Emergent self-organization is a hallmark of natural bacterial communities, whose spatial structures and dynamic gradients are shaped by the complex interplay among bacterial diversity, oxygen and its consumption, and motility and by hydrodynamic flows. Key aspects of the interplay between oxygen gradients and bacterial community spatial structure remain obscure. Here we aim to elucidate the role that oxygen plays in the self-organization of multispecies bacteria in the water column, focusing on oxytactic bioconvection suspensions of naturally coexisting aerobic bacteria and on self-organization near air-water interfaces. Combining microscopy, mapping of the oxygen field and controlled external oxygen levels, we show that species-specific oxygen affinities and consumption rates induce the formation of distinct bacterial layers near air-water interfaces, resulting in dynamic segregation during multispecies bioconvection, and play a key role in determining the nature of bioconvective patterns in single species suspensions. We further find that oxygen and bacterial fields are tightly coupled and fluctuate with similar spatiotemporal scales, giving rise to oxygen advection and a well-defined oxic-anoxic boundary. Together, our results illuminate the fundamental role that oxygen gradients play in multispecies bacterial active matter and its spatial self-organization, with implications for the formation and stability of ecological niches in aquatic and sedimentary environments.

Patterns in time and space spanning a wide range of scales abound in nature, from the circadian oscillations in cyanobacteria to epidemic cycles, from the exquisite skins in zebras to the spatial structure of ecological habitats. For many of these systems, deterministic descriptions may necessitate fine tuning of parameters in order for patterns to form, in a way that is incompatible with their observed natural robustness. Furthermore, endogenous stochasticity such as that coming from fluctuations in small copy numbers of the interacting basic variables may be significant for these systems, in addition to external sources of noise that act as a macroscopic bias to deterministic, reproducible dynamics. Here we survey natural systems for which demographic noise has been shown to be important, and for which an individual-based description that is intrinsically stochastic is needed. This amounts to characterizing their microscopic dynamics via master equations, which return the probability for observing a system in a given state, at a given time. The application of perturbative expansions to obtain approximate analytical and numerical solutions of the master equations describing these specific systems, shows how the effects of demographic noise emerge in a perturbative fashion, and how endogenous stochasticity can be self-consistently amplified, yielding almost regular oscillations, termed quasi-cycles. These quasi-cycles appear for parameter values that lie outside the regions in parameter space where deterministic oscillations are predicted, thereby enhancing robustness. Thus, counter intuitively, noise can organize in regular quasi-periodic orbits, thereby building macroscopic order from microscopic disorder. This intriguing concept, introduced for the temporal domain, can be extended to the realm of spatial systems: demographic noise can seed the emergence of stochastic Turing patterns, which have been observed in systems ranging from developmental patterns, hallucinations to microbial biofilms. We illustrate the analytical approach by making reference to simple toy models and highlight the constructive role that demographic noise can play in seeding stochastic self-organized patterns in specific systems, both in time and space.

Chromosomal sites in rod-like bacteria move sub-diffusely, driven by the out-of-equilibrium nature of the viscoelastic, crowded intracellular environment. Furthermore, it has been shown that there is a pronounced dynamical asymmetry between longitudinal (long-axis) and transverse (radial) motions, manifested in different exponents in the mean-square displacements MSD(t) ∼t^α , α_𝓁 and α_r respectively. Here, using polymer simulations and experimental observations of a locus in Bacillus subtilis, we substantiate the notion that asymmetric dynamics are a result of the asymmetric structure of the bacterial chromosome modeled as a bottlebrush polymer in a poor solvent. Our simulations recapitulate the observed asymmetry as well as the range of the observed exponent values that are related to scaling models of polymer dynamics, lending support to the notion that asymmetric dynamics is a consequence of the bottlebrush structure of the bacterial chromosome. Anisotropic sub-diffusion of chromosomal loci in Bacillus subtilis arises from the bottlebrush chromosomal architecture, as shown by simulations and experiments, consistent with polymer scaling predictions in crowded, viscoelastic media.

Active matter, from motile bacteria to animals, can exhibit striking collective and coherent behavior. Despite significant advances in understanding the behavior of homogeneous systems, little is known about the self-organization and dynamics of heterogeneous active matter, such as complex and diverse bacterial communities. Under oxygen gradients, many bacterial species swim towards air-liquid interfaces in auto-organized, directional bioconvective flows, whose spatial scales exceed the cell size by orders of magnitude. Here we show that multispecies bacterial suspensions undergoing oxytactic-driven bioconvection exhibit dynamically driven spatial segregation, despite the enhanced mixing of bioconvective flows, and the fact that these species coexist in their natural habitat. Segregation is observed as patterns of spatially interlocked domains, with local dominance of one of the constituent species in the suspension. Our findings suggest that segregation mechanisms are driven by species-specific motile behaviors under conditions of hydrodynamic flow, rather than biochemical repulsion. Thus, species with different motile characteristics in the same ecological context can enhance their access to limiting resources. This work provides novel insights on the role of heterogeneity in active matter, as well as on the dynamics of complex microbial communities, their spatial organization and their collective behavior.

Temperate phage-mediated horizontal gene transfer is a potent driver of genetic diversity in the evolution of bacteria. Most lambdoid prophages in Escherichia coli are integrated into the chromosome with the same orientation with respect to the direction of chromosomal replication, and their location on the chromosome is far from homogeneous. To better understand these features, we studied the interplay between lysogenic and lytic states of phage lambda in both native and inverted integration orientations at the wild-type integration site as well as at other sites on the bacterial chromosome. Measurements of free phage released by spontaneous induction showed that the stability of lysogenic states is affected by location and orientation along the chromosome, with stronger effects near the origin of replication. Competition experiments and range expansions between lysogenic strains with opposite orientations and insertion loci indicated that there are no major differences in growth. Moreover, measurements of the level of transcriptional bursts of the cI gene coding for the lambda phage repressor using single-molecule fluorescence in situ hybridization resulted in similar levels of transcription for both orientations and prophage location. We postulate that the preference for a given orientation and location is a result of a balance between the maintenance of lysogeny and the ability to lyse.IMPORTANCEThe integration of genetic material of temperate bacterial viruses (phages) into the chromosomes of bacteria is a potent evolutionary force, allowing bacteria to acquire in one stroke new traits and restructure the information in their chromosomes. Puzzlingly, this genetic material is preferentially integrated in a particular orientation and at non-random sites on the bacterial chromosome. The work described here reveals that the interplay between the maintenance of the stability of the integrated phage, its ability to excise, and its localization along the chromosome plays a key role in setting chromosomal organization in Escherichia coli.

Integrative and conjugative elements (ICEs) are mobile genetic elements that can transfer by conjugation to recipient cells. Some ICEs integrate into a unique site in the genome of their hosts. We studied quantitatively the process by which an ICE searches for its unique integration site in the Bacillus subtilis chromosome. We followed the motion of both ICEBs1 and the chromosomal integration site in real time within individual cells. ICEBs1 exhibited a wide spectrum of dynamical behaviors, ranging from rapid sub-diffusive displacements crisscrossing the cell, to kinetically trapped states. The chromosomal integration site moved sub-diffusively and exhibited pronounced dynamical asymmetry between longitudinal and transversal motions, highlighting the role of chromosomal structure and the heterogeneity of the bacterial interior in the search. The successful search for and subsequent recombination into the integration site is a key step in the acquisition of integrating mobile genetic elements. Our findings provide new insights into intracellular transport processes involving large DNA molecules.

Biological patterns that emerge during the morphogenesis of multicellular organisms can display high precision at large scales, while at cellular scales, cells exhibit large fluctuations stemming from cell-cell differences in molecular copy numbers also called demographic noise. We study the conflicting interplay between high precision and demographic noise in trichome patterns on the epidermis of wild-type Arabidopsis thaliana leaves, as a two-dimensional model system. We carry out a statistical characterization of these patterns and show that their power spectra display fat tails-a signature compatible with noise-driven stochastic Turing patterns-which are absent in power spectra of patterns driven by deterministic instabilities. We then present a theoretical model that includes demographic noise stemming from birth-death processes of genetic regulators which we study analytically and by stochastic simulations. The model captures the observed experimental features of trichome patterns.

ABSTRACTCircadian clock arrays in multicellular filaments of the heterocyst-forming cyanobacterium Anabaena sp. strain PCC 7120 display remarkable spatio-temporal coherence under nitrogen-replete conditions. To shed light on the interplay between circadian clocks and the formation of developmental patterns, we followed the expression of a clock-controlled gene under nitrogen deprivation, at the level of individual cells. Our experiments showed that differentiation into heterocysts took place preferentially within a limited interval of the circadian clock cycle, that gene expression in different vegetative intervals along a developed filament was discoordinated, and that the circadian clock was active in individual heterocysts. Furthermore, Anabaena mutants lacking the kaiABC genes encoding the circadian clock core components produced heterocysts but failed in diazotrophy. Therefore, genes related to some aspect of nitrogen fixation, rather than early or mid-heterocyst differentiation genes, are likely affected by the absence of the clock. A bioinformatics analysis supports the notion that RpaA may play a role as master regulator of clock outputs in Anabaena, the temporal control of differentiation by the circadian clock and the involvement of the clock in proper diazotrophic growth. Together, these results suggest that under nitrogen-deficient conditions, the clock coherent unit in Anabaena is reduced from a full filament under nitrogen-rich conditions to the vegetative cell interval between heterocysts.IMPORTANCECircadian clocks, from unicellular organisms to animals, temporally align biological processes to day and night cycles. We study the dynamics of a circadian clock-controlled gene at the individual cell level in the multicellular filamentous cyanobacterium Anabaena, under nitrogen-stress conditions. Under these conditions, some cells along filaments differentiate to carry out atmospheric nitrogen fixation and lose their capability for oxygenic photosynthesis. We found that clock synchronization is limited to organismic units of contiguous photosynthetic cells, contrary to nitrogen-replete conditions in which clocks are synchronized over a whole filament. We provided evidence that the circadian clock regulates the process of differentiation, allowing it to occur preferentially within a limited time window during the circadian clock period. Lastly, we present evidence that the signal from the core clock to clock-regulated genes is conveyed in Anabaena as in unicellular cyanobacteria.

Abstract Circadian clock arrays in multicellular filaments of the heterocyst-forming cyanobacterium Anabaena sp. strain PCC 7120 display remarkable spatio-temporal coherence under nitrogen-replete conditions. To shed light on the interplay between circadian clocks and the formation of developmental patterns, we followed the expression of a clock-controlled gene under nitrogen deprivation, at the level of individual cells. Our experiments showed that differentiation into heterocysts took place preferentially within a limited interval of the circadian clock cycle; that gene expression in different vegetative intervals along a developed filament was discoordinated; and that the circadian clock was active in individual heterocysts. Furthermore, Anabaena mutants lacking the kaiABC genes encoding the circadian clock core components produced heterocysts but failed in diazotrophy. Therefore, genes related to some aspect of nitrogen fixation, rather than early or mid-heterocyst differentiation genes, are likely affected by the absence of the clock. A bioinformatics analysis supports the notion that RpaA may play a role as master regulator of clock outputs in Anabaena , the gating of differentiation by the circadian clock and the involvement of the clock in proper diazotrophic growth. Together, these results suggest that under nitrogen deficient conditions, the functional unit in Anabaena is reduced from a full filament under nitrogen-rich conditions, to the vegetative cell interval between heterocysts.

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.

Cell-cell communication is an essential attribute of multicellular organisms. The effects of perturbed communication were studied in septal protein mutants of the heterocyst-forming filamentous cyanobacterium Anabaena sp. PCC 7120 model organism. Strains bearing sepJ and sepJ/fraC/fraD deletions showed differences in growth, pigment absorption spectra, and spatial patterns of expression of the hetR gene encoding a heterocyst differentiation master regulator. Global changes in gene expression resulting from deletion of those genes were mapped by RNA sequencing analysis of wild-type and mutant strains, both under nitrogen-replete and nitrogen-poor conditions. The effects of sepJ and fraC/fraD deletions were non-additive, and perturbed cell-cell communication led to significant changes in global gene expression. Most significant effects, related to carbon metabolism, included increased expression of genes encoding carbon uptake systems and components of the photosynthetic apparatus, as well as decreased expression of genes encoding cell wall components related to heterocyst differentiation and to polysaccharide export.

Article Figures and data Abstract Introduction Results Discussion Materials and methods Appendix 1 Data availability References Decision letter Author response Article and author information Metrics 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 Ph⁢e⁢t⁢R-g⁢f⁢p 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). Figure 1 Download asset Open asset Circadian oscillation in Anabaena. (A) GFP fluorescence in a filament of an Anabaena strain bearing a Pp⁢e⁢c⁢B-g⁢f⁢p 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 Download asset Open asset Characterization of a clock-controlled gene in Anabaena. (A) Average cell fluorescence intensity from Pp⁢e⁢c⁢B-g⁢f⁢p 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 Ph⁢e⁢t⁢R-g⁢f⁢p (cyan line); and temporal dependence of the cell-cell variability C⁢V2 in expression of Pp⁢e⁢c⁢B-g⁢f⁢p (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 Pp⁢e⁢c⁢B-g⁢f⁢p fusion in the lineages of two contiguous cells as a function of time. (D) Average spatial autocorrelation function of Pp⁢e⁢c⁢B-g⁢f⁢p 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 2⁢π 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 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 strainp-ValueWTContiguous0.89 ± 0.043WT (separate)0.117WTSeparate0.85 ± 0.012WTDifferent filaments0.75 ± 0.042WT (separate)0.026*Δ⁢s⁢e⁢p⁢J⁢Δ⁢f⁢r⁢a⁢C⁢DContiguous0.73 ± 0.054WT (contiguous)0.001*Δ⁢k⁢a⁢i⁢A⁢B⁢CContiguous0.71 ± 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 Pp⁢e⁢c⁢B-g⁢f⁢p-specific, but clock-independent, as it was also observed in filaments of the Δ⁢k⁢a⁢i⁢A⁢B⁢C 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 Δ⁢k⁢a⁢i⁢A⁢B⁢C background, in which Pp⁢e⁢c⁢B-g⁢f⁢p 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 Pp⁢e⁢c⁢B-g⁢f⁢p. To quantify these variations, we calculated the square of the coefficient of variation C⁢V=σ/μ, where σ denotes the standard deviation of the fluorescence intensity of contiguous cells along a filament. A plot of C⁢V2 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 Pp⁢e⁢c⁢B-g⁢f⁢p 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 Download asset Open asset 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 (n≥3) 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 Δ⁢k⁢a⁢i⁢A⁢B⁢C 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 kX⁢Y 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). Figure 4 Download asset Open asset 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 Download asset Open asset 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 kx⁢y. 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

Under nitrogen-poor conditions, multicellular cyanobacteria such as Anabaena sp. PCC 7120 undergo a process of differentiation, forming nearly regular, developmental patterns of individual nitrogen-fixing cells, called heterocysts, interspersed between intervals of vegetative cells that carry out photosynthesis. Developmental pattern formation is mediated by morphogen species that can act as activators and inhibitors, some of which can diffuse along filaments. We survey the limitations of the classical, deterministic Turing mechanism that has been often invoked to explain pattern formation in these systems, and then, focusing on a simpler system governed by birth-death processes, we illustrate pedagogically a recently proposed paradigm that provides a much more robust description of pattern formation: stochastic Turing patterns. We emphasize the essential role that cell-to-cell differences in molecular numbers—caused by inevitable fluctuations in gene expression—play, the so called demographic noise, in seeding the formation of stochastic Turing patterns over a much larger region of parameter space, compared to their deterministic counterparts.

Under nitrogen deprivation, the one-dimensional cyanobacterial organism Anabaena sp. PCC 7120 develops patterns of single, nitrogen-fixing cells separated by nearly regular intervals of photosynthetic vegetative cells. We study a minimal, stochastic model of developmental patterns in Anabaena that includes a nondiffusing activator, two diffusing inhibitor morphogens, demographic fluctuations in the number of morphogen molecules, and filament growth. By tracking developing filaments, we provide experimental evidence for different spatiotemporal roles of the two inhibitors during pattern maintenance and for small molecular copy numbers, justifying a stochastic approach. In the deterministic limit, the model yields Turing patterns within a region of parameter space that shrinks markedly as the inhibitor diffusivities become equal. Transient, noise-driven, stochastic Turing patterns are produced outside this region, which can then be fixed by downstream genetic commitment pathways, dramatically enhancing the robustness of pattern formation, also in the biologically relevant situation in which the inhibitors' diffusivities may be comparable.

Brouwer’s fixed point theorem, a fundamental theorem in algebraic topology proved more than a hundred years ago, states that given any continuous map from a closed, simply connected set into itself, there is a point that is mapped unto itself. Here we point out the connection between a one-dimensional application of Brouwer’s fixed point theorem and a mechanism proposed to explain how extension of single-stranded DNA substrates by recombinases of the RecA superfamily facilitates significantly the search for homologous sequences on long chromosomes.

Brouwer's fixed point theorem, a fundamental theorem in algebraic topology proved more than a hundred years ago, states that given any continuous map from a closed, simply connected set into itself, there is a point that is mapped unto itself. Here we point out the connection between a one-dimensional application of Brouwer's fixed point theorem and a mechanism proposed to explain how extension of single-stranded DNA substrates by recombinases of the RecA superfamily facilitates significantly the search for homologous sequences on long chromosomes.

Small non-coding RNAs can exert significant regulatory activity on gene expression in bacteria. In recent years, substantial progress has been made in understanding bacterial gene expression by sRNAs. However, recent findings that demonstrate that families of mRNAs show non-trivial subcellular distributions raise the question of how localization may affect the regulatory activity of sRNAs. Here we address this question within a simple mathematical model. We show that the non-uniform spatial distributions of mRNA can alter the threshold-linear response that characterizes sRNAs that act stoichiometrically, and modulate the hierarchy among targets co-regulated by the same sRNA. We also identify conditions where the sub-cellular organization of cofactors in the sRNA pathway can induce spatial heterogeneity on sRNA targets. Our results suggest that under certain conditions, interpretation and modeling of natural and synthetic gene regulatory circuits need to take into account the spatial organization of the transcripts of participating genes.

A method is described for labeling individual messenger RNA (mRNA) transcripts in fixed bacteria for use in single-molecule fluorescence in situ hybridization (smFISH) experiments in E. coli. smFISH allows the measurement of cell-to-cell variability in mRNA copy number of genes of interest, as well as the subcellular location of the transcripts. The main steps involved are fixation of the bacterial cell culture, permeabilization of cell membranes, and hybridization of the target transcripts with sets of commercially available short fluorescently-labeled oligonucleotide probes. smFISH can allow the imaging of the transcripts of multiple genes in the same cell, with limitations imposed by the spectral overlap between different fluorescent markers. Following completion of the protocol illustrated below, cells can be readily imaged using a microscope coupled with a camera suitable for low-intensity fluorescence. These images, together with cell contours obtained from segmentation of phase contrast frames, or from cell membrane staining, allow the calculation of the mRNA copy number distribution of a sample of cells using open-source or custom-written software. The labeling method described here can also be applied to image transcripts with stochastic optical reconstruction microscopy (STORM).

Cyanobacteria carry out oxygenic photosynthesis, play a key role in the cycling of carbon and nitrogen in the biosphere, and have had a large impact on the evolution of life and the Earth itself. Many cyanobacterial strains exhibit a multicellular lifestyle, growing as filaments that can be hundreds of cells long and endowed with intercellular communication. Furthermore, under depletion of combined nitrogen, filament growth requires the activity of two interdependent cell types: vegetative cells that fix CO2 and heterocysts that fix N2. Intercellular molecular transfer is essential for signaling involved in the regulation of heterocyst differentiation and for reciprocal nutrition of heterocysts and vegetative cells. Here we review various aspects of multicellularity in cyanobacterial filaments and their differentiation, including filament architecture with emphasis on the structures used for intercellular communication; we survey theoretical models that have been put forward to understand heterocyst patterning and discuss the factors that need to be considered for these models to reflect the biological entity; and finally, since cell division in filamentous cyanobacteria has the peculiarity of producing linked instead of independent cells, we review distinct aspects of cell division in these organisms.