1. Physics of Living Systems
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Robust, coherent, and synchronized circadian clock-controlled oscillations along Anabaena filaments

  1. Rinat Arbel-Goren
  2. Valentina Buonfiglio
  3. Francesca Di Patti
  4. Sergio Camargo
  5. Anna Zhitnitsky
  6. Ana Valladares
  7. Enrique Flores
  8. Antonia Herrero
  9. Duccio Fanelli
  10. Joel Stavans  Is a corresponding author
  1. Department of Physics of Complex Systems, Weizmann Institute of Science, Israel
  2. Dipartimento di Fisica e Astronomia, Università di Firenze, INFN and CSDC, Italy
  3. Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Italy
  4. Instituto de Bioquímica Vegetal y Fotosíntesis, CSIC and Universidad de Sevilla, Spain
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Cite this article as: eLife 2021;10:e64348 doi: 10.7554/eLife.64348

Abstract

Circadian clocks display remarkable reliability despite significant stochasticity in biomolecular reactions. We study the dynamics of a circadian clock-controlled gene at the individual cell level in Anabaena sp. PCC 7120, a multicellular filamentous cyanobacterium. We found significant synchronization and spatial coherence along filaments, clock coupling due to cell-cell communication, and gating of the cell cycle. Furthermore, we observed low-amplitude circadian oscillatory transcription of kai genes encoding the post-transcriptional core oscillatory circuit and high-amplitude oscillations of rpaA coding for the master regulator transducing the core clock output. Transcriptional oscillations of rpaA suggest an additional level of regulation. A stochastic one-dimensional toy model of coupled clock cores and their phosphorylation states shows that demographic noise can seed stochastic oscillations outside the region where deterministic limit cycles with circadian periods occur. The model reproduces the observed spatio-temporal coherence along filaments and provides a robust description of coupled circadian clocks in a multicellular organism.

Introduction

Endogenous circadian clocks allow the alignment of cellular physiology with diurnal light/darkness cycles on Earth, endowing organisms, from unicellular cyanobacteria to multicellular plants and mammals, with a selective fitness advantage (Cohen and Golden, 2015). Significant progress has been achieved in understanding circadian clock architectures and function in cyanobacteria, which are arguably the simplest organisms exhibiting self-sustained oscillations (Hasty et al., 2010; Rust et al., 2007; Lambert et al., 2016; Dong et al., 2010; Teng et al., 2013; Gan and O'Shea, 2017). The molecular mechanisms behind autonomous circadian clocks have been elucidated primarily in the unicellular Synechococcus elongatus. These investigations have shown that the core of the circadian clock consists of three proteins, KaiA, KaiB, and KaiC (Ishiura et al., 1998), whose oscillating behavior can be reconstituted in vitro (Nakajima et al., 2005). The basic mechanism behind the clock is based on the stimulation of KaiC autophosphorylation by the binding of KaiA and the autodephosphorylation that ensues when KaiB binds to KaiC, blocking KaiA’s stimulatory function. A salient feature of the circadian clock in Synechococcus is the high temporal precision it can exhibit, despite the fact that biochemical reactions in a cell are stochastic events and that clock components may be subject to variations in molecular copy numbers between cells, variations known as demographic noise (Tsimring, 2014). Many studies have addressed the robustness of circadian rhythms to demographic noise in Kai proteins (Mihalcescu et al., 2004; Chabot et al., 2007; Teng et al., 2013; Pittayakanchit et al., 2018; Chew et al., 2018), but copy number variations in KaiC phosphoforms, which impact directly on clock function, have not been previously considered.

The multicellular character of higher organisms and of some cyanobacterial species (Herrero et al., 2016) naturally prompts the question of how do ensembles of noisy circadian clocks perform in a multicellular organismal setting. Theoretical considerations have motivated the notion that reliable collective oscillations may result from the coupling of ‘sloppy’, noisy clocks (Enright, 1980). It has been suggested that coupling of circadian clocks in unicellular organisms by quorum sensing interactions may result in emergent synchronization (Garcia-Ojalvo et al., 2004), and experimental evidence in support of this notion has been reported in a synthetic system (Danino et al., 2010).

Anabaena sp. strain PCC 7120 (henceforth Anabaena) is a multicellular cyanobacterium in which cells are arranged in a one-dimensional configuration, with local, nearest-neighbor cell-cell coupling through septal junctions (Herrero et al., 2016). Evidence of coupling of metabolic pathways along a filament due to cell-cell communication has been reported (Mullineaux et al., 2008). In contrast to Synechococcus, information about the mechanism of the circadian clock of Anabaena is scant. Sequence BLAST analysis has shown that Anabaena contains homologs of the kaiA, kaiB, and kaiC genes of Synechococcus, and structural studies indicate that the interactions between the respective proteins are similar (Garces et al., 2004). Furthermore, Anabaena also contains factors coupling the Kai post-transcriptional oscillator to input signals and to output factors such as RpaA, CikA, and SasA that couple the clock to the genes it regulates (Schmelling et al., 2017). The roles of these genes in Anabaena remain to be elucidated. A first characterization of the circadian rhythms in bulk cultures of Anabaena has shown that the clock is autonomous, running freely under constant light conditions following exposure to two 12 hr light-dark cycles, similarly to Synechococcus (Kushige et al., 2013). However, this study also showed that, in contrast to Synechococcus, none of the kai genes are expressed with a large amplitude. Nonetheless, about 80 kai-controlled genes that exhibit high-amplitude circadian oscillations were identified using DNA microarray analysis, a behavior that was abolished in a kaiABC deletion mutant.

Here, we present an experimental and theoretical study of circadian clocks in multicellular Anabaena. Its one-dimensional character allowed us to follow clocks in each and every cell along a filament, and shed light on the interplay between demographic noise and cell-cell communication in setting synchrony and spatial coherence along filaments. In our experiments, we followed in vivo the output of clocks in individual cells by monitoring the expression from the promoter of pecB, a clock-controlled gene of high-amplitude oscillations (Kushige et al., 2013). This gene is part of the pecBACEF operon and codes for the beta subunits of phycoerythrocyanin, a structural component of the phycobilisome rod that plays a major role in light harvesting for photosynthesis (Swanson et al., 1992). On the theoretical side, we first incorporated the effects of demographic noise into a three-component model of a single clock describing the phosphorylation states of KaiC, as in Synechococcus. Next, we extended this single-cell stochastic model to describe a one-dimensional array of coupled clocks, as in multicellular Anabaena, allowing us also to analyze the spatio-temporal coherence properties of noisy oscillations in Anabaena filaments.

Results

Circadian clocks of individual cells in growing filaments are highly synchronized

We followed circadian oscillations in Anabaena from a chromosomal gfp fusion to the N-terminus of the clock-controlled protein, PecB, here denoted as PpecB-gfp at the level of individual cells along filaments. Prior to and during the experiments, filaments were grown under constant light conditions. Typical images of wild-type (WT) filaments at different time points are shown in Figure 1A (see also Figure 1, Video 1). One salient feature in the images is significant synchrony, that is, cellular oscillations progressed in individual cells along filaments together and with a similar period. The images in Figure 1A correspond to successive maxima and minima of the mean cell fluorescence intensity, µ, as a function of time, which we plot in Figure 2A. Similar experiments with a strain in which the kaiABC genes were deleted (ΔkaiABC) resulted in a low-level, non-oscillatory signal (Figure 2A), confirming the regulation of pecB expression by the circadian clock genes. As a control, we monitored expression from the promoter of hetR, a gene coding for the master regulator of development in Anabaena. The expression from the hetR promoter PhetR-gfp did not exhibit an oscillatory behavior (Figure 2A), a result consistent with previous microarray experiments (Kushige et al., 2013). This does not preclude a possible interaction between the circadian clock and differentiation. On the other hand, the autofluorescence from photosynthetic pigments did not display oscillatory behavior (see Figure 1B, Figure 2A).

Circadian oscillation in Anabaena.

(A) GFP fluorescence in a filament of an Anabaena strain bearing a PpecB-gfp promoter fusion, growing under nitrogen-replete conditions. The snapshots were chosen near maxima and minima of the circadian oscillations. (B) Autofluorescence as a function of time in Anabaena. Snapshots correspond to those in (A), and time 0 corresponds to the time at which filaments were placed in a device for microscope observation (for details, see Materials and methods). For a time-lapse movie, see Video 1 (taken over 6 days).

Figure 2 with 1 supplement see all
Characterization of a clock-controlled gene in Anabaena.

(A) Average cell fluorescence intensity from PpecB-gfp in a filament as a function of time for a wild-type genetic background (full black circles) and for a ΔkaiABC background (empty black circles); intensity of autofluorescence as a function of time (blue circles); average cell fluorescence intensity from PhetR-gfp (cyan line); and temporal dependence of the cell-cell variability CV2 in expression of PpecB-gfp (red line). Data were taken from at least 50 contiguous cells along a filament. (B) Average fluorescence intensity as a function of time of filaments in different fields of view from the same experimental run. Each trace was obtained from at least 50 contiguous cells along each filament. (C) Expression from a PpecB-gfp fusion in the lineages of two contiguous cells as a function of time. (D) Average spatial autocorrelation function of PpecB-gfp expression along filaments of wild-type (blue), ΔsepJ/ΔfraCD (green), and ΔkaiABC (orange) genetic backgrounds. Error bars represent standard errors. Magenta line: contribution to the spatial autocorrelation function of fluctuations from the wild-type data set, induced by binomial partitioning of molecules between daughter cells, following each of three consecutive cell divisions. Prior to divisions, the cell order in each filament was reshuffled. (E) Histogram of the phase of cell-division events along the circadian cycles, with 0 and 2π denoting two consecutive minima, from two independent experiments. For additional data similar to (A) and (C), corresponding to filaments of the sepJ/fraCD genetic background, see Figure 2—figure supplement 1.

Video 1
Real-time expression of a clock-controlled gene and filament autofluorescence during circadian oscillations in WT Anabaena.

To characterize quantitatively the degree of synchronization between cellular clocks along a filament, we used the synchronization index R (Garcia-Ojalvo et al., 2004) (Materials and methods), which can be readily calculated from the measured fluorescence intensity in each cell and which varies between 0 (no synchronization) and 1 (full synchronization). To compute R, several cells along a filament were selected, and their fluorescence intensity was followed over a full period of oscillation. Contiguous cells, which include sister cells from the same mother, are highly synchronized, as shown by the large value of R (0.89±0.04) (Table 1). The value of R for cells initially separated by intervals of 10 other cells – chosen in an attempt to avoid initial correlations between their clocks – was significantly indistinguishable from that computed for contiguous cells, underscoring the large degree of synchronization of clocks along a filament.

Table 1
Synchronization of expression of a clock-controlled gene in cells within and between Anabaena filaments.

The synchronization index R for strains with the indicated genotypes (Materials and methods) was measured from the fluorescence intensities of PpecB-gfp expression in the same cells followed over a full circadian period in a filament, either in clusters of contiguous cells (contiguous) or for cells separated by intervals of 10 cells (separate). To measure synchronization between filaments, R was computed from about 10 cells, each belonging to a different filament. For each genetic background, the mean and standard error of the mean (SEM) of R was determined from a number of independent repeats (Rust et al., 2007; Lambert et al., 2016; Dong et al., 2010; Teng et al., 2013), carried out in n different experimental runs. Significance (p-value) in interstrain comparisons was established by the Mann–Whitney U-test, and * represents rejection of the null hypothesis that samples come from distributions with equal medians. WT: wild type.

GenotypeCell clusterR
(mean ± SEM)
nComparison
with strain
p-Value
WTContiguous0.89 ± 0.043WT (separate)0.117
WTSeparate0.85 ± 0.012
WTDifferent filaments0.75 ± 0.042WT (separate)0.026*
ΔsepJΔfraCDContiguous0.73 ± 0.054WT (contiguous)0.001*
ΔkaiABCContiguous0.71 ± 0.033WT0.001*

To test the extent to which different filaments are synchronized under the same conditions, we measured the average fluorescence intensity per cell in different filaments as a function of time (Figure 2B). The expression from PpecB-gfp in different filaments oscillated nearly in phase mainly during the first oscillations. To evaluate quantitatively the degree of phase synchronization between filaments, we calculated R by choosing one cell per filament in a number of filaments measured simultaneously (Materials and methods). We obtained R=0.75±0.04, a value that is significantly smaller than that obtained for cells within the same filament (Figure 2C, Table 1). We surmise that initial synchronization may be due to phase resetting following the change in conditions, for example, illumination, upon transfer of cells from bulk culture to the microscope for real-time measurements. This change also could account for the decay in fluorescence intensity observed during the first circa 24 hr of our experiments (Figure 2A, B). Furthermore, this decay is PpecB-gfp-specific, but clock-independent, as it was also observed in filaments of the ΔkaiABC background (Figure 2A).

Circadian clocks along filaments are coupled by cell-cell communication

Another salient feature in the snapshots of Figure 1A is the high spatial coherence of the expression from PpecB-gfp along filaments, that is, all cells have nearly the same phase along their circadian cycle. To quantify the extent to which clocks are actually correlated, we calculated the spatial autocorrelation function of fluorescence intensity g as a function of distance along a filament (see Materials and methods). We found that g decays to zero for separations of five cells or more (Figure 2D).

To evaluate the contribution of phase inheritance following cell division to the observed autocorrelation, a simulated filament was generated from each measured filament by dividing each cell into two for three generations, partitioning the fluorescence of a mother cell binomially between the two daughters, and then multiplying the results by two, in order to conserve the average fluorescence per cell. The autocorrelation functions of these simulated filaments were then computed and averaged. The resulting mean autocorrelation (Figure 2D, magenta line) decreased to zero already at the second neighbor, suggesting that another factor, for example, cell-cell communication, contributes significantly to the coupling of fluctuations of pecB expression along a filament. To support this notion further, we calculated the spatial autocorrelation function of PpecB-gfp expression in filaments of a ΔsepJ/ΔfraCD strain in which genes coding for three septal proteins SepJ, FraC, and FraD involved in cell-cell communication were deleted (Nürnberg et al., 2015). Circadian oscillations in expression from PpecB-gfp in filaments of this strain were observed (Figure 2—figure supplement 1A). The resulting spatial autocorrelation function decreased over significantly shorter lengthscales (about two cells) than the WT (Figure 2D). Therefore, we calculated the synchronization index between contiguous cells in this genetic background and found that R was significantly smaller than that measured for contiguous cells within filaments of the WT background, but comparable to that obtained for cells in different filaments (Table 1). Consistently with this smaller value of R, traces of individual cells and their respective lineages were considerably more noisy (see Figure 2—figure supplement 1B) than lineages in WT filaments (Figure 2C). These findings indicate that the correlation in PpecB-gfp expression is due primarily to significant coupling between the clocks in neighboring cells due to cell-cell communication. Of note, the value of R calculated for a ΔkaiABC background, in which PpecB-gfp expression is clock-independent, was similar to that for ΔsepJ/ΔfraCD, in which cell-cell communication is impaired (Table 1).

Cell-cell variability oscillates out of phase with the circadian rhythm

Variations in gene expression between cells along a filament may limit both synchrony and spatial coherence. These variations are evident in Figure 1A even though their amplitude is small relative to the clock-modulated activity of PpecB-gfp. To quantify these variations, we calculated the square of the coefficient of variation CV=σ/μ, where σ denotes the standard deviation of the fluorescence intensity of contiguous cells along a filament. A plot of CV2 as a function of time showed that the cell-cell variability itself displays oscillatory behavior, attaining maxima approximately in the middle of periods during which the mean fluorescence intensity increases (Figure 2A).

Coupling between cell cycle and clocks

In a cellular setting, circadian and cell cycle oscillations take place concurrently. In a variety of organisms, from prokaryotes to mammals, the circadian clock has been observed to gate cell division, enabling cell division to take place at some phases of the circadian cycle but inhibiting at others (Mori, 2009; Yang et al., 2010). To test whether the cell cycle and circadian clocks are coupled in Anabaena cells, we monitored the time at which cell division takes place along the circadian cycle in individual cell traces, under conditions of constant illumination. The fluorescence intensity traces of two contiguous ancestor cells bearing the PpecB-gfp fusion and their respective lineage are shown in Figure 2C. In Figure 2E, we show a histogram of the timing of cell division events as a function of the phase at which they occur along the circadian clock, obtained from traces similar to those in Figure 2C. Far from being equiprobable along the circadian cycle, cell division events showed a marked tendency to occur near the beginning or the end of a circadian cycle (minima in µ) as for Synechococcus (Yang et al., 2010). Figure 2C also shows that the clock phase was faithfully inherited by any two daughters following cell division.

Transcriptional oscillations of kai genes and the master transducer regulator rpaA

Previous northern blot measurements and microarray analysis of kai genes showed no reliable, high-amplitude oscillatory expression of any of the kai genes in Anabaena. To study with higher sensitivity the expression of kai genes, we carried out real-time quantitative polymerase chain reaction (RT-qPCR) measurements of WT and ΔkaiABC strains in bulk cultures. Since the value of the R index was smaller between filaments than within (Table 1), the experiments were carried out under constant light conditions following two 12 hr/12 hr light-dark cycles, to enhance synchronization (see Materials and methods). The relative expression of the three kai genes indeed showed noisy, low-amplitude temporal modulations (Figure 3A). The oscillations were largely in phase, and transcription occurred mainly during subjective day. To assess quantitatively the extent of coordination, we calculated the synchronization index R for the different pairs and obtained RA,B=0.85±0.05, RA,C=0.79±0.09, and RB,C=0.87±0.06. Error bars were obtained by bootstrap methods (Materials and methods). Since the differences between these values are not significant, we conclude that transcription of the three genes is coordinated. In fact, the value of R evaluated for the three genes was 0.78 ± 0.08.

Figure 3 with 2 supplements see all
Transcriptional oscillations in the core clock genes, rpaA and pecB.

(A) Relative expression of kaiA (green), kaiB (red), and kaiC (blue) as a function of time measured by RT-qPCR (Materials and methods). A persistence homology analysis of these data is presented in Figure 3—figure supplement 1. (B, C) Relative expression levels of rpaA and pecB, respectively, in wild-type (full circles) and ΔkaiABC strains (empty circles). Curves have been normalized by their temporal mean. Error bars represent the standard error of the mean of three independent experiments (see Materials and methods). Gray shades represent periods of subjective night. For additional information about regulatory sequences of the kaiABC, rpaA, pecB promoter regions and RpaA binding sites in Anabaena, see (C) (Figure 3—figure supplement 2).

To expose an underlying oscillatory behavior in the kai genes data and support the notion that the oscillations are circadian, we applied persistent homology methods (Pereira and de Mello, 2015; Otter et al., 2017) to two-dimensional phase portraits of their respective time series (see Appendix 1 – supplemental methods). The delay τ for each phase portrait corresponds to the first minimum of the auto-mutual information of the time series (Fraser and Swinney, 1986), and for a periodic, nearly sinusoidal signal, corresponds to a quarter of the signal's period (Kennedy et al., 2018). We obtained τ=7.1±1.2 hr, τ=6.7±1.2 hr, and τ=7.3±1.1 (n3) hr for kaiA, kaiB, and kaiC, respectively, all consistent with a circadian period of oscillation (Appendix 1). A Vietoris–Rips filtration of the clouds of points in the phase portraits (Appendix 1) indeed showed evidence for a persistent cycle in the transcription of each of the kai genes (Figure 3—figure supplement 1).

To shed light on how the state of the clock is relayed to the genes it controls, for example, pecB, we measured the relative expression of rpaA in WT and ΔkaiABC strains by RT-qPCR. RpaA has been shown to transduce the phosphorylation state of KaiC to clock-controlled genes in other cyanobacteria (Markson et al., 2013; Iijima et al., 2015) and is highly conserved (Schmelling et al., 2017). The role RpaA plays in the circadian oscillations of Anabaena is unknown. We found that the relative expression of rpaA displays high-amplitude oscillatory behavior (Figure 3B). Furthermore, pecB displays oscillatory behavior, with similar amplitude and phase. The oscillatory behavior of both genes was abolished in the ΔkaiABC mutant (Figure 3B, C). Note that the expression of both rpaA and pecB peaks during subjective night.

Incorporating demographic noise into a model of a single clock

In order to develop a model of an array of coupled noisy clocks as in Anabaena, we first characterized the spectral properties of uncoupled clocks in individual cells of Synechococcus. We carried out experiments following the expression of YFP from the kaiBC promoter in single cells of Synechococcus growing within patterned gels (Figure 4A). Circadian oscillations in the lineages of two sister cells are shown in Figure 4B (see also Figure 5B for the associated power spectrum). Expression from the kaiBC promoter exhibited circadian oscillations with a period of about 25 hr, similar to those observed previously (Teng et al., 2013). Our next goal was to generalize a deterministic model for individual clocks in Synechococcus (Rust et al., 2007) to include the effects of demographic stochasticity. We followed the interaction network depicted in Figure 5A (adapted from Rust et al., 2007). Our mathematical model takes into account the dynamics of three forms of KaiC, namely, the single-phosphorylated forms S-KaiC (phosphorylated at serine 431), T-KaiC (phosphorylated at threonine 432), and the double-phosphorylated form D-KaiC, while the unphosphorylated U-KaiC can be deduced from the conservation of the total number of molecules of KaiC. Crucial for the appearance of oscillations is the negative feedback mediated by S-KaiC through inactivation of KaiA via KaiB function. Furthermore, the condition on the active KaiA monomers (see Equation S4 in Rust et al., 2007) is modeled here by a continuous function (Appendix 1 – Equation 6) that makes analytical progress possible. The parameter γ in our nonlinear phosphorylation and dephosphorylation rates kXY for X,Y={U,T,D,S} sets the steepness of the inverted sigmoidal dependence of the rates on KaiA (Figure 5—figure supplement 1). The dynamics of the stochastic model is described by a master equation accounting for discrete variations in molecular copy numbers of phosphoforms instead of deterministic, ordinary differential equations (Rust et al., 2007; Brettschneider et al., 2010). We then use the van Kampen system-size expansion to carry out a linear noise approximation that yields, to leading order, an extended set of ordinary differential equations for the concentrations of S (ϕS=[S-KaiC]), T (ϕT=[T-KaiC]), and D (ϕD=[D-KaiC]). The analysis of these equations allows us to determine the region in parameter space within which the model exhibits sustained deterministic oscillations (Figure 5C). The parameters of the model have been set to the values determined in vitro in Rust et al., 2007 and reported in Appendix 1—table 1, except for γ and [KaiA], which were allowed to change freely. Note that deterministic oscillations with a circadian period were limited only to a small strip near the stability boundary. At the next-to-leading order, the expansion allows to evaluate the effects of demographic noise and calculate the power spectrum of fluctuations for each species’ abundance due to finite size effects (see Appendix 1). A fit of the theoretical power spectrum to the experimentally measured one provides an adequate interpolation upon adjusting the two fitting parameters, γ and [KaiA] (see Figure 5B). In drawing the comparison between theory and experiments, we assumed that the fluorescence intensity is an immediate proxy of the phosphoform T-KaiC (Teng et al., 2013). The fitted parameters position the system outside the region of deterministic oscillations (circle in Figure 5C), suggesting that circadian rhythms can be a manifestation of noise-driven oscillations (Figure 5D). Remarkably, the fitted value for [KaiA]=1.308 µM matches the concentration reported previously ([KaiA]=1.3 µM; see Rust et al., 2007).

Circadian oscillations in Synechococcus.

(A) Growth and lineage of a cell in patterned agarose, expressing YFP from the kaiBC promoter. The snapshots were chosen near maxima and minima of the circadian oscillations. (B) Fluorescence intensity of YFP of individual cells obtained from two independent cell lineages (red and blue).

Figure 5 with 1 supplement see all
Stochastic model for circadian oscillations in Synechococcus.

(A) Schematic representation of interconversion between KaiC phosphoforms modulated by the activity of KaiA in an individual clock. The different phosphoform states of KaiC are denoted by U (unphosphorylated, U-KaiC), T (phosphorylated at threonine, T-KaiC), S (phosphorylated at serine, S-KaiC), and D (phosphorylated at both sites). Arrows denote transitions between the different phosphoforms X,Y with the indicated rates kxy. KaiB mediates the inactivation of KaiA by S, as described by the continuous function f (see Figure 5—figure supplement 1). (B) Average power spectrum of single-cell fluorescence (red symbols) fit to the data with the prediction from the stochastic model (blue line). (C) γ-[KaiA] plane where deterministic limit cycle oscillations in individual clocks occur. The color corresponds to the period of oscillations (in hours). The boundary of the colored region corresponds to a Hopf bifurcation. Note that deterministic oscillations with a circadian period are limited only to a small strip near the stability boundary at the bottom right. The circle identifies the values of γ and KaiA that we obtain by fitting experimental power spectra in (B). The diamond stands for best fit parameters obtained for Anabaena (Figure 6C). (D) Comparison between damped deterministic oscillations (blue line) and quasi-cycles, both at the circle point outside the region of the deterministic oscillations in (C).

Theoretical model of arrays of coupled noisy clocks

Next, we generalized the single-clock model above to an array of coupled circadian clocks as in Anabaena, which is endowed with cell-cell communication via septal proteins (Herrero et al., 2016; Figure 6A). We postulate that the intercellular transfer of factors such as sugars (Mullineaux et al., 2008; Nürnberg et al., 2015) may affect the behavior of neighboring clocks, yielding an effective long-ranged coupling between clock inputs across the filament. The interaction is here modeled by an exponential kernel that extends over a few cells. We assumed that the core clock mechanisms of Synechococcus and Anabaena are similar (Kushige et al., 2013; Schmelling et al., 2017; Garces et al., 2004) and followed the interaction network depicted in Figure 5A. We further assumed that the rates of interconversion between KaiC phosphoforms for Anabaena are similar to those measured previously (Rust et al., 2007) in a reconstituted in vitro system consisting of KaiA, KaiB, and KaiC of Synechococcus (Appendix 1—table 2). This is supported by the fact that individual cells in both organisms exhibit oscillations with circadian periods (Figure 2C, Figure 4B) that in the case of Synechococcus are rather insensitive to changes in KaiC concentrations (Chew et al., 2018), constraining the values of these rates. Furthermore, KaiA of Anabaena, which is about two-thirds shorter than KaiA of Synechococcus, is similarly able to dimerize and enhance the phosphorylation of KaiC of other cyanobacteria in vitro, as well as elicit oscillations when its gene is transferred to Synechococcus cells, despite the evolutionary divergence between both organisms (Uzumaki et al., 2004). Individual cells can therefore display self-sustained oscillations only if the parameters γ and [KaiA] take values inside the colored region of Figure 5C. These oscillations are characterized by a circadian period only within a narrow strip near the stability boundary. The fluorescence intensity is assumed to reflect the output of the clock, with a phase difference as shown previously (Kushige et al., 2013). We hence calculated the power spectrum of the fluorescence signal on every cell along the filament and averaged together the results. The experimentally computed power spectrum shows a clear peak, which is nicely interpolated by the theoretically predicted curve (Figure 6C) upon adjusting the two fitting parameters, γ and [KaiA]. Again, the fitted values position the system (diamond in Figure 5C) outside the region of deterministic order, suggesting that demographic noise could trigger the observed oscillations. To test the robustness of the fit, we have implemented a bootstrap procedure (see Materials and methods) by perturbing each kinetic parameter with respect to the values reported in Rust et al., 2007. The imposed perturbation was about 10% for each individual kinetic parameter. The obtained best fit estimates for [KaiA] and γ were found to display a degree of variability of approximately 20%, with reference to averaged values reported in Figure 5C. The bifurcation line that sets the separation between the deterministic limit cycle and the stationary stable fixed point became modulated depending on the set of assigned kinetic constants. Each pair of fitted γ and [KaiA] falls by definition in the domain where the deterministic oscillations are impeded and the stochastic finite size corrections drive the emergence of the resonant quasi-cycles.

Stochastic model for circadian oscillations in Anabaena.

(A) Schematic representation of the Anabaena filament showing coupling of circadian clocks via cell-cell communication (red arrows). (B) Gillespie simulations of quasi-cycles of T-KaiC in a continuous stretch of 10 cells along a filament. The reaction parameters correspond to the diamond plotted in Figure 5C. The total amount of KaiC phosphoforms was set to 5000, and the number of steps of the Gillespie algorithm was 1.6×107. (C) Average power spectrum of single-cell fluorescence intensity along filaments (red symbols) fit to the data with the prediction from the stochastic model (blue line). The best fit values correspond to the diamond shown in Figure 5C. (D) Complex coherence function measuring the correlation expression from PpecB-gfp in 35 cell segments at the frequency of temporal oscillations. Red full circles correspond to experimental data, blue squares represent the fit to the experimental data with the prediction of the stochastic model, and empty red circles represent the coherence function of the experimental data in which cells have been reshuffled. Lines between symbols are a guide to the eye. The fit was carried out by adjusting two parameters, the strength of the imposed spatial coupling and the characteristic scale of the exponential kernel, see Appendix 1. Remarkably, the range of the interaction as obtained from the fit is compatible with that estimated from the spatial autocorrelation depicted in Figure 2D. The intercell coupling was obtained from the fit in (C).

In Anabaena, the oscillations displayed by different cells along a filament are synchronized, most likely by cell-cell communication. To support further the notion of clock coupling, we studied the spatial coherence of noisy oscillations along an Anabaena filament. We assume that PpecB-gfp expression reports faithfully the output of the clock, albeit with a delay due to the transduction of KaiC’s phosphorylation state up to activation of the promoter, probably by phosphorylated RpaA. To study spatial coherence, we calculated the complex coherence function (Marple, 1987), whose magnitude measures the degree of correlation between cells within a filament at different distances. As shown in Figure 6D, the coherence at the characteristic frequency of oscillations was a monotonically decaying function of distance, an observation that points to the existence of non-local interactions by cell-cell communication along the filament. As complementary evidence, we notice that the magnitude of the coherence function took a constant value when spatial correlations were broken by randomly reshuffling the position of the cells along the filament. Remarkably, the prediction of the stochastic model captures the experimental behavior, as shown in Figure 6C. The range of interaction of the exponential kernel is self-consistently quantified by the fitting strategy and yields a result that agrees with that obtained from the spatial autocorrelation analysis (Figure 2D). More specifically, this conclusion is consistent with the reduction of spatial autocorrelation in the expression of a clock-regulated gene when either cell-cell communication is perturbed or clock genes are deleted, and with the notion that inheritance following cell division alone cannot account for all the spatial variation of expression along filaments (Figure 2D). In Figure 6C, quasi-cycles recorded on different cells of the stochastic Anabaena model are depicted showing a remarkable degree of synchronization. In Appendix 1, the analysis is extended so as to account for both a constant and a decaying power-law kernel of interaction.

Discussion

Anabaena, a model organism in which each cell has a well-defined number of neighbors with which it communicates, has enabled us to study one-dimensional arrays of circadian clocks in a multicellular organism in space and time by interrogating each cell individually. Remarkably, our experiments using a fluorescence reporter fused to the N-terminus of a clock-controlled protein show that filaments display large-amplitude oscillations, consistently with previous studies carried out in bulk cultures (Kushige et al., 2013). These oscillations, which run freely under constant light conditions and are therefore autonomous, are characterized by high spatial coherence and synchrony. In addition to inheritance following cell division as in unicellular Synechococcus (Amdaoud et al., 2007), the behaviors of both the spatial autocorrelation (Figure 2D) and the complex coherence function (Figure 6D) represent strong evidence that these two characteristics result from local coupling between clocks of neighboring cells due to cell-cell communication via septal junctions. It is unlikely that this coupling results from the direct cell-to-cell transfer of KaiA, KaiB, or KaiC clock components as septal junctions in Anabaena are known to serve as conduits of only small molecules including metabolites, nutrients, and small peptides (e.g., PatS-derived peptides involved in lateral inhibition during developmental pattern formation, mediated by SepJ; Corrales-Guerrero et al., 2015). We postulate that metabolic factors such as sugars, which are transferred between cells via FraC and FraD proteins (Mullineaux et al., 2008; Nürnberg et al., 2015), may affect the behavior of neighboring clocks, yielding an effective long-ranged coupling between clock inputs (e.g., redox state, glycogen abundance, and ATP/ADP ratio) across the filament (Cohen and Golden, 2015; Golden, 2020). This is consistent with the shorter decay of the spatial autocorrelation function of PpecB-gfp fluorescence intensity, and with the reduced value of the synchronization index, measured in filaments of a strain in which the transport of sugars and/or other coupling factors is impaired (ΔsepJ/ΔfraCD). Of note, the value of the synchronization index in this strain is similar to that obtained for cells from different filaments. Thus, we have found that cells within a filament behave coordinately, whereas different filaments appear to be in different oscillatory phases. Nonetheless, coherent oscillations at the whole culture level were observed after light/darkness training of the cultures, showing that input signals can coordinate the circadian rhythm in cells from different filaments.

The high synchrony and spatial coherence observed in our experiments further suggest that circadian rhythms in Anabaena are not centrally coordinated as in higher multicellular organisms (Bell-Pedersen et al., 2005). In mammals, for instance, clocks in peripheral tissues are centrally coordinated from a central pacemaker in the brain, the suprachiasmatic nucleus (Reppert and Weaver, 2002). In plants such as Arabidopsis, recent studies revealed the existence of waves of gene expression across the whole plant and the response of cells to positional information (Gould et al., 2018), consistent with weak coupling between cells and supporting a more decentralized model of clock coordination in plants (Endo, 2016). High synchrony and spatial coherence may be viewed as a necessary adaptation following the transition from a unicellular to a multicellular lifestyle (Masuda et al., 2017), while preserving key architectural features, gating of the cell cycle by the circadian clock and robust response to stresses, supporting the notion that a filament represents the organismic unit in Anabaena.

Our single-cell measurements of gene expression from PpecB-gfp indicated that the cell-cell variability along filaments as measured by CV2 is oscillating, achieving maxima approximately half-way during periods of increase of the fluorescence intensity. The phase difference between the oscillation in fluorescence intensity and its cell-cell variability is consistent with the phase of rpaA transcriptional oscillations observed in our RT-qPCR measurements. The oscillatory nature of CV2 in PpecB-gfp expression may be due to the relay of the signal of the core clock by the transcriptional oscillations of a transcription factor, here RpaA (Heltberg et al., 2019).

Our experiments, carried out under constant light conditions, that is, non-varying external cues, provide clear evidence of gating of the cell cycle by the circadian clock, as for Synechococcus and many other organisms (Mori, 2009). In Synechococcus, cell doubling times vary considerably with light intensity (Teng et al., 2013), and the cell cycle has no effect on the circadian clock, irrespective of the cell cycle rate (Mori et al., 1996; Mori and Johnson, 2001). Furthermore, as for Synechococcus (Mihalcescu et al., 2004; Mori et al., 1996; Yang et al., 2010), clock phase is faithfully inherited by any two daughters following cell division as illustrated in Figure 2C. In fact, the post-transcriptional design of cyanobacterial circadian circuits has been suggested to provide insulation from effects due to variable cell division rates (Paijmans et al., 2016). Cell doubling times in Anabaena can also vary significantly with light intensity (Zhao et al., 2007). The mechanism of how the circadian clock controls the timing of cell division has been studied in Synechococcus as well as in Synechocystis, and it has been suggested that phosphorylated RpaA regulates the bacterial tubulin-analog FtsZ, inhibiting the formation of the cytokinetic ring (Dong et al., 2010; Kizawa and Osanai, 2020). While the precise genetic and biochemical differences between the circadian clocks of Anabaena and Synechococcus remain to be elucidated, the presence of homologs of core clock components, output coupling factors such as RpaA (Schmelling et al., 2017), and cell division promoters such as FtsZ in Anabaena (Corrales-Guerrero et al., 2018) suggests that the mechanisms for gating in these two organisms may be similar. Note that ftsZ has an upstream putative RpaA binding site motif (see Figure 3—figure supplement 2). Interestingly, nitrogen-fixing cells – heterocysts – that are formed under nitrogen deprivation in Anabaena lose the ability to divide, and yet, the heterocyst-enriched fraction in bulk experiments has been shown to exhibit clear circadian oscillations (Kushige et al., 2013). This supports the notion that cell division does not affect the circadian clock in Anabaena as in Synechococcus.

The RT-qPCR results and their analysis provide conclusive and quantitative evidence for circadian oscillations in the expression of kai genes in Anabaena. The calculation of the synchronization index R for different pairs of kai genes yields large and similar values, suggesting that the expression of the three genes is highly coordinated, consistent with their possible expression as an operon. Furthermore, a persistent homology analysis of the time series of the three genes yields clear evidence of oscillatory behavior, consistent with a circadian period. In agreement with microarray measurements (Kushige et al., 2013), the oscillations we detect are of small amplitude, similar to those of Synechocystis (Kucho et al., 2005; Beck et al., 2014), but unlike those of Synechococcus. Given that the oscillations in the expression of kai genes are small, but that of pecB and many other targets are large (Kushige et al., 2013), we reasoned that a possible way of transducing the core clock signal to clock-controlled genes may be furnished by the downstream transcription factor RpaA. The kai-dependent transcriptional oscillations of rpaA we observed in Anabaena suggest an additional level of regulation of clock-controlled genes, which differs from the post-translational mechanism observed in Synechococcus. In particular, the oscillations in kai genes are consistent with the existence of a transcription-translation feedback loop via RpaA (Markson et al., 2013). Indeed, a well-defined candidate binding site motif of RpaA is located upstream of the kai genes (Figure 3—figure supplement 2). Additional candidate binding site motifs of RpaA are found upstream of the rpaA locus itself as well as the pecBACEF operon. The mechanism behind the large-amplitude oscillatory behavior of rpaA transcription we observed remains to be elucidated.

In order to describe theoretically circadian oscillations in individual Anabaena filaments, we started by incorporating demographic noise into a deterministic model of a single clock that includes only the kai genes as a core, such as the one describing oscillations in Synechococcus, and then extended the model to one describing a one-dimensional array of coupled noisy clocks as in Anabaena. The stochastic models capture well the peaks of finite width in the power spectra of fluctuations observed in our experiments (Figure 5B, Figure 6C). Importantly, the parameters we obtain from the fits to the experimental power spectra lie well outside the region in parameter space where deterministic oscillations occur, and in particular, the smaller region corresponding to oscillations characterized by circadian periods. This strongly suggests that the oscillatory behavior may correspond to noise-seeded oscillations as observed in other systems (McKane and Newman, 2005), without the need of training by light-dark cycles. The noise-seeded enlargement of the range of biological parameters over which circadian oscillatory behavior can be observed may confer robustness to clocks, a decisive biological advantage. Robustness may enable proper circadian clock function of Anabaena under a variety of stresses, including those involved in metabolic rewiring and the ensuing differentiation in response to combined nitrogen deprivation (Herrero et al., 2016; Di Patti et al., 2018). In this context, note that Anabaena filaments display circadian oscillations under nitrogen-deprived conditions, as shown previously (Kushige et al., 2013). Lastly, the model also reproduces the spatial coherence of oscillations in filaments, providing independent evidence of clock coupling through cell-cell communication.

In spite of the fact that Kai protein copy numbers run in the thousands in Synechococcus, phase fluctuations between cells are readily visible both in our experiments and in previous ones (Chew et al., 2018; Chabot et al., 2007). Stochastic modeling indicates that these numbers may be needed to compensate for noise amplification introduced by the post-translational feedback loop provided by KaiA, and reducing the numbers of Kai proteins leads to lower clock precision (Chew et al., 2018). We point out that demographic noise may be more significant than the copy numbers of Kai proteins may suggest: the functional form of KaiC is hexameric, which further partitions into its different phosphoforms (Chew et al., 2018). In addition, it has been shown that KaiB spends most of its time in an inactive tetrameric state, preventing its interaction with other Kai proteins, and that KaiA also oscillates between active and inactive states by binding to KaiB (Tseng et al., 2017). Together, these regulatory contributions might reduce active KaiC-effective numbers, thereby increasing fluctuations.

While the copy numbers of Kai proteins in Anabaena have not been measured, it is likely that they are significantly smaller than in Synechococcus, given the low transcriptional levels observed in our RT-qPCR experiments, as well as in DNA microarrays (Kushige et al., 2013). The expression of kai genes in Synechocystis is weak (Kucho et al., 2005), and protein copy numbers are correspondingly small (Zavřel et al., 2019). Thus, demographic noise effects may be indeed at least as important in Anabaena as they are for Synechococcus. One may hypothesize that cell-cell communication and the resulting coupling of clocks compensate for the smaller number of Kai proteins, setting the high synchrony and spatial coherence we observe in Anabaena. The picture that emerges may be more general and applicable to other multicellular organisms (Gould et al., 2018).

The transition from unicellular to multicellular organisms demanded coordination between physiological processes in different cells in order to enhance organismal fitness and adaptation to stresses. This is true in particular of circadian clocks and the genes they regulate. By analyzing filaments at the individual cell level, we found concrete evidence supporting coordination mediated by cell-cell communication, allowing clocks in different cells to be coupled. Furthermore, the high synchrony and spatial coherence of cell clocks along filaments observed in the experiments suggest that cell-cell communication contributes significantly to these two properties. Our theoretical model of single core clocks and arrays thereof shows that far from being detrimental demographic noise may seed oscillations that can be synchronized by clock coupling. This provides a robust description of circadian oscillations in a multicellular organism such as Anabaena.

Materials and methods

Key resources table
Reagent type (species)
or resource
DesignationSource or referenceIdentifiersAdditional information
Strain, strain background
(Anabaena)
PpecBgfp,
WT
This paperAnabaena PCC 7120 WT, bearing a
pecB promoter fusion to gfp
Strain, strain background
(Anabaena)
PpecBgfp,
ΔkaiABC
This paperAnabaena PCC 7120 deletion mutant
of the kaiABC genes,
bearing a pecB promoter fusion to gfp
Strain, strain background
(Anabaena)
PhetRgfpdoi: 10.1371/journal.pgen.1005031CSL64Anabaena PCC 7120 WT, bearing a
hetR promoter fusion to gfp
Strain, strain background
(Anabaena)
PpecBgfp,
ΔsepJ/ΔfraC/ΔfraD
This paperAnabaena PCC 7120 deletion mutant
of the sepJ, fraC, fraD
genes (CSVM141), bearing a
pecB promoter fusion to gfp
Strain, strain background
(Synechococcus elongatus)
YFP-SsrAThis paperPCC 7942Synechococcus elongatus PCC
7942 (wild-type) expressing
YFP- SsrA
Recombinant
DNA reagent
EB2316 (plasmid)Addgene plasmid87753http://n2t.net/addgene: 87753
Recombinant
DNA reagent
pSpark (plasmid)CanvaxC0001https://lifescience.canvaxbiotech.com/wpcontent/uploads/sites/2/2015/08pSpark-DNA-Cloning.pdf
Recombinant
DNA reagent
pCSRO sacB- containing
cloning vector
doi: 10.1128/JB.00181-13
Commercial
assay or kit
Fast SYBR Green
Master Mix
Applied Biosystems4385612
Commercial
assay or kit
QuantiTect Reverse
Transcription kit
QIAGEN205311

Strains

Strains bearing a chromosomally encoded PpecB-gfp, were obtained by conjugation with the following backgrounds: WT Anabaena sp. (also known as Nostoc sp.) strain PCC 7120; ΔsepJ/ΔfraCD, strain CSVM141 in which sepJ, fraC, and fraD were deleted (Nürnberg et al., 2015); ΔkaiABC in which the kaiABC genes were deleted (for details, see Appendix 1 – Supplemental methods), as recipients. Strain CSL64 bearing a chromosomally encoded PhetR-gfp transcriptional fusion in a WT background has been reported previously (Corrales-Guerrero et al., 2015). The S. elongatus strain containing the gene encoding a YFP-SsrA reporter whose expression is driven by the kaiBC promoter at neutral site II was constructed by transforming Synechococcus sp. PCC 7942 with plasmid EB2316 (Addgene plasmid # 87753; http://n2t.net/addgene:87753; RRID: Addgene 87753).

Culture conditions

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Strains and derived strains were grown photoautotrophically in BG11 medium containing NaNO3, supplemented with 20 mM HEPES (pH 7.5) with shaking at 180 rpm, at 30°C, as described previously (Corrales-Guerrero et al., 2013; Di Patti et al., 2018). Growth took place under constant illumination (10 µmol of photons [spectrum centered at 450 nm]) from a cool-white LED array. When required, streptomycin sulfate (Sm) and spectinomycin dihydrochloride pentahydrate (Sp) were added to the media at final concentrations of 2 µg/mL for liquid and 5 µg/mL for solid media (1% Difco agar). The densities of the cultures were adjusted so as to have a chlorophyll content of 2–4 µg/mL 24 hr prior to the experiment, following published procedures (Di Patti et al., 2018). For time-lapse measurements, filaments in cultures were harvested and concentrated 50-fold. Synechococcus cultures were grown as above, and when required, 7.5 µg chloramphenicol (Cm) was added.

Samples for time-lapse microscopy

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Strains were grown as described previously (Di Patti et al., 2018). When required, antibiotics, Sm and Sp were added to the media at final concentrations of 2 µg/mL for liquid and 5 µg/mL for solid media. The densities of the cultures, grown under an external LED array (15 µmol for about 5 days), were adjusted so as to have a chlorophyll a content of 2–4 µg/mL, 24 hr prior to the experiment following published procedures (Di Patti et al., 2018). For time-lapse, single-cell measurements of Anabaena, 5 µL of culture concentrated 100-fold were pipetted onto an agarose low-melting gel pad (1.5%) in BG11 medium containing NaNO3 and 10 mM NaHCO3, which was placed on a microscope slide. The pad with the cells was then covered with a #0 mm coverslip and then placed on the microscope at 30°C. The cells grew under light from both an external LED array (15 µmol) and tungsten halogen light (10 µmol, 3000K color). Under these illumination conditions, the doubling time of cells is similar to that in bulk cultures (Di Patti et al., 2018). The change in illumination conditions when transferring cells from bulk cultures to the microscope results in high synchronization within filaments. Images of about 10 different fields of view were taken every 30 min on a Nikon Eclipse Ti-E microscope controlled by the NIS-Elements software using a 60 N.A. 1.40 oil immersion phase contrast objective lens (Nikon plan-apochromat 60 1.40) and an Andor iXon X3 EMCCD camera. Focus was maintained throughout the experiment using a Perfect Focus System (Nikon). All the filters used are from Chroma. The filters used were ET480/40X for excitation, T510 as dichroic mirror, ET535/50M for emission (GFP set), ET500/20x for excitation, T515lp as dichroic mirror, and ET535/30m for emission (EYFP set), and ET430/24x for excitation, 505dcxt as dichroic mirror, and HQ600lp for emission (chlorophyll set). Samples were excited with a pE-2 fluorescence LED illumination system (CoolLED).

For measurements of Synechococcus, the cultures were grown as above to a OD750 of 0.3–0.4. Samples were then entrained by two light-dark cycles (12 hr–12 hr) before measurements commenced. The cultures were then diluted to a OD750 of 0.1 using BG11 medium, and 2 µL of the culture was pipetted onto a glass-bottom culture dish. A patterned pad prepared as above but solidified on an optical grating (Hadizadeh Yazdi et al., 2012) was placed atop the cell suspension. Then, 1.5% agarose melted in BG11, cooled to 37°C, was poured on top of the well to cover the pad. After solidifying this last layer, 5 mL of BG11 was added to the culture dishes to maintain the moisture level of the agarose pad during the experiment. This device was placed in the microscope, and time-lapse measurements were carried out as for Anabaena.

RT-qPCR measurements

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For RT-qPCR measurements, strains were grown under the same conditions as described above. When the cultures reached the beginning of their exponential phase, about 1.8 µg/mL of chlorophyll a content, they were entrained by two light-dark cycles (12 hr–12 hr) and then grown under constant light. Then, measurements were started 24 hr after, and total RNA was extracted from the PCC 7120 and ΔkaiABC cultures in two biological replicates, every 4 hr. For each sample, 20 mL of cells were collected by filtration and washed with buffer TE50 (50 mM Tris-HCl at pH 8.0, 100 mM EDTA). Cells were resuspended in 2 mL TE50 buffer, and then they were centrifuged at 11,500 g for 2 min at 4°C. Supernatant was then removed and cell pellets were flash-frozen in nitrogen liquid for storage at 80°C. Total RNA was isolated by using hot phenol as described (Mohamed and Jansson, 1989) with modifications. All samples were treated with DNase I, and RNA quantity and purity were assessed with a NanoDrop One spectrophotometer as well as by agarose gel electrophoresis.

For cDNA synthesis, 600 ng of total RNA of each sample were used for reverse transcription with the QuantiTect Reverse Transcription kit (QIAGEN). Then, PCR amplification of 17.4 ng of each cDNA was carried out using Fast SYBR Green Master Mix (Applied Biosystems) that includes an internal reference based on the ROX dye and specific primers (Appendix 1—table 2). The amplification protocol was one cycle at 95°C for 10 min, 40 cycles of 95°C for 15 s, and 60°C for 60 s. RT-qPCR was carried out using an Applied Biosystems StepOnePlus instrument equipped with the StepOne Software v2.3. After the amplification was completed, a melting point calculation protocol was carried out in order to check that only the correct product was amplified in each reaction. Reactions were run in triplicate in 3–4 independent experiments from two biological replicates. The transcript levels of kaiA (alr2884), kaiB (alr2885), kaiC (alr2886), pecB (alr0523), and rpaA (all0129) were normalized to the transcript levels of the housekeeping genes rnpB and all5167 (ispD) to obtain the Δct value. Relative gene expression was calculated as 2-ΔΔct (Pfaffl, 2001).

Image segmentation

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All image processing and data analysis were carried out using MATLAB (MathWorks). Filament and individual cell recognition was performed on phase contrast images using an algorithm developed in our laboratory. The program’s segmentation was checked in all experiments and corrected manually for errors in recognition. The total fluorescence from GFP (for Anabaena) and chlorophyll (autofluorescence) channels of each cell, as well as the cell area, was obtained as output for further statistical analysis.

Analysis of synchronization and spatial correlations along filaments

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Synchronization was measured by the order parameter proposed by Garcia-Ojalvo et al., 2004:

(1) R=μ2μ2fi2fi2¯

where denotes a time average, .¯ indicates an average over all cells, and µ denotes the average of the fluorescence intensity of each cell fi. Hence, R is defined as the ratio of the standard deviation of μ(t) to the standard deviation of fi, averaged over all cells. For the measurement of synchronization within the same filament, groups of 8–11 cells were chosen, whether separated or contiguous (sharing a common ancestor as determined from a lineage analysis). For evaluation of inter-filament synchronization, one cell per filament was chosen randomly in different fields of view. R was then calculated, and this procedure was repeated for different choices of cells, at least three times for each experiment. All the evaluations of R were carried out over a full period of oscillation in either one of the first two oscillations, except for the ΔkaiABC background, for which R was calculated for an interval of 24 hr, during which other strains display the first full oscillation. The final result comprises the mean of at least three independent repeats in at least two independent experiments. Errors in the quoted values of R therefore represent standard errors (SEM). Statistical analyses were performed in MATLAB using Mann–Whitney U-test.

The spatial autocorrelation function of fluorescence intensities along filaments was calculated using the autocorr MATLAB command with 30 lags from at least 25 filaments of 50 cells each and at least two independent experiments. For WT and ΔsepJ/ΔfraCD backgrounds, autocorrelation functions of individual filaments were calculated at maxima and minima of circadian oscillations and the results were averaged. For filaments of the ΔkaiABC background, autocorrelation functions were calculated at about 30 and 50 hr, corresponding to the first minimum and maximum of oscillations observed in filaments of WT background. The spatial autocorrelation function was calculated from the formula

(2) gk=ckc0

where

(3) ck=1Ni=1Nk(fif¯) (fi+kf¯)

where fi denotes the fluorescence intensity in cell i in a filament of N cells.

Robustness of Anabaena dataset fit to the Synechococcus parameters

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Each kinetic parameter (k) was randomly selected from a Gaussian distribution centered at the nominal value (k¯) estimated by Rust et al., 2007 using the normrnd MATLAB function. The standard deviation is assigned as σ=k¯/10. For every complete set of (randomly selected) kinetic constants, we proceeded with the fit of γ and [KaiA] to interpolate the experimental power spectrum. Each pair of fitted values was stored to eventually compute averaged estimates, together with the error, as quantified by the associated standard deviation. By averaging over 200 independent realizations of the implemented procedure yields γ=7.2±1.4 and [KaiA] =1.3±0.24 (mean ± SD).

Appendix 1

Persistent homology analysis of periodic behavior

To apply persistent homology methods for the detection of oscillatory behavior in the noisy low-amplitude time series of kaiA, kaiB, and kaiC, a two-dimensional representation of each time series was constructed by a time-delayed embedding, that is, a plot of the signal versus itself at two different time points separated by a delay τ (see Figure 3—figure supplement 1). To overcome the sparsity of data (12 time points) when plotting the time-delayed embedding, each time series was first padded by intermediate points determined by linear interpolation. The delay τ was then evaluated as the first minimum of the mutual information of the series, as determined by a fifth degree polynomial fit, following previously published procedures (Fraser and Swinney, 1986). The existence of an underlying oscillation in the data was assessed by using a MATLAB algorithm based on the JavaPlex package (Tausz et al., 2014), which yields as output a barcode diagram in which each bar represents the range of scales over which a cycle persists as observed in the phase portrait, regarded as a simplicial complex. Note that for a periodic signal the delay τ corresponds to a quarter of the signal's period (Kennedy et al., 2018).

Stochastic model

We model the Anabaena filament as a one-dimensional chain, where each node represents a cell with its individual circadian clock. We use the letter α to label the node and denote by Ω the length of the chain, so that α=1,,Ω (see Appendix 1—figure 1). As explained in the main text, the coupling between clocks that result from metabolic factors yields a long-range interaction between clocks along the filament. To account for this crucial phenomenon, we schematize the chain as a network and, therefore, introduce the adjacency matrix W whose elements Wαβ are 1 if nodes α and β are connected, 0 otherwise. The connectivity of each node is assumed to be constant for each node and is denoted by kα.

Appendix 1—figure 1
Schematic representation of an Anabaena filament.

The parameter kα measures the connectivity of each node.

Referring back to Figure 3A, we consider, in each cell, the cyclic interconversion between phosphoforms of KaiC, T, S, D (ST), and U encapsulated in the following chemical equations:

(4) UαkUTTαTαkTUUαDαkDTTαSαkSUUαUαkUSSαTαkTDDαDαkDSSαSαkSDDα

where the superscript α indicates that the species belongs to the cell α. The interconversion rates between phosphoforms are given by

(5) kXY(nSαN)=kXY0+kXYAf(nSαN) for (X,Y){U, T, D, S}

with

(6) f(nSαN)=Γ1+exp[γ[KaiA](nSα[KaiC]N[KaiA]2)]

where [KaiA] (resp. [KaiC]) is the concentration of KaiA (resp. [KaiC]). The values of the rates are set as those of Lambert et al., 2016 and are listed in Appendix 1—table 1. The variable nSα stands for the total number of S molecules in cell α at time t, and, in the same way, we will use nXα to denote the total number of X molecules for X=T,D,U. The total number of KaiC phosphoforms remains constant in each cell of the filament, and we set this value equal to N. Note that KaiB does not appear explicitly in the equations as it is assumed that it is in excess at all times and it is further assumed that it interacts with S-KaiC instantaneously (Lambert et al., 2016). The typical sigmoidal shape of the nonlinear function (Teng et al., 2013) is plotted in Figure 5—figure supplement 1 (blue dashed line), where we also plotted the corresponding function used in the paper by Lambert et al., 2016 (red line).

Appendix 1—table 1
Parameters used in the simulations.
kUT00 h−1kUTA0.479077 h−1
kTD00 h−1kTDA0.212923 h−1
kSD00 h−1kSDA0.505692 h−1
kUS00 h−1kUSA0.0532308 h−1
kTU00.21 h−1kTUA0.0798462 h−1
kDT00 h−1kDTA0.1730000 h−1
kDS00.31 h−1kDSA−0.319885 h−1
kSU00.11 h−1kSUA−0.133077 h−1
k1/20.43 µM[KaiC]3.4 µM
  1. All the values are from Lambert et al., 2016.

To model cell-cell communication, we postulate a constant kernel that extends over few cells, as depicted in Appendix 1—figure 1. This hypothesis will be relaxed later (see Accounting for a smooth kernel of interactions). More specifically, and despite the fact that there is no experimental evidence for intercellular transport of KaiC, we assume that coupling between clocks is metabolic and describe this effect through long-range interactions between molecules at different nodes. The parameter kα, as shown in Appendix 1—figure 1, counts the number of neighbors of any given cell. The chemical equations corresponding to the envisaged coupling read

(7) TανTβ DανDβ SανSβ

where α and β label connected nodes.

Since the total number of molecules within each cell of the filament remains constant, the variable nUα can be neglected because it can be recovered from the conservation law N=nUα+nSα+nTα+nDα. For this reason, at each time t the state of the system is specified by the 3Ω-dimensional vector 𝒏(t)

(8) n(t)=(nT1(t),nD1(t),nS1(t);;nTΩ,nDΩ,nSΩ)=(n1(t),,nΩ(t))

The microscopic dynamics described by rules (Equations 4 and 7) is Markovian, and thus the configuration of the system only depends on that at the previous time. We introduce the transition rates 𝕋(𝒏|𝒏) from state 𝒏 to a new state 𝒏 for all the chemical equations that constitute the stochastic model. To denote the new state 𝒏, we only explicitly write the components of the vector that change. Accordingly, the phosphorylation and dephosphorylation transition rates in each cell α read

(9) T(niα+1|n)=nUαNkUi(α)i{T, S}
(10) T(niα1|n)=niαNkiU(α)i{T, S}

while the interconversion transitions are

(11) T(niα1,njα+1|n)=niαNkij(α)(i,j){(T,D), (D,T), (D,S), (S,D)}

To account for cell-cell communication, we also add the transitions for long-range interaction between two connected cells α and β:

(12) T(niα1,niβ+1|n)=niαNνi{T, D, S}

The dynamics is fully described in terms of the following master equation:

(13) P(𝒏,t)t=𝒏𝒏[𝕋(𝒏|𝒏)P(𝒏,t)-𝕋(𝒏|𝒏)P(𝒏,t)]

whose explicit version reads

P(n,t)t=α=1Ω{T(n|nTα1)P(nTα1;t)T(nTα1|n)P(n,t)++T(n|nTα1,nDα+1)P(nTα1,nDα+1;t)T(nTα1,nDα+1|n)P(n,t)++T(n|nTα+1,nDα1)P(nTα+1,nDα1;t)T(nTα+1,nDα1|n)P(n,t)++T(n|nDα1,nSα+1)P(nD1α,nSα+1;t)T(nDα1,nSα+1|n)P(n,t)++T(n|nSα1)P(nSα1;t)T(nSα1|n)P(n,t)++T(n|nDα+1,nSα1)P(nDα+1,nSα1;t)T(nDα+1,nSα1|n)P(n,t)++T(n|nTα+1)P(nTα+1;t)T(nTα+1|n)P(n,t)++T(n|nSα+1)P(nSα+1;t)T(nSα+1|n)P(n,t)+
(14) +β=1ΩWαβ[T(n|nTα+1,nTβ1)P(nTα+1,nTβ1;t)T(nTα+1,nTβ1|n)P(n,t)++T(n|nDα+1,nDβ1)P(nDα+1,nDβ1;t)T(nDα+1,nDβ1|n)P(n,t)++T(n|nSα+1,nSβ1)P(nSα+1,nSβ1;t)T(nTα+1,nTβ1|n)P(n,t)++T(n|nTα1,nTβ+1)P(nTα1,nTβ+1;t)T(nTα1,nTβ+1|n)P(n,t)++T(n|nDα1,nDβ+1)P(nDα1,nDβ+1;t)T(nDα1,nDβ+1|n)P(n,t)++T(n|nSα1,nSβ+1)P(nSα1,nSβ+1;t)T(nTα1,nTβ+1|n)P(n,t)]}

The master equation can be written in a more compact form though the use of the step operators ϵi±(α) that act on a generic function g(𝒏) as ϵi±(α)g(niα)=g(niα±1) i{T, D, S} and α=1,,Ω:

(15) P(n,t)t=α=1Ω{[(ϵT(α)1)T(nTα+1|n)+(ϵS(α)1)T(nSα+1|n)++(ϵT+(α)1)T(nTα1|n)+(ϵT+(α)ϵD(α)1)T(nTα1,nDα+1|n)++(ϵT(α)ϵD+(α)1)T(nTα+1,nDα1|n)+(ϵD+(α)ϵS(α)1)T(nDα1,nSα+1|n)++(ϵS+(α)1)T(nSα1|n)+(ϵD(α)ϵS+(α)1)T(nDα+1,nSα1|n)]P(n,t)++β=1ΩWαβ[(ϵT+(α)ϵT(β)1)T(nTα1,nTβ+1|n)++(ϵD+(α)ϵD(β)1)T(nDα1,nDβ+1|n)++(ϵS+(α)ϵS(β)1)T(nSα1,nSβ+1|n)]P(n,t)}

The master equation cannot be solved analytically, and therefore, we apply the van Kampen system-size expansion (van Kampen, 2014) to analyze both the deterministic behavior and the role of demographic noise. For this purpose, we split each discrete variable nXα(t) into two contributions, the deterministic one, which is obtained taking the limit N denoted by ϕX(α), and the fluctuations ξX(α) whose amplitude scales as 1/N, as 

(16) nTα(t)N=ϕTα(t)+ξTαNnDα(t)N=ϕDα(t)+ξDαNα=1,,ΩnSα(t)N=ϕSα(t)+ξSαN

We introduce a new probability function Π(𝝃,t)P(𝒏(ϕ(t),𝝃);t), where the 𝝃 is a 3Ω-dimensional vector whose components are

(17) ξ=(ξT1,ξD1,ξS1,,ξTΩ,ξDΩ,ξSΩ)(ξ1,,ξΩ)

so that the left-hand side of the master Equation (15) can be recast in the form

(18) Pt=α=1Ω[1NΠτ1Ni{T, D, S}Πξiαdϕiαdτ]

with τ=t/N.

Operating an expansion of the step operators with respect to the small parameter 1/N, performing the change of variable (Equation 16) into the right-hand side of the master equation, and collecting terms proportional to 1/N, we get

(19) dϕiαdτ=hi(α)(ϕα)+νβ=1ΩΔαβϕiβ

where

(20) hT(α)(ϕα)=ϕUαkUT+ϕDαkDTϕTαkTUϕTαkTD
(21) hD(α)(ϕα)=ϕTαkTD+ϕSαkSDϕDαkDTϕDαkDS
(22) hS(α)(ϕα)=ϕUαkUS+ϕDαkDSϕSαkSUϕSαkSD

and the matrix 𝚫 denotes the discrete Laplacian operator. The concentration ϕU reads [KaiC]-ϕT-ϕD-ϕS and [KaiC]ϕXϕX for the ease of notation.

Stability analysis

We set to zero the coupling parameter ν, which amounts to neglecting the spatial terms. In this way, system (Equation 19) simplifies to

(23) dϕidτ=hi(ϕα)

We fix all the parameters to the values specified in Appendix 1—table 1, while [KaiA] and γ are used as control parameters and denote by ϕ*=(ϕT*,ϕD*,ϕS*) the homogeneous stationary solution of system (Equation 23). To characterize the nature of the fixed points, we perform a linear stability analysis. For this purpose, for each positive ϕ* we evaluate the Jacobian matrix J

(24) J=(-kUT-kTD-kTU-kUT+kDTJ1kTD-kDT-kDSJ2-kUS-kUS+kDSJ3)

where

J1=kUT+f(ϕS)[ϕUkUTA+ϕT(kTUAkTDA)+ϕDkDTA]J2=kSD+f(ϕS)[ϕTkTDA+ϕD(kDTAkDSA)+ϕSkSDA]J3=kUSkSDkSU+f(ϕS)[ϕUkUSA+ϕDkDSA+ϕS(kSUAkSDA)]

and calculate its three eigenvalues. The nature of the equilibrium points and their stability is summarized in Appendix 1—figure 2, where we plot the equilibrium points to varying γ. Continuous lines denote stable equilibrium points, while dashed lines denote unstable solutions. The diagram shows three different regions: in region I, the system admits only one stable complex equilibrium point that becomes unstable as long as γ>γ1. The value γ1 marks a supercritical Hopf bifurcation though which we enter in region II. Here, we have a couple of complex eigenvalues with positive real part and the trajectories approach a stable limit cycle. For γ>γ2, the system undergoes a saddle-node bifurcation with the sudden creation of two additional fixed points, one stable and the other unstable (region III).

Appendix 1—table 2
Oligodeoxynucleotide primers used in this work.
NameSequence 5′–3′
alr0523-EcoRI-FwTTTTGAATTCGCTTATAAACAGCAGTTAACAGGCT
alr0523-RevTGCTACCTCCACCGCCTGCCTGTTCAACTACTTTGGA
4G-GFP-FwGCGGTGGAGGTAGCAAAGGAGAAGAACTTTTCAC
GFP-RevGCCTGAATTCTTATTTGTATAGTTCATCCATGCC
alr0523-1-FwGAATTCGCTTATAAACAGCAGTTAACAGGCT
alr0523-1-RevCTAGCACCTCCACCGCCTGCCTGTTCAACTACTTTGGA
4G-GFP-Fw in plasmid PCSV3ATTTGAAACTGCGCCACGGATC
Rev plasmid PCSV3GACCATGACGGATTAGCTCAGTAG
alr0523(7120)–1CGT GAG TCT CCA ACG GAG GC
Kai-1GAAACTGCAGGCAGAATAGGAAATCTCTAC
Kai-2CCAAATGATATCGTGCTGACAAACCTACAGTGC
Kai-3CAGCACGATATCATTTGGTATCGTACTATATTC
Kai-4CTTTCTGCAGGTTGTCCAGCCAGCAGGGTAG
kaiA-2CAGGGTGAGGCGATAATCCAT
kaiA-1GCCAGAGTACTTGTTTCTAAGCAAC
CK1-RCGATTCCGAAGCCCAACCT
kaiC-4CGAGCTACCAACCGAAAG
kaiB-1CGGCAATACTCCAAACTCAG
PkaiBC-1GGTCTATCCCACGAGAAACC
YFP-2GGTAGCTCAGGTAGTGGTTGTC
all5167-1q (forward)GCTCAAGCAATTCGTCACTGTTCC
all5167-2q (reverse)AAAGATTGCGTCGGTCTGGTGT
rnpB-1q (forward)CTCTTGGTAAGGGTGCAAAGGTG
rnpB-4q (reverse)GGCTCTCTGATAGCGGAACTGG
kaiC-3q (forward)ATGAAGCAGTGGGAGTGGTG
kaiC-8q (reverse)ACGTTACGGGCTATGACCAC
kaiB-1q (forward)ACCAAATTCAGTCAGGGCGT
kaiB-4q (reverse)GCCAATCAGAACTCTTTCCCG
kaiA-3q (forward)CAACTCAAATCAGATTATCGCCA
kaiA-4q (reverse)CTGCCCTCTAGTCGTAGCTG
pecB-1q (forward)ATATTTAATGCTGGTGGTGCTTGTT
pecB-4q (reverse)GCAGCGATCGTCCATGACACTAC
rpaA-1q (forward)TTTAACGCCGGAGCAGATGA
rpaA-4q (reverse)TGTCCGTGACGTTGTAGCAA
Appendix 1—figure 2
Stability diagram.

Stability of the equilibrium points for the three species as a function of γ and [KaiA]. The values of all the other parameters are specified in Appendix 1—table 1. Continuous lines denote stable equilibrium points, and dashed lines denote unstable equilibrium points.

To better investigate the region of the parameter's space where the deterministic system exhibits regular oscillations similar to those observed in experiments, we focused on an interval of γ including the two critical points γ1 and γ2. In Appendix 1—figure 3, we delimit with blue line the portion of the parameter plane (γ,[KaiA]) where the concentrations of three phosphoforms of KaiC exhibit regular oscillations. The period of oscillations corresponding to the smaller selected region delimited by the black rectangle is reported in Figure 3B. From Figure 3B, it appears that the region with period T24 hr is close to the separatrix between the stable fixed point dynamics and the limit cycle regions. A typical result of a numerical integration of system (Equation 23) for a set of parameters outside the limit cycle region, but close to the border, is shown in Appendix 1—figure 4. Concentrations exhibit damped oscillations that eventually reach the stationary state. Performing stochastic simulations for the same parameters as those in Appendix 1—figure 4 yields the results displayed in Appendix 1—figure 5. Quasi-regular oscillations develop when the deterministic solutions attain the corresponding equilibrium value (McKane and Newman, 2005). To shed light onto this phenomenon is entirely devoted the forthcoming discussion.

Appendix 1—figure 3
Deterministic limit cycle region.

The portion of the plane delimited by the blue line marks the region of the parameter's space where deterministic regular oscillations occur.

Appendix 1—figure 4
Deterministic simulation.

Results of the numerical integration of system (Equation 23) for γ=8 and [KaiA]=1.2. The other parameters are assigned as specified in Appendix 1—table 1. ϕX for X=T,D,S stands here for the relative abundance of the phosphoforms in units of [KaiC].

Appendix 1—figure 5
Comparison between deterministic and stochastic simulations.

The blue lines denote the result of the numerical integration of Equation (23) while the noisy red lines represent the stochastic simulation of the system through the Gillespie, 1977 algorithm. For all the plots for γ=8 and [KaiA]=1.2 while the other parameters are set as those in Appendix 1—table 1. ϕX for X=T,D,S stands here for the relative abundance of the phosphoforms in units of [KaiC].

Analysis of fluctuations

Considering the next-to-leading order of the van Kampen expansion and collecting together terms proportional to 1/N, we find the differential equation for the probability distribution Π(𝝃,τ), which results in the following Fokker–Planck equation:

(25) Π(𝝃,t)τ=α=1Ω{-i=13ξiα[𝒜iα(𝝃)Π(𝝃,τ)]+12i,j=13β=1Ω2ξiαξjβ[ij,αβΠ(𝝃,τ)]}

where 𝒜 is

(26) 𝒜iα=j=13β=1Ωij,αβξjβ

The two matrices and have dimension 3Ω×3Ω, and they can be expressed as the sum of two distinct terms, one due to the spatial contribution that we label with the superscript (SP), and the other related to the interactions within each cell that we label as (NS). Moreover, we assume that all the entries of the two matrices are evaluated at the equilibrium point ϕ*. Thus, we have

(27) ij,αβ=ij(NS)δαβ+ij(SP)Δαβ

with

(28a) (NS)=J
(28b) (SP)=(ν000ν000ν)

where J is the Jacobian matrix relative to system (Equation 19) evaluated at the steady state. For , we have

(29) ij,αβ=ij(NS)δαβ+ij(SP)Δαβ

with

(30a) (NS)(ϕUkUT+ϕDkDT+ϕTkTDϕDkDT0+ϕTkTU+ϕTkTDϕTkTDϕDkDTϕTkTD+ϕSkSD+ϕDkDSϕSkSD+ϕDkDT+ϕDkDS0ϕDkDSϕSkSDϕUkUS+ϕDkDS++ϕSkSU+ϕSkSD)
(30b) (SP)=(2νϕT0002νϕD0002νϕS)

Spatial correlations

The Fokker–Planck Equation (25) describes the fluctuations about the equilibrium point ϕ*, and it is equivalent to the following coupled Langevin equations:

(31) dξiαdτ=j=13β=1Ωij,αβξjβ+ηiα

where ξiα is the component of vector 𝝃α defined in Equation (17) while ηiα is the ith component of the noise 𝜼α with the following properties:

(32) ηiα(τ)=0
(33) ηiα(τ)ηjβ(τ)=ij,αβδ(ττ)

Starting from this equation, it is possible to derive an analytic expression for the so-called power spectrum density matrix (PSDM), a diagnostic tool that allows to highlight the presence of correlations between species at different distances along the chain (Challenger and McKane, 2013; Zankoc et al., 2019; Zankoc et al., 2017). The (PSDM) Pij,αβ(ω) is defined by

(34) Pij,αβ(ω)=ξi~α(ω)ξj~β(ω)

where ξ~iα(ω) is the temporal Fourier transform of the ith component of 𝝃α(τ). The degree of correlation between species i in cell α and species j in cell β can therefore be quantified through the complex coherence function (CCF), a 3Ω×3Ω matrix C, whose entries are the ratio of the elements of PSDM to a normalization coefficient:

(35) Cij,αβ(ω)=Pij,αβPii,ααPjj,ββ

To obtain an analytic expression of Pij,αβ, we solve the Langevin Equation (31) performing a temporal Fourier transform. In this way, for each component of the fluctuations vector on cell α we get

(36) ξ~iα(ω)=j=13β=1Ω(-iωδij,αβ-ij,αβ)-1ηjβ(ω)=j=13β=1ΩΦij,αβ-1(ω)ηjβ(ω)

with Φ(ω)=iω𝟙, where 𝟙 denotes the 3Ω×3Ω identity matrix. Substituting Equation (36) into Equation (34), one eventually obtains

(37) Pij,αβ(ω)=ξi~α(ω)ξj~β(ω)==l,m=13γ,δ=1ΩΦil,αγ1(ω)η~lγ(ω)η~mδ(ω)(Φmj,δβ)1(ω)==l,m=13γ,δ=1ΩΦil,αγ1(ω)(lm,γδ)(Φmj,δβ)1(ω)==(Φ1(Φ)1)ij,αβ

where Φ=(Φ*)T is the complex conjugate transpose of Φ and use has been made of the following properties:

(38a) η~iα(ω)=0
(38b) η~iα(ω)η~jβ(ω)=ij,αβ

Accounting for a smooth kernel of interactions

In this section, we will extend the analysis so as to account for a generalized kernel of interaction, which decays with the inter-particle distance. More concretely, each node is coupled with all the other nodes along the chain, but the strength of the coupling is modulated as a monotonous decreasing function of the distance, namely, f(|α-β|). Adjacent nodes are hence prone to mutual influences as compared to nodes that are far apart from each other. The newly imposed modulation reflects in the transition rates (12) as follows:

(39) T(niα-1,niβ+1|𝒏)=niNνf(|α-β|)

where ν is a constant and f(|α-β|) is a decreasing function of the distance between nodes α and β. This results in a modification of the diffusive-like term in the master equation, which can be reabsorbed in a new definition of the adjacency matrix Wαβ as

(40) Wαβ=Wαβf(|α-β|)

and the corresponding Laplacian matrix Δ now becomes

(41) Δαβ=Wαβkαδαβ

Accordingly, the two matrices (Equations 27 and 29) appearing in the Fokker–Planck Equation (25) must be replaced by

(42) ij,αβ=ij(NS)δαβ+ij(SP)Δαβ
(43) ij,αβ=ij(NS)δαβ+ij(SP)Δαβ

In the following, we will focus on two different choices for the function f. Our first choice is to consider an exponentially decaying kernel

(44) f(|αβ|)=eσ|αβ|

were σ>0. The second option that we set to explore assumes a power-law decay in the form

(45) f(|αβ|)=1|αβ|ρ

with ρ>0. In the next section, we will challenge different scenarios against experimental data.

Comparison with experiments

Setting ν=0 in Equation (37) and considering the diagonal terms α=β and i=j returns N identical temporal power spectra of the fluctuations. The analytical prediction can be used to fit experimental data for both Anabaena (see Figure 4C) and Synechococcus (see Figure 3C). In both cases, the values of γ and [KaiA] predicted by the fit lay outside the region where deterministic oscillations occur.

Further, the CCF can be invoked to quantitatively explain the correlation of the fluorescence signal recorded in different cells of the Anabaena filaments. More specifically, we will adjust the theoretically predicted CCF to the analogous quantity estimated experimentally by varying the coupling parameter ν and the connectivity kα when postulating a uniform range of interaction or by varying ν and the characteristic parameter of the decay function f(|α-β|), a smoothly decaying kernel of interaction.

To calculate the experimental spatial correlation, we proceeded as follows. We focused on a given portion made of Ω cells of the full Anabaena filament. We recall that this latter is growing in time. Hence, as time progresses, the cells within the examined portion of filament get replaced by their offspring as follows cellular duplication. However, for the analysis of spatial correlations, we are only interested in the relative positioning of the cells. In other words, we analyze the fluorescence content of each cell in relation to the position occupied within the filament irrespectively of the fact that the monitored population of Ω cells is constantly renewed in time by new offspring generation. By following this procedure, we obtained Ω time series for the fluorescence on each spot along the chain. Then these temporal signals are transformed via a standard FFT in time. We further select the signal corresponding to ω*, the frequency at which the temporal power spectrum exhibits a peak, and insert it in Equation (34).

The results of the comparison between theory and experiments are plotted in Figure 5 and Appendix 1—figure 6. The plot in the main text corresponds to the exponential long-range cell-cell communication (Equation 44), where the best-fit parameters are ν=0.61 and σ=0.35. Notice that 1/σ3 set the characteristic scale of the spatial interaction (in units of cells). In Appendix 1—figure 6, we compare the fit of experimental data (red circles) obtained by using a fixed interaction kernel with connectivity kα=8 (black squares) and a decaying function (Equation 45) (blue squares). In all cases, the experimental points are nicely interpolated by the theoretical curves. Accounting for a smooth kernel of interaction yields more accurate fittings.

Appendix 1—figure 6
The complex coherence function: comparison between theory and experiments.

Complex coherence function measuring the correlation of 35 cell segments at the frequency of temporal oscillations. Red circles correspond to experimental data, and the squares represent fits to the experimental data with the prediction of the stochastic model using the constant kernel (black squares) and the power-law kernel (blue squares). Lines between symbols are a guide to the eye. The fitting procedure gives ν=0.49 for the fixed connectivity kα=8, while it returns ν=1.11 and ρ=1.96 for the power-law function.

Supplemental methods

Cloning and strain construction

For construction of a fusion of the promoter of pecB to the gfp-mut2 gene in Anabaena sp. (also known as Nostoc sp.) strain PCC 7120, a custom DNA sequence containing an EcorRI restriction site preceded by the pecB sequence of Anabaena was synthesized and inserted into the pUC57 plasmid (Bio Basic) (region alr0523). The pecB construct was amplified with the primer pair alr0523-EcoRI-Fw/alr0523-Rev. A pCSAL33 plasmid, which bears the gfp-mut2 sequence, was amplified with the primer pair 4G-GFP-Fw/alr0523-GFP-Rev. The PCR-amplified sequence products gfp-mut2 and pecB were mixed and amplified with the primer pair (alr0523 PCR-Fw/GFP-PCR2-Rev). The merged PCR product and a CSV3 plasmid were digested with EcoRI and ligated to produce pSVRG4. The transferred sequence was confirmed by sequencing with the primers (alr0523-1-Fw, alr0523-1-Rev, 4G-gfp-Fw in plasmid CSV3, and Rev plasmid CSV3). The final construct bears a sequence of 1337 bp that comprises the pecB promoter and the 5′ end of the pecB gene linked to the gfp-mut2 gene by a sequence encoding four glycines. The pSVRG4 plasmid was transferred to Anabaena by conjugation, which was performed as described (Elhai et al., 1997). Clones resistant to streptomycin (Sm) and spectinomycin dihydrochloride pentahydrate (Sp) were selected, and their genomic structure was tested by PCR (alr0523(7120)–1/GFP-Rev). Periodically performed PCR analysis indicated that the strains were stable, not showing WT chromosomes, in BG11 medium supplemented with antibiotics. The Anabaena strains used as recipients in the conjugations were the WT PCC 7120, ΔsepJ/ΔfraCD, in which sepJ, fraC, and fraD were deleted (Nürnberg et al., 2015), and ΔkaiABC, in which the kaiABC genes were deleted (see Appendix 1—figure 7). Strain CSL64 bearing a chromosomally encoded PhetR-gfp transcriptional fusion in a WT background has been reported previously (Corrales-Guerrero et al., 2013).

Appendix 1—figure 7
Construction of an AnabaenaΔkaiABC deletion mutant in the PpecB-gfp genetic background.

PCR, DNA restriction/ligation, and transformation into Escherichia coli were performed by standard techniques. Conjugation from E. coli to Anabaena was performed as described by Elhai et al., 1997, and sucrose (sacB)-based positive selection for double recombinants was performed as described by Cai and Wolk, 1990. AnabaenaΔkaiABC homozygous mutants containing the kai deletion with insertion of the C.K1 gene cassette in direct orientation were obtained and confirmed by PCR analysis. Kai-1 to Kai-4 are oligodeoxynucleotide primers.

The Synechococcus strain containing the gene encoding a YFP-SsrA reporter whose expression is driven by the kaiBC promoter at neutral site II was constructed by transforming Synechococcus sp. PCC 7942 with plasmid EB2316, which was a gift from Erin O’Shea (Addgene plasmid # 87753; http://n2t.net/addgene:87753; RRID:Addgene 87753). The transformation of Synechococcus was carried out as previously described (Golden and Sherman, 1984). The cells were incubated for 48 hr at 30°C under illumination on nitrocellulose filters, and transformants were selected on Cm-containing BG11 plates. Confirmation of genomic structure was carried out by PCR with the primers kaiC-4, kaiB-1, PkaiBC-1, and YFP-2. All oligodeoxynucleotide sequences used for construction are listed in Appendix 1—table 2.

Culture conditions

Strains and derived strains were grown photoautotrophically in BG11 medium containing NaNO3, supplemented with 20 mM HEPES (pH 7.5) with shaking at 180 rpm, at 30°C, as described previously (Corrales-Guerrero et al., 2013; Di Patti et al., 2018). Growth took place under constant illumination (10 µmol of photons [spectrum centered at 450 nm]) from a cool-white LED array. When required, Sm and Sp were added to the media at final concentrations of 2 µg/mL for liquid and 5 µg/mL for solid media (1% Difco agar). The densities of the cultures were adjusted so as to have a chlorophyll content of 2–4 µg/mL 24 hr prior to the experiment, following published procedures (Di Patti et al., 2018). For time-lapse measurements, filaments in cultures were harvested and concentrated 50-fold. Synechococcus cultures were grown as above, and when required, 7.5 µg Cm was added.

Sequence

Region alr0523

ttttGAAttcGCTTATAAACAGCAGTTAACAGGCTATTAATAATTATGATGAATATTAGTCTTGAAAATTATTATCATTTTTAGCCACAAAATAAAAACCGAAGATTGTTATCTAAAACACAGATTTTTTTTGATAAAAATCCTGCTTTAGCTTTGTTTATCAATTTCTACTGTCAGCATGAAAGTATATATAAGCTAGATTTGCTACCGCCGTAGAAATTATACTCAAGCATTTATTAAGTAAAGTAAACAATATTATTTAATTGTTAAACGGTTTTTCACCTAAAGGCATTACAACCTTTATAAAACAATAATTTTTAGAGAAGTCTGTATGGATTAACACTATCTTACAAAACTTAATAATAAATTCACTCAAGCCATATATAAAATACGAACTAAGGTTTTCAAAGCAATTAAAAAACAGAAAACAAATTGCTGTTTTCCTAATAAGAAATCGCACATGATTGCAGTCCCCCAAGATAATATATGGGAGTTAATTCTGCTTGAATATTTGTTTTCTAAATAGTAAGAATAATTGCAATCGACCTTATAAAAAGCTGCAATGACCTTTAGGAGGAAAGAAAGATGCTCGATGCTTTTTCCAAAGTAGTTGAACAGGCAggcggtggaggtagca

Search of putative RpaA regulatory sequences of the kaiABC, rpaA, pecB, and ftsZ promoter regions in Anabaena

The positions for all transcription start sites were extracted from Mitschke et al., 2011. For in silico analysis of the Anabaena promoter regions, we use a position-specific probability matrix that defines the RpaA binding motif in Synechococcus (Markson et al., 2013) in FIMO tool (Grant et al., 2011).

Data availability

Source data files, Video 1 and Key resources table have been deposited in Dryad (https://doi.org/10.5061/dryad.sxksn031n).

The following data sets were generated
    1. Arbel-Goren R
    2. Buonfiglio V
    3. Patti FD
    4. Camargo S
    5. Zhitnitsky A
    6. Valladares A
    7. Flores E
    8. Herrero A
    9. Fanelli D
    10. Stavans J
    (2020) Dryad Digital Repository
    Demographic noise seeds robust synchronized oscillations in the circadian clock of Anabaena.
    https://doi.org/10.5061/dryad.sxksn031n
The following previously published data sets were used
    1. Markson JS
    2. Piechura JR
    3. Puszynska AM
    4. O'Shea EK
    (2013) NCBI Gene Expression Omnibus
    ID GSE50922. Circadian Control of Global Gene Expression by the Cyanobacterial Master Regulator RpaA.

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Decision letter

  1. Aleksandra M Walczak
    Senior Editor; École Normale Supérieure, France
  2. Sandeep Krishna
    Reviewing Editor; National Centre for Biological Sciences‐Tata Institute of Fundamental Research, India
  3. Mogens H Jensen
    Reviewer

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Acceptance summary:

This paper applies a combination of experimental and theoretical approaches to study the circadian clock of Anabaena, a cyanobacterium which exists as multicellular filaments. The system offers a rare natural example of a one dimensional system of coupled oscillators which have been the subject of much theoretical investigation. The authors demonstrate an interesting role of rpaA as a regulator of the clock and cell-cell communication to synchronize oscillations across cells in the same filament. They also show that inherent demographic noise can expand the parameter range over which oscillations with circadian periods occur, possibly enhancing the robustness of the clock.

Decision letter after peer review:

Thank you for submitting your article "Robust, coherent and synchronized circadian clock-controlled oscillations along Anabaena filaments" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Aleksandra Walczak as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Mogens H Jensen (Reviewer #2).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

All three reviewers found the Anabaena system interesting and the analysis of a circadian clock that depends on cell-cell communication to be of general interest. However, they had a series of concerns about the claims made in the paper.

Essential revisions:

1. On the experimental front, please pay particular attention to address points 1a-d raised by Reviewer 1.

2. On the theoretical front, please give a clearer explanation of Figures 5 and 6 (see comments by Reviewers 2 and 3) and provide analyses to show that the claims about the model are robust to parameter choices (points 1 and 2 of Reviewer 3).

Please go through all three reviews appended below and try to address all the comments in a revised manuscript.

We would like to draw your attention to changes in our policy on revisions we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, when editors judge that a submitted work as a whole belongs in eLife but that some conclusions require a modest amount of additional new data, as they do with your paper, we are asking that the manuscript be revised to either limit claims to those supported by data in hand, or to explicitly state that the relevant conclusions require additional supporting data.

Our expectation is that the authors will eventually carry out the additional experiments and report on how they affect the relevant conclusions either in a preprint on bioRxiv or medRxiv, or if appropriate, as a Research Advance in eLife, either of which would be linked to the original paper.

Reviewer #1:

Arbel-Goren et al., apply a combination of experimental and theoretical approaches to study the circadian clock of Anabaena sp. Unlike Synechococcus which is unicellular, Anabaena exists as multicellular filaments. The authors follow the expression dynamics of gfp expressed from a clock-controlled promoter (PpecB) and show that expression follows an oscillatory pattern. They find that cells within the same filament (up to 10 cells apart) are synchronous, but cells across unrelated filaments are not. Synchrony is inherited upon division. The oscillatory patten is not consistent with the expression pattern of kai genes. Instead, the authors show that expression is in phase with rpaA expression, but dependent on Kai. To strengthen the idea that cell-cell communication is essential for the oscillations, the authors delete septal proteins, which breaks the pattern observed. Theoretical model of coupled clocks fits the experimental observations. The authors conclude that this clock might provide robustness in case of fluctuating or stressful conditions.

I have focused my comments on the experimental aspects of this manuscript.

1. The main conclusion of the present work is the importance of cell-cell communication in establishing the circadian clock of Anabaena, which is in contrast to that observed for the unicellular Synechococcus. This is an interesting observation. However, I think the authors must provide more compelling evidence to strengthen this conclusion. For example:

a. The effects of filament lengths are unclear to me. In Figure 2, the authors compare the gene expression profiles for cells up to 10 cells apart in the same filament and state that these are comparable. However, according to their model, there seems to be a length-dependent effect. To test the same, the authors can compare profiles of cells in increasing intervals and provide plots as shown in Figure 2A for the same. In general, while the authors use synchronization index to characterize degree of synchronization in various conditions, the representation of expression profiles as in Figure 2A is better to follow. This analysis for cells deleted for sepJ/ fraCD must be included.

b. In addition to the analysis above, the authors can block cell division (using drug treatment such as cephalexin) to assess how division affects the oscillatory patterns observed.

c. The authors make an argument in the Discussion section that levels of Kai proteins may not be limiting to trigger the coupled clock behaviour they observe. In order to support this conclusion, they must perturb expression of these genes (overexpress the kai operon).

d. The authors propose that this cell-cell communication must involve transport of small molecules. Could the authors please elaborate on how these molecules feed directly into regulation of gene expression patterns? Are they produced in the cell or freely diffusing in the growth media? The current Discussion section does not explain this idea with clarity. It would be nice to include the influence of such a molecule in their theoretical framework as well.

2. The physiological relevance of the mechanism described in this study is unclear to me. How do the authors envision such clock coupling provides robustness and a biological advantage under stress conditions? Could such robustness be assessed under various sugar availabilities or under starvation for example?

Reviewer #2:

This is a nice paper on measurements on circadian oscillations in Anabaena filaments. There has been a lot of investigations on circadian rhythms in various systems. I am not an experimentalist and I do not work on circadian systems so it is hard for me to judge whether this paper presents fundamental new measurements on the circadian clock and its relations to the cell cycle. However, the paper appears very comprehensive and presents to the best of my knowledge new nice results.

Let me concentrate on the model presented in Figure 5A. This is defined by different interactions between various phosphorylated states of KaiC gene. I should like the authors to explain in more details the different links in this diagram. It probably makes sense that there are transitions between the phosphorylated states but please explain in more details. It is well know that oscillations are generated from negative feed-back loops.

In the diagram there are two negative links, from S state to Kai A and back again. This by itself defines a positive feed-back loop which will give rise to a switch not to oscillations. Therefore the only negative loop I can identify is the one S -> KaiA -> D(ST) -> S. Is that the loop that is responsible for all the oscillations? I might doubt it but please explain.

The authors have 'exported' all equations to the supporting information but I would have liked whether it is possible to include the equations for this basic loop as a diagram in the figure?

From the underlying deterministic model of this diagram the authors obtain the phase diagram in Figure 5C. I presume the line to the colored region is defined through a Hopf bifurcation, am I right? As the authors mention, there is 'only' a circadian time scale at the lower right boundary. But is that not surprising?

Next, the authors presents a model for an array of coupled circadian clocks which is supposed to model Anabaena through cell-cell communications via septul proteins. The full structure of the model is nicely presented in the supporting information. Well, it is not a simple model that is developed and I cannot claim I understand all the involved steps. Some of the results are presented in Figure 6. Figure 6B shows results from Gillespie on how the cells are correlated. Bit I am missing a little more explanation. What is the noise level (volume/number of molecules in Gillespie?), max/min of what. And in the text there is an error where Figure 6B says it is the power spectrum. Altogether, I find it a nice paper.

Reviewer #3:

The system is very interesting and offers a rare natural example of a system of coupled oscillators which have been the subject of much theoretical investigation.

I think the results certainly merit consideration in eLife. However, I think some of the theoretical claims made are not sufficiently supported in the paper in its current version.

1. A central argument that the system does not display deterministic oscillations seems to rely heavily on Figure 5C. However, it is a little difficult to see how much that conclusion is robust to the very specific model and parameters used. For example, it would seem that a factor of 2 in [Kai C] and a factor of 10% in [KaiA] would completely change that conclusion. Surely, this is possible given that there is evidence that Anabaena differs from Synechococcus as the authors themselves claim. Perhaps there is no parameter combination (including parameters other than the two the authors chose to vary) which would show deterministic oscillations without making the Figure 5B fit significantly worse but this is a little unclear from the current text.

2. Following on that point, the authors do not seem to sufficiently discuss the applicability of Rust et al. model and parameters to Anabaena. I think the paper would be considerably strengthened by a clear discussion on how robust the conclusion is to different parameter variations and the applicability of this choice, as well as the possibility of deterministic vs noise driven oscillations.

3. There are three different Kernels used for the cell-to-cell communication, about which little is known experimentally. Of these, the exponential seems most plausible in the absence of other knowledge. The caption of Figure 6D says the complex coherence function is fit using an exponential kernel but the sharp cutoff in the fit seems to suggest a constant Kernel with a sharp cutoff. The SI suggests a power law Kernel fits well but it is unclear what the justification of such a Kernel would be. As an aside, is it possible to simplify the equations in a mean-field kind of way?

4. It is a little unclear in Table 1 why R is so high even for different filaments particularly since it was originally introduced in Garcia-Ojalvo as an order parameter that sharply distinguishes between synchronization and lack of.

I think the paper is not suitable for eLife in its current form but a clearer and stronger discussion of model choice and fitting could remedy that.

https://doi.org/10.7554/eLife.64348.sa1

Author response

Reviewer #1:

[…] I have focused my comments on the experimental aspects of this manuscript.

1. The main conclusion of the present work is the importance of cell-cell communication in establishing the circadian clock of Anabaena, which is in contrast to that observed for the unicellular Synechococcus. This is an interesting observation. However, I think the authors must provide more compelling evidence to strengthen this conclusion. For example:

a. The effects of filament lengths are unclear to me. In Figure 2, the authors compare the gene expression profiles for cells up to 10 cells apart in the same filament and state that these are comparable. However, according to their model, there seems to be a length-dependent effect. To test the same, the authors can compare profiles of cells in increasing intervals and provide plots as shown in Figure 2A for the same. In general, while the authors use synchronization index to characterize degree of synchronization in various conditions, the representation of expression profiles as in Figure 2A is better to follow. This analysis for cells deleted for sepJ/ fraCD must be included.

In all our experiments, filaments were long enough so that any possible end effects were negligible. Consequently, the behavior we observed throughout ALL experiments was essentially independent of filament length. Likewise, in the theoretical model, we first introduced demographic noise for a single oscillator and then generalized to an array of oscillators, without addressing any length dependence. Rather than <milestone-start />“<milestone-end />filament length”, we assume the reviewer meant distance between cells along a single filament. To highlight the high synchronization within filaments, we calculated the synchronization index R in two situations: one in which cells are sufficiently separated (by intervals of at least 10 other cells), and one in which cells are adjacent. As demonstrated by the first two lines in Table 1, both values of R are statistically indistinguishable. We believe that this is compelling evidence for very high synchronization. We also wish to stress that the reviewer<milestone-start />’<milestone-end />s suggestion of plotting individual traces for increasing intervals is already implemented in Figure 2C, where the lineage of two cells are plotted for five cycles. As the reviewer will note, after five cycles cell division results in 32 contiguous cells, whose individual traces largely overlap.

As per the reviewer<milestone-start />’<milestone-end />s suggestion, we have added what is now Figure S1, which includes two panels. In A, we plot the mean fluorescence per cell in a ∆sepJfraCD filament, as well as the respective autofluorescence, analogously to Figure 2A. Note that the traces extend for only two complete cycles instead of five as in Figure 2A. Deletion of septal junction genes ∆sepJfraCD turns out to result in filament fragmentation, preventing us from following filaments under the microscope from more than 2-3 days (see also Nurnberg et al., mBio, 2015).

In panel B we plot traces of four adjacent individual cells in a ∆sepJfraCD filament and their respective lineages, as you requested. The noisier nature of these traces as compared to cell traces in the WT (Figure 2C), highlight the smaller extent of synchronization when cell-cell communication is perturbed, consistently with the smaller value of R in Table 1. The value of R in Table 1 represents the statistics of various such experiments (n=4).

b. In addition to the analysis above, the authors can block cell division (using drug treatment such as cephalexin) to assess how division affects the oscillatory patterns observed.

In our study, we presented evidence that the circadian clock gates cell division, allowing division at some phases but not at others. Whether cell division affects the circadian clock in Anabaena is unknown. We point out that in organisms for which the interplay between these two oscillators has been examined (including unicellular Synechococcus and eukaryotes), the circadian clock is independent of cell division, and the cell cycle has no effect on the circadian clock (Mori T, Cell division cycles and circadian rhythms. Bact Circadian Programs (2009); Mori et al., J Bacteriol. 183:2439-2444 (2001); Mori et al., PNAS 93:10183 (1996)).

While it would be interesting in future studies to test whether blocking cell division has any effect on the circadian clock of Anabaena, cephalexin and similar compounds are not the way to go: cephalexin is an inhibitor of cell wall biosynthesis in Anabaena and it would kill growing cells rather than stop cell division.

In this connection, it is important to note that heterocyst cells that develop under nitrogen deprivation in Anabaena lose the ability to divide, and yet, the heterocyst-enriched fraction in the bulk experiments of Kushige et al. (J. Bact, 2013) exhibited clear circadian oscillations. This supports the notion that cell division does not affect the circadian clock in Anabaena, and a sentence was added to the Discussion about this issue (line 345).

c. The authors make an argument in the Discussion section that levels of Kai proteins may not be limiting to trigger the coupled clock behaviour they observe. In order to support this conclusion, they must perturb expression of these genes (overexpress the kai operon).

We stress that we assumed in our Discussion that the copy number of Kai proteins is likely to be small and unknown (lines 386-396), and we hypothesized that cell-cell communication may compensate for these small numbers in synchronizing clocks. Supporting this hypothesis remains a topic for future studies.

Typically overexpression is carried out and is most informative in cases when deletion of function or mutation (inactivation) proves impossible (e.g. in the case of an essential gene). This is not the case of our experiments, as shown by our results with ΔkaiABC filaments. We point out in passing that overexpression of kai genes from plasmids would also be problematic, as cell-cell fluctuations in plasmid copy numbers would introduce an unwanted, extraneous source of variability. In addition, the synchrony we observe in wild-type filaments is so high, that we do not expect that over-expression of Kai proteins will enhance it.

d. The authors propose that this cell-cell communication must involve transport of small molecules. Could the authors please elaborate on how these molecules feed directly into regulation of gene expression patterns? Are they produced in the cell or freely diffusing in the growth media? The current Discussion section does not explain this idea with clarity. It would be nice to include the influence of such a molecule in their theoretical framework as well.

In the present work, we deal exclusively with within-filament communication through septal junctions, and not through freely diffusing molecules in the growth medium. We have made clear in the Discussion that communication between cells takes place through septal junctions (line 297). This said, the precise identity of cell-to-cell signals that couple clocks in different cells has not been established, and remains a topic for future research. Nonetheless, this has been discussed in the first paragraph of the Discussion, including mention of sugars, which may be key in synchronization.

Interestingly, coupling of clocks in different cells in plants as Arabidopsis through plasmodesmata increases synchrony of circadian rhythms and the identity of cell-to-cell signals coupling clocks has not been identified either (Greenwood et al. PLOS Biol. 17:e3000407 (2019)).

2. The physiological relevance of the mechanism described in this study is unclear to me. How do the authors envision such clock coupling provides robustness and a biological advantage under stress conditions? Could such robustness be assessed under various sugar availabilities or under starvation for example?

Synchronization of clocks along a filament through coupling endows the filament with the property to function as an organismic unit, allowing it to respond as a unit to external cues that promote responses at the gene expression level. The opposite situation, i.e., one in which every cell displays a different phase could lead to discoordinated gene expression responses, and consequently, to different fitness in each cell. In the more extreme situation, big differences in transcriptional responses could lead to parts of the filament being viable and others that are not. A sentence was revised in the Discussion about this issue (line 322). In the particular case of responses to nitrogen scarcity, a coordinated transcriptional response along the filament could be required for the establishment of a spatial pattern of nitrogen-fixing cells (heterocysts) distribution along the filament. Note that even heterocyst-enriched fractions display circadian oscillations as demonstrated in bulk cultures by the work of Kushige et al. (2013). We added a sentence to the Discussion stating this fact (lines 382-384).

Reviewer #2:

This is a nice paper on measurements on circadian oscillations in Anabaena filaments. There has been a lot of investigations on circadian rhythms in various systems. I am not an experimentalist and I do not work on circadian systems so it is hard for me to judge whether this paper presents fundamental new measurements on the circadian clock and its relations to the cell cycle. However, the paper appears very comprehensive and presents to the best of my knowledge new nice results.

Let me concentrate on the model presented in Figure 5A. This is defined by different interactions between various phosphorylated states of KaiC gene. I should like the authors to explain in more details the different links in this diagram. It probably makes sense that there are transitions between the phosphorylated states but please explain in more details. It is well know that oscillations are generated from negative feed-back loops.

We thank the reviewer for pointing this out. While the basic scheme was already introduced starting in line 206, we have defined the different symbols in the scheme in the figure legend. In addition, we have added to this panel the precise definition of the rates between the different phosphoforms of KaiC. Regarding the remark on feedback loops, please see our answer to the next comment.

In the diagram there are two negative links, from S state to Kai A and back again. This by itself defines a positive feed-back loop which will give rise to a switch not to oscillations. Therefore the only negative loop I can identify is the one S -> KaiA -> D(ST) -> S. Is that the loop that is responsible for all the oscillations? I might doubt it but please explain.

The crucial negative feedback loop in the core oscillator is mediated by S-KaiC, through inactivation of KaiA via KaiB function, as explained previously by Rust et al. 2007, who analyzed the deterministic model embodied by this diagram. We have introduced a short sentence with this statement in line 209 to make this clear. A detailed analysis of the molecular steps in this feedback loop has recently been reported (Hong et al. Mol. Sys. Biol. 16:e9355 (2020)).

We note in passing that the minimal core circuit represented in the diagram results in measurable oscillations, as demonstrated when only the KaiABC proteins are reconstituted in vitro together with ATP (Nakajima et al. Science 308:414 (2005)).

The authors have 'exported' all equations to the supporting information but I would have liked whether it is possible to include the equations for this basic loop as a diagram in the figure?

Length limitations prevent us from displaying the full deterministic equations, which have been presented previously (in Rust et al., Science 318:809 (2007)). However, following their work, we have added to Figure 5A the interconversion rates between KaiC phosphoforms X and Y, and their analytical dependence on active KaiA monomers, which itself depends on the concentration of the S-KaiC phosphoform. This dependence is different from that of Rust et al.

From the underlying deterministic model of this diagram the authors obtain the phase diagram in Figure 5C. I presume the line to the colored region is defined through a Hopf bifurcation, am I right? As the authors mention, there is 'only' a circadian time scale at the lower right boundary. But is that not surprising?

The reviewer is right. The boundary of the colored region corresponds to the onset of a Hopf bifurcation from a (deterministic) non-oscillating state to limit cycle oscillations (clearly reviewed in Markson, J.S., & O<milestone-start />’<milestone-end />Shea, E.K. FEBS Letters, 583(24), 3938-3947 (2009)). We have made this clear in the revised legend of Figure 5. In the colored region, the period of oscillations varies with variations in parameter values. We do not deem this variation surprising; for different fixed points the oscillation period may be different.

Next, the authors presents a model for an array of coupled circadian clocks which is supposed to model Anabaena through cell-cell communications via septal proteins. The full structure of the model is nicely presented in the supporting information. Well, it is not a simple model that is developed and I cannot claim I understand all the involved steps. Some of the results are presented in Figure 6. Figure 6B shows results from Gillespie on how the cells are correlated. Bit I am missing a little more explanation. What is the noise level (volume/number of molecules in Gillespie?), max/min of what. And in the text there is an error where Figure 6B says it is the power spectrum. Altogether, I find it a nice paper.

We thank the reviewer for pointing out the lack of information relative to Figure 6B. Figure 6B reports the concentration of T-KaiC in a filament of 10 cells obtained from a single realization of the Gillespie algorithm run for 16 million steps. The strength of noise is controlled by the parameter N, the total number of KaiC phosphoforms, that we have set to 5000. We have included all those relevant details in the caption of Figure 6B. The typo in the text has been corrected: from Figure 6B to Figure 6C.

Reviewer #3:

The system is very interesting and offers a rare natural example of a system of coupled oscillators which have been the subject of much theoretical investigation.

I think the results certainly merit consideration in eLife. However, I think some of the theoretical claims made are not sufficiently supported in the paper in its current version.

1. A central argument that the system does not display deterministic oscillations seems to rely heavily on Figure 5C. However, it is a little difficult to see how much that conclusion is robust to the very specific model and parameters used. For example, it would seem that a factor of 2 in [Kai C] and a factor of 10% in [KaiA] would completely change that conclusion. Surely, this is possible given that there is evidence that Anabaena differs from Synechococcus as the authors themselves claim. Perhaps there is no parameter combination (including parameters other than the two the authors chose to vary) which would show deterministic oscillations without making the Figure 5B fit significantly worse but this is a little unclear from the current text.

We thank the reviewer for raising this important point. In the former version of the paper, when carrying out the fit, all parameters (except for γ and the concentration [KaiA]) were frozen to the nominal values as determined in Rust et al. for the reference case of Synechococcus. Following the reviewer’s remark, we have implemented a bootstrap procedure to account for a degree of variability of the aforementioned kinetic parameters. More precisely, each kinetic parameter (k) is randomly selected from a Gaussian distribution centered at the nominal value k¯ estimated by Rust et al. (2007). The standard deviation of the distribution is assigned as: σ=k¯10. For every complete set of (randomly selected) kinetic constants, we proceed with a two-parameter fit (γ and [KaiA] let free to change) to interpolate the experimental power spectrum. Each pair of fitted values is stored to compute eventually averaged estimates, together with the error, as quantified by the associated standard deviation. By averaging over 200 independent realizations of the implemented procedure yields γ=7.2±1.4 (mean ± std) and [KaiA] =1.3±0.24, with a degree of the variability which is hence quantified in approximately 20%. It is worth emphasizing that each fit rests on a stochastic description of the dynamics. The bifurcation line that sets the separation between the deterministic limit cycle and the stationary stable fixed point is modulated, depending on the set of assigned kinetic constants. Each pair of parameters γ and [KaiA] from the fit falls by definition in the domain were deterministic oscillations do not occur and the stochastic finite size corrections drive the emergence of the resonant quasi cycles.

Our results suggests that circadian clocks could have a stochastic origin. The broad power spectrum profile as recorded in the experiments is a clear indication that noise plays a role, and available data are compatible with a stochastic cause for the onset of the observed oscillation. We have added a detailed discussion of this point from line 258 and a subsection at the end of Materials and methods explaining these procedures (line 517).

2. Following on that point, the authors do not seem to sufficiently discuss the applicability of Rust et al. model and parameters to Anabaena. I think the paper would be considerably strengthened by a clear discussion on how robust the conclusion is to different parameter variations and the applicability of this choice, as well as the possibility of deterministic vs noise driven oscillations.

To answer this point we have refined the analysis, by modulating the parameters as compared to their reference values estimated by Rust et al. The adopted procedure has been discussed above. For all sets of kinetic parameters, the experimental power spectrum is nicely interpolated by the corresponding (fitted) theoretical profile, which assumes that the observed oscillations have a stochastic origin. Based on this, we argue that circadian oscillations could result from a resonant amplification of the endogenous component of noise, as stemming from demographic fluctuations. As a matter of fact, we do not claim that circadian clocks are definitely stochastic oscillators. We only cast on solid grounds the observation that they could be adequately explained by resorting to a stochastic scenario, that successfully captures experimental data. In this regard, it is worth stressing that a fully deterministic mechanism for cycle generation would yield a δ-like profile in the frequency domain, in stark contrast with the broad spectrum that we have experimentally recorded.

3. There are three different Kernels used for the cell-to-cell communication, about which little is known experimentally. Of these, the exponential seems most plausible in the absence of other knowledge. The caption of Figure 6D says the complex coherence function is fit using an exponential kernel but the sharp cutoff in the fit seems to suggest a constant Kernel with a sharp cutoff. The SI suggests a power law Kernel fits well but it is unclear what the justification of such a Kernel would be. As an aside, is it possible to simplify the equations in a mean-field kind of way?

The model that we propose is stochastic in nature and the mean-field approximation yields the set of coupled ordinary differential equations that govern the dynamical evolution of the interacting species in the deterministic limit. In Figure 6D we analyze the fluctuation beyond the mean field limit, so as to bring into evidence the role played by fluctuations in cell-cell communication. Unfortunately, we could not find any viable strategy to simplify further the equations that yield the complex coherence function, under different couplings. However, we agree with the referee that further insight would be useful to discriminate eventually between the different kernels proposed.

4. It is a little unclear in Table 1 why R is so high even for different filaments particularly since it was originally introduced in Garcia-Ojalvo as an order parameter that sharply distinguishes between synchronization and lack of.

As detailed in the Materials and methods section, filaments are initially grown in flasks under specific conditions of light intensity and spectrum, prior to being put in a device for microscope, long-term observation. The process of putting filaments in devices and in the microscope setup involves manifold changes in illumination properties, temperature and adjustment to the constrained space of the device, where filaments are fed exclusively by nutrients in the gel pad. All these changes induce synchronization between filaments, yielding large values of R.

I think the paper is not suitable for eLife in its current form but a clearer and stronger discussion of model choice and fitting could remedy that.

It is our hope that the changes we have introduced in the revised manuscript have remedied the problems highlighted by the reviewer.

https://doi.org/10.7554/eLife.64348.sa2

Article and author information

Author details

  1. Rinat Arbel-Goren

    Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel
    Contribution
    Conceptualization, Resources, Data curation, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - original draft, Project administration, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-7253-2036
  2. Valentina Buonfiglio

    Dipartimento di Fisica e Astronomia, Università di Firenze, INFN and CSDC, Sesto Fiorentino, Italy
    Contribution
    Software, Formal analysis
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-1528-2955
  3. Francesca Di Patti

    Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Sesto Fiorentino, Italy
    Contribution
    Conceptualization, Resources, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-1288-0079
  4. Sergio Camargo

    Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel
    Contribution
    Resources, Formal analysis, Investigation, Visualization
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-5688-3215
  5. Anna Zhitnitsky

    Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel
    Contribution
    Formal analysis
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-9598-4812
  6. Ana Valladares

    Instituto de Bioquímica Vegetal y Fotosíntesis, CSIC and Universidad de Sevilla, Sevilla, Spain
    Contribution
    Resources, Validation, Investigation
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-6683-1355
  7. Enrique Flores

    Instituto de Bioquímica Vegetal y Fotosíntesis, CSIC and Universidad de Sevilla, Sevilla, Spain
    Contribution
    Conceptualization, Resources, Supervision, Funding acquisition, Methodology, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-7605-7343
  8. Antonia Herrero

    Instituto de Bioquímica Vegetal y Fotosíntesis, CSIC and Universidad de Sevilla, Sevilla, Spain
    Contribution
    Conceptualization, Resources, Supervision, Funding acquisition, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-1071-6590
  9. Duccio Fanelli

    Dipartimento di Fisica e Astronomia, Università di Firenze, INFN and CSDC, Sesto Fiorentino, Italy
    Contribution
    Conceptualization, Resources, Formal analysis, Supervision, Funding acquisition, Visualization, Methodology, Project administration, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-8545-9424
  10. Joel Stavans

    Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel
    Contribution
    Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing - original draft, Project administration, Writing - review and editing
    For correspondence
    joel.stavans@weizmann.ac.il
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-0396-7797

Funding

Minerva Foundation

  • Joel Stavans

Fondazione Ente Cassa di Risparmio di Firenze

  • Duccio Fanelli

European Regional Development Fund (BUF2016-77097-P)

  • Antonia Herrero

European Regional Development Fund (BFU2017-88202-P)

  • Enrique Flores

Ministry of Foreign Affairs (EXPLICS)

  • Francesca Di Patti

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

We thank M Feingold, M Camarena, T Unger, JE Frías, N Rosenthal, and O Vardi for help, and M Rust for strains. This study was supported by the Minerva Foundation (JS); grant numbers BFU2017-88202-P (EF) and BUF2016-77097-P (AH) from Plan Nacional de Investigación, Spain, co-financed by the European Regional Development Fund; Fondazione Ente Cassa di Risparmio, Florence, Italy (DF, FDP, VB); Project EXPLICS granted by the Italian Ministry of Foreign Affairs and International Cooperation (FDP).

Senior Editor

  1. Aleksandra M Walczak, École Normale Supérieure, France

Reviewing Editor

  1. Sandeep Krishna, National Centre for Biological Sciences‐Tata Institute of Fundamental Research, India

Reviewer

  1. Mogens H Jensen

Publication history

  1. Received: October 26, 2020
  2. Accepted: March 20, 2021
  3. Accepted Manuscript published: March 22, 2021 (version 1)
  4. Version of Record published: April 23, 2021 (version 2)

Copyright

© 2021, Arbel-Goren et al.

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Further reading

    1. Microbiology and Infectious Disease
    2. Physics of Living Systems
    Brandon H Schlomann, Raghuveer Parthasarathy
    Research Article Updated

    The spatial organization of gut microbiota influences both microbial abundances and host-microbe interactions, but the underlying rules relating bacterial dynamics to large-scale structure remain unclear. To this end, we studied experimentally and theoretically the formation of three-dimensional bacterial clusters, a key parameter controlling susceptibility to intestinal transport and access to the epithelium. Inspired by models of structure formation in soft materials, we sought to understand how the distribution of gut bacterial cluster sizes emerges from bacterial-scale kinetics. Analyzing imaging-derived data on cluster sizes for eight different bacterial strains in the larval zebrafish gut, we find a common family of size distributions that decay approximately as power laws with exponents close to −2, becoming shallower for large clusters in a strain-dependent manner. We show that this type of distribution arises naturally from a Yule-Simons-type process in which bacteria grow within clusters and can escape from them, coupled to an aggregation process that tends to condense the system toward a single massive cluster, reminiscent of gel formation. Together, these results point to the existence of general, biophysical principles governing the spatial organization of the gut microbiome that may be useful for inferring fast-timescale dynamics that are experimentally inaccessible.

    1. Evolutionary Biology
    2. Physics of Living Systems
    Yuanxiao Gao et al.
    Research Article

    A key innovation emerging in complex animals is irreversible somatic differentiation: daughters of a vegetative cell perform a vegetative function as well, thus, forming a somatic lineage that can no longer be directly involved in reproduction. Primitive species use a different strategy: vegetative and reproductive tasks are separated in time rather than in space. Starting from such a strategy, how is it possible to evolve life forms which use some of their cells exclusively for vegetative functions? Here, we develop an evolutionary model of development of a simple multicellular organism and find that three components are necessary for the evolution of irreversible somatic differentiation: (i) costly cell differentiation, (ii) vegetative cells that significantly improve the organism’s performance even if present in small numbers, and (iii) large enough organism size. Our findings demonstrate how an egalitarian development typical for loose cell colonies can evolve into germ-soma differentiation dominating metazoans.