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

BDEtools: A MATLAB Package for Boolean Delay Equation Modeling

Boolean Delay Equations (BDEs) can simulate surprisingly complex behavior, despite their relative simplicity. In addition to steady-state dynamics, BDEs can also generate periodic and quasiperiodic oscillations, m:n frequency locking, and even chaos. Further, the enumerability of Boolean update functions and their compact parametrization means that BDEs can be leveraged to generate low-level descriptions of biological networks, from which more detailed formulations (e.g., differential equation models) can be constructed. However, although several studies have demonstrated the utility of BDE modeling in computational biology, a current barrier to the wider adoption of the BDE approach is the absence of freely available simulation software. In this work, we present BDEtools-an open-source MATLAB package for numerically solving BDE models. After giving a brief introduction to BDE modeling, we describe the package's solver algorithms, together with several utility functions that can be used to provide solver inputs and to process solver outputs. We also demonstrate the functionality of BDEtools by illustrating its application to an established model of a gene regulatory network that controls circadian rhythms. BDEtools makes it straightforward for researchers to quickly build reliable BDE models of biological networks. We hope that its ease of use and free availability will encourage more researchers to explore BDE formulations of their systems of interest. Through the continued use of BDEs by the computational biology community, we will, no doubt, discover their potential applicability to a broader class of biological networks.

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.

Spatiotemporal limitations in plant biology research

The way we currently capture biological processes in space and time often limits our understanding of plant development and stress responses, leading to an incomplete picture of plant life. Choosing the correct time frame for the study of every biological process, from seed germination to senescence or in plant stress responses, is essential, despite methodological limitations. A greater effort is needed in current plant biology studies to incorporate spatiotemporal approaches so that scientific knowledge meets the possibilities technological advances currently provide. From molecular, biochemical, and cellular approaches to (eco)physiological and population studies scaled up to the ecosystem level, there is an urgent need to link space and time using integrative and scalable data.

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.

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.

A Holistic Approach to Study Photosynthetic Acclimation Responses of Plants to Fluctuating Light

Plants in natural environments receive light through sunflecks, the duration and distribution of these being highly variable across the day. Consequently, plants need to adjust their photosynthetic processes to avoid photoinhibition and maximize yield. Changes in the composition of the photosynthetic apparatus in response to sustained changes in the environment are referred to as photosynthetic acclimation, a process that involves changes in protein content and composition. Considering this definition, acclimation differs from regulation, which involves processes that alter the activity of individual proteins over short-time periods, without changing the abundance of those proteins. The interconnection and overlapping of the short- and long-term photosynthetic responses, which can occur simultaneously or/and sequentially over time, make the study of long-term acclimation to fluctuating light in plants challenging. In this review we identify short-term responses of plants to fluctuating light that could act as sensors and signals for acclimation responses, with the aim of understanding how plants integrate environmental fluctuations over time and tailor their responses accordingly. Mathematical modeling has the potential to integrate physiological processes over different timescales and to help disentangle short-term regulatory responses from long-term acclimation responses. We review existing mathematical modeling techniques for studying photosynthetic responses to fluctuating light and propose new methods for addressing the topic from a holistic point of view.

A Magnesium Transport Protein Related to Mammalian SLC41 and Bacterial MgtE Contributes to Circadian Timekeeping in a Unicellular Green Alga

Circadian clocks in eukaryotes involve both transcriptional-translational feedback loops, post-translational regulation, and metabolic, non-transcriptional oscillations. We recently identified the involvement of circadian oscillations in the intracellular concentrations of magnesium ions (Mg²⁺_i) that were conserved in three eukaryotic kingdoms. Mg²⁺_i in turn contributes to transcriptional clock properties of period and amplitude, and can function as a zeitgeber to define phase. However, the mechanism-or mechanisms-responsible for the generation of Mg²⁺_i oscillations, and whether these are functionally conserved across taxonomic groups, remain elusive. We employed the cellular clock model Ostreococcustauri to provide a first study of an MgtE domain-containing protein in the green lineage. OtMgtE shares homology with the mammalian SLC41A1 magnesium/sodium antiporter, which has previously been implicated in maintaining clock period. Using genetic overexpression, we found that OtMgtE contributes to both timekeeping and daily changes in Mg²⁺_i. However, pharmacological experiments and protein sequence analyses indicated that critical differences exist between OtMgtE and either the ancestral MgtE channel or the mammalian SLC41 antiporters. We concluded that even though MgtE domain-containing proteins are only distantly related, these proteins retain a shared role in contributing to cellular timekeeping and the regulation of Mg²⁺_i.

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.

Heterogeneity of cellular circadian clocks in intact plants and its correction under light-dark cycles

Recent advances in single-cell analysis have revealed the stochasticity and nongenetic heterogeneity inherent to cellular processes. However, our knowledge of the actual cellular behaviors in a living multicellular organism is still limited. By using a single-cell bioluminescence imaging technique on duckweed, Lemna gibba, we demonstrate that, under constant conditions, cells in the intact plant work as individual circadian clocks that oscillate with their own frequencies and respond independently to external stimuli. Quantitative analysis uncovered the heterogeneity and instability of cellular clocks and partial synchronization between neighboring cells. Furthermore, we found that cellular clocks in the plant body under light-dark cycles showed a centrifugal phase pattern in which the effect of cell-to-cell heterogeneity in period lengths was almost masked. The inherent heterogeneity in the properties of cellular clocks observed under constant conditions is corrected under light-dark cycles to coordinate the daily rhythms of the plant body. These findings provide a novel perspective of spatiotemporal architectures in the plant circadian system.

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.

Label-free quantitative analysis of the casein kinase 2-responsive phosphoproteome of the marine minimal model species Ostreococcus tauri

Casein kinase 2 (CK2) is a protein kinase that phosphorylates a plethora of cellular target proteins involved in processes including DNA repair, cell cycle control, and circadian timekeeping. CK2 is functionally conserved across eukaryotes, although the substrate proteins identified in a range of complex tissues are often different. The marine alga Ostreococcus tauri is a unicellular eukaryotic model organism ideally suited to efficiently study generic roles of CK2 in the cellular circadian clock. Overexpression of CK2 leads to a slow circadian rhythm, verifying functional conservation of CK2 in timekeeping. The proteome was analysed in wild-type and CK2-overexpressing algae at dawn and dusk, revealing that differential abundance of the global proteome across the day is largely unaffected by overexpression. However, CK2 activity contributed more strongly to timekeeping at dusk than at dawn. The phosphoproteome of a CK2 overexpression line and cells treated with CK2 inhibitor was therefore analysed and compared to control cells at dusk. We report an extensive catalogue of 447 unique CK2-responsive differential phosphopeptide motifs to inform future studies into CK2 activity in the circadian clock of more complex tissues. All MS data have been deposited in the ProteomeXchange with identifier PXD000975 (http://proteomecentral.proteomexchange.org/dataset/PXD000975).