=4$ than for $N=1$. (B) The gene density $G\left(t\right)=g\left(t\right)/V\left(t\right)$, normalized to its time average, for $N=4$ (thick curve) and $N=1$ (thin curve). At a higher gene copy number, the deviations from the average gene density become smaller. The width of the 1:1 locking region (C) and the square root of the average variance in the peak-to-peak time (D) as a function of the SD in the gene replication time ${\sigma }_{\mathrm{rep}}$ in a model where the times of replication events vary stochastically about their means (Supporting Information), for the NTFO (solid line) and the TTC–PPC (36) (dashed line). Increasing the chromosome copy number N reduces both the width of the locking region (C) and the variance in the peak-to-peak time (D). In contrast, whereas increasing ${\sigma }_{\mathrm{dup}}$ reduces the former, it increases the latter. See also Figs. S3 and S9.

Discussion

Given the pleiotropic roles of both the cell cycle and the circadian clock, it is natural to ask whether they also influence each other. Our central observation is that such influence need not involve specific interactions between the core genes or proteins of the two systems (2, 10, 11); rather, the simple fact that the number of cellular copies of a given gene necessarily experiences discrete jumps during DNA replication (Fig. 1) implies that clocks must in general feel a periodic driving from the cell cycle (12). Whereas some genetic circuits can simply average over this time-varying input, oscillators—including biological clocks—are known to be especially sensitive to rhythmic forcing. Indeed, an NTFO either locks to the cell cycle or shows erratic oscillations for a range of doubling times $T_{\text{d}}$ (Fig. 2), losing its ability to function as a clock in either case.

In light of this strong and detrimental coupling between the cell cycle and a simple transcriptional clock, it is all the more striking that the S. elongatus clock is so stable. Our analysis highlights two features of the cyanobacterial clock that are predicted to allow the necessary decoupling from the cell cycle. First, a time-varying gene dosage influences a clock with an autonomous posttranslational oscillator less than it does a purely transcriptional clock. This observation that a PPC is able to protect a TTC complements our previous finding that in a rapidly growing cell a PPC cannot function without a TTC (36). Together, our results show that a robust clock requires both a TTC and a PPC.

Even within the combined TTC–PPC, the oscillations of the KaiC phosphorylation fraction $p\left(t\right)$ are less affected by periodic gene replication than are those of the total KaiC concentration $C_{\mathrm{tot}}\left(t\right)$ (Fig. 4 and Fig. S2C). Strikingly, the RpaA-based push–pull network that communicates the clock state to the rest of the cell responds to p while ignoring the more strongly fluctuating $C_{\mathrm{tot}}$ [somewhat in the spirit of mechanisms that improve the robustness of bacterial chemotaxis to gene expression noise (44)]. This filtering function of the push–pull architecture could help explain why the S. elongatus clock has a relatively complex output mechanism requiring both CikA and SasA rather than a simpler linear design (45).

The second feature of the S. elongatus clock that we predict mitigates perturbations from the cell cycle is the presence of multiple, identical, asynchronously replicating chromosome copies (35, 40, 42, 43). This reduces the importance of each individual gene replication event: Rather than seeing a single doubling of the number of gene copies each cell cycle, a cell with many chromosomes instead sees a number of smaller jumps that it can more easily ignore (Fig. 5). This adaptation may thus have evolved in part to protect the S. elongatus clock from cell-cycle effects.

Whereas we have argued that the cell cycle generically affects any transcriptional clock, no comparably general mechanisms exist in the other direction. Moreover, although in many eukaryotic systems the clock is known to regulate key cell-cycle genes (2–9, 46), no similar, specific connections have yet been characterized in S. elongatus. In particular, clock-dependent cell-cycle gating (4), because it acts on cell division but not on growth or DNA replication, does not allow the clock to block the discrete gene replication events that underlie the driving. Nonetheless, because the majority of S. elongatus genes show some degree of clock-dependent expression (47), it is possible that the cyanobacterium’s clock does regulate its cell cycle in some as-yet-undiscovered way. Any such coupling would, however, have to be weak enough to be consistent with the observation that the rhythm of DNA replication does not depend on clock phase (4, 35, 40, 42, 43). Because phase locking between two oscillators has strong similarities to the locking of a single oscillator to periodic driving (19), most of our qualitative conclusions would remain unchanged in this case.

To isolate the behavior of the core, autonomous circadian oscillator, studies in the laboratory are typically performed at constant light levels. In keeping with this tradition, we have limited ourselves here to models of free-running clocks, without any diurnal environmental variation. In nature, however, the circadian clock is exposed to many additional entrainment signals, most notably the 24-h light–dark cycle. In fact, the environmental and cell-cycle entrainment signals are intricately intertwined, because DNA replication and the synthesis of most proteins, including clock components, come to a standstill in the dark in a clock-independent fashion (43, 48). We leave the effects of this complex interplay for future work.

Although we have focused on interactions between the cell cycle and the clock in S. elongatus, the basic idea that periodic gene replications must influence biological oscillators is more general and should apply to a wide range of prokaryotic and eukaryotic species. Indeed, cell-cycle-dependent changes in gene copy number have clearly observable effects on gene expression in eukaryotic cells (16), and recent experiments in cultured metazoan cells strongly suggest that the cell cycle exerts a considerable influence on the circadian clock, generally leading to phase-locking of the two oscillators (10, 11). Other generic forms of driving from the cell cycle may also play a role here: For example, in contrast to prokaryotes, eukaryotes typically shut down transcription around mitosis, thereby introducing another source of periodic, cell-cycle-dependent variation in protein synthesis (2, 10, 11). Our analysis thus highlights an important constraint on the design of circadian clocks in organisms from bacteria to humans.

Further, there is no reason for the effects of regular, discrete gene replications to be limited to circadian clocks; they should be observable in any cellular oscillator that depends on transcription and has a period on the same order as that of the cell cycle. Thus, our results may be relevant to phenomena such as coupling between the cell cycle and the segmentation clock in vertebrate development (49). Similarly, in Supporting Information (Figs. S6 and S7) we show that two well-known synthetic circuits (50, 51) can also lock to the cell cycle and that the strength of locking depends sensitively on the oscillator architecture.

Fig. S6.

The repressilator (50) can strongly lock to the cell cycle, and the strength of locking depends sensitively on the temporal order in which the respective genes are replicated during the cell cycle. (A) Average (solid line) and SD (shaded region) of the peak-to-peak time $T_{\text{PtP}}$ as a function of the division time $T_{\text{d}}$ for a repressilator with initial gene copy number $N=1$. The repressilator has an intrinsic period of $T_{\text{int}}=125$ min and the three genes are replicated simultaneously. The locking regions around $T_{\text{int}}$ and $2T_{\text{int}}$ are almost absent. (B and C) Representative time traces of the concentrations of the three repressilator proteins, $p_1\left(t\right)$ (red), $p_2\left(t\right)$ (blue), and $p_3\left(t\right)$ (orange), for the cell-division times indicated by the arrows in A. (B) When $T_{\text{d}}=T_{\text{int}}$, the oscillations are very regular (almost no variance in the PtP times), but each protein concentration has a different amplitude. (C) At $T_{\text{d}}=2T_{\text{int}}$, all three protein concentrations switch between a small and a large amplitude in successive oscillation cycles. (D and E) The effect of varying the timing of replication of the three genes. For clarity, we only show the average peak-to-peak time as a function of $T_{\text{d}}$, not the standard deivation. We assume that the $p_1$ gene is always replicated halfway through the cell cycle, $d_1=0$, and that the $p_2$ and $p_3$ genes are replicated with delays $d_2$ and $d_3=-d_2$, respectively. The gray line gives the situation where all genes are replicated simultaneously, $d_1=d_2=d_3=0$. Other values of $d_2$ are given in the legend and are written as a fraction of the intrinsic period $T_{\text{int}}$ of the oscillator. D shows scenarios for which $d_3<d_1<d_2$, meaning that the chronological order of replication is $p_3$ before $p_1$ before $p_2$. E shows situations for which $d_2<d_1<d_3$. Remarkably, for all $d_2=-d_3\ne 0$, there is significant locking. For the scenarios in E, however, the width of the 1:1 locking region is the largest; this is because the temporal order the genes’ replication during the cell cycle is the same as that of their expression in the oscillator (i.e., $p_2,p_1,p_3$) (B and C). 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.

Fig. S7.

The dual-feedback oscillator (51) 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 $T_{\text{int}}=73$ min. (A) Average (solid line) and SD (shaded region) of the peak-to-peak time $T_{\text{PtP}}$ as a function of the division time $T_{\text{d}}$ for a dual-feedback oscillator with initial gene copy number $N=1$ and all genes replicated simultaneously. The region of 1:1 locking (around $T_{\text{d}}$ $=73$ min) with the cell cycle (left dashed line) has widened considerably compared with the NTFO model (compare with Fig. 2 of the main text). (B and C) Representative time traces for the division times indicated by the arrows in A. Shown are the activator and repressor concentrations $a\left(t\right)$ (green line) and $r\left(t\right)$ (green line), respectively. At a cell-division time of $T_{\text{d}}=96$ 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 $T_{\text{d}}=2T_{\text{int}}$ (C), the oscillations switch between a small and a large amplitude in successive oscillation cycles, just as in the NTFO. (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 $T_{\text{d}}$ is shown, not the SD. We assume that the activator gene is always replicated halfway through the cell cycle, $d_{\text{a}}=0$, and that the repressor gene is replicated with a delay $d_{\text{r}}$. In both figures, the gray line gives the situation where the genes are replicated simultaneously, $d_{\text{r}}=0$. Other values of $d_{\text{r}}$ are given in the legend and are written as a fraction of the intrinsic period $T_{\text{int}}$ of the oscillator. (D) Positive $d_{\text{r}}$; the repressor gene is replicated after the activator gene. (E) Negative $d_{\text{r}}$; the repressor gene is replicated before the activator gene. D shows that locking decreases as $d_{\text{r}}$ becomes more positive, whereas E shows that the size of the 1:1 locking region depends nonmonotonically on $d_{\text{r}}$ for $d_{\text{r}}<0$. Comparing the behavior of the dual-feedback oscillator, which exhibits the strongest entrainment when the genes are replicated simulateneously, with that of the repressilator, which shows the weakest coupling when the genes are replicated together, shows 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.

Because we have argued that S. elongatus possesses particular adaptations that decouple its circadian clock from the cell cycle, the most obvious experimental test of our ideas would be to observe the consequences of blocking or removing these features. Several strains already exist that might allow just such experiments. Mutants of S. elongatus are known with significantly fewer chromosomes per cell than the wild type (52); moreover, in some other Synechococcus strains, cells are always monoploid (41). We find that in cells where the number of chromosomes goes from one to two over the course of a single division cycle, it should be possible to observe clear signatures of driving by the cell cycle in plots of KaiC’s abundance—but not its phosphorylation level—as a function of time (Fig. 4). We predict that this effect will be further strengthened if the PPC is removed entirely. It is well-established that this can be accomplished by hyperphosphorylating KaiC (53, 54). In all cases, one could study forcing by the cell cycle at a variety of different doubling times. We suggest, however, that a doubling time near 48 h offers a particularly unambiguous signature of the cell cycle’s influence: The KaiC abundance as a function of time should then rise and fall every 24 h, with successive peaks strictly alternating between higher and lower levels (Fig. 4C).

Overview of Models and Nomenclature

Supporting Information discusses three different models of a circadian clock driven by periodic gene replication, which we list here to summarize our naming conventions and the distinctions among the models:

• •The negative transcriptional feedback oscillator, or NTFO, consists of a single gene that negatively regulates its own transcription with a delay. Because of gene replication, the number of copies of this gene in each cell doubles over the course of a cell cycle (as described in the next section and in the main text).
• •We refer to our central model of the S. elongatus circadian clock as either the TTC–PPC model or the TTC–(PPC_Zwicker) model, with the latter name reserved for those parts of Supporting Information where there is potential for confusion with the TTC–(PPC_Rust) model (next item). The TTC–(PPC_Zwicker) model consists of the PPC model of Van Zon et al. (25) joined to a TTC as in the work of Zwicker et al. (36) and subject to periodic driving from the cell cycle because of gene replication, as discussed in the main text.
• •We also present results for the TTC–(PPC_Rust) model, which couples the description of the PPC proposed by Rust et al. (26) to the same models of the TTC and of forcing from the cell cycle used in the TTC–(PPC_Zwicker) model.

Of these models, the first two [NTFO and TTC–(PPC_Zwicker)] are discussed in the main text, and the third [TTC–(PPC_Rust)] is introduced in Supporting Information to demonstrate that our major qualitative conclusions do not depend on our specific assumptions about the PPC.

In addition to models of a circadian clock coupled to a cell cycle, Supporting Information also considers the influence of periodic gene replication on two common synthetic clocks [Supporting Information, Two Synthetic Oscillators: The Repressilator and the Dual-Feedback Oscillator (Figs. S6 and S7)].

The Deterministic NTFO (Figs. 2 and 4)

Here we describe the NTFO studied in the main text, together with its parameters. The model consists of a single variable, $C\left(t\right)$, describing the concentration of proteins that inhibit their own production:

$$\frac{dC\left(t\right)}{dt}=\beta \hspace{0.17em}\tilde{G}\left(t\right)\hspace{0.17em}\frac{K_{\text{c}}^n}{K_{\text{c}}^n+C{\left(t-\mathrm{\Delta }\right)}^n}-{\mu }_{\mathrm{tot}}C\left(t\right).$$ [S1]

We impose a fixed delay $\mathrm{\Delta }$ between the initiation of transcription and the appearance of functional proteins. Therefore, protein production at time t is proportional to the gene copy number $g\left(t-\mathrm{\Delta }\right)$ at time $t-\mathrm{\Delta }$. These proteins enter the cell volume $V\left(t\right)$ at time t. Combining these two effects, the protein synthesis rate per unit volume at time t is thus proportional to the protein production density $\tilde{G}\left(t\right)\equiv g\left(t-\mathrm{\Delta }\right)/V\left(t\right)$. Because the production at time t depends on the state of the promoter at $t-\mathrm{\Delta }$, the Hill function describing autoregulation with Hill coefficient n and concentration of half-maximal repression $K_c$ is evaluated with the delayed concentration $C\left(t-\mathrm{\Delta }\right)$. Proteins are degraded with a total rate ${\mu }_{\mathrm{tot}}={\mu }_{\text{d}}+{\mu }_{\mathrm{act}}$, where ${\mu }_{\text{d}}$ describes dilution due to cell growth and ${\mu }_{\mathrm{act}}$ describes possible active degradation. Including both terms allows us to vary the doubling time $T_{\text{d}}$ while holding ${\mu }_{\mathrm{tot}}$ constant and hence to distinguish the trivial influence of the cell cycle on the clock through the dilution rate ${\mu }_{\text{d}}$ from other effects. This model is a deterministic one, based on mean-field chemical rate equations. A stochastic version that takes into account the intrinsic stochasticity of biochemical reactions is introduced further below.

Parameters used in the simulations are $\beta =6.0\cdot {10}^3{\text{h}}^{-1}$, $K_c=1.0\mathrm{\mu }\text{M}$, $n=2$, ${\mu }_{\mathrm{tot}}=0.2{\text{h}}^{-1}$, and $\mathrm{\Delta }=8\text{h}$. Details regarding the simulations are given in Supporting Information, Methods. The principal results of this model are presented in Figs. 2 and 4 of the main text.

The TTC–(PPC_Zwicker) Model (Figs. 4 and 5)

The TTC–PPC model of the main text is built on the TTC–PPC model of Zwicker et al. (36). It consists of a PPC combined with a TTC. The PPC model is based on that of Van Zon et al. (25). In this model, each KaiC hexamer has an intrinsic tendency to progress through a phosphorylation cycle, whereas the phosphorylation cycles of the individual hexamers are synchronized via the mechanism of differential affinity: KaiA stimulates KaiC phosphorylation, but the limited supply of KaiA binds preferentially to those KaiC hexamers that are falling behind in the cycle, forcing the front runners to slow down and allowing the laggards to catch up. The model includes the following reactions (25):

$${\text{C}}_i\underset{b_i}{\stackrel{f_i}{\rightleftarrows }}{\tilde{\text{C}}}_i,\hspace{1em}{\text{C}}_i+\text{A}\underset{k_i^{\mathrm{Ab}}}{\overset{k_i^{\mathrm{Af}}}{\rightleftarrows }}{\mathrm{AC}}_i\stackrel{k_{\mathrm{pf}}}{\to }{\text{C}}_{i+1}+\text{A},$$ [S2]

$${\tilde{\text{C}}}_i+\text{B}\underset{{\tilde{k}}_i^{\mathrm{Bb}}}{\overset{2{\tilde{k}}_i^{\mathrm{Bf}}}{\rightleftarrows }}{\text{B}\tilde{\text{C}}}_i,\hspace{1em}{\text{B}\tilde{\text{C}}}_i+\text{B}\underset{2{\tilde{k}}_i^{\mathrm{Bb}}}{\overset{{\tilde{k}}_i^{\mathrm{Bf}}}{\rightleftarrows }}{\text{B}}_2{\tilde{\text{C}}}_i,\hspace{1em}\text{AB}{\tilde{\mathrm{C}}}_i+\text{B}\underset{2{\tilde{k}}_i^{\text{Bb}}}{\overset{{\tilde{k}}_i^{\text{Bf}}}{\rightleftarrows }}{\text{AB}}_2{\tilde{\mathrm{C}}}_i,$$ [S3]

$${\text{B}}_x{\tilde{\text{C}}}_i+\text{A}\underset{{\tilde{k}}_i^{\mathrm{Ab}}}{\stackrel{x{\tilde{k}}_i^{\mathrm{Af}}}{\rightleftarrows }}{\mathrm{AB}}_x{\tilde{\text{C}}}_i,\hspace{1em}{\mathrm{AB}}_2{\tilde{\text{C}}}_i+\text{A}\underset{2{\tilde{k}}_i^{\mathrm{Ab}}}{\stackrel{{\tilde{k}}_i^{\mathrm{Af}}}{\rightleftarrows }}{\text{A}}_2{\text{B}}_2{\tilde{\text{C}}}_i,$$ [S4]

$${\text{C}}_i\underset{k_{\mathrm{dps}}}{\overset{k_{\mathrm{ps}}}{\rightleftarrows }}{\text{C}}_{i+1},\hspace{1em}{\tilde{\text{C}}}_i\underset{{\tilde{k}}_{\mathrm{dps}}}{\stackrel{{\tilde{k}}_{\mathrm{ps}}}{\rightleftarrows }}{\tilde{\text{C}}}_{i+1},$$ [S5]

$${\text{B}}_x{\tilde{\text{C}}}_i\underset{{\tilde{k}}_{\mathrm{dps}}}{\stackrel{{\tilde{k}}_{\mathrm{ps}}}{\rightleftarrows }}{\text{B}}_x{\tilde{\text{C}}}_{i+1},\hspace{1em}{\text{A}}_y{\text{B}}_x{\tilde{\text{C}}}_i\underset{{\tilde{k}}_{\mathrm{dps}}}{\stackrel{{\tilde{k}}_{\mathrm{ps}}}{\rightleftarrows }}{\text{A}}_y{\text{B}}_x{\tilde{\text{C}}}_{i+1}.$$ [S6]

Here, ${\text{C}}_i$ denotes a KaiC hexamer in the active conformational state, in which the number i of phosphorylated monomers tends to increase, and ${\tilde{\text{C}}}_i$ denotes a KaiC hexamer in the inactive conformational state in which i tends to decrease; $\text{A}$ denotes a KaiA dimer, and $\text{B}$ denotes a KaiB dimer. The reactions ${\text{C}}_i\rightleftarrows {\tilde{\text{C}}}_i$ in Eq. S2 model the conformational transitions between active and inactive KaiC; the second set of reactions in Eq. S2 describe phosphorylation of active KaiC that is stimulated by KaiA. The reactions in Eq. S3 model the binding of KaiB to inactive KaiC, and those in S4 model the sequestration of KaiA by inactive KaiC that is bound to KaiB; note that in this model an inactive KaiC hexamer can bind up to two KaiA dimers. The reactions in Eqs. S5 and S6 model spontaneous phosphorylation and dephosphorylation of active and inactive KaiC. For a more detailed discussion of the model, we refer to ref. 25.

The TTC and the coupling between the PPC and TTC are described by the following reactions:

$$\tilde{\text{R}}+\text{X}\stackrel{k_{\text{a}}}{\to }\text{R}+\text{X},\hspace{1em}\text{R}+\tilde{\text{X}}\stackrel{k_{\text{i}}}{\to }\tilde{\text{R}}+\tilde{\text{X}},$$ [S7]

$$\mathrm{\varnothing }\underset{\tau }{\stackrel{\beta \tilde{G}\left(t\right){\left[\text{R}\right]}^4/\left(K^4+{\left[\text{R}\right]}^4\right)}{\Rightarrow }}{\text{C}}_6+3\text{B},\hspace{1em}\mathrm{\varnothing }\stackrel{{\beta }_{\text{a}}}{\to }\text{A},\hspace{0.17em}\mathrm{\varnothing }\stackrel{{\beta }_{\text{r}}}{\to }\tilde{\text{R}},$$ [S8]

$$\text{R},\tilde{\text{R}},\text{A},\text{B},{\mathrm{AC}}_i,{\text{B}}_x{\tilde{\text{C}}}_i,{\text{A}}_y{\text{B}}_x{\tilde{\text{C}}}_i\stackrel{\mu }{\to }\mathrm{\varnothing }.$$ [S9]

Here, R and $\tilde{\text{R}}$ denote the RpaA protein in its active and its inactive form, respectively. The $\text{X}$ and $\tilde{\text{X}}$ in Eq. S7 denote any of the phosphoforms of KaiC that mediate the activation and repression of RpaA, respectively; as described in more detail in ref. 36, KaiC in the phosphorylation phase activates RpaA, whereas KaiC in the dephosphorylation phase deactivates it. The double arrow indicates a reaction with a fixed delay τ; we use τ rather than $\mathrm{\Delta }$ to denote the delay to agree with the notation of ref. 36. We thus assume that kaiBC expression is activated by RpaA and that the activity of RpaA is modulated by the PPC. In contrast, the expression of KaiA and RpaA is taken to occur constitutively. The effect of gene replication is included by making the rate of ${\text{C}}_6$ production dependent on $\tilde{G}\left(t\right)$, just as in the NTFO. We use the parameters given in the supplementary information of ref. 36, except for $\beta =1.02\cdot {10}^2{\text{h}}^{-1}$ and ${\mu }_{\mathrm{tot}}=0.1\hspace{0.5em}{\text{h}}^{-1}$. As for the NTFO, we keep the total degradation rate constant, so that we can distinguish the effects of protein dilution from those due specifically to periodic gene replication events.

The TTC–(PPC_Rust) Model (Fig. S4)

We argue in the main text that entrainment of circadian clocks by the cell cycle should be a robust, generic phenomenon that does not depend on the precise model studied. In support of this claim, we here present a model of the Kai system, the TTC–(PPC_Rust) model, that combines the TTC of Zwicker et al. (36) with a model of the PPC developed by Rust et al. (26); this model is here extended to include a description of the cell cycle with periodic gene replication. The PPC model of Rust et al. (26) describes the PPC at the level of KaiC monomers, rather than of hexamers, as in the models of Van Zon et al. (25) and Zwicker et al. (36) considered in the preceding section and in the main text. The monomers go through a sequence of four different phosphorylation states—unphosphorylated (U), phosphorylated only on threonine 432 (T), doubly phosphorylated (D), and phosphorylated only on serine 431 (S)—in a 24-h cycle (38). As detailed in the supplementary information of ref. 36, we extended the original Rust model, which contains only the PPC, to include a transcription–translation cycle. Here, we briefly describe the model and discuss how it responds to forcing from the cell cycle.

This PPC is described by the reactions

$$\text{U}\leftrightarrow \text{T},\hspace{1em}\text{T}\leftrightarrow \text{ST},\hspace{1em}\text{ST}\leftrightarrow \text{S},\hspace{1em}\text{S}\leftrightarrow \text{U}$$ [S10]

with reaction rates given by equation 5 of the supplementary material of Rust et al. (26). These rates depend on the concentration of free KaiA, which is sequestered by KaiC in the S-state. We model KaiA sequestration explicitly:

$$\text{S}+\text{A}\leftrightarrow \text{AS},\hspace{1em}\text{AS}+\text{A}\leftrightarrow {\text{A}}_2\text{S}.$$ [S11]

Dephosphorylation of KaiC in the S-state is allowed to occur even when KaiA is bound, in which case the KaiA protein is released from the complex. We define the output signal as

$$p\left(t\right)=\frac{\left[\text{T}\right]+\left[\text{ST}\right]+\left[\text{S}\right]+\left[\text{AS}\right]+\left[{\text{A}}_2\text{S}\right]}{\left[\text{U}\right]+\left[\text{T}\right]+\left[\text{ST}\right]+\left[\text{S}\right]+\left[\text{AS}\right]+\left[{\text{A}}_2\text{S}\right]},$$ [S12]

which resembles the phosphorylation ratio in the case where we cannot distinguish between singly and doubly phosphorylated KaiC. The denominator in the above expression is also the total KaiC monomer concentration. We use the same total protein concentrations as Rust et al. (26), $\left[\text{KaiA}\right]=\left[1.3\right]\text{μM}$ (active KaiA monomers) and $\left[\text{KaiC}\right]=\left[3.4\right]\text{μM}$ (KaiC monomers).

The TTC is modeled as

$$\mathrm{\varnothing }\underset{\tau }{\stackrel{\beta \tilde{G}\left(t\right){\left[\text{R}\right]}^4/\left(K^4+{\left[\text{R}\right]}^4\right)}{\Rightarrow }}\text{ST},\hspace{1em}\mathrm{\varnothing }\stackrel{{\beta }_a}{\to }\text{A},\hspace{1em}\mathrm{\varnothing }\stackrel{{\beta }_r}{\to }\tilde{\text{R}},$$ [S13]

$$\tilde{\text{R}}+\text{T}\stackrel{k_a}{\to }\text{R}+\text{T},\hspace{1em}\text{R}+{\text{A}}_x\text{S}\stackrel{k_i}{\to }\tilde{\text{R}}+{\text{A}}_x\text{S},\hspace{1em}x\in \left\{\mathrm{0,1,2}\right\},$$ [S14]

$$\text{A},\text{R},\tilde{\text{R}},\text{U},\text{S},{\text{A}}_x\text{S},\text{T},\text{ST}\stackrel{\mu }{\to }\mathrm{\varnothing }.$$ [S15]

The first line describes protein production and decay; as in Eq. S9, the double arrow indicates a delayed reaction. The second line summarizes the RpaA signaling pathway, where KaiC that is phosphorylated at the T site activates RpaA and KaiC that is phosphorylated at the S site represses RpaA activation.

As in our other models, we introduced the effect of gene replication by making the production rate of KaiC proportional to the production density $\tilde{G}\left(t\right)$. We use the parameters given in ref. 26 and in section S5.2 of the supplementary information of ref. 36, except for $\beta =1.29\cdot {10}^3{\text{h}}^{-1}$ and ${\mu }_{\mathrm{tot}}=0.1\hspace{0.5em}{\text{h}}^{-1}$.

Fig. S4 shows the peak-to-peak times of the phosphorylation fraction of this model as a function of the cell-division time $T_{\text{d}}$. Clearly, this TTC–(PPC_Rust) model, which combines a TTC with a PPC based on the model of Rust et al. (26), responds to forcing from the cell cycle in essentially the same way as the TTC–(PPC_Zwicker) model (36), which couples the same TTC with a PPC based on the model of Van Zon et al. (25) and Zwicker et al. (36). This supports the idea that the benefit of a self-contained protein-modification oscillator is a generic feature of biological clocks.

The NTFO with Intrinsic Noise (Fig. S1)

To test the effect on the locking mechanism of intrinsic noise in the chemical reactions that constitute the circadian clock, we simulate the NTFO using kinetic Monte Carlo simulations of the chemical master equation. Eq. 2 of the main text implies that when this intrinsic noise is neglected, the number of proteins $N_{\text{C}}$ in the cell obeys

$$\frac{d{N}_{\text{C}}\left(t\right)}{dt}=\beta g\left(t-\mathrm{\Delta }\right)\frac{K_{\text{C}}^2}{K_{\text{C}}^2+{\left(\frac{N_{\text{C}}\left(t-\mathrm{\Delta }\right)}{V\left(t\right)}\right)}^2}-{\mu }_{\mathrm{act}}{N}_{\text{C}}\left(t\right).$$ [S16]

Here, $V\left(t\right)$ is the cell volume, which grows exponentially as described in the main text, and ${\mu }_{\mathrm{act}}$ is the active contribution to the total decay rate ${\mu }_{\mathrm{tot}}={\mu }_{\mathrm{act}}+{\mu }_{\text{d}}$ in Eq. 2; the term ${\mu }_{\text{d}}$ contributes to the total apparent decay rate for the protein concentration, but not for the protein number. To include the effects of intrinsic noise on the evolution of $N_{\text{C}}$, we adapted the standard kinetic Monte Carlo algorithm to take into account delayed reactions, volume growth, and gene replication, as described in ref. 36: The cell volume is increased at discrete time intervals, and reaction propensities are recalculated after each volume update. After each cell division, the protein number $N_{\text{C}}$ is chosen from a binomial distribution, with $N_{\text{C}}$ halved on average. Gene replication is included through the time dependence of the gene copy number $g\left(t\right)$, whose behavior is detailed in the main text.

The parameters used in the simulation are $\beta =6.0\cdot {10}^3{\text{h}}^{-1}$, $K_c=1.0\hspace{0.5em}\mathrm{\mu }\text{M}$, $n=2$, and $\mathrm{\Delta }=8\hspace{0.5em}\text{h}$. The time average $\overline{V}$ of the cell volume is chosen to be 1$\mathrm{\mu }{\text{m}}^3$. For different cell division times $T_{\text{d}}$, we change the active degradation rate such that the total decay rate is kept constant at $0.2\hspace{0.5em}{\text{h}}^{-1}$: ${\mu }_{\mathrm{act}}=0.2\hspace{0.5em}{\text{h}}^{-1}-\text{log}\left(2\right)/T_{\text{d}}$. As in the main text, this allows us to disentangle the effect of gene replication on locking from that of simple protein dilution.

We vary the cell division time $T_{\text{d}}$ from 6 to 52 h in 0.1-h intervals and simulate a single trajectory of 10,000 h for each $T_{\text{d}}$. From the trajectories, we extract the peak-to-peak times of the oscillations (Supporting Information, Methods). Fig. S1 shows the average peak-to-peak time of the stochastic NTFO model for different values of the cell division time, with initial gene copy number $N=1$. Two differences from Fig. 2 of the main text, where the system obeys deterministic equations, are worthy of note. First, the 1:1 locking region is much larger when intrinsic noise is included, whereas the width of the 2:1 locking region has decreased. Intrinsic noise can thus dramatically change the extent of locking. We leave a full investigation of the origins of this effect for future work. Second, because of intrinsic noise, the amplitude is considerably more variable, and the variances in the peak-to-peak times outside of the locking regions are much larger. As in the mean-field model, the variances are very small around $T_{\text{d}}=24$ h, due to locking.

The Effect of Noise in the Timing of Gene Replication Events (Fig. S2)

In the model of the cell cycle described in the main text, gene replication events occur at perfectly regular, evenly spaced intervals in time. In this section, we explore how stochastic variability in the timing of these replications affects the interaction between the cell cycle and other oscillators. We consider a model in which each gene replicates exactly once in each cell division cycle, at a time $t_{\text{g}}$ between 0 and $T_{\text{d}}$, where $T_{\text{d}}$ is the length of the cell cycle and does not vary. The times $t_{\text{g}}$ are drawn in each cell cycle independently from a Gaussian distribution of mean ${\overline{t}}_{\text{rep}}=T_{\text{d}}/2$ and variance ${\sigma }_{\text{rep}}^2$. Replication times $t_{\text{g}}\prime$ that fall outside the interval [0,$T_{\text{d}}$), are mapped back onto it via $t_{\text{g}}=\text{Mod}\left(t_{\text{g}}\prime ,T_{\text{d}}\right)$.

We assume that the SD in the gene replication times, ${\sigma }_{\text{rep}}$, is proportional to the cell division time: ${\sigma }_{\text{rep}}\propto T_{\text{d}}$. Because this quantity is, to our knowledge, not known for S. elongatus, we varied this quantity between zero and a value that corresponds to replication times being chosen randomly from a uniform distribution, ${\sigma }_{\text{rep}}/T_{\text{d}}=1/\sqrt{12}\approx 0.3$.

Fig. S2 shows the effects of introducing this variability on the behavior of our clock models. For the NTFO (Fig. S2A), the locking regions are reduced in size but still clearly noticeable. The variance in the peak-to-peak times is typically larger than in deterministic case. For the TTC–(PPC_Zwicker) model (36) (Fig. S2B), the locking regions have almost disappeared, but the variance in $T_{\text{PtP}}$ is nearly unaffected by the noise in the timing of gene replication. Fig. S2C shows representative time traces of the total KaiC concentration, $C_{\mathrm{tot}}\left(t\right)$ and the phosphorylation fraction $p\left(t\right)$, for $T_{\text{d}}=48$ h. Clearly, $C_{\mathrm{tot}}\left(t\right)$ shows much more variability in the height of its peaks than does $p\left(t\right)$, further demonstrating the value of a posttranslational oscillator in insulating a circadian clock from influences from the cell cycle. We also note that, as in the deterministic limit, the amplitude of the $C_{\mathrm{tot}}$ oscillation cycles still (despite the noise) alternates between a high and a low value when gene replications occur only once every 48 h. It should thus be possible to observe experimentally that $C_{\mathrm{tot}}\left(t\right)$ is markedly affected by periodic gene replication, whereas $p\left(t\right)$ is much less so.

Phase Diagrams for the NTFO, TTC–(PPC_Zwicker), and TTC–(PPC_Rust) Models (Fig. S3)

To get a better understanding of how the cell cycle perturbs the clock, we have made phase diagrams for both the width of the 1:1 locking region and the average variance of the peak-to-peak times, as a function of the two key variables that affect locking, the number of gene copies N and the SD in their replication times, ${\sigma }_{\text{rep}}$, for all models, that is, the NTFO model, the TTC–(PPC_Zwicker) model of Zwicker et al. (36), and the TTC–(PPC_Rust) model (26, 36).

To make phase diagrams as a function N and ${\sigma }_{\text{rep}}$, we need to extend our model of stochasticity in the timing of gene replication introduced earlier in this document for $N=1$, to also allow for a higher gene copy number, $N>1$. Each cell cycle has N gene replication events that occur at times $t_{\text{g}}^i$ within the interval [0,$T_{\text{d}}$). Each replication time is drawn from a Gaussian distribution with a SD ${\sigma }_{\text{rep}}$ that is proportional to $T_{\text{d}}$, and a mean ${\overline{t}}_{\text{g}}^i=T_{\text{d}}/N\left(\frac{1}{2}+i\right)$, where $i\in \left\{0,\dots ,N-1\right\}$ for each cell cycle. Replication times $t_{\text{g}}^{\prime i}$ that fall outside the interval [0,$T_{\text{d}}$), are mapped back onto it via $t_{\text{g}}^i=\text{Mod}\left(t_{\text{g}}^{\prime i},T_{\text{d}}\right)$. Note that this mapping slightly decreases the variance in the replication times; the real variance will therefore be slightly less than ${\sigma }_{\text{rep}}^2$ (at most 10%).

We run simulations with values for ${\sigma }_{\text{rep}}/T_{\text{d}}$ in the range $\left[\mathrm{0,0.3}\right]$, which, as mentioned above, corresponds to a scenario where the replication time is essentially chosen at random between 0 and $T_{\text{d}}$. We consider initial gene copy numbers of $N\in \left\{\mathrm{1,2,3,4}\right\}$.

Fig. S3 A, E, and I show the phase diagrams of the width of the 1:1 locking regions as a function of N and ${\sigma }_{\text{rep}}/T_{\text{d}}$, for the NTFO, TTC–(PPC_Zwicker), and TTC–(PPC_Rust) models, respectively. Fig. S3 B, F, and J, respectively, show cuts through these phase diagrams. Here, we consider an oscillator to be 1:1 locked to the cell cycle when the difference between its average PtP-time, $\langle T_{\text{PtP}}\rangle$, and the cell-division time $T_{\text{d}}$ is less than 0.05 h: $\langle T_{\text{PtP}}\left(T_{\text{d}}\right)\rangle -T_{\text{d}}<0.05$.

Fig. S3 C, G, and K show the phase diagrams of the average variance in the peak-to-peak times, $\langle {\sigma }_{\text{PtP}}^2\rangle$, again as a function of N and ${\sigma }_{\text{rep}}/T_{\text{d}}$, for the NTFO, TTC–(PPC_Zwicker), and TTC–(PPC_Rust) models, respectively. Here, the variance is averaged over a range of cell-division times $6<T_{\text{d}}<52$:

$$\langle {\sigma }_{\text{PtP}}^2\rangle =\frac{1}{N_{\text{sim}}}{\displaystyle \sum _{i=1}^{N_{\text{sim}}}{\sigma }_{\text{PtP}}^2\left(T_{\text{d}}^i\right)},$$ [S17]

where $N_{\text{sim}}$ is the number of evenly spaced simulations performed in the range $6<T_{\text{d}}<52$ in steps of 0.1 h, and ${\sigma }_{\text{PtP}}^2\left(T_{\text{d}}^i\right)$ is the variance in the PtP-times for 5,000 h of simulated time at cell-divistion time $T_{\text{d}}^i$. This quantity is a measure for the erratic behavior induced by the coupling to the cell cycle outside the locking region.

It can be seen that for all models both the width of the 1:1 locking region (Fig. S3 A and B, E and F, and I and J) and the variance in the peak-to-peak times (Fig. S3 C and D, G and H, and K and L) initially rapidly decreases with N but then reaches a plateau. Even in the limit that $N\to \infty$, there is still some weak, residual driving by the cell cycle, because N rises linearly whereas the volume V rises exponentially during the cell cycle, leading to periodic variations in the gene density $G\left(t\right)=N\left(t\right)/V\left(t\right)$. It is also seen that the width of the 1:1 locking region decreases with increasing ${\sigma }_{\text{rep}}$, especially when N is small. However, although increasing the noise in the timing of replication reduces the width of the locking region, it also strongly increases the variance in the peak-to-peak times. A reliable clock requires not only a small locking window but also a small peak-to-peak variance. Clearly, allowing for stochasticity in the timing of replication is not a solution to the locking problem.

Comparing the different models, it is seen that building the clock around a PPC reduces not only the width of the locking region, but also the variance in the peak-to-peak times, both in the TTC–(PPC_Zwicker) (Fig. S3 E–H) and in the TTC–(PPC_Rust) model (Fig. S3 I–L). However, with $N=1$ chromosome copy at the beginning of the cell cycle, the introduction of the PPC is not sufficient to fully eliminate locking. A TTC–PPC model still requires multiple chromosome copies that are replicated asynchronously.

Allowing the Protein Decay Rate to Vary with Growth Rate (Fig. S8)

The total protein decay rate ${\mu }_{\mathrm{tot}}$ depends on the rate of active protein degradation ${\mu }_{\mathrm{act}}$ and on the rate of dilution due to cell growth ${\mu }_{\text{d}}$. In the results shown in the main text, we adjusted the active degradation rate with the growth rate so that the total protein decay rate remained constant—this allowed us to zoom in on the effect of locking that is due to periodic gene replication while ignoring confounding changes to clock behavior that might arise because of the variation of ${\mu }_{\mathrm{tot}}$.

Fig. S8.

The effect of allowing the total protein decay rate ${\mu }_{\mathrm{tot}}$ to vary with the cell-division time $T_{\text{d}}$, both for an NTFO (A) and for the TTC–(PPC_Zwicker) model (36) of a clock, which combines a TTC with a PPC (B; note the vertical scale is different from that in A). The blue lines correspond to the scenario in which the total degradation rate is kept constant at ${\mu }_{\mathrm{tot}}=0.1/\text{h}$ (as in Figs. 2A and 4A of the main text), and the red lines to the scenario in which the total degradation rate depends on the division time as ${\mu }_{\mathrm{tot}}=0.1\hspace{0.5em}{\text{h}}^{-1}+\text{log}\left(2\right)/T_{\text{d}}\left[1/\text{h}\right]$. When ${\mu }_{\mathrm{tot}}$ depends on $T_{\text{d}}$, we adjust the KaiC production rate β such that the intrinsic period of the clock remains 24 h. For both clock models and both choices of ${\mu }_{\mathrm{tot}}$, the clock tends to lock to the cell cycle. The difference between the results of the two protein-decay scenarios in the case of the NTFO can be understood by noticing that we have chosen our rates so that ${\mu }_{\mathrm{tot}}$ never drops below 0.1 ${\text{h}}^{-1}$ in either case, but can become larger than this bound when it is allowed to depend on $T_{\text{d}}$. The protein synthesis rate β then becomes higher than when ${\mu }_{\mathrm{tot}}$ is constant. The higher synthesis and decay rate raises the amplitude of the concentration oscillations, making the clock more stable. As the inset indicates, the width of the locking region decreases with increasing ${\mu }_{\mathrm{tot}}$ in a similar fashion when ${\mu }_{\mathrm{tot}}$ does not depend on $T_{\text{d}}$. The TTC–PPC is almost insensitive to the higher degradation and production rates.

It is entirely possible, however, that the real clock system in S. elongatus is actually closer to the opposite limit, in which ${\mu }_{\mathrm{act}}$ is fixed and ${\mu }_{\text{d}}$ and ${\mu }_{\mathrm{tot}}$ vary together with the division time $T_{\text{d}}$. To investigate how this affects the locking mechanism, we performed simulations in which we kept the active degradation rate ${\mu }_{\mathrm{act}}$ constant but allowed the total degradation rate ${\mu }_{\mathrm{tot}}$ to vary with the cell-division time: ${\mu }_{\mathrm{tot}}={\mu }_{\mathrm{act}}+\text{ln}\left(2\right)/T_{\text{d}}$; upon varying ${\mu }_{\mathrm{tot}}$, we adjusted the protein synthesis rate β to keep the oscillation period at 24 h. The result is shown in Fig. S8. Just as in the case in which ${\mu }_{\mathrm{tot}}$ is fixed, the effect of locking is very pronounced, both for a simple NTFO (Fig. S8A) and for a clock incorporating both a TTC and a PPC (Fig. S8B).

How Does the Cell Read Out the Phosphorylation Fraction Instead of the Concentration of KaiC? (Fig. S5)

Both Fig. 4 and a comparison of the blue lines in Fig. S2 A and B show that a circadian clock with a posttranslational oscillator is much more robust to a time-varying gene density than is an NTFO. Both the peak-to-peak time and the amplitude of the phosphorylation fraction $p\left(t\right)$ of the TTC–PPC vary considerably less not just than the corresponding quantities for the NTFO, but even than those for the total protein concentration $C_{\mathrm{tot}}\left(t\right)$ of the TTC–PPC itself. To take advantage of the relative stability of the oscillation in $p\left(t\right)$, the cell needs to read out the phosphorylation fraction in a way that is insensitive both to the total concentration of KaiC and to the absolute concentrations of its specific phosphorylation states. How does it accomplish this?

To find out, we looked at the architecture of the biochemical network in S. elongatus that allows the clock to regulate the transcription of downstream genes. The temporal information encoded in the dynamics of the clock proteins is transmitted via a central node, the response regulator RpaA. This protein can be phosphorylated, and it is known that the phosphorylation level of RpaA controls the expression not only of core clock components but also of a small set of genes that, in turn, direct genome-wide circadian rhythms (30, 33). The phosphorylation state of RpaA is regulated by a push–pull network consisting of the histidine kinase SasA, which acts primarily to phosphorylate RpaA, and CikA, whose primary function is to dephosphorylate it (30–32). The activities of SasA and CikA are in turn controlled by the different phosphorylation states of KaiC, such that KaiC proteins that are in the phosphorylation phase of the clock tend to push up the phosphorylation level of RpaA (30, 34), whereas those that are in the clock’s dephosphorylation phase tend to pull down RpaA’s phosphorylation level (31, 34). Recent experiments suggest that a structural change in KaiB, which binds KaiC during its dephosphorylation phase, might play a key role in this switch, by simultaneously blocking SasA from binding to KaiC and engaging CikA (34).

To show that such a push–pull network makes the clock readout, the RpaA phosphorylation level, sensitive to the KaiC phosphorylation fraction $p\left(t\right)$, but not to its concentration $C_{\mathrm{tot}}\left(t\right)$, we adapted the canonical model of Goldbeter and Koshland (39). A cartoon of the model is shown in Fig. S5B. It consists of a substrate S, playing the role of RpaA, which can be phosphorylated and dephosphorylated by the antagonistic enzymes K and P, corresponding, respectively, to SasA and CikA, each of which can be in active (${\text{K}}^{\ast }$ and ${\text{P}}^{\ast }$) or inactive states. Transitions between these states are governed by the time-dependent forward rates $k_{\text{K}}\left(t\right)$ and $k_{\text{P}}\left(t\right)$, which mimic the effects of time-varying concentrations of different KaiC phosphoforms. In our model, KaiC in the phosphorylation phase increases $k_{\text{K}}\left(t\right)$, whereas KaiC in the dephosphorylation phase increases $k_{\text{P}}\left(t\right)$. The substrate-modification reactions follow the standard Michaelis–Menten schemes ${\text{K}}^{\ast }+\text{S}\rightleftharpoons {\text{K}}^{*}\text{S}\to {\text{K}}^{*}+{\mathrm{S}}_{\text{p}}$ and ${\text{P}}^{\ast }+{\mathrm{S}}_{\text{p}}\rightleftharpoons {\text{P}}^{*}{\mathrm{S}}_{\text{p}}\to {\text{P}}^{*}+\text{S}$.

The key question is whether the phosphorylation level of RpaA depends only on the phosphorylation fraction of KaiC, $p\left(t\right)$, or whether it is also significantly affected by the total concentration of KaiC. In terms of the model just presented, we must thus ask whether $\left[{\mathrm{S}}_{\text{p}}\right]/\left[{\text{S}}_{\mathrm{tot}}\right]$ (the fraction of phosphorylated RpaA) is sensitive only to the ratio $k_{\text{K}}\left(t\right)/k_{\text{P}}\left(t\right)$ or whether it depends on the two rates individually. We begin by considering this question for the case of time-independent rates, in which case the ratio of total concentrations of the active enzyme forms $\left[{\text{K}}^{\ast }\right]/\left[{\text{P}}^{\ast }\right]$ is proportional to $k_{\text{K}}\left(t\right)/k_{\text{P}}\left(t\right)$ and may be used in its place. In qualitative terms, it is easy to imagine that, because ${\text{K}}^{\ast }$ and ${\text{P}}^{\ast }$ have opposite effects, increasing the concentration of each one by the same factor might speed up the reaction kinetics but would not affect the steady-state ratio $\left[{\mathrm{S}}_{\text{p}}\right]/\left[{\text{S}}_{\mathrm{tot}}\right]$. Indeed, in the regime that the concentrations of the complexes ${\text{K}}^{\ast }\text{S}$ and ${\text{P}}^{\ast }{\mathrm{S}}_{\text{p}}$ are negligible compared with the concentrations of free S and ${\mathrm{S}}_{\text{p}}$, we can equate the rates of S phosphorylation and dephosphorylation in steady state to arrive at a relation of the form

$$\frac{k_{+}\left[{\text{K}}^{*}\right]\left[\text{S}\right]}{K_{M,+}+\left[\text{S}\right]}=\frac{k_{-}\left[{\text{P}}^{*}\right]\left[{\mathrm{S}}_{\text{p}}\right]}{K_{M,-}+\left[{\mathrm{S}}_{\text{p}}\right]}.$$ [S18]

Together with the conservation law $\left[\text{S}\right]+\left[{\mathrm{S}}_{\text{p}}\right]=\left[{\text{S}}_{\mathrm{tot}}\right]$, this equation can easily be solved to give $\left[{\mathrm{S}}_{\text{p}}\right]/\left[{\text{S}}_{\mathrm{tot}}\right]$ as a function of $\left[{\text{K}}^{\ast }\right]/\left[{\text{P}}^{\ast }\right]$ only. It is possible, however, that in the RpaA system the concentrations of the intermediate complexes ${\text{K}}^{\ast }\text{S}$ and ${\text{P}}^{\ast }{\mathrm{S}}_{\text{p}}$ cannot be neglected. Fig. S5A shows that even in this case, to a very good approximation the steady-state phosphorylation fraction of S depends only on the ratio $\left[{\text{K}}^{\ast }\right]/\left[{\text{P}}^{\ast }\right]$. This mechanism, by which the output of a push–pull network depends on the ratio of the concentrations of the two antagonistic enzymes, but not on their absolute values, has previously been invoked to explain the robustness of the Escherichia coli chemotaxis pathway to concerted variations in the expression levels of the chemotaxis proteins (44).

To extend these results to the case of time-varying activation rates that is more directly relevant to a situation in which the concentrations of the KaiC phosphoforms rise and fall, we allow $k_{\text{K}}\left(t\right)$ and $k_{\text{P}}\left(t\right)$ to vary with time as shown in Fig. S5C, Upper: Similarly to the total KaiC concentration at $T_{\text{d}}=48$ h with $N=1$ (Fig. 4C of the main text), each rate changes periodically with a period of 48 h, with alternating higher and lower peaks. Fig. S5C, Lower, with time traces of $\left[{\mathrm{S}}_{\text{p}}\right]$, $\left[{\text{K}}^{\ast }\right]$ and $\left[{\text{P}}^{\ast }\right]$, shows that even though the amplitude of $\left[{\text{K}}^{\ast }\right]\left(t\right)$ and $\left[{\text{P}}^{\ast }\right]\left(t\right)$ varies between oscillation cycles, the amplitude of $S_{\text{p}}\left(t\right)$ is constant. Comparable behavior is expected as long as the rates associated with the enzymatic reactions are faster than the timescale of variation of $k_{\text{K}}\left(t\right)$ and $k_{\text{P}}\left(t\right)$. Provided this is the case, the push–pull system built around RpaA will allow the bacterium to take as its clock readout the phosphorylation fraction $p\left(t\right)$—which we have argued is robust to perturbations associated with the cell cycle—rather than the absolute concentration of KaiC or one of its phosphoforms.

Multiple Chromosomes Reduce the Coupling Between Clock and Cell Cycle in TTC–PPC and NTFO Models (Fig. S9)

Fig. 5 C and D show, respectively, the width of the 1:1 locking region and the square root of the average variance in the peak-to-peak time as a function of the standard deviation in the gene replication time σ_rep, both for N = 1 and N = 4, and for both the NTFO and the TTC–(PPC_Zwicker) model (36). For the same model, Fig. S9A shows the average peak-to-peak time and its SD, as a function of the cell-division time T_d, both for N = 1 and for N = 4. Underneath it are shown, for $N=4$, time traces for $p\left(t\right)$ and $C_{\mathrm{tot}}\left(t\right)$ at $T_{\text{d}}=24.5$ h (Fig. S9B) and $T_{\text{d}}=48$ h (Fig. S9C), as also indicated by the arrows in Fig. S9A. As discussed in the main text, the effects of the cell cycle have almost completely disappeared, even at $T_{\text{d}}=24.5$ h, immediately outside the locking regime, where the effects are usually most visible.

Fig. S9.

Multiple chromosome copies reduce the effect of periodic gene replications in both the TTC–(PPC_Zwicker) model (36) (A–C) and in the simple NTFO (D–G). A shows for the TTC–PPC model the average peak-to-peak time as a function of the cell-division time $T_{\text{d}}$ for initial gene copy numbers $N=1$ (blue) and $N=4$ (red). Both the SD in the peak-to-peak times (given by the shaded regions) and the regions where the oscillator is locked to the cell cycle are strongly reduced for $N=4$. (B) The total KaiC concentration $C_{\mathrm{tot}}\left(t\right)$ (dashed line) and phosphorylation fraction $p\left(t\right)$ (solid line) for the TTC–PPC model at $T_{\text{d}}=24.5$ h, for $N=4$. (C) $C_{\mathrm{tot}}\left(t\right)$ and $p\left(t\right)$ at $T_{\text{d}}=48$ h for the same model, again for $N=4$. Note that, even at $T_{\text{d}}=24.5$ h, immediately outside the locking region where the variance in $T_{\text{PtP}}$ is generally largest, the effects of the cell cycle have almost completely disappeared. (D) The average peak-to-peak time and its SD for the NTFO model, for $N=1$ (blue line) and $N=4$ (red line). It is seen that as in the Zwicker model, multiple chromosome copies dramatically reduce the strength of locking. (E–G) NTFO time traces of the protein-concentration oscillations $C\left(t\right)$ (blue lines) and the production density $\tilde{G}\left(t\right)$ (red lines), both normalized to their time average values, for $N=4$, and for cell-division times indicated by the arrows in D. With $N=4$, only at, or very close to, $T_{\text{d}}=24$ h (E) is the NTFO locked to the cell cycle. Even at $T_{\text{d}}=27$ h (F) and $T_{\text{d}}=48$ h (G), where for $N=1$ the NTFO shows irregular behavior and large amplitude variations, respectively (see Fig. 2 C and D), the time courses are much less perturbed by the cell cycle when $N=4$. However, albeit greatly reduced, the effect of driving by the cell cycle can nonetheless still be observed, both in the persistence of small regions of locking and in the still appreciable variance in $T_{\text{PtP}}$ when the oscillators are not locked (D). A PPC must be added to more fully attenuate the cell cycle’s influence. (Compare D with A, taking into account the difference in scale of the y axis; the SD of $T_{\text{PtP}}$ is about three times larger for the NTFO than for the full TTC–PPC model.)

Fig. S9 D–G show the effect of multiple chromosomes on the locking behavior of the NTFO model. Fig. S9D shows that with $N=4$ chromosome copies at the beginning of the cell cycle, both the variance and the width of the locking regions have decreased to no more than 1 h. The time traces (Fig. S9 E–G) of the protein concentration at cell division times of 24, 27, and 48 h confirm that the NTFO has indeed become very stable. At $T_{\text{d}}=27$ h, where the protein concentration had showed irregular behavior and large-amplitude variations for $N=1$, its behavior has now become much more regular. At a division time of 48 h, the marked amplitude variations present when $N=1$ have disappeared. Indeed, the beneficial effect of a higher gene-copy number is larger than that of adding a PPC, as can be seen by comparing Fig. S9D with Fig. 4 of the main text. Fig. S3 shows that essentially the same effect is also found in the TTC–(PPC_Zwicker) model. Of course, the most stable clock is obtained when a higher gene copy number is combined with a PPC (Fig. 5), suggesting that both are needed for circadian rhythms that are maximally resilient against perturbations from the cell cycle.

Two Synthetic Oscillators: The Repressilator and the Dual-Feedback Oscillator (Figs. S6 and S7)

Although in the main text we focus on circadian clocks, the mechanism of entrainment via periodic gene replication is very generic and should thus pertain to any cellular oscillator, including synthetic oscillators. To investigate this, we study how strongly two canonical synthetic oscillators, both constructed in E. coli, can lock to the cell cycle: the repressilator, developed by Elowitz and Leibler (50), and the dual-feedback oscillator, developed by Hasty and coworkers (51). Importantly, these synthetic oscillators were originally constructed on plasmids, which often occur in large copy numbers: The plasmid copy number of the dual-feedback oscillator, for example, has been estimated to be around 25 (51). Moreover, experiments indicate that these plasmids are copied at random times during the major part of the cell cycle (55). In line with our observation that multiple chromosome copies that are replicated asynchronously strongly reduce the strength of locking (Fig. 5 of the main text and Fig. S3), we find that with multiple (i.e., $N=25$) asynchronously replicating plasmids these synthetic oscillators lock to the cell cycle only very weakly: Both the width of the 1:1 locking region and the variance in the peak-to-peak time are less than a percent of their respective average.* We thus expect that with multiple asynchronously replicating plasmids, locking of these synthetic oscillators to the cell cycle is difficult, if not impossible, to detect experimentally.

However, synthetic networks are increasingly being constructed directly onto the chromosome. If the repressilator and dual-feedback oscillator were similarly integrated into the chromosome of E. coli, then strong coupling to the cell cycle can be expected, as we will show below. E. coli typically has one chromosome at the beginning of the cell cycle, in which case the gene copy number goes from 1 to 2 over the course of the cell cycle. At high growth rates, corresponding to cell division times shorter than the replication time of the DNA (on the order of 40 min), the chromosome can have multiple replication forks, which means that the gene copy number can be larger. Here, however, we only consider the regime that the cell division time is on the order of the DNA replication time or longer, such that the gene copy number rises from $N=1$ at the beginning of the cell cycle to $2N=2$ at the end. Interestingly, both synthetic oscillators consist of more than one gene, in contrast to the NTFO of the main text. Moreover, the period of both synthetic oscillators can be on the order of the cell-cycle time and the DNA replication time. This means that the spatial distribution of the genes on the chromosome can become critically important, as we will show below.

Repressilator (Fig. S6).

The repressilator consists of three genes, where the first gene represses the expression of the second, which represses the third gene, which in turn represses the expression of the first again (50). To take into account gene replication, we change the model of Elowitz and Leibler (50) by making the expression of the mRNA proportional to the gene density $G_i\left(t\right)$

$$\begin{array}{l}\frac{d{m}_i\left(t\right)}{dt}=-m_i\left(t\right)+\frac{G_i\left(t\right)}{\overline{G_i}}\frac{\alpha }{1+{\left(p{\left(t\right)}_j\right)}^n}+{\alpha }_0\\ \frac{d{p}_i\left(t\right)}{dt}=-{\mu }_p{p}_i\left(t\right)+\gamma m_i\left(t\right).\end{array}$$ [S19]

Here, $m_i$ and $p_i$ are the concentrations of mRNA and proteins ($i\in \left\{\mathrm{1,2,3}\right\}$), both rescaled with the constant of half-maximum repression $K_{\mathrm{m}}$. Because transcription is fast compared with the clock period, the delay $\mathrm{\Delta }$ in the expression of the mRNA is ignored—the transcription rate is thus assumed to be proportional to the instantaneous gene density $G_i\left(t\right)$; importantly, the gene density can differ between the three genes when they are positioned differently on the chromosome. ${\overline{G}}_i$ is the time-averaged gene density, which depends on the phase of the cell cycle at which the gene is duplicated. The mRNA expression has a basal rate ${\alpha }_0$ and an enhanced rate α, which is repressed by protein $p_j$, where $j\in \left\{\mathrm{3,1,2}\right\}$, with a Hill coefficient n; here, following the original paper (50), time is rescaled in units of the mRNA lifetime and protein concentrations are in units of the concentration necessary for half-maximal repression. In the second equation, ${\mu }_p$ is the protein decay rate over the mRNA decay rate and γ is the translation efficiency, that is, the average number of proteins produced per mRNA molecule. We used the parameters given in box 1 of ref. 50.

We first consider the scenario in which the three genes are close together on the chromosome, such that we can assume that their genes are replicated at the same time. Fig. S6 shows that in this case the locking is not very strong—the locking windows are very small; the only effect of locking is that in these very small windows the variance in the peak-to-peak time is strongly reduced. The reason why locking is weak is that although the genes are replicated at the same time, they are expressed at different times. This means that gene replication has a different effect on the expression level of each of the three genes. Hence, even when the cell-cycle period is approximately equal to the oscillator’s intrinsic period, $T_{\text{d}}\approx T_{\text{int}}$, the oscillation of each protein concentration has a different amplitude, as shown in Fig. S6B. This makes it harder for all three protein oscillations to get the same period as that of the cell cycle, and become locked to it. Interestingly, Fig. S6C shows that when the cell-cycle time is twice the intrinsic clock period, the pattern of alternating smaller and larger oscillation amplitudes can still be observed for each of the respective protein concentration profiles. As for the circadian clocks studied in the main text, we believe that this observation can be used to detect the effect of periodic gene replication experimentally.

We now consider a scenario in which the different genes are replicated at different times during the cell cycle, which corresponds to a situation where the genes are located at different positions on the chromosome. We assume that the gene for protein $p_1$ is replicated halfway through the cell cycle, at a time $d_1=0$; the gene for $p_2$ is replicated with a delay $d_2$ with respect to the replication of $p_1$; the gene for $p_3$ is replicated at a time $d_3=-d_2$. Thus, the gene for $p_3$ is replicated before that for $p_1$ when $d_2>0$, and after that for $p_1$ when $d_2<0$. Interestingly, whereas the locking regions are very small when the genes are replicated at the same time ($d_2=0$, Fig. S6A; gray lines, Fig. S6 D and E), replicating them at different times introduces marked locking: both for $d_2>0$ (Fig. S6D) and $d_2<0$ Fig. S6E) strong locking is observed. Even more strikingly, the 1:1 locking region is largest when $d_2<0$, meaning $p_2$ is replicated before $p_1$, which is replicated before $p_3$ (Fig. S6E). This can be understood by noting that in this case the genes are replicated in the same temporal order as that in which they are expressed in the oscillator (Fig. S6B): Shifting the phase of the clock with respect to that of the cell cycle has then the strongest effect on the amplitude and hence the period of the clock oscillations, which underlies the phenomenon of locking (see Fig. 3 of the main text).

In summary, the repressilator can strongly lock to the cell cycle. Moreover, the strength of locking depends sensitively on when the respective genes are replicated, and hence on their spatial position on the chromosome.

Dual-Feedback Oscillator (Fig. S7).

The dual-feedback oscillator consists of two genes, one encoding for an activator and one for a repressor (51). The activator enhances the expression of both genes, whereas the repressor represses the expression of both genes. Because the genes have identical promoters, the temporal expression of the two proteins is similar. The model we use is presented in the supporting information of ref. 51, but to take into account the periodic variations in the gene density, we have modified the equations describing the transcription of mRNA of the activator and repressor:

$$\begin{array}{l}{P}_{\mathrm{0,0}}^{a/r}\stackrel{b_{a/r}{G}_{a/r}\left(t\right)/G_{a/r}^{-}}{\to }{P}_{\mathrm{0,0}}^{a/r}+m_{a/r}\\ P_{\mathrm{1,0}}^{a/r}\stackrel{\alpha b_{a/r}{G}_{a/r}\left(t\right)/G_{a/r}^{-}}{\to }{P}_{\mathrm{1,0}}^{a/r}+m_{a/r}.\end{array}$$ [S20]

Here $P_{m,n}^{a/r}$ denotes the promoter of the (a)ctivator/(r)epressor gene, with $m=\mathrm{0,1}$ activator protein and $n=0$ repressor protein bound to it, respectively. The mRNA $m_{a/r}$ of the activator (a) and repressor (r) is transcribed with a rate $\left(\alpha \right)b_{a/r}G\left(t\right)$, which depends on the state of the promoter and on the gene density $G_{a/r}\left(t\right)$. Because the transcription of mRNA is fast compared with the period of the clock, we neglect the delay $\mathrm{\Delta }$ between the beginning and end of transcription. Parameters are given in the supporting information of ref. 51, with an arabinose inducer level of 0.7% and an isopropyl β-d-1-thiogalactopyranoside concentration of 2 nM. The intrinsic period of this oscillator without the driving by the gene density is ∼40 min. To study the importance of the timing of gene replication, we want an intrinsic period that is longer than the replication time of the DNA, which is also around 40 min. To obtain a longer clock period, we use the experimental observation in ref. 51 that the clock period scales with temperature via the Arrhenius law. To this end, we scale all rate constants, $k_i$, in the model, using

$$k_i=k_{\text{ref}}\hspace{0.17em}\text{exp}\left(-{\mathrm{\Theta }}_{\text{cc}}\left[1/T-1/T_{\text{ref}}\right]\right),$$ [S21]

where $k_{\text{ref}}$ is the rate constant at the reference temperature $T_{\text{ref}}$ of 310 K and ${\mathrm{\Theta }}_{\text{cc}}\approx \mathrm{8,300}$ K is a constant (see the supporting information of ref. 51). We will evaluate the model at a temperature of 303 K where the clock has an intrinsic period of 73 min.

Fig. S7A shows the locking of the dual-feedback oscillator with $N=1$ copy of each gene at the beginning of the cell cycle. We assume here that the genes are located next to each other on the chromosome, so that their time-varying gene-densities are the same. It is seen that the width of the locking regions is much larger than those of the NTFO. In Fig. S7B we show a time trace of the irregular oscillations around a cell-division time of $T_{\text{d}}=96$ min. Fig. S7C shows that, as observed for the circadian clocks studied in the main text, the amplitude of the oscillations alternates between a high and a low value when the cell-division time $T_{\text{d}}=146$ min is about twice the intrinsic clock period, due to periodic gene replication every other clock period. We thus conclude that also the dual-feedback oscillator can strongly lock to the cell cycle when it is constructed onto the chromosome, and that this effect should be observable experimentally.

Fig. S7 D and E shows the result of varying the moment of gene replication for the two genes. We let the activator gene always replicate halfway through the cell cycle and vary the time delay $d_r$ between the replication of the two genes, as $d_{\text{r}}=0,T_{\text{int}}/12,T_{\text{int}}/8$ and $T_{\text{int}}/4$ (Fig. S7D) and minus these values (Fig. S7E), where $T_{\text{int}}$ is the intrinsic clock period; Fig. S7D corresponds to the case in which the repressor is replicated after the activator, and Fig. S7E corresponds to the opposite scenario. It is seen that in both scenarios the strength of locking decreases: The strongest entrainment is observed when the genes are replicated at the same time during the cell cycle (gray lines), in stark contrast to the behavior of the repressilator. Combining the observations on the behavior of the repressilator and the dual-feedback oscillator, the conclusion is that also synthetic circuits can strongly lock to the cell cycle, and that the strength of locking depends very sensitively on both the architecture of the oscillator and on the timing of gene replication and hence on the spatial distribution of the genes on the chromosome.

Methods

The delay-differential equations (DDE) describing our models in the absence of noise were propagated using the numerical differential equation solver of Mathematica 8 (Wolfram Research). For each value of $T_{\text{d}}$, we generated a single time trace of 2,000 h. To allow the oscillations to settle down to a steady state, we discarded the first 500 h of each simulation and analyzed the remaining 1,500 h.

To find the peak-to-peak times $T_{\text{PtP}}$ in the deterministic DDE simulations (including those with noise in the gene replication times), we used the built-in methods of Mathematica to return all local extrema in the concentration and phosphorylation fraction, respectively; these extrema correspond to the time points $t_i$ where $C_i$ or, respectively, $p_i$, is higher, in the case of a maximum, or lower, in the case of a minimum, than its two immediate neighbors. As is standard for numerical solution of differential equations, the spacing $t_i-t_{i-1}$ between successive time points was determined adaptively by the algorithm to meet imposed precision bounds but never exceeded 0.2 h. We then checked if a given local minimum was the lowest point within an interval of ±18 h centered on the minimum; if so, we defined this point as the global minimum of a single oscillation cycle. If there did exist a local extremum with a lower value, we repeated this procedure around the lower point until we found a point which was the lowest within a time interval of ±18 h. The same procedure was followed for the local maxima. The peak-to-peak time was then calculated by subtracting the times of two consecutive minima; we verified that substracting the times of two consecutive maxima gave essentially the same results.

To find the peak to peak times in the kinetic Monte Carlo simulation of the NTFO, we record the protein concentration $c_i\left(t_i\right)$ every 0.01 h of simulated time. We then use a sliding window of 18 h over these concentrations to find the extrema of the oscillations. Specifically, to find the time $t_j$ of the next local minimum, starting from the maximum of the preceding oscillation cycle at time $t_i$ we find the smallest concentration $c_j$, with $j>i$, in the window $t_j-t_i\le 18$ h. (Given $c_i$ was a local maxmium, there must exist a concentration $c_j<c_i$.) We then check whether there exists a concentration $c_k<c_j$ for $k>j$ and $t_k-t_j\le 18$ h. If there is, we replace $c_j$ by $c_k$ and again search for a deeper minimum within 18 h; otherwise, $c_j$ is the minimum of the next oscillation cycle. A completely analogous procedure is used to identify the maxima of successive oscillation cycles.