Stochastic properties of the plant circadian clock

Self-organization underlies developmental robustness in plants

Development is a self-organized process that builds on cells and their interactions. Cells are heterogeneous in gene expression, growth, and division; yet how development is robust despite such heterogeneity is a fascinating question. Here, we review recent progress on this topic, highlighting how developmental robustness is achieved through self-organization. We will first discuss sources of heterogeneity, including stochastic gene expression, heterogeneity in growth rate and direction, and heterogeneity in division rate and precision. We then discuss cellular mechanisms that buffer against such noise, including Paf1C- and miRNA-mediated denoising, spatiotemporal growth averaging and compensation, mechanisms to improve cell division precision, and coordination of growth rate and developmental timing between different parts of an organ. We also discuss cases where such heterogeneity is not buffered but utilized for development. Finally, we highlight potential directions for future studies of noise and developmental robustness.

Red Light Resets the Expression Pattern, Phase, and Period of the Circadian Clock in Plants: A Computational Approach

Simple Summary

Progress in computational biology has provided a comprehensive understanding of the dynamics of the plant circadian clock. Previously proposed models of the plant circadian clock have intended to model its entrainment using white-light/dark cycles. However, these models have failed to take into account the effect of light quality on circadian rhythms, which has been experimentally observed. In this work, we developed a computational approach to characterizing the effects of light quality on plant circadian rhythms. The results demonstrated that red light can reset the expression patterns, phases, and periods of clock component genes. The circadian period, amplitude, and phase can be co-optimized for high-quality and efficient breeding.

Abstract

Recent research in the fields of biochemistry and molecular biology has shown that different light qualities have extremely different effects on plant development, and optimizing light quality conditions can speed up plant growth. Clock-regulated red-light signaling, can enhance hypocotyl elongation, and increase seedling height and flower and fruit productivity. In order to investigate the effect of red light on circadian clocks in plants, a novel computational model was established. The expression profiles of the circadian element CCA1 from previous related studies were used to fit the model. The simulation results were validated by the expression patterns of CCA1 in Arabidopsis, including wild types and mutants, and by the phase shifts of CCA1 after red-light pulse. The model was used to further explore the complex responses to various photoperiods, such as the natural white-light/dark cycles, red/white/dark cycles, and extreme 24 h photoperiods. These results demonstrated that red light can reset the expression pattern, period, and phase of the circadian clock. Finally, we identified the dependence of phase shifts on the length of red-light pulse and the minimum red-light pulse length required for producing an observable phase shift. This work provides a promising computational approach to investigating the response of the circadian clock to other light qualities.

Cut the noise or couple up: Coordinating circadian and synthetic clocks

Circadian clocks are important to much of life on Earth and are of inherent interest to humanity, implicated in fields ranging from agriculture and ecology to developmental biology and medicine. New techniques show that it is not simply the presence of clocks, but coordination between them that is critical for complex physiological processes across the kingdoms of life. Recent years have also seen impressive advances in synthetic biology to the point where parallels can be drawn between synthetic biological and circadian oscillators. This review will emphasize theoretical and experimental studies that have revealed a fascinating dichotomy of coupling and heterogeneity among circadian clocks. We will also consolidate the fields of chronobiology and synthetic biology, discussing key design principles of their respective oscillators.

Stochastic simulation of a model for circadian rhythms in plants

Circadian clocks allow living organisms to anticipate and adapt to the daily variations of the environment. The interlocked feedback loops of the transcription factors network in the plant clock generate oscillations with expression peaks at specific times of the day. In this work, we explore the effect of molecular noise on the behavior of the plant circadian clock through numerical simulations. The influence of system size, photoperiod, and mutations of clock genes on the robustness of the oscillations are discussed. Our simulations show that the oscillations remain robust when the mRNA and protein levels are in the range of a few hundreds molecules. Entrainment by light-dark cycles enhances the robustness compared to constant conditions. Multiple light inputs and inter-cellular coupling also contribute to the robustness of the oscillations. The comparison between deterministic and stochastic simulations of single and double mutants shows that stochasticity does not qualitatively affect the behaviour of mutants but that they do not have the same robustness to noise. Finally, the model shows that noise can induce transitions between two limit cycles in a birhythmic clock mutant.

What is quantitative plant biology?

Quantitative plant biology is an interdisciplinary field that builds on a long history of biomathematics and biophysics. Today, thanks to high spatiotemporal resolution tools and computational modelling, it sets a new standard in plant science. Acquired data, whether molecular, geometric or mechanical, are quantified, statistically assessed and integrated at multiple scales and across fields. They feed testable predictions that, in turn, guide further experimental tests. Quantitative features such as variability, noise, robustness, delays or feedback loops are included to account for the inner dynamics of plants and their interactions with the environment. Here, we present the main features of this ongoing revolution, through new questions around signalling networks, tissue topology, shape plasticity, biomechanics, bioenergetics, ecology and engineering. In the end, quantitative plant biology allows us to question and better understand our interactions with plants. In turn, this field opens the door to transdisciplinary projects with the society, notably through citizen science.

Signaling pathways have an inherent need for noise to acquire information

BACKGROUND

All living systems acquire information about their environment. At the cellular level, they do so through signaling pathways. Such pathways rely on reversible binding interactions between molecules that detect and transmit the presence of an extracellular cue or signal to the cell's interior. These interactions are inherently stochastic and thus noisy. On the one hand, noise can cause a signaling pathway to produce the same response for different stimuli, which reduces the amount of information a pathway acquires. On the other hand, in processes such as stochastic resonance, noise can improve the detection of weak stimuli and thus the acquisition of information. It is not clear whether the kinetic parameters that determine a pathway's operation cause noise to reduce or increase the acquisition of information.

RESULTS

We analyze how the kinetic properties of the reversible binding interactions used by signaling pathways affect the relationship between noise, the response to a signal, and information acquisition. Our results show that, under a wide range of biologically sensible parameter values, a noisy dynamic of reversible binding interactions is necessary to produce distinct responses to different stimuli. As a consequence, noise is indispensable for the acquisition of information in signaling pathways.

CONCLUSIONS

Our observations go beyond previous work by showing that noise plays a positive role in signaling pathways, demonstrating that noise is essential when such pathways acquire information.

Circadian gating of dark-induced increases in chloroplast- and cytosolic-free calcium in Arabidopsis

Changes in the spatiotemporal concentration of free Ca²⁺ ([Ca²⁺ ]) in different organelles of the cell contribute to responses of plants to physiological and environmental stimuli. One example are [Ca²⁺ ] increases in the stroma of chloroplasts during light-to-dark transitions; however, the function and mechanisms responsible are unknown, in part because there is a disagreement in the literature concerning whether corresponding dark-induced changes in cytosolic [Ca²⁺ ] ([Ca²⁺ ]_cyt ) can be detected. We have measured changes in [Ca²⁺ ]_cyt upon darkness in addition to the already known dark-induced increases in [Ca²⁺ ]_stroma in the aerial part of the Arabidopsis thaliana plant. These [Ca²⁺ ]_cyt transients depend on the photoperiod and time of day, peaking at anticipated dusk, and are superimposed on daily 24 h oscillations in [Ca²⁺ ]_cyt . We also find that the magnitude of the dark-induced increases in Ca²⁺ in both the cytosol and chloroplasts are gated by the nuclear circadian oscillator. The modulation of the magnitude of dark-induced increases in [Ca²⁺ ]_stroma and [Ca²⁺ ]_cyt by transcriptional regulators in the nucleus that are part of the circadian oscillator demonstrates a new role for the circadian system in subcellular Ca²⁺ signalling, in addition to its role in driving circadian oscillations of [Ca²⁺ ] in the cytosol and chloroplasts.

Overview and Evaluation of Recent Methods for Statistical Inference of Gene Regulatory Networks from Time Series Data

A challenging problem in systems biology is the reconstruction of gene regulatory networks from postgenomic data. A variety of reverse engineering methods from machine learning and computational statistics have been proposed in the literature. However, deciding on the best method to adopt for a particular application or data set might be a confusing task. The present chapter provides a broad overview of state-of-the-art methods with an emphasis on conceptual understanding rather than a deluge of mathematical details, and the pros and cons of the various approaches are discussed. Guidance on practical applications with pointers to publicly available software implementations are included. The chapter concludes with a comprehensive comparative benchmark study on simulated data and a real-work application taken from the current plant systems biology.

Practical steps to digital organism models, from laboratory model species to 'Crops in silico

A recent initiative named 'Crops in silico' proposes that multi-scale models 'have the potential to fill in missing mechanistic details and generate new hypotheses to prioritize directed engineering efforts' in plant science, particularly directed to crop species. To that end, the group called for 'a paradigm shift in plant modelling, from largely isolated efforts to a connected community'. 'Wet' (experimental) research has been especially productive in plant science, since the adoption of Arabidopsis thaliana as a laboratory model species allowed the emergence of an Arabidopsis research community. Parts of this community invested in 'dry' (theoretical) research, under the rubric of Systems Biology. Our past research combined concepts from Systems Biology and crop modelling. Here we outline the approaches that seem most relevant to connected, 'digital organism' initiatives. We illustrate the scale of experimental research required, by collecting the kinetic parameter values that are required for a quantitative, dynamic model of a gene regulatory network. By comparison with the Systems Biology Markup Language (SBML) community, we note computational resources and community structures that will help to realize the potential for plant Systems Biology to connect with a broader crop science community.

Arabidopsis JMJD5/JMJ30 Acts Independently of LUX ARRHYTHMO Within the Plant Circadian Clock to Enable Temperature Compensation

The circadian system ensures that plants respond appropriately to environmental change by predicting regular transitions that occur during diel cycles. In order to be most useful, the circadian system needs to be compensated against daily and seasonal changes in temperature that would otherwise alter the pace of this biological oscillator. We demonstrate that an evening-phased protein, the putative histone demethylase JMJD5, contributes to temperature compensation. JMJD5 is co-expressed with components of the Evening Complex, an agglomeration of proteins including EARLY FLOWERING3 (ELF3), ELF4, and LUX ARRHYTHYMO (LUX), which also integrates temperature changes into the molecular clockwork. One role of the Evening Complex is to regulate expression of PSEUDORESPONSE REGULATOR9 (PRR9) and PRR7, important components of the temperature compensation mechanism. Surprisingly we find that LUX, but not other Evening Complex components, is dispensable for clock function at low temperatures. Further genetic analysis suggests JMJD5 acts in a parallel pathway to LUX within the circadian system. Although an intact JMJD5 catalytic domain is required for its function within the clock, our findings suggest JMJD5 does not directly regulate H3K36 methylation at circadian loci. Such data refine our understanding of how JMDJ5 acts within the Arabidopsis circadian system.

Dynamical differential expression (DyDE) reveals the period control mechanisms of the Arabidopsis circadian oscillator

The circadian oscillator, an internal time-keeping device found in most organisms, enables timely regulation of daily biological activities by maintaining synchrony with the external environment. The mechanistic basis underlying the adjustment of circadian rhythms to changing external conditions, however, has yet to be clearly elucidated. We explored the mechanism of action of nicotinamide in Arabidopsis thaliana, a metabolite that lengthens the period of circadian rhythms, to understand the regulation of circadian period. To identify the key mechanisms involved in the circadian response to nicotinamide, we developed a systematic and practical modeling framework based on the identification and comparison of gene regulatory dynamics. Our mathematical predictions, confirmed by experimentation, identified key transcriptional regulatory mechanisms of circadian period and uncovered the role of blue light in the response of the circadian oscillator to nicotinamide. We suggest that our methodology could be adapted to predict mechanisms of drug action in complex biological systems.

Coordination of robust single cell rhythms in the Arabidopsis circadian clock via spatial waves of gene expression

The Arabidopsis circadian clock orchestrates gene regulation across the day/night cycle. Although a multiple feedback loop circuit has been shown to generate the 24-hr rhythm, it remains unclear how robust the clock is in individual cells, or how clock timing is coordinated across the plant. Here we examine clock activity at the single cell level across Arabidopsis seedlings over several days under constant environmental conditions. Our data reveal robust single cell oscillations, albeit desynchronised. In particular, we observe two waves of clock activity; one going down, and one up the root. We also find evidence of cell-to-cell coupling of the clock, especially in the root tip. A simple model shows that cell-to-cell coupling and our measured period differences between cells can generate the observed waves. Our results reveal the spatial structure of the plant clock and suggest that unlike the centralised mammalian clock, the Arabidopsis clock has multiple coordination points.

Circadian Rhythms and Redox State in Plants: Till Stress Do Us Part

A growing body of evidence demonstrates a significant relationship between cellular redox state and circadian rhythms. Each day these two vital components of plant biology influence one another, dictating the pace for metabolism and physiology. Diverse environmental stressors can disrupt this condition and, although plant scientists have made significant progress in re-constructing functional networks of plant stress responses, stress impacts on the clock-redox crosstalk is poorly understood. Inter-connected phenomena such as redox state and metabolism, internal and external environments, cellular homeostasis and rhythms can impede predictive understanding of coordinated regulation of plant stress response. The integration of circadian clock effects into predictive network models is likely to increase final yield and better predict plant responses to stress. To achieve such integrated understanding, it is necessary to consider the internal clock not only as a gatekeeper of environmental responses but also as a target of stress syndromes. Using chlorophyll fluorescence as a reliable and high-throughput probe of stress coupled to functional genomics and metabolomics will provide insights on the crosstalk across a wide range of stress severity and duration, including potential insights into oxidative stress response and signaling. We suggest the efficiency of photosystem II in light conditions (Fv'/Fm') to be the most dynamic of the fluorescence variables and therefore the most reliable parameter to follow the stress response from early sensing to mortality.

Shining a light on the Arabidopsis circadian clock

The circadian clock provides essential timing information to ensure optimal growth to prevailing external environmental conditions. A major time-setting mechanism (zeitgeber) in clock synchronization is light. Differing light wavelengths, intensities, and photoperiodic duration are processed for the clock-setting mechanism. Many studies on light-input pathways to the clock have focused on Arabidopsis thaliana. Photoreceptors are specific chromic proteins that detect light signals and transmit this information to the central circadian oscillator through a number of different signalling mechanisms. The most well-characterized clock-mediating photoreceptors are cryptochromes and phytochromes, detecting blue, red, and far-red wavelengths of light. Ultraviolet and shaded light are also processed signals to the oscillator. Notably, the clock reciprocally generates rhythms of photoreceptor action leading to so-called gating of light responses. Intermediate proteins, such as Phytochrome interacting factors (PIFs), constitutive photomorphogenic 1 (COP1) and EARLY FLOWERING 3 (ELF3), have been established in signalling pathways downstream of photoreceptor activation. However, the precise details for these signalling mechanisms are not fully established. This review highlights both historical and recent efforts made to understand overall light input to the oscillator, first looking at how each wavelength of light is detected, this is then related to known input mechanisms and their interactions.

Targeting Bayes factors with direct-path non-equilibrium thermodynamic integration

Thermodynamic integration (TI) for computing marginal likelihoods is based on an inverse annealing path from the prior to the posterior distribution. In many cases, the resulting estimator suffers from high variability, which particularly stems from the prior regime. When comparing complex models with differences in a comparatively small number of parameters, intrinsic errors from sampling fluctuations may outweigh the differences in the log marginal likelihood estimates. In the present article, we propose a TI scheme that directly targets the log Bayes factor. The method is based on a modified annealing path between the posterior distributions of the two models compared, which systematically avoids the high variance prior regime. We combine this scheme with the concept of non-equilibrium TI to minimise discretisation errors from numerical integration. Results obtained on Bayesian regression models applied to standard benchmark data, and a complex hierarchical model applied to biopathway inference, demonstrate a significant reduction in estimator variance over state-of-the-art TI methods.

Approximate Bayesian inference in semi-mechanistic models

Inference of interaction networks represented by systems of differential equations is a challenging problem in many scientific disciplines. In the present article, we follow a semi-mechanistic modelling approach based on gradient matching. We investigate the extent to which key factors, including the kinetic model, statistical formulation and numerical methods, impact upon performance at network reconstruction. We emphasize general lessons for computational statisticians when faced with the challenge of model selection, and we assess the accuracy of various alternative paradigms, including recent widely applicable information criteria and different numerical procedures for approximating Bayes factors. We conduct the comparative evaluation with a novel inferential pipeline that systematically disambiguates confounding factors via an ANOVA scheme.

Stochastic Simulation of Biomolecular Networks in Dynamic Environments

Simulation of biomolecular networks is now indispensable for studying biological systems, from small reaction networks to large ensembles of cells. Here we present a novel approach for stochastic simulation of networks embedded in the dynamic environment of the cell and its surroundings. We thus sample trajectories of the stochastic process described by the chemical master equation with time-varying propensities. A comparative analysis shows that existing approaches can either fail dramatically, or else can impose impractical computational burdens due to numerical integration of reaction propensities, especially when cell ensembles are studied. Here we introduce the Extrande method which, given a simulated time course of dynamic network inputs, provides a conditionally exact and several orders-of-magnitude faster simulation solution. The new approach makes it feasible to demonstrate—using decision-making by a large population of quorum sensing bacteria—that robustness to fluctuations from upstream signaling places strong constraints on the design of networks determining cell fate. Our approach has the potential to significantly advance both understanding of molecular systems biology and design of synthetic circuits.

Simulation algorithms have become indispensable tools in modern quantitative biology, providing deep insight into many biochemical systems, including gene regulatory networks. However, current stochastic simulation approaches handle the effects of fluctuating extracellular signals and upstream processes poorly, either failing to give qualitatively reliable predictions or being very inefficient computationally. Here we introduce the Extrande method, a novel approach for simulation of biomolecular networks embedded in the dynamic environment of the cell and its surroundings. The method is accurate and computationally efficient, and hence fills an important gap in the field of stochastic simulation. In particular, we employ it to study a bacterial decision-making network and demonstrate that robustness to fluctuations from upstream signaling places strong constraints on the design of networks determining cell fate.

A Compact Model for the Complex Plant Circadian Clock

The circadian clock is an endogenous timekeeper that allows organisms to anticipate and adapt to the daily variations of their environment. The plant clock is an intricate network of interlocked feedback loops, in which transcription factors regulate each other to generate oscillations with expression peaks at specific times of the day. Over the last decade, mathematical modeling approaches have been used to understand the inner workings of the clock in the model plant Arabidopsis thaliana. Those efforts have produced a number of models of ever increasing complexity. Here, we present an alternative model that combines a low number of equations and parameters, similar to the very earliest models, with the complex network structure found in more recent ones. This simple model describes the temporal evolution of the abundance of eight clock gene mRNA/protein and captures key features of the clock on a qualitative level, namely the entrained and free-running behaviors of the wild type clock, as well as the defects found in knockout mutants (such as altered free-running periods, lack of entrainment, or changes in the expression of other clock genes). Additionally, our model produces complex responses to various light cues, such as extreme photoperiods and non-24 h environmental cycles, and can describe the control of hypocotyl growth by the clock. Our model constitutes a useful tool to probe dynamical properties of the core clock as well as clock-dependent processes.

Computational Modeling, Formal Analysis, and Tools for Systems Biology

As the amount of biological data in the public domain grows, so does the range of modeling and analysis techniques employed in systems biology. In recent years, a number of theoretical computer science developments have enabled modeling methodology to keep pace. The growing interest in systems biology in executable models and their analysis has necessitated the borrowing of terms and methods from computer science, such as formal analysis, model checking, static analysis, and runtime verification. Here, we discuss the most important and exciting computational methods and tools currently available to systems biologists. We believe that a deeper understanding of the concepts and theory highlighted in this review will produce better software practice, improved investigation of complex biological processes, and even new ideas and better feedback into computer science.

Dynamic properties of the segmentation clock mediated by microRNA

Somites are embryonic precursors that give rise to the axial skeleton and skeletal muscles and form the segmental vertebrate body plan. Somitogenesis is controlled by the "segmentation clock", which contains multiple oscillator genes that must be tightly regulated at both the transcriptional and post-transcriptional levels for proper clock function. However, how the segmentation clock governs the formation of the somites at post-transcriptional level, remains unclear. In this work, we develop an integrated model with three modules for the segmentation clock and explore the mechanism for somite segmentation based on the dynamics of the network. By numerical simulations, we find that the amplitude and period of the somite segmentation clock are sensitive to Notch activity, which is fine-tuned by Lunatic fringe (Lfng) and microRNA (miRNA), and Lfng and miRNA are essential for forming the proper segmentation during somitogenesis. Moreover, miRNA is found to have a crucial role in minimizing the fluctuation period and amplitude to maintain coherent oscillation. Introduction of stochasticity in the model enables us to explain the available experimental data with dampening of oscillations. These findings uncover a fresh mechanism for regulation of the segmentation clock at a post-transcriptional level and provide important insights into how the relatively subtle effects of miRNAs on target genes can have broad effects in developmental situations that have critical requirements for tight posttranscriptional regulation.

Stochastic models of cellular circadian rhythms in plants help to understand the impact of noise on robustness and clock structure

Rhythmic behavior is essential for plants; for example, daily (circadian) rhythms control photosynthesis and seasonal rhythms regulate their life cycle. The core of the circadian clock is a genetic network that coordinates the expression of specific clock genes in a circadian rhythm reflecting the 24-h day/night cycle. Circadian clocks exhibit stochastic noise due to the low copy numbers of clock genes and the consequent cell-to-cell variation: this intrinsic noise plays a major role in circadian clocks by inducing more robust oscillatory behavior. Another source of noise is the environment, which causes variation in temperature and light intensity: this extrinsic noise is part of the requirement for the structural complexity of clock networks. Advances in experimental techniques now permit single-cell measurements and the development of single-cell models. Here we present some modeling studies showing the importance of considering both types of noise in understanding how plants adapt to regular and irregular light variations. Stochastic models have proven useful for understanding the effect of regular variations. By contrast, the impact of irregular variations and the interaction of different noise sources are less well studied.

Conservation of Arabidopsis thaliana circadian clock genes in Chrysanthemum lavandulifolium

In Arabidopsis, circadian clock genes play important roles in photoperiod pathway by regulating the daytime expression of CONSTANS (CO), but related reports for chrysanthemum are notably limited. In this study, we isolated eleven circadian clock genes, which lie in the three interconnected negative and positive feedback loops in a wild diploid chrysanthemum, Chrysanthemum lavandulifolium. With the exception of ClELF3, ClPRR1 and ClPRR73, most of the circadian clock genes are expressed more highly in leaves than in other tested tissues. The diurnal rhythms of these circadian clock genes are similar to those of their homologs in Arabidopsis. ClELF3 and ClZTL are constitutively expressed at all time points in both assessed photoperiods. The expression succession from morning to night of the PSEUDO RESPONSE REGULATOR (PRR) gene family occurs in the order ClPRR73/ClPRR37, ClPRR5, and then ClPRR1. ClLHY is expressed during the dawn period, and ClGIs is expressed during the dusk period. The peak expression levels of ClFKF1 and ClGIs are synchronous in the inductive photoperiod. However, in the non-inductive night break (NB) condition or non-24 h photoperiod, the peak expression level of ClFKF1 is significantly changed, indicating that ClFKF1 itself or the synchronous expression of ClFKF1 and ClGIs might be essential to initiate the flowering of C. lavandulifolium. This study provides the first extensive evaluation of circadian clock genes, and it presents a useful foundation for dissecting the functions of circadian clock genes in C. lavandulifolium.

De novo transcriptome assembly and analyses of gene expression during photomorphogenesis in diploid wheat Triticum monococcum

BACKGROUND

Triticum monococcum (2n) is a close ancestor of T. urartu, the A-genome progenitor of cultivated hexaploid wheat, and is therefore a useful model for the study of components regulating photomorphogenesis in diploid wheat. In order to develop genetic and genomic resources for such a study, we constructed genome-wide transcriptomes of two Triticum monococcum subspecies, the wild winter wheat T. monococcum ssp. aegilopoides (accession G3116) and the domesticated spring wheat T. monococcum ssp. monococcum (accession DV92) by generating de novo assemblies of RNA-Seq data derived from both etiolated and green seedlings.

PRINCIPAL FINDINGS

The de novo transcriptome assemblies of DV92 and G3116 represent 120,911 and 117,969 transcripts, respectively. We successfully mapped ∼90% of these transcripts from each accession to barley and ∼95% of the transcripts to T. urartu genomes. However, only ∼77% transcripts mapped to the annotated barley genes and ∼85% transcripts mapped to the annotated T. urartu genes. Differential gene expression analyses revealed 22% more light up-regulated and 35% more light down-regulated transcripts in the G3116 transcriptome compared to DV92. The DV92 and G3116 mRNA sequence reads aligned against the reference barley genome led to the identification of ∼500,000 single nucleotide polymorphism (SNP) and ∼22,000 simple sequence repeat (SSR) sites.

CONCLUSIONS

De novo transcriptome assemblies of two accessions of the diploid wheat T. monococcum provide new empirical transcriptome references for improving Triticeae genome annotations, and insights into transcriptional programming during photomorphogenesis. The SNP and SSR sites identified in our analysis provide additional resources for the development of molecular markers.

No time for spruce: rapid dampening of circadian rhythms in Picea abies (L. Karst)

The identification and cloning of full-length homologs of circadian clock genes from Picea abies represent a first step to study the function and evolution of the circadian clock in gymnosperms. Phylogenetic analyses suggest that the sequences of key circadian clock genes are conserved between angiosperms and gymnosperms. though fewer homologous copies were found for most gene families in P. abies. We detected diurnal cycling of circadian clock genes in P. abies using quantitative real-time PCR; however, cycling appeared to be rapidly dampened under free-running conditions. Given the unexpected absence of transcriptional cycling during constant conditions, we employed a complementary method to assay circadian rhythmic outputs and measured delayed fluorescence in seedlings of Norway spruce. Neither of the two approaches to study circadian rhythms in Norway spruce could detect robust ∼24 h cycling behavior under constant conditions. These data suggest gene conservation but fundamental differences in clock function between gymnosperms and other plant taxa.

Mathematical models light up plant signaling

Plants respond to changes in the environment by triggering a suite of regulatory networks that control and synchronize molecular signaling in different tissues, organs, and the whole plant. Molecular studies through genetic and environmental perturbations, particularly in the model plant Arabidopsis thaliana, have revealed many of the mechanisms by which these responses are actuated. In recent years, mathematical modeling has become a complementary tool to the experimental approach that has furthered our understanding of biological mechanisms. In this review, we present modeling examples encompassing a range of different biological processes, in particular those regulated by light. Current issues and future directions in the modeling of plant systems are discussed.

Modeling and simulating the Arabidopsis thaliana circadian clock using XPP-AUTO

Circadian clocks are endogenous timekeepers that produce oscillations with a period of about one day. Their rhythmicity originates from complex gene regulatory networks at the cellular level. In the last decades, computational models have been proven to be a powerful tool in order to understand the dynamics and design principles of the complex regulatory circuitries underlying the circadian clocks of different organisms. We present the process of model development using a small and simplified two-gene regulatory network of the Arabidopsis circadian clock. Subsequently, we discuss important numerical techniques to analyze such a mathematical model using XPP-AUTO. We show how to solve deterministic and stochastic ordinary differential equations and how to compute bifurcation diagrams or simulate phase-shift experiments. We finally discuss the contributions of modeling to the understanding and dissection of the Arabidopsis circadian system.

A neuropeptide speeds circadian entrainment by reducing intercellular synchrony

Shift work or transmeridian travel can desynchronize the body's circadian rhythms from local light-dark cycles. The mammalian suprachiasmatic nucleus (SCN) generates and entrains daily rhythms in physiology and behavior. Paradoxically, we found that vasoactive intestinal polypeptide (VIP), a neuropeptide implicated in synchrony among SCN cells, can also desynchronize them. The degree and duration of desynchronization among SCN neurons depended on both the phase and the dose of VIP. A model of the SCN consisting of coupled stochastic cells predicted both the phase- and the dose-dependent response to VIP and that the transient phase desynchronization, or "phase tumbling", could arise from intrinsic, stochastic noise in small populations of key molecules (notably, Period mRNA near its daily minimum). The model also predicted that phase tumbling following brief VIP treatment would accelerate entrainment to shifted environmental cycles. We tested this using a prepulse of VIP during the day before a shift in either a light cycle in vivo or a temperature cycle in vitro. Although VIP during the day does not shift circadian rhythms, the VIP pretreatment approximately halved the time required for mice to reentrain to an 8-h shifted light schedule and for SCN cultures to reentrain to a 10-h shifted temperature cycle. We conclude that VIP below 100 nM synchronizes SCN cells and above 100 nM reduces synchrony in the SCN. We show that exploiting these mechanisms that transiently reduce cellular synchrony before a large shift in the schedule of daily environmental cues has the potential to reduce jet lag.

Mathematical modeling of an oscillating gene circuit to unravel the circadian clock network of Arabidopsis thaliana

The Arabidopsis thaliana circadian clock is an interconnected network highly tractable to systems approaches. Most elements in the transcriptional-translational oscillator were identified by genetic means and the expression of clock genes in various mutants led to the founding hypothesis of a positive-negative feedback loop being the core clock. The identification of additional clock genes beyond those defined in the core led to the use of systems approaches to decipher this angiosperm oscillator circuit. Kinetic modeling was first used to explain periodicity effects of various circadian mutants. This conformed in a flexible way to experimental details. Such observations allowed a recursive use of hypothesis generating from modeling, followed by experimental corroboration. More recently, the biochemical finding of new description of a DNA-binding activity for one class of clock components directed improvements in feature generation, one of which revealed that the core of the oscillator is a negative-negative feedback loop. The recursive use of modeling and experimental validation has thus revealed many essential transcriptional components that drive negative arms in the circadian oscillator. What awaits is to more fully describe the positive arms and an understanding of how additional pathways converge on the clock.

Modeling regulatory networks to understand plant development: small is beautiful

We now have unprecedented capability to generate large data sets on the myriad genes and molecular players that regulate plant development. Networks of interactions between systems components can be derived from that data in various ways and can be used to develop mathematical models of various degrees of sophistication. Here, we discuss why, in many cases, it is productive to focus on small networks. We provide a brief and accessible introduction to relevant mathematical and computational approaches to model regulatory networks and discuss examples of small network models that have helped generate new insights into plant biology (where small is beautiful), such as in circadian rhythms, hormone signaling, and tissue patterning. We conclude by outlining some of the key technical and modeling challenges for the future.

Spontaneous spatiotemporal waves of gene expression from biological clocks in the leaf

The circadian clocks that drive daily rhythms in animals are tightly coupled among the cells of some tissues. The coupling profoundly affects cellular rhythmicity and is central to contemporary understanding of circadian physiology and behavior. In contrast, studies of the clock in plant cells have largely ignored intercellular coupling, which is reported to be very weak or absent. We used luciferase reporter gene imaging to monitor circadian rhythms in leaves of Arabidopsis thaliana plants, achieving resolution close to the cellular level. Leaves grown without environmental cycles for up to 3 wk reproducibly showed spatiotemporal waves of gene expression consistent with intercellular coupling, using several reporter genes. Within individual leaves, different regions differed in phase by up to 17 h. A broad range of patterns was observed among leaves, rather than a common spatial distribution of circadian properties. Leaves exposed to light-dark cycles always had fully synchronized rhythms, which could desynchronize rapidly. After 4 d in constant light, some leaves were as desynchronized as leaves grown without any rhythmic input. Applying light-dark cycles to such a leaf resulted in full synchronization within 2-4 d. Thus, the rhythms of all cells were coupled to external light-dark cycles far more strongly than the cellular clocks were coupled to each other. Spontaneous desynchronization under constant conditions was limited, consistent with weak intercellular coupling among heterogeneous clocks. Both the weakness of coupling and the heterogeneity among cells are relevant to interpret molecular studies and to understand the physiological functions of the plant circadian clock.

Communication: limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error between the predictions of the two master equations for the Michaelis-Menten reaction with substrate input. The reduced approach is predicted to overestimate the variance of the substrate concentration fluctuations by as much as 30%. The theoretical results are validated by stochastic simulations using experimental parameter values for enzymes involved in proteolysis, gluconeogenesis, and fermentation.