Robust stochastic Turing patterns in the development of a one-dimensional cyanobacterial organism

Spatio-temporal coherence of circadian clocks and temporal control of differentiation in Anabaena filaments

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 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.

PatU3 plays a central role in coordinating cell division and differentiation in pattern formation of filamentous cyanobacterium Nostoc sp. PCC 7120

Spatial periodic signal for cell differentiation in some multicellular organisms is generated according to Turing's principle for pattern formation. How a dividing cell responds to the signal of differentiation is addressed with the filamentous cyanobacterium Nostoc sp. PCC 7120, which forms the patterned distribution of heterocysts. We show that differentiation of a dividing cell was delayed until its division was completed and only one daughter cell became heterocyst. A mutant of patU3, which encodes an inhibitor of heterocyst formation, showed no such delay and formed heterocyst pairs from the daughter cells of cell division or dumbbell-shaped heterocysts from the cells undergoing cytokinesis. The patA mutant, which forms heterocysts only at the filament ends, restored intercalary heterocysts by a single nucleotide mutation of patU3, and double mutants of patU3/patA and patU3/hetF had the phenotypes of the patU3 mutant. We provide evidence that HetF, which can degrade PatU3, is recruited to cell divisome through its C-terminal domain. A HetF mutant with its N-terminal peptidase domain but lacking the C-terminal domain could not prevent the formation of heterocyst pairs, suggesting that the divisome recruitment of HetF is needed to sequester HetF for the delay of differentiation in dividing cells. Our study demonstrates that PatU3 plays a key role in cell-division coupled control of differentiation.

Cyanobacterial bioreporter of nitrate bioavailability in aquatic ecosystems

The water eutrophication, resulting from the discharge of industrial and agricultural wastewater, leads to ecological degradation. However, to date, how to assess and manage the risks of water pollution, especially nitrogen pollution, remains a particularly noteworthy issue. Nitrate, the most important nitrogen compound, has become a bottleneck restricting total nitrogen management. The development of bioreporters monitoring nitrate pollution contributes to the estimation of water quality, especially the availability of nutrients. In this study, we obtained 9 bioreporters from 40 cyanobacterial derivatives which were constructed based on different hosts, copy numbers, and sensing elements and evaluated the performance of bioreporters. The results showed that single-celled Synechocystis was more sensitive to nitrate than filamentous Anabaena, that the reporter gene luxABCDE responded faster than sfgfp in most bioreporters, and that relatively medium-copy plasmid improved the performance of sensing elements. Nine bioreporters performed well in bioavailable nitrate detection, of which AD-AS-X and AR-NI-X, activated by nitrate repletion, had the shortest response time (2 h) and the widest response range (20-800 μM), respectively. Moreover, SR-GLN-SG, activated by nitrate deficiency, exhibited the best linear response (R2 = 0.998). After parameter optimization, exponential growth phase bioreporters, culture temperature of 30 °C, sample volume of 200 μL were determined as optimal monitoring conditions. We found that common water contaminants (copper, cadmium, and phosphorus) had no impact on the performance of bioreporters, indicating the stability of bioreporters. Six out of 9 bioreporters, especially the SR-NB-X, were highly effective in detecting the bioavailable nitrate in wastewater sample. This study provides valuable references for developing more cyanobacterial bioreporters and their practical application in nitrate detection.

Stochastic Turing patterns of trichomes in Arabidopsis leaves

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.

Mathematical models of nitrogen-fixing cell patterns in filamentous cyanobacteria

The Anabaena genus is a model organism of filamentous cyanobacteria whose vegetative cells can differentiate under nitrogen-limited conditions into a type of cell called a heterocyst. These heterocysts lose the possibility to divide and are necessary for the filament because they can fix and share environmental nitrogen. In order to distribute the nitrogen efficiently, heterocysts are arranged to form a quasi-regular pattern whose features are maintained as the filament grows. Recent efforts have allowed advances in the understanding of the interactions and genetic mechanisms underlying this dynamic pattern. Here, we present a systematic review of the existing theoretical models of nitrogen-fixing cell differentiation in filamentous cyanobacteria. These filaments constitute one of the simplest forms of multicellular organization, and this allows for several modeling scales of this emergent pattern. The system has been approached at three different levels. From bigger to smaller scale, the system has been considered as follows: at the population level, by defining a mean-field simplified system to study the ratio of heterocysts and vegetative cells; at the filament level, with a continuous simplification as a reaction-diffusion system; and at the cellular level, by studying the genetic regulation that produces the patterning for each cell. In this review, we compare these different approaches noting both the virtues and shortcomings of each one of them.

Terminal heterocyst differentiation in the Anabaena patA mutant as a result of post-transcriptional modifications and molecular leakage

The Anabaena genus is a model organism of filamentous cyanobacteria whose vegetative cells can differentiate under nitrogen-limited conditions into a type of cell called heterocyst. These heterocysts lose the possibility to divide and are necessary for the colony because they can fix and share environmental nitrogen. In order to distribute the nitrogen efficiently, heterocysts are arranged to form a quasi-regular pattern whose features are maintained as the filament grows. Recent efforts have allowed advances in the understanding of the interactions and genetic mechanisms underlying this dynamic pattern. However, the main role of the patA and hetF genes are yet to be clarified; in particular, the patA mutant forms heterocysts almost exclusively in the terminal cells of the filament. In this work, we investigate the function of these genes and provide a theoretical model that explains how they interact within the broader genetic network, reproducing their knock-out phenotypes in several genetic backgrounds, including a nearly uniform concentration of HetR along the filament for the patA mutant. Our results suggest a role of hetF and patA in a post-transcriptional modification of HetR which is essential for its regulatory function. In addition, the existence of molecular leakage out of the filament in its boundary cells is enough to explain the preferential appearance of terminal heterocysts, without any need for a distinct regulatory pathway.

Understanding multicellular pattern formation is key for the study of both natural and synthetic developmental processes. Arguably one of the simplest model systems for this is the filamentous cyanobacterium Anabaena, that in conditions of nitrogen deprivation undergoes a dynamical differentiation process that differentiates roughly one in every ten cells into nitrogen-fixing heterocysts, in a quasi-regular pattern that is maintained as the filament keeps growing. One of the most characteristic mutations affecting this process forms heterocysts mostly constrained to the terminal cells of the filament. We have used experimental observations to propose a mathematical model of heterocyst differentiation able to reproduce this striking phenotype. The model extends our understanding of the regulations in this pattern-forming system and makes several predictions on molecular interactions. Importantly, a key aspect is the boundary condition at the filament’s ends: inhibitors of differentiation should be able to leak out of the filament, or otherwise the terminal cells would not differentiate. This highlights, in a very clear example, the importance of considering physical constraints in developmental processes.

The Making of a Heterocyst in Cyanobacteria

Heterocyst differentiation that occurs in some filamentous cyanobacteria, such as Anabaena sp. PCC 7120, provides a unique model for prokaryotic developmental biology. Heterocyst cells are formed in response to combined-nitrogen deprivation and possess a microoxic environment suitable for nitrogen fixation following extensive morphological and physiological reorganization. A filament of Anabaena is a true multicellular organism, as nitrogen and carbon sources are exchanged among different cells and cell types through septal junctions to ensure filament growth. Because heterocysts are terminally differentiated cells and unable to divide, their activity is an altruistic behavior dedicated to providing fixed nitrogen for neighboring vegetative cells. Heterocyst development is also a process of one-dimensional pattern formation, as heterocysts are semiregularly intercalated among vegetative cells. Morphogens form gradients along the filament and interact with each other in a fashion that fits well into the Turing model, a mathematical framework to explain biological pattern formation. Expected final online publication date for the Annual Review of Microbiology, Volume 76 is September 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Turing pattern design principles and their robustness

Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

The design principles of discrete turing patterning systems

The formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-species networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find these Turing pattern topologies to be substantially more robust to changes in the parameters of the model. We also find that diffusion-driven instabilities are substantially weaker predictors for Turing patterns in our discrete modelling framework in comparison to the continuous case, in the sense that the presence of an instability does not guarantee a pattern emerging in simulations. We show that a more refined criterion constitutes a stronger predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.

Turing's Diffusive Threshold in Random Reaction-Diffusion Systems

Turing instabilities of reaction-diffusion systems can only arise if the diffusivities of the chemical species are sufficiently different. This threshold is unphysical in most systems with N=2 diffusing species, forcing experimental realizations of the instability to rely on fluctuations or additional nondiffusing species. Here, we ask whether this diffusive threshold lowers for N>2 to allow "true" Turing instabilities. Inspired by May's analysis of the stability of random ecological communities, we analyze the probability distribution of the diffusive threshold in reaction-diffusion systems defined by random matrices describing linearized dynamics near a homogeneous fixed point. In the numerically tractable cases N⩽6, we find that the diffusive threshold becomes more likely to be smaller and physical as N increases, and that most of these many-species instabilities cannot be described by reduced models with fewer diffusing species.

Robust, coherent, and synchronized circadian clock-controlled oscillations along Anabaena filaments

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.

Impaired cell-cell communication in the multicellular cyanobacterium Anabaena affects carbon uptake, photosynthesis, and the cell wall

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.

Stochastic Turing Pattern Formation in a Model with Active and Passive Transport

We investigate Turing pattern formation in a stochastic and spatially discretized version of a reaction-diffusion-advection (RDA) equation, which was previously introduced to model synaptogenesis in C. elegans. The model describes the interactions between a passively diffusing molecular species and an advecting species that switches between anterograde and retrograde motor-driven transport (bidirectional transport). Within the context of synaptogenesis, the diffusing molecules can be identified with the protein kinase CaMKII and the advecting molecules as glutamate receptors. The stochastic dynamics evolves according to an RDA master equation, in which advection and diffusion are both modeled as hopping reactions along a one-dimensional array of chemical compartments. Carrying out a linear noise approximation of the RDA master equation leads to an effective Langevin equation, whose power spectrum provides a means of extending the definition of a Turing instability to stochastic systems, namely in terms of the existence of a peak in the power spectrum at a nonzero spatial frequency. We thus show how noise can significantly extend the range over which spontaneous patterns occur, which is consistent with previous studies of RD systems.

Precision and dissipation of a stochastic Turing pattern

Spontaneous pattern formation is a fundamental scientific problem that has received much attention since the seminal theoretical work of Turing on reaction-diffusion systems. In molecular biophysics, this phenomenon often takes place under the influence of large fluctuations. It is then natural to inquire about the precision of such pattern. In particular, spontaneous pattern formation is a nonequilibrium phenomenon, and the relation between the precision of a pattern and the thermodynamic cost associated with it remains largely unexplored. Here, we analyze this relation with a paradigmatic stochastic reaction-diffusion model, i.e., the Brusselator in one spatial dimension. We find that the precision of the pattern is maximized for an intermediate thermodynamic cost, i.e., increasing the thermodynamic cost beyond this value makes the pattern less precise. Even though fluctuations get less pronounced with an increase in thermodynamic cost, we argue that larger fluctuations can also have a positive effect on the precision of the pattern.

Nontrivial amplification below the threshold for excitable cell signaling

In many asymptotically stable fluid systems, arbitrarily small fluctuations can grow by orders of magnitude before eventually decaying, dramatically enhancing the fluctuation variance beyond the minimum predicted by linear stability theory. Here using influential quantitative models drawn from the mathematical biology literature, we establish that dramatic amplification of arbitrarily small fluctuations is found in excitable cell signaling systems as well. Our analysis highlights how positive and negative feedback, proximity to bifurcations, and strong separation of timescales can generate nontrivial fluctuations without nudging these systems across their excitation thresholds. These insights, in turn, are relevant for a broader range of related oscillatory, bistable, and pattern-forming systems that share these features. The common thread connecting all of these systems with fluid dynamical examples of noise amplification is non-normality.

HetL, HetR and PatS form a reaction-diffusion system to control pattern formation in the cyanobacterium nostoc PCC 7120

Local activation and long-range inhibition are mechanisms conserved in self-organizing systems leading to biological patterns. A number of them involve the production by the developing cell of an inhibitory morphogen, but how this cell becomes immune to self-inhibition is rather unknown. Under combined nitrogen starvation, the multicellular cyanobacterium Nostoc PCC 7120 develops nitrogen-fixing heterocysts with a pattern of one heterocyst every 10-12 vegetative cells. Cell differentiation is regulated by HetR which activates the synthesis of its own inhibitory morphogens, diffusion of which establishes the differentiation pattern. Here, we show that HetR interacts with HetL at the same interface as PatS, and that this interaction is necessary to suppress inhibition and to differentiate heterocysts. hetL expression is induced under nitrogen-starvation and is activated by HetR, suggesting that HetL provides immunity to the heterocyst. This protective mechanism might be conserved in other differentiating cyanobacteria as HetL homologues are spread across the phylum.

Expression from DIF1-motif promoters of hetR and patS is dependent on HetZ and modulated by PatU3 during heterocyst differentiation

HetR and PatS/PatX-derived peptides are the activator and diffusible inhibitor for cell differentiation and patterning in heterocyst-forming cyanobacteria. HetR regulates target genes via HetR-recognition sites. However, some genes (such as patS/patX) upregulated at the early stage of heterocyst differentiation possess DIF1 (or DIF⁺) motif (TCCGGA) promoters rather than HetR-recognition sites; hetR possesses both predicted regulatory elements. How HetR controls heterocyst-specific expression from DIF1 motif promoters remains to be answered. This study presents evidence that the expression from DIF1 motif promoters of hetR, patS and patX is more directly dependent on hetZ, a gene regulated by HetR via a HetR-recognition site. The HetR-binding site upstream of hetR is not required for the autoregulation of hetR. PatU3 (3′ portion of PatU) that interacts with HetZ may modulate the expression of hetR, hetZ and patS. These findings contribute to understanding of the mutual regulation of hetR, hetZ-patU and patS/patX in a large group of multicellular cyanobacteria.

Stochastic fluctuations and quasipattern formation in reaction-diffusion systems with anomalous transport

Many approaches to modeling reaction-diffusion systems with anomalous transport rely on deterministic equations which ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasipatterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the formation of both deterministic and stochastic patterns. Our analysis shows that the combination of subdiffusion and intrinsic stochasticity can reduce the threshold ratio of the effective diffusion coefficients required for pattern formation to a greater degree than either effect on its own.

Noise⁻Seeded Developmental Pattern Formation in Filamentous Cyanobacteria

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.

Stochastic Turing patterns in a synthetic bacterial population

The origin of biological morphology and form is one of the deepest problems in science, underlying our understanding of development and the functioning of living systems. In 1952, Alan Turing showed that chemical morphogenesis could arise from a linear instability of a spatially uniform state, giving rise to periodic pattern formation in reaction-diffusion systems but only those with a rapidly diffusing inhibitor and a slowly diffusing activator. These conditions are disappointingly hard to achieve in nature, and the role of Turing instabilities in biological pattern formation has been called into question. Recently, the theory was extended to include noisy activator-inhibitor birth and death processes. Surprisingly, this stochastic Turing theory predicts the existence of patterns over a wide range of parameters, in particular with no severe requirement on the ratio of activator-inhibitor diffusion coefficients. To explore whether this mechanism is viable in practice, we have genetically engineered a synthetic bacterial population in which the signaling molecules form a stochastic activator-inhibitor system. The synthetic pattern-forming gene circuit destabilizes an initially homogenous lawn of genetically engineered bacteria, producing disordered patterns with tunable features on a spatial scale much larger than that of a single cell. Spatial correlations of the experimental patterns agree quantitatively with the signature predicted by theory. These results show that Turing-type pattern-forming mechanisms, if driven by stochasticity, can potentially underlie a broad range of biological patterns. These findings provide the groundwork for a unified picture of biological morphogenesis, arising from a combination of stochastic gene expression and dynamical instabilities.