A novel computational model of the circadian clock in Arabidopsis that incorporates PRR7 and PRR9

Decoding the plant clock: a review of mathematical models for the circadian regulatory network

Most organisms have evolved specific mechanisms to respond to changes in environmental conditions such as light and temperature over the course of day. These periodic changes in the physiology and behaviour of organisms, referred to as circadian rhythms, are a consequence of intricate molecular mechanisms in the form of transcription and translational feedback loops. The plant circadian regulatory network is a complex web of interconnected feedback loops involving various transcription factors such as CCA1, LHY, PRRs, TOC1, LUX, ELF3, ELF4, RVE8, and more. This network enables plants to adapt and thrive in diverse environmental conditions. It responds to entrainment signals, including light, temperature, and nutrient concentrations and interacts with most of the physiological functions such as flowering, growth and stress response. Mathematical modelling of these gene regulatory networks enables a deeper understanding of not only the function but also the perturbations that may affect the plant growth and function with changing climate. Over the years, numerous mathematical models have been developed to understand the diverse aspects of plant circadian regulation. In this review, we have delved into the systematic development of these models, outlining the model components and refinements over time. We have also highlighted strengths and limitations of each of the models developed so far. Finally, we conclude the review by describing the prospects for investigation and advancement of these models for better understanding of plant circadian regulation.

The σB alternative sigma factor circuit modulates noise to generate different types of pulsing dynamics

Single-cell approaches are revealing a high degree of heterogeneity, or noise, in gene expression in isogenic bacteria. How gene circuits modulate this noise in gene expression to generate robust output dynamics is unclear. Here we use the Bacillus subtilis alternative sigma factor σB as a model system for understanding the role of noise in generating circuit output dynamics. σB controls the general stress response in B. subtilis and is activated by a range of energy and environmental stresses. Recent single-cell studies have revealed that the circuit can generate two distinct outputs, stochastic pulsing and a single pulse response, but the conditions under which each response is generated are under debate. We implement a stochastic mathematical model of the σB circuit to investigate this and find that the system's core circuit can generate both response types. This is despite one response (stochastic pulsing) being stochastic in nature, and the other (single response pulse) being deterministic. We demonstrate that the main determinant for whichever response is generated is the degree with which the input pathway activates the core circuit, although the noise properties of the input pathway also biases the system towards one or the other type of output. Thus, our work shows how stochastic modelling can reveal the mechanisms behind non-intuitive gene circuit output dynamics.

The evolution and function of the PSEUDO RESPONSE REGULATOR gene family in the plant circadian clock

PSEUDO-RESPONSE PROTEINS (PRRs) are a gene family vital for the generation of rhythms by the circadian clock. Plants have circadian clocks, or circadian oscillators, to adapt to a rhythmic environment. The circadian clock system can be divided into three parts: the core oscillator, the input pathways, and the output pathways. The PRRs have a role in all three parts. These nuclear proteins have an N-terminal pseudo receiver domain and a C-terminal CONSTANS, CONSTANS-LIKE, and TOC1 (CCT) domain. The PRRs can be identified from green algae to monocots, ranging from one to >5 genes per species. Arabidopsis thaliana, for example, has five genes: PRR9, PRR7, PRR5, PRR3 and TOC1/PRR1. The PRR genes can be divided into three clades using protein homology: TOC1/PRR1, PRR7/3, and PRR9/5 expanded independently in eudicots and monocots. The PRRs can make protein complexes and bind to DNA, and the wide variety of protein-protein interactions are essential for the multiple roles in the circadian clock. In this review, the history of PRR research is briefly recapitulated, and the diversity of PRR genes in green and recent works about their role in the circadian clock are discussed.

Systems Medicine Design based on Systems Biology Approaches and Deep Neural Network for Gastric Cancer

Gastric cancer (GC) is the third leading cause of cancer death in the world. It is associated with the stimulation of microenvironment, aberrant epigenetic modification, and chronic inflammation. However, few researches discuss the GC molecular progression mechanisms from the perspective of the system level. In this study, we proposed a systems medicine design procedure to identify essential biomarkers and find corresponding drugs for GC. At first, we did big database mining to construct candidate protein-protein interaction network (PPIN) and candidate gene regulation network (GRN). Second, by leveraging the next-generation sequencing (NGS) data, we performed system modeling and applied system identification and model selection to obtain real genome-wide genetic and epigenetic networks (GWGENs). To make the real GWGENs easy to analyze, the principal network projection method was used to extract the core signaling pathways denoted by KEGG pathways. Subsequently, based on the identified biomarkers, we trained a deep neural network of drug-target interaction (DeepDTI) with supervised learning and filtered our candidate drugs considering drug regulation ability and drug sensitivity. With the proposed systematic strategy, we not only shed the light on the progression of GC but also suggested potential multiple-molecule drugs efficiently.

Environment-mediated mutagenetic interference on genetic stabilization and circadian rhythm in plants

Many mortal organisms on this planet have developed the potential to merge all internal as well as external environmental cues to regulate various processes running inside organisms and in turn make them adaptive to the environment through the circadian clock. This moving rotator controls processes like activation of hormonal, metabolic, or defense pathways, initiation of flowering at an accurate period, and developmental processes in plants to ensure their stability in the environment. All these processes that are under the control of this rotating wheel can be changed either by external environmental factors or by an unpredictable phenomenon called mutation that can be generated by either physical mutagens, chemical mutagens, or by internal genetic interruption during metabolic processes, which alters normal functionality of organisms like innate immune responses, entrainment of the clock, biomass reduction, chlorophyll formation, and hormonal signaling, despite its fewer positive roles in plants like changing plant type, loss of vernalization treatment to make them survivable in different latitudes, and defense responses during stress. In addition, with mutation, overexpression of gene components sometimes supresses mutation effect and promote normal circadian genes abundance in the cell, while sometimes it affects circadian functionality by generating arrhythmicity and shows that not only mutation but overexpression also effects normal functional activities of plant. Therefore, this review mainly summarizes the role of each circadian clock genes in regulating rhythmicity, and shows that how circadian outputs are controlled by mutations as well as overexpression phenomenon.

A spatial model of the plant circadian clock reveals design principles for coordinated timing

Individual plant cells possess a genetic network, the circadian clock, that times internal processes to the day-night cycle. Mathematical models of the clock are typically either "whole-plant" that ignore tissue or cell type-specific clock behavior, or "phase-only" that do not include molecular components. To address the complex spatial coordination observed in experiments, here we implemented a clock network model on a template of a seedling. In our model, the sensitivity to light varies across the plant, and cells communicate their timing via local or long-distance sharing of clock components, causing their rhythms to couple. We found that both varied light sensitivity and long-distance coupling could generate period differences between organs, while local coupling was required to generate the spatial waves of clock gene expression observed experimentally. We then examined our model under noisy light-dark cycles and found that local coupling minimized timing errors caused by the noise while allowing each plant region to maintain a different clock phase. Thus, local sensitivity to environmental inputs combined with local coupling enables flexible yet robust circadian timing.

Modelling of plant circadian clock for characterizing hypocotyl growth under different light quality conditions

To meet the ever-increasing global food demand, the food production rate needs to be increased significantly in the near future. Speed breeding is considered as a promising agricultural technology solution to achieve the zero-hunger vision as specified in the United Nations Sustainable Development Goal 2. In speed breeding, the photoperiod of the artificial light has been manipulated to enhance crop productivity. In particular, regulating the photoperiod of different light qualities rather than solely white light can further improve speed breading. However, identifying the optimal light quality and the associated photoperiod simultaneously remains a challenging open problem due to complex interactions between multiple photoreceptors and proteins controlling plant growth. To tackle this, we develop a first comprehensive model describing the profound effect of multiple light qualities with different photoperiods on plant growth (i.e. hypocotyl growth). The model predicts that hypocotyls elongated more under red light compared to both red and blue light. Drawing similar findings from previous related studies, we propose that this might result from the competitive binding of red and blue light receptors, primarily Phytochrome B (phyB) and Cryptochrome 1 (cry1) for the core photomorphogenic regulator, CONSTITUTIVE PHOTOMORPHOGENIC 1 (COP1). This prediction is validated through an experimental study on Arabidopsis thaliana. Our work proposes a potential molecular mechanism underlying plant growth under different light qualities and ultimately suggests an optimal breeding protocol that takes into account light quality.

Universality in kinetic models of circadian rhythms in [Formula: see text]

Biological evolution has endowed the plant Arabidopsis thaliana with genetically regulated circadian rhythms. A number of authors have published kinetic models for these oscillating chemical reactions based on a network of interacting genes. To investigate the hypothesis that the Arabidopsis circadian dynamical system is poised near a Hopf bifurcation like some other biological oscillators, we varied the kinetic parameters in the models and searched for bifurcations. Finding that each model does exhibit a supercritical Hopf bifurcation, we performed a weakly nonlinear analysis near the bifurcation points to derive the Stuart-Landau amplitude equation. To illustrate a common dynamical structure, we scaled the numerical solutions to the models with the asymptotic solutions to the Stuart-Landau equation to collapse the circadian oscillations onto two universal curves-one for amplitude, and one for frequency. However, some models are close to bifurcation while others are far, some models are post-bifurcation while others are pre-bifurcation, and kinetic parameters that lead to a bifurcation in some models do not lead to a bifurcation in others. Future kinetic modeling can make use of our analysis to ensure models are consistent with each other and with the dynamics of the Arabidopsis circadian rhythm.

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.

Layers of crosstalk between circadian regulation and environmental signalling in plants

Circadian regulation has a pervasive influence upon plant development, physiology and metabolism, impacting upon components of fitness and traits of agricultural importance. Circadian regulation is inextricably connected to the responses of plants to their abiotic environments, from the cellular to whole plant scales. Here, we review the crosstalk that occurs between circadian regulation and responses to the abiotic environment from the intracellular scale through to naturally fluctuating environments. We examine the spatial crosstalk that forms part of plant circadian regulation, at the subcellular, tissue, organ and whole-plant scales. This includes a focus on chloroplast and mitochondrial signalling, alternative splicing, long-distance circadian signalling and circadian regulation within natural environments. We also consider mathematical models for plant circadian regulation, to suggest future areas for advancing understanding of roles for circadian regulation in plant responses to environmental cues.

Systems Medicine Design for Triple-Negative Breast Cancer and Non-Triple-Negative Breast Cancer Based on Systems Identification and Carcinogenic Mechanisms

Triple-negative breast cancer (TNBC) is a heterogeneous subtype of breast cancers with poor prognosis. The etiology of triple-negative breast cancer (TNBC) is involved in various biological signal cascades and multifactorial aberrations of genetic, epigenetic and microenvironment. New therapeutic for TNBC is urgently needed because surgery and chemotherapy are the only available modalities nowadays. A better understanding of the molecular mechanisms would be a great challenge because they are triggered by cascade signaling pathways, genetic and epigenetic regulations, and drug–target interactions. This would allow the design of multi-molecule drugs for the TNBC and non-TNBC. In this study, in terms of systems biology approaches, we proposed a systematic procedure for systems medicine design toward TNBC and non-TNBC. For systems biology approaches, we constructed a candidate genome-wide genetic and epigenetic network (GWGEN) by big databases mining and identified real GWGENs of TNBC and non-TNBC assisting with corresponding microarray data by system identification and model order selection methods. After that, we applied the principal network projection (PNP) approach to obtain the core signaling pathways denoted by KEGG pathway of TNBC and non-TNBC. Comparing core signaling pathways of TNBC and non-TNBC, essential carcinogenic biomarkers resulting in multiple cellular dysfunctions including cell proliferation, autophagy, immune response, apoptosis, metastasis, angiogenesis, epithelial-mesenchymal transition (EMT), and cell differentiation could be found. In order to propose potential candidate drugs for the selected biomarkers, we designed filters considering toxicity and regulation ability. With the proposed systematic procedure, we not only shed a light on the differences between carcinogenetic molecular mechanisms of TNBC and non-TNBC but also efficiently proposed candidate multi-molecule drugs including resveratrol, sirolimus, and prednisolone for TNBC and resveratrol, sirolimus, carbamazepine, and verapamil for non-TNBC.

Circadian Rhythm and Pain: Mathematical Model based on Multiagent Simulation

The circadian rhythm is responsible for the daily variations in metabolism, and circadian rhythm disorders have direct implications for many diseases, such as obesity and mental disorders. The regulation of sleep time is the most common example of the importance of the circadian rhythm for the functioning of the human body. In this sense, this work aims to study a mathematical and computational model based on multiagent simulation that simulates the synchronization and desynchronization of the circadian rhythm in relation to the pain variables. The results from the multiagent simulation of circadian rhythms show that in relation to pain, sleep, especially its biological rhythms, is directly affected by pain. In this way, our mathematical model was able to show that pain causes changes in the circadian rhythm and it can contribute to the medical field analysis.

Chemical Perturbation of Chloroplast-Related Processes Affects Circadian Rhythms of Gene Expression in Arabidopsis: Salicylic Acid Application Can Entrain the Clock

The plant circadian system reciprocally interacts with metabolic processes. To investigate entrainment features in metabolic–circadian interactions, we used a chemical approach to perturb metabolism and monitored the pace of nuclear-driven circadian oscillations. We found that chemicals that alter chloroplast-related functions modified the circadian rhythms. Both vitamin C and paraquat altered the circadian period in a light-quality-dependent manner, whereas rifampicin lengthened the circadian period under darkness. Salicylic acid (SA) increased oscillatory robustness and shortened the period. The latter was attenuated by sucrose addition and was also gated, taking place during the first 3 h of the subjective day. Furthermore, the effect of SA on period length was dependent on light quality and genotype. Period lengthening or shortening by these chemicals was correlated to their inferred impact on photosynthetic electron transport activity and the redox state of plastoquinone (PQ). Based on these data and on previous publications on circadian effects that alter the redox state of PQ, we propose that the photosynthetic electron transport and the redox state of PQ participate in circadian periodicity. Moreover, coupling between chloroplast-derived signals and nuclear oscillations, as observed in our chemical and genetic assays, produces traits that are predicted by previous models. SA signaling or a related process forms a rhythmic input loop to drive robust nuclear oscillations in the context predicted by the zeitnehmer model, which was previously developed for Neurospora. We further discuss the possibility that electron transport chains (ETCs) are part of this mechanism.

Photoperiod sensing of the circadian clock is controlled by EARLY FLOWERING 3 and GIGANTEA

ELF3 and GI are two important components of the Arabidopsis circadian clock. They are not only essential for the oscillator function but are also pivotal in mediating light inputs to the oscillator. Lack of either results in a defective oscillator causing severely compromised output pathways, such as photoperiodic flowering and hypocotyl elongation. Although single loss of function mutants of ELF3 and GI have been well studied, their genetic interaction remains unclear. We generated an elf3 gi double mutant to study their genetic relationship in clock-controlled growth and phase transition phenotypes. We found that ELF3 and GI repress growth differentially during the night and the day, respectively. Circadian clock assays revealed that ELF3 and GI are essential that enable the oscillator to synchronize the endogenous cellular mechanisms to external environmental signals. In their absence, the circadian oscillator fails to synchronize to the light-dark cycles even under diurnal conditions. Consequently, clock-mediated photoperiod-responsive growth and development are completely lost in plants lacking both genes, suggesting that ELF3 and GI together convey photoperiod sensing to the central oscillator. Since ELF3 and GI are conserved across flowering plants and represent important breeding and domestication targets, our data highlight the possibility of developing photoperiod-insensitive crops by adjusting the allelic combination of these two key genes.

Photoperiod sensing of the circadian clock is controlled by EARLY FLOWERING 3 and GIGANTEA

ELF3 and GI are two important components of the Arabidopsis circadian clock. They are not only essential for the oscillator function but are also pivotal in mediating light inputs to the oscillator. Lack of either results in a defective oscillator causing severely compromised output pathways, such as photoperiodic flowering and hypocotyl elongation. Although single loss of function mutants of ELF3 and GI have been well studied, their genetic interaction remains unclear. We generated an elf3 gi double mutant to study their genetic relationship in clock-controlled growth and phase transition phenotypes. We found that ELF3 and GI repress growth differentially during the night and the day, respectively. Circadian clock assays revealed that ELF3 and GI are essential that enable the oscillator to synchronize the endogenous cellular mechanisms to external environmental signals. In their absence, the circadian oscillator fails to synchronize to the light-dark cycles even under diurnal conditions. Consequently, clock-mediated photoperiod-responsive growth and development are completely lost in plants lacking both genes, suggesting that ELF3 and GI together convey photoperiod sensing to the central oscillator. Since ELF3 and GI are conserved across flowering plants and represent important breeding and domestication targets, our data highlight the possibility of developing photoperiod-insensitive crops by adjusting the allelic combination of these two key genes.

Alternative Splicing of Circadian Clock Genes Correlates With Temperature in Field-Grown Sugarcane

Alternative Splicing (AS) is a mechanism that generates different mature transcripts from precursor mRNAs (pre-mRNAs) of the same gene. In plants, a wide range of physiological and metabolic events are related to AS, as well as fast responses to changes in temperature. AS is present in around 60% of intron-containing genes in Arabidopsis, 46% in rice, and 38% in maize and it is widespread among the circadian clock genes. Little is known about how AS influences the circadian clock of C4 plants, like commercial sugarcane, a C4 crop with a complex hybrid genome. This work aims to test if the daily dynamics of AS forms of circadian clock genes are regulated by environmental factors, such as temperature, in the field. A systematic search for AS in five sugarcane clock genes, ScLHY, ScPRR37, ScPRR73, ScPRR95, and ScTOC1 using different organs of sugarcane sampled during winter, with 4 months old plants, and during summer, with 9 months old plants, revealed temperature- and organ-dependent expression of at least one alternatively spliced isoform in all genes. Expression of AS isoforms varied according to the season. Our results suggest that AS events in circadian clock genes are correlated with temperature.

Heat the Clock: Entrainment and Compensation in Arabidopsis Circadian Rhythms

The circadian clock is a biological mechanism that permits some organisms to anticipate daily environmental variations. This clock generates biological rhythms, which can be reset by environmental cues such as cycles of light or temperature, a process known as entrainment. After entrainment, circadian rhythms typically persist with approximately 24 hours periodicity in free-running conditions, i.e. in the absence of environmental cues. Experimental evidence also shows that a free-running period close to 24 hours is maintained across a range of temperatures, a process known as temperature compensation. In the plant Arabidopsis, the effect of light on the circadian system has been widely studied and successfully modelled mathematically. However, the role of temperature in periodicity, and the relationship between entrainment and compensation, are not fully understood. Here we adapt recent models to incorporate temperature dependence by applying Arrhenius equations to the parameters of the models that characterize transcription, translation, and degradation rates. We show that the resulting models can exhibit thermal entrainment and temperature compensation, but that these phenomena emerge from physiologically different sets of processes. Further simulations combining thermal and photic forcing in more realistic scenarios clearly distinguish between the processes of entrainment and compensation, and reveal temperature compensation as an emergent property which can arise as a result of multiple temperature-dependent interactions. Our results consistently point to the thermal sensitivity of degradation rates as driving compensation and entrainment across a range of conditions.

Circadian oscillator proteins across the kingdoms of life: structural aspects

Circadian oscillators are networks of biochemical feedback loops that generate 24-hour rhythms in organisms from bacteria to animals. These periodic rhythms result from a complex interplay among clock components that are specific to the organism, but share molecular mechanisms across kingdoms. A full understanding of these processes requires detailed knowledge, not only of the biochemical properties of clock proteins and their interactions, but also of the three-dimensional structure of clockwork components. Posttranslational modifications and protein–protein interactions have become a recent focus, in particular the complex interactions mediated by the phosphorylation of clock proteins and the formation of multimeric protein complexes that regulate clock genes at transcriptional and translational levels. This review covers the structural aspects of circadian oscillators, and serves as a primer for this exciting realm of structural biology.

Computational and experimental insights into the circadian effects of SIRT1

The circadian clock orchestrates 24-h rhythms in physiology in most living organisms. At the molecular level, the dogma is that circadian oscillations are based on a negative transcriptional feedback loop. Recent studies found the NAD+-dependent histone deacetylase, SIRT1, directly regulates acetylation status of clock components and influences circadian amplitude in cells. While Nakahata et al. [Nakahata Y, Kaluzova M (2008) Cell 134:329-340] reported that loss of SIRT1 increases amplitude through BMAL1 acetylation, Asher et al. [Asher G, Gatfield D (2008) Cell 134:317-328] reported that loss of SIRT1 decreases amplitude through an increase in acetylated PER2. To address this SIRT1 paradox, we developed a circadian enzymatic model. Predictions from this model and experimental validation strongly align with the findings of Asher et al., with PER2 as the primary target of SIRT1. Further, the model suggested SIRT1 influences BMAL1 expression through actions on PGC1α. We validated this finding experimentally. Thus, our computational and experimental approaches suggest SIRT1 positively regulates clock function through actions on PER2 and PGC1α.

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.

Organ specificity in the plant circadian system is explained by different light inputs to the shoot and root clocks

Circadian clocks allow the temporal compartmentalization of biological processes. In Arabidopsis, circadian rhythms display organ specificity but the underlying molecular causes have not been identified. We investigated the mechanisms responsible for the similarities and differences between the clocks of mature shoots and roots in constant conditions and in light : dark cycles. We developed an imaging system to monitor clock gene expression in shoots and light- or dark-grown roots, modified a recent mathematical model of the Arabidopsis clock and used this to simulate our new data. We showed that the shoot and root circadian clocks have different rhythmic properties (period and amplitude) and respond differently to light quality. The root clock was entrained by direct exposure to low-intensity light, even in antiphase to the illumination of shoots. Differences between the clocks were more pronounced in conditions where light was present than in constant darkness, and persisted in the presence of sucrose. We simulated the data successfully by modifying those parameters of a clock model that are related to light inputs. We conclude that differences and similarities between the shoot and root clocks can largely be explained by organ-specific light inputs. This provides mechanistic insight into the developing field of organ-specific clocks.

Transcriptional and post-transcriptional control of the plant circadian gene regulatory network

The circadian clock drives rhythms in multiple physiological processes allowing plants to anticipate and adjust to periodic changes in environmental conditions. These physiological rhythms are associated with robust oscillations in the expression of thousands of genes linked to the control of photosynthesis, cell elongation, biotic and abiotic stress responses, developmental processes such as flowering, and the clock itself. Given its pervasive effects on plant physiology, it is not surprising that circadian clock genes have played an important role in the domestication of crop plants and in the improvement of crop productivity. Therefore, identifying the principles governing the dynamics of the circadian gene regulatory network in plants could strongly contribute to further speed up crop improvement. Here we provide an historical as well as a current description of our knowledge of the molecular mechanisms underlying circadian rhythms in plants. This work focuses on the transcriptional and post-transcriptional regulatory layers that control the very core of the circadian clock, and some of its complex interactions with signaling pathways that help synchronize plant growth and development to daily and seasonal changes in the environment. This article is part of a Special Issue entitled: Plant Gene Regulatory Mechanisms and Networks, edited by Dr. Erich Grotewold and Dr. Nathan Springer.

Phase Sensitivity Analysis of Circadian Rhythm Entrainment

As a biological clock, circadian rhythms evolve to accomplish a stable (robust) entrainment to environmental cycles, of which light is the most obvious. The mechanism of photic entrainment is not known, but two models of entrainment have been proposed based on whether light has a continuous (parametric) or discrete (nonparametric) effect on the circadian pacemaker. A novel sensitivity analysis is developed to study the circadian entrainment in silico based on a limit cycle approach and applied to a model of Drosophila circadian rhythm. The comparative analyses of complete and skeleton photoperiods suggest a trade-off between the contribution of period modulation (parametric effect) and phase shift (nonparametric effect) in Drosophila circadian entrainment. The results also give suggestions for an experimental study to (in)validate the two models of entrainment.

Age-associated circadian period changes in Arabidopsis leaves

As most organisms age, their appearance, physiology, and behaviour alters as part of a life history strategy that maximizes their fitness over their lifetime. The passage of time is measured by organisms and is used to modulate these age-related changes. Organisms have an endogenous time measurement system called the circadian clock. This endogenous clock regulates many physiological responses throughout the life history of organisms to enhance their fitness. However, little is known about the relation between ageing and the circadian clock in plants. Here, we investigate the association of leaf ageing with circadian rhythm changes to better understand the regulation of life-history strategy in Arabidopsis. The circadian periods of clock output genes were approximately 1h shorter in older leaves than younger leaves. The periods of the core clock genes were also consistently shorter in older leaves, indicating an effect of ageing on regulation of the circadian period. Shortening of the circadian period with leaf age occurred faster in plants grown under a long photoperiod compared with a short photoperiod. We screened for a regulatory gene that links ageing and the circadian clock among multiple clock gene mutants. Only mutants for the clock oscillator TOC1 did not show a shortened circadian period during leaf ageing, suggesting that TOC1 may link age to changes in the circadian clock period. Our findings suggest that age-related information is incorporated into the regulation of the circadian period and that TOC1 is necessary for this integrative process.

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.

Circadian systems biology: When time matters

The circadian clock is a powerful endogenous timing system, which allows organisms to fine-tune their physiology and behaviour to the geophysical time. The interplay of a distinct set of core-clock genes and proteins generates oscillations in expression of output target genes which temporally regulate numerous molecular and cellular processes. The study of the circadian timing at the organismal as well as at the cellular level outlines the field of chronobiology, which has been highly interdisciplinary ever since its origins. The development of high-throughput approaches enables the study of the clock at a systems level. In addition to experimental approaches, computational clock models exist which allow the analysis of rhythmic properties of the clock network. Such mathematical models aid mechanistic understanding and can be used to predict outcomes of distinct perturbations in clock components, thereby generating new hypotheses regarding the putative function of particular clock genes. Perturbations in the circadian timing system are linked to numerous molecular dysfunctions and may result in severe pathologies including cancer. A comprehensive knowledge regarding the mechanistic of the circadian system is crucial to develop new procedures to investigate pathologies associated with a deregulated clock.

In this manuscript we review the combination of experimental methodologies, bioinformatics and theoretical models that have been essential to explore this remarkable timing-system. Such an integrative and interdisciplinary approach may provide new strategies with regard to chronotherapeutic treatment and new insights concerning the restoration of the circadian timing in clock-associated diseases.

Evolutionary relationships among barley and Arabidopsis core circadian clock and clock-associated genes

The circadian clock regulates a multitude of plant developmental and metabolic processes. In crop species, it contributes significantly to plant performance and productivity and to the adaptation and geographical range over which crops can be grown. To understand the clock in barley and how it relates to the components in the Arabidopsis thaliana clock, we have performed a systematic analysis of core circadian clock and clock-associated genes in barley, Arabidopsis and another eight species including tomato, potato, a range of monocotyledonous species and the moss, Physcomitrella patens. We have identified orthologues and paralogues of Arabidopsis genes which are conserved in all species, monocot/dicot differences, species-specific differences and variation in gene copy number (e.g. gene duplications among the various species). We propose that the common ancestor of barley and Arabidopsis had two-thirds of the key clock components identified in Arabidopsis prior to the separation of the monocot/dicot groups. After this separation, multiple independent gene duplication events took place in both monocot and dicot ancestors.

The effects of phytochrome-mediated light signals on the developmental acquisition of photoperiod sensitivity in rice

Plants commonly rely on photoperiodism to control flowering time. Rice development before floral initiation is divided into two successive phases: the basic vegetative growth phase (BVP, photoperiod-insensitive phase) and the photoperiod-sensitive phase (PSP). The mechanism responsible for the transition of rice plants into their photoperiod-sensitive state remains elusive. Here, we show that se13, a mutation detected in the extremely early flowering mutant X61 is a nonsense mutant gene of OsHY2, which encodes phytochromobilin (PΦB) synthase, as evidenced by spectrometric and photomorphogenic analyses. We demonstrated that some flowering time and circadian clock genes harbor different expression profiles in BVP as opposed to PSP, and that this phenomenon is chiefly caused by different phytochrome-mediated light signal requirements: in BVP, phytochrome-mediated light signals directly suppress Ehd2, while in PSP, phytochrome-mediated light signals activate Hd1 and Ghd7 expression through the circadian clock genes' expression. These findings indicate that light receptivity through the phytochromes is different between two distinct developmental phases corresponding to the BVP and PSP in the rice flowering process. Our results suggest that these differences might be involved in the acquisition of photoperiod sensitivity in rice.

ELF3-PIF4 interaction regulates plant growth independently of the Evening Complex

The circadian clock plays a pivotal role in the control of Arabidopsis hypocotyl elongation by regulating rhythmic expression of the bHLH factors PHYTOCHROME INTERACTING FACTOR 4 and 5 (PIF4 and 5). Coincidence of increased PIF4/PIF5 transcript levels with the dark period allows nuclear accumulation of these factors, and in short days it phases maximal hypocotyl growth at dawn. During early night, PIF4 and PIF5 transcription is repressed by the Evening Complex (EC) proteins EARLY FLOWERING3 (ELF3), EARLY FLOWERING4 (ELF4), and LUX ARRHYTHMO (LUX). While ELF3 has an essential role in EC complex assembly, several lines of evidence indicate that this protein controls plant growth via other mechanisms that are presently unknown. Here, we show that the ELF3 and PIF4 proteins interact in an EC-independent manner, and that this interaction prevents PIF4 from activating its transcriptional targets. We also show that PIF4 overexpression leads to ELF3 protein destabilization, and that this effect is mediated indirectly by negative feedback regulation of photoactive PHYTOCHROME B (phyB). Physical interaction of the phyB photoreceptor with ELF3 has been reported, but its functional relevance remains poorly understood. Our findings establish that phyB is needed for ELF3 accumulation in the light, most likely by competing for CONSTITUTIVELY PHOTOMORPHOGENIC1 (COP1)-mediated ubiquitination and the proteasomal degradation of ELF3. Our results explain the short hypocotyl phenotype of ELF3 overexpressors, despite their normal clock function, and provide a molecular framework for understanding how warm temperatures promote hypocotyl elongation and affect the endogenous clock.

SRR1 is essential to repress flowering in non-inductive conditions in Arabidopsis thaliana

Timing of flowering is determined by environmental and developmental signals, leading to promotion or repression of key floral integrators. SENSITIVITY TO RED LIGHT REDUCED (SRR1) is a pioneer protein previously shown to be involved in regulation of the circadian clock and phytochrome B signalling in Arabidopsis thaliana. This report has examined the role of SRR1 in flowering time control. Loss-of-function srr1-1 plants flowered very early compared with the wild type under short-day conditions and had a weak flowering response to increasing daylength. Furthermore, FLOWERING LOCUS T (FT) transcript levels were elevated already in short days in srr1-1 compared with the wild type. This correlated with elevated end of day levels of CONSTANS (CO), whereas levels of CYCLING DOF FACTOR 1 (CDF1), a repressor of CO transcription, were reduced. srr1-1 gi-2 and srr1-1 co-9 double mutants showed that SRR1 can also repress flowering independently of the photoperiodic pathway. srr1-1 flowered consistently early between 16 °C and 27 °C, showing that SRR1 prevents premature flowering over a wide temperature range. SRR1 also promotes expression of the repressors TEMPRANILLO 1 (TEM1) and TEM2. Consequently their targets in the gibberellin biosynthesis pathway were elevated in srr1-1. SRR1 is thus an important focal point of both photoperiodic and photoperiod-independent regulation of flowering. By stimulating expression of the FT-binding repressors CDF1, TEM1 and TEM2, and FLC, flowering is inhibited in non-inductive conditions.

Bayesian parameter inference and model selection by population annealing in systems biology

Parameter inference and model selection are very important for mathematical modeling in systems biology. Bayesian statistics can be used to conduct both parameter inference and model selection. Especially, the framework named approximate Bayesian computation is often used for parameter inference and model selection in systems biology. However, Monte Carlo methods needs to be used to compute Bayesian posterior distributions. In addition, the posterior distributions of parameters are sometimes almost uniform or very similar to their prior distributions. In such cases, it is difficult to choose one specific value of parameter with high credibility as the representative value of the distribution. To overcome the problems, we introduced one of the population Monte Carlo algorithms, population annealing. Although population annealing is usually used in statistical mechanics, we showed that population annealing can be used to compute Bayesian posterior distributions in the approximate Bayesian computation framework. To deal with un-identifiability of the representative values of parameters, we proposed to run the simulations with the parameter ensemble sampled from the posterior distribution, named “posterior parameter ensemble”. We showed that population annealing is an efficient and convenient algorithm to generate posterior parameter ensemble. We also showed that the simulations with the posterior parameter ensemble can, not only reproduce the data used for parameter inference, but also capture and predict the data which was not used for parameter inference. Lastly, we introduced the marginal likelihood in the approximate Bayesian computation framework for Bayesian model selection. We showed that population annealing enables us to compute the marginal likelihood in the approximate Bayesian computation framework and conduct model selection depending on the Bayes factor.

Light and circadian regulation of clock components aids flexible responses to environmental signals

The circadian clock measures time across a 24 h period, increasing fitness by phasing biological processes to the most appropriate time of day. The interlocking feedback loop mechanism of the clock is conserved across species; however, the number of loops varies. Mathematical and computational analyses have suggested that loop complexity affects the overall flexibility of the oscillator, including its responses to entrainment signals. We used a discriminating experimental assay, at the transition between different photoperiods, in order to test this proposal in a minimal circadian network (in Ostreococcus tauri) and a more complex network (in Arabidopsis thaliana). Transcriptional and translational reporters in O. tauri primarily tracked dawn or dusk, whereas in A. thaliana, a wider range of responses were observed, consistent with its more flexible clock. Model analysis supported the requirement for this diversity of responses among the components of the more complex network. However, these and earlier data showed that the O. tauri network retains surprising flexibility, despite its simple circuit. We found that models constructed from experimental data can show flexibility either from multiple loops and/or from multiple light inputs. Our results suggest that O. tauri has adopted the latter strategy, possibly as a consequence of genomic reduction.

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.

Natural variation reveals that intracellular distribution of ELF3 protein is associated with function in the circadian clock

Natural selection of variants within the Arabidopsis thaliana circadian clock can be attributed to adaptation to varying environments. To define a basis for such variation, we examined clock speed in a reporter-modified Bay-0 x Shakdara recombinant inbred line and localized heritable variation. Extensive variation led us to identify EARLY FLOWERING3 (ELF3) as a major quantitative trait locus (QTL). The causal nucleotide polymorphism caused a short-period phenotype under light and severely dampened rhythm generation in darkness, and entrainment alterations resulted. We found that ELF3-Sha protein failed to properly localize to the nucleus, and its ability to accumulate in darkness was compromised. Evidence was provided that the ELF3-Sha allele originated in Central Asia. Collectively, we showed that ELF3 protein plays a vital role in defining its light-repressor action in the circadian clock and that its functional abilities are largely dependent on its cellular localization.

DOI:http://dx.doi.org/10.7554/eLife.02206.001

Life on Earth tends to follow a daily rhythm: some animals are awake during the day and asleep at night, whilst others are more active at night, or during the twilight around dawn and dusk. For many living things, these cycles of activity are driven by an internal body clock that helps the organism to adapt to the daily cycle of light and dark—and similar internal clocks also exist in plants.

These internal clocks define daily—or circadian—cycles whereby multiple genes are switched ‘on’ or ‘off’ at different time points in every 24-hr period. And, because light and ambient temperatures also vary with time of the day, many organisms use these external signals as cues to reset their own internal clocks. Moreover, the hours of daylight and temperature vary around the world, and also with the seasons, so plants and animals must be able to change how these external signals influence their internal clocks so that they stay in tune with the day/night cycle. However, it is not clear how they do this.

To explore this question, Anwer et al. grew plants that were from a cross between two types of the model plant Arabidopsis thaliana from different environments: one from Germany, and the other from Tajikistan in Central Asia. These offspring were also genetically engineered so that an enzyme that could give off light was produced under the control of the internal clock. Anwer et al. found that the plants continued to glow and fade with an almost daily rhythm even after external cues, such as changes in temperature or light, had been removed.

Different offspring plants consistently glowed and faded with different rhythms such that some had, for example, a 21-hr day and others a 28-hr day. These differences were caused by many genes that differed from the original German and Tajikistan parent plants, and Anwer et al. ‘mapped’ one of these genetic differences to a single gene. Offspring that inherited a version of a gene called ELF3 from the Tajikistan parent had internal clocks that ran faster when the plant was under the light. These plants also gradually stopped glowing as brightly as the German parent when they were kept in the dark, suggesting that their internal clocks were ‘ticking more softly’. It was already known that the ELF3 gene affected the circadian clock in plants, and Anwer et al. thus concluded that the plants with Tajikistan version of this gene, called ELF3-Sha, were also less able to reset their internal clocks to synchronize in response to external cues.

Anwer et al. also showed that the normal ELF3 protein is more likely to be found in the nucleus of a plant cell than the ELF3-Sha version, which might suggest that this protein is involved in switching genes off. Further research is now needed to uncover exactly how the ELF3 protein does this to keep the plant's internal clock ‘ticking’ correctly.

DOI:http://dx.doi.org/10.7554/eLife.02206.002

Is transcriptomic regulation of berry development more important at night than during the day?

Diurnal changes in gene expression occur in all living organisms and have been studied on model plants such as Arabidopsis thaliana. To our knowledge the impact of the nycthemeral cycle on the genetic program of fleshly fruit development has been hitherto overlooked. In order to circumvent environmental changes throughout fruit development, young and ripening berries were sampled simultaneously on continuously flowering microvines acclimated to controlled circadian light and temperature changes. Gene expression profiles along fruit development were monitored during both day and night with whole genome microarrays (Nimblegen® vitis 12x), yielding a total number of 9273 developmentally modulated probesets. All day-detected transcripts were modulated at night, whereas 1843 genes were night-specific. Very similar developmental patterns of gene expression were observed using independent hierarchical clustering of day and night data, whereas functional categories of allocated transcripts varied according to time of day. Many transcripts within pathways, known to be up-regulated during ripening, in particular those linked to secondary metabolism exhibited a clearer developmental regulation at night than during the day. Functional enrichment analysis also indicated that diurnally modulated genes considerably varied during fruit development, with a shift from cellular organization and photosynthesis in green berries to secondary metabolism and stress-related genes in ripening berries. These results reveal critical changes in gene expression during night development that differ from daytime development, which have not been observed in other transcriptomic studies on fruit development thus far.

Wheels within wheels: new transcriptional feedback loops in the Arabidopsis circadian clock

The circadian clock allows organisms to temporally coordinate their biology with the diurnal oscillation of the environment, which enhances plant performance. Accordingly, a fuller understanding of the circadian clock mechanism may contribute to efforts to optimize plant performance. One recurring theme in clock mechanism is coupled transcription-translation feedback loops. To date, the majority of plant transcription factors constituting these loops, including the central oscillator components CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), LATE ELONGATED HYPOCOTYL (LHY), and TIMING OF CAB2 EXPRESSION 1 (TOC1), and the related PSEUDO-RESPONSE REGULATORS (PRRs), are transcriptional repressors, leading to a model of the clock emphasizing repressive interactions. Recent work, however, has revealed that a subset of the REVEILLE (RVE) family of Myb transcription factors closely related to CCA1 and LHY are transcriptional activators in novel feedback transcription-translation feedback loops. Other recently identified transcriptional activators that contribute to clock function include LIGHT-REGULATED WD 1 (LWD1) and LWD2 and night light-inducible and clock-regulated transcription factors NIGHT LIGHT-INDUCIBLE AND CLOCK-REGULATED1 (LNK1) and LNK2. Collectively, these advances permit a substantial reconfiguration of the clock model.

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.

Bayesian parameter inference by Markov chain Monte Carlo with hybrid fitness measures: theory and test in apoptosis signal transduction network

When exact values of model parameters in systems biology are not available from experiments, they need to be inferred so that the resulting simulation reproduces the experimentally known phenomena. For the purpose, Bayesian statistics with Markov chain Monte Carlo (MCMC) is a useful method. Biological experiments are often performed with cell population, and the results are represented by histograms. On another front, experiments sometimes indicate the existence of a specific bifurcation pattern. In this study, to deal with both type of such experimental results and information for parameter inference, we introduced functions to evaluate fitness to both type of experimental results, named quantitative and qualitative fitness measures respectively. We formulated Bayesian formula for those hybrid fitness measures (HFM), and implemented it to MCMC (MCMC-HFM). We tested MCMC-HFM first for a kinetic toy model with a positive feedback. Inferring kinetic parameters mainly related to the positive feedback, we found that MCMC-HFM reliably infer them with both qualitative and quantitative fitness measures. Then, we applied the MCMC-HFM to an apoptosis signal transduction network previously proposed. For kinetic parameters related to implicit positive feedbacks, which are important for bistability and irreversibility of the output, the MCMC-HFM reliably inferred these kinetic parameters. In particular, some kinetic parameters that have the experimental estimates were inferred without these data and the results were consistent with the experiments. Moreover, for some parameters, the mixed use of quantitative and qualitative fitness measures narrowed down the acceptable range of parameters. Taken together, our approach could reliably infer the kinetic parameters of the target systems.

Approximating subcellular organisation of carbohydrate metabolism during cold acclimation in different natural accessions of Arabidopsis thaliana

Accessions of Arabidopsis thaliana originating from climatically different habitats show different levels of cold acclimation when exposed to low temperatures. The central carbohydrate metabolism plays a crucial role during this acclimation. Subcellular distribution of carbohydrates over the compartments cytosol, vacuole and plastids, and putative interactions of the compartments, are analyzed in three differentially cold-tolerant accessions of Arabidopsis thaliana, originating from the Iberian Peninsula (C24), Russia (Rschew) and Scandinavia (Tenela), respectively. Subcellular carbohydrate concentrations were determined by applying the nonaqueous fractionation technique. Mathematical modeling and steady-state simulation was used to analyse the metabolic homeostasis during cold exposure. In all accessions, the initial response to cold exposure was a significant increase of plastidial and cytosolic sucrose concentrations. Raffinose accumulated in all cellular compartments of cold-tolerant accessions with a delay of 3 d, indicating that raffinose accumulation is a long-term component of cold acclimation. Minimal rates of metabolite transport permitting steady-state simulations of metabolite concentrations correlated with cold tolerance, indicating an important role of subcellular re-distribution of metabolites during cold acclimation. A highly regulated interplay of enzymatic reactions and intracellular transport processes appears to be a prerequisite for maintaining carbohydrate homeostasis during cold exposure and allowing cold acclimation in Arabidopsis thaliana.

A circadian clock-regulated toggle switch explains AtGRP7 and AtGRP8 oscillations in Arabidopsis thaliana

The circadian clock controls many physiological processes in higher plants and causes a large fraction of the genome to be expressed with a 24h rhythm. The transcripts encoding the RNA-binding proteins AtGRP7 (Arabidopsis thaliana Glycine Rich Protein 7) and AtGRP8 oscillate with evening peaks. The circadian clock components CCA1 and LHY negatively affect AtGRP7 expression at the level of transcription. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate post-transcriptionally: high protein levels promote the generation of an alternative splice form that is rapidly degraded. This clock-regulated feedback loop has been proposed to act as a molecular slave oscillator in clock output. While mathematical models describing the circadian core oscillator in Arabidopsis thaliana were introduced recently, we propose here the first model of a circadian slave oscillator. We define the slave oscillator in terms of ordinary differential equations and identify the model's parameters by an optimization procedure based on experimental results. The model successfully reproduces the pertinent experimental findings such as waveforms, phases, and half-lives of the time-dependent concentrations. Furthermore, we obtain insights into possible mechanisms underlying the observed experimental dynamics: the negative auto-regulation and reciprocal cross-regulation via alternative splicing could be responsible for the sharply peaking waveforms of the AtGRP7 and AtGRP8 mRNA. Moreover, our results suggest that the AtGRP8 transcript oscillations are subordinated to those of AtGRP7 due to a higher impact of AtGRP7 protein on alternative splicing of its own and of the AtGRP8 pre-mRNA compared to the impact of AtGRP8 protein. Importantly, a bifurcation analysis provides theoretical evidence that the slave oscillator could be a toggle switch, arising from the reciprocal cross-regulation at the post-transcriptional level. In view of this, transcriptional repression of AtGRP7 and AtGRP8 by LHY and CCA1 induces oscillations of the toggle switch, leading to the observed high-amplitude oscillations of AtGRP7 mRNA.

The circadian clock organizes the day in the life of a plant by causing 24h rhythms in gene expression. For example, the core clockwork of the model plant Arabidopsis thaliana causes the transcripts encoding the RNA-binding proteins AtGRP7 and AtGRP8 to undergo high amplitude oscillations with a peak at the end of the day. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate their own expression by causing alternative splicing of their pre-mRNAs, followed by rapid degradation of the alternatively spliced transcripts. This has led to the suggestion that they represent molecular slave oscillators downstream of the core clock. Using a mathematical model we obtain insights into possible mechanisms underlying the experimentally observed dynamics, e.g. a higher impact of AtGRP7 protein compared to the impact of AtGRP8 protein on the alternative splicing explains the experimentally observed phases of their transcript. Previously, components that reciprocally repress their own transcription (double negative loops) have been shown to potentially act as a toggle switch between two states. We provide theoretical evidence that the slave oscillator could be a bistable toggle switch as well, operating at the post-transcriptional level.

Chromatin remodeling and alternative splicing: pre- and post-transcriptional regulation of the Arabidopsis circadian clock

Circadian clocks are endogenous mechanisms that translate environmental cues into temporal information to generate the 24-h rhythms in metabolism and physiology. The circadian function relies on the precise regulation of rhythmic gene expression at the core of the oscillator, which temporally modulates the genome transcriptional activity in virtually all multicellular organisms examined to date. Emerging evidence in plants suggests a highly sophisticated interplay between the circadian patterns of gene expression and the rhythmic changes in chromatin remodeling and histone modifications. Alternative precursor messenger RNA (pre-mRNA) splicing has also been recently defined as a fundamental pillar within the circadian system, providing the required plasticity and specificity for fine-tuning the circadian clock. This review highlights the relationship between the plant circadian clock with both chromatin remodeling and alternative splicing and compares the similarities and divergences with analogous studies in animal circadian systems.

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.

Florigenic and antiflorigenic signaling in plants

The evidence that FLOWERING LOCUS T (FT) protein, and its paralog TWIN SISTER OF FT, act as the long-distance floral stimulus, or at least that they are part of it in diverse plant species, has attracted much attention in recent years. Studies to understand the physiological and molecular apparatuses that integrate spatial and temporal signals to regulate developmental transitions in plants have occupied countless scientists and have resulted in an unmanageably large amount of research data. Analysis of these data has helped to identify multiple systemic florigenic and antiflorigenic regulators. This study gives an overview of the recent research on gene products, phytohormones and other metabolites that have been demonstrated to have florigenic or antiflorigenic functions in plants.

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.

The impact of photoperiod insensitive Ppd-1a mutations on the photoperiod pathway across the three genomes of hexaploid wheat (Triticum aestivum)

Flowering time is a trait that has been extensively altered during wheat domestication, enabling it to be highly productive in diverse environments and providing a rich source of variation for studying adaptation mechanisms. Hexaploid wheat is ancestrally a long-day plant, but many environments require varieties with photoperiod insensitivity (PI) that can flower in short days. PI results from mutations in the Ppd-1 gene on the A, B or D genomes, with individual mutations conferring different degrees of earliness. The basis of this is poorly understood. Using a common genetic background, the effects of A, B and D genome PI mutations on genes of the circadian clock and photoperiod pathway were studied using genome-specific expression assays. Ppd-1 PI mutations did not affect the clock or immediate clock outputs, but affected TaCO1 and TaFT1, with a reduction in TaCO1 expression as TaFT1 expression increased. Therefore, although Ppd-1 is related to PRR genes of the Arabidopsis circadian clock, Ppd-1 affects flowering by an alternative route, most likely by upregulating TaFT1 with a feedback effect that reduces TaCO1 expression. Individual genes in the circadian clock and photoperiod pathway were predominantly expressed from one genome, and there was no genome specificity in Ppd-1 action. Lines combining PI mutations on two or three genomes had enhanced earliness with higher levels, but not earlier induction, of TaFT1, showing that there is a direct quantitative relationship between Ppd-1 mutations, TaFT1 expression and flowering.

A novel cost function to estimate parameters of oscillatory biochemical systems

Oscillatory pathways are among the most important classes of biochemical systems with examples ranging from circadian rhythms and cell cycle maintenance. Mathematical modeling of these highly interconnected biochemical networks is needed to meet numerous objectives such as investigating, predicting and controlling the dynamics of these systems. Identifying the kinetic rate parameters is essential for fully modeling these and other biological processes. These kinetic parameters, however, are not usually available from measurements and most of them have to be estimated by parameter fitting techniques. One of the issues with estimating kinetic parameters in oscillatory systems is the irregularities in the least square (LS) cost function surface used to estimate these parameters, which is caused by the periodicity of the measurements. These irregularities result in numerous local minima, which limit the performance of even some of the most robust global optimization algorithms. We proposed a parameter estimation framework to address these issues that integrates temporal information with periodic information embedded in the measurements used to estimate these parameters. This periodic information is used to build a proposed cost function with better surface properties leading to fewer local minima and better performance of global optimization algorithms. We verified for three oscillatory biochemical systems that our proposed cost function results in an increased ability to estimate accurate kinetic parameters as compared to the traditional LS cost function. We combine this cost function with an improved noise removal approach that leverages periodic characteristics embedded in the measurements to effectively reduce noise. The results provide strong evidence on the efficacy of this noise removal approach over the previous commonly used wavelet hard-thresholding noise removal methods. This proposed optimization framework results in more accurate kinetic parameters that will eventually lead to biochemical models that are more precise, predictable, and controllable.