A proposal for robust temperature compensation of circadian rhythms

A Compensated Clock: Temperature and Nutritional Compensation Mechanisms Across Circadian Systems

Circadian rhythms are ∼24-h biological oscillations that enable organisms to anticipate daily environmental cycles, so that they may designate appropriate day/night functions that align with these changes. The molecular clock in animals and fungi consists of a transcription-translation feedback loop, the plant clock is comprised of multiple interlocking feedback-loops, and the cyanobacterial clock is driven by a phosphorylation cycle involving three main proteins. Despite the divergent core clock mechanisms across these systems, all circadian clocks are able to buffer period length against changes in the ambient growth environment, such as temperature and nutrients. This defining capability, termed compensation, is critical to proper timekeeping, yet the underlying mechanism(s) remain elusive. Here we examine the known players in, and the current models for, compensation across five circadian systems. While compensation models across these systems are not yet unified, common themes exist across them, including regulation via temperature-dependent changes in post-translational modifications.

A mathematical model of melatonin synthesis and interactions with the circadian clock

A new mathematical model of melatonin synthesis in pineal cells is created and connected to a slightly modified previously created model of the circadian clock in the suprachiasmatic nucleus (SCN). The SCN influences the production of melatonin by upregulating two key enzymes in the pineal. The melatonin produced enters the blood and the cerebrospinal fluid and thus the SCN, influencing the circadian clock. We show that the model of melatonin synthesis corresponds well with extant experimental data and responds similarly to clinical experiments on bright light in the middle of the night. Melatonin is widely used to treat jet lag and sleep disorders. We show how the feedback from the pineal to the SCN causes phase resetting of the circadian clock. Melatonin doses early in the evening advance the clock and doses late at night delay the clock with a dead zone in between where the phase of the clock does not change.

Temperature compensation through kinetic regulation in biochemical oscillators

Nearly all circadian clocks maintain a period that is insensitive to temperature changes, a phenomenon known as temperature compensation (TC). Yet, it is unclear whether there is any common feature among different systems that exhibit TC. From a general timescale invariance, we show that TC relies on the existence of certain period-lengthening reactions wherein the period of the system increases strongly with the rates in these reactions. By studying several generic oscillator models, we show that this counterintuitive dependence is nonetheless a common feature of oscillators in the nonlinear (far-from-onset) regime where the oscillation can be separated into fast and slow phases. The increase of the period with the period-lengthening reaction rates occurs when the amplitude of the slow phase in the oscillation increases with these rates while the progression speed in the slow phase is controlled by other rates of the system. The positive dependence of the period on the period-lengthening rates balances its inverse dependence on other kinetic rates in the system, which gives rise to robust TC in a wide range of parameters. We demonstrate the existence of such period-lengthening reactions and their relevance for TC in all four model systems we considered. Theoretical results for a model of the Kai system are supported by experimental data. A study of the energy dissipation also shows that better TC performance requires higher energy consumption. Our study unveils a general mechanism by which a biochemical oscillator achieves TC by operating in parameter regimes far from the onset where period-lengthening reactions exist.

Temperature-driven coordination of circadian transcriptional regulation

The circadian clock is an evolutionarily-conserved molecular oscillator that enables species to anticipate rhythmic changes in their environment. At a molecular level, the core clock genes induce circadian oscillations in thousands of genes in a tissue-specific manner, orchestrating myriad biological processes. While previous studies have investigated how the core clock circuit responds to environmental perturbations such as temperature, the downstream effects of such perturbations on circadian regulation remain poorly understood. By analyzing bulk-RNA sequencing of Drosophila fat bodies harvested from flies subjected to different environmental conditions, we demonstrate a highly condition-specific circadian transcriptome: genes are cycling in a temperature-specific manner, and the distributions of their phases also differ between the two conditions. Further employing a reference-based gene regulatory network (Reactome), we find evidence of increased gene-gene coordination at low temperatures and synchronization of rhythmic genes that are network neighbors. We report that the phase differences between cycling genes increase as a function of geodesic distance in the low temperature condition, suggesting increased coordination of cycling on the gene regulatory network. Our results suggest a potential mechanism whereby the circadian clock mediates the fly's response to seasonal changes in temperature.

Alternative polyadenylation factor CPSF6 regulates temperature compensation of the mammalian circadian clock

A defining property of circadian clocks is temperature compensation, characterized by the resilience of their near 24-hour free-running periods against changes in environmental temperature within the physiological range. While temperature compensation is evolutionary conserved across different taxa of life and has been studied within many model organisms, its molecular underpinnings remain elusive. Posttranscriptional regulations such as temperature-sensitive alternative splicing or phosphorylation have been described as underlying reactions. Here, we show that knockdown of cleavage and polyadenylation specificity factor subunit 6 (CPSF6), a key regulator of 3'-end cleavage and polyadenylation, significantly alters circadian temperature compensation in human U-2 OS cells. We apply a combination of 3'-end-RNA-seq and mass spectrometry-based proteomics to globally quantify changes in 3' UTR length as well as gene and protein expression between wild-type and CPSF6 knockdown cells and their dependency on temperature. Since changes in temperature compensation behavior should be reflected in alterations of temperature responses within one or all of the 3 regulatory layers, we statistically assess differential responses upon changes in ambient temperature between wild-type and CPSF6 knockdown cells. By this means, we reveal candidate genes underlying circadian temperature compensation, including eukaryotic translation initiation factor 2 subunit 1 (EIF2S1).

Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms

Temperature compensation and robustness to biological noise are two key characteristics of the circadian clock. These features allow the circadian pacemaker to maintain a steady oscillation in a wide range of environmental conditions. The presence of a time-delayed negative feedback loop in the regulatory network generates autonomous circadian oscillations in eukaryotic systems. In comparison, the circadian clock of cyanobacteria is controlled by a strong positive feedback loop. Positive feedback loops with substrate depletion can also generate oscillations, inspiring other circadian clock models. What makes a circadian oscillatory network robust to extrinsic noise is unclear. We investigated four basic circadian oscillators with negative, positive, and combinations of positive and negative feedback loops to explore network features necessary for circadian clock resilience. We discovered that the negative feedback loop system performs the best in compensating temperature changes. We also show that a positive feedback loop can reduce extrinsic noise in periods of circadian oscillators, while intrinsic noise is reduced by negative feedback loops.

Role of the reaction-structure coupling in temperature compensation of the KaiABC circadian rhythm

When the mixture solution of cyanobacterial proteins, KaiA, KaiB, and KaiC, is incubated with ATP in vitro, the phosphorylation level of KaiC shows stable oscillations with the temperature-compensated circadian period. Elucidating this temperature compensation is essential for understanding the KaiABC circadian clock, but its mechanism has remained a mystery. We analyzed the KaiABC temperature compensation by developing a theoretical model describing the feedback relations among reactions and structural transitions in the KaiC molecule. The model showed that the reduced structural cooperativity should weaken the negative feedback coupling among reactions and structural transitions, which enlarges the oscillation amplitude and period, explaining the observed significant period extension upon single amino-acid residue substitution. We propose that an increase in thermal fluctuations similarly attenuates the reaction-structure feedback, explaining the temperature compensation in the KaiABC clock. The model explained the experimentally observed responses of the oscillation phase to the temperature shift or the ADP-concentration change and suggested that the ATPase reactions in the CI domain of KaiC affect the period depending on how the reaction rates are modulated. The KaiABC clock provides a unique opportunity to analyze how the reaction-structure coupling regulates the system-level synchronized oscillations of molecules.

Temperature Controls Onset and Period of NF-κB Oscillations and can Lead to Chaotic Dynamics

The transcription factor NF-κB plays a vital role in the control of the immune system, and following stimulation with TNF-α its nuclear concentration shows oscillatory behaviour. How environmental factors, in particular temperature, can control the oscillations and thereby affect gene stimulation is still remains to be resolved question. In this work, we reveal that the period of the oscillations decreases with increasing temperature. We investigate this using a mathematical model, and by applying results from statistical physics, we introduce temperature dependency to all rates, resulting in a remarkable correspondence between model and experiments. Our model predicts how temperature affects downstream protein production and find a crossover, where high affinity genes upregulates at high temperatures. Finally, we show how or that oscillatory temperatures can entrain NF-κB oscillations and lead to chaotic dynamics presenting a simple path to chaotic conditions in cellular biology.

Uncovering the dynamics of a circadian-dopamine model influenced by the light-dark cycle

The neurotransmitter dopamine (DA) is known to be influenced by the circadian timekeeping system in the mammalian brain. We have previously created a single-cell differential equations model to understand the mechanisms behind circadian rhythms of extracellular DA. In this paper, we investigate the dynamics in our model and study different behaviors such as entrainment to the 24-hour light-dark cycle and robust periodicity versus decoupling, quasiperiodicity, and chaos. Imbalances in DA are often accompanied by disrupted circadian rhythms, such as in Parkinson's disease, hyperactivity, and mood disorders. Our model provides new insights into the links between the circadian clock and DA. We show that the daily rhythmicity of DA can be disrupted by decoupling between interlocked loops of the clock circuitry or by quasiperiodic clock behaviors caused by misalignment with the light-dark cycle. The model can be used to further study how the circadian clock affects the dopaminergic system, and to help develop therapeutic strategies for disrupted DA rhythms. .

FRQ-CK1 Interaction Underlies Temperature Compensation of the Neurospora Circadian Clock

Temperature compensation is a fundamental property of all circadian clocks; temperature compensation results in a relatively constant period length at different physiological temperatures, but its mechanism is unclear. Formation of a stable complex between clock proteins and casein kinase 1 (CK1) is a conserved feature in eukaryotic circadian mechanisms. Here, we show that the FRQ-CK1 interaction and CK1-mediated FRQ phosphorylation, not FRQ stability, are main mechanisms responsible for the circadian temperature compensation phenotypes in Neurospora. Inhibition of CK1 kinase activity impaired the temperature compensation profile. Importantly, both the loss of temperature compensation and temperature overcompensation phenotypes of the wild-type and different clock mutant strains can be explained by temperature-dependent alterations of the FRQ-CK1 interaction. Furthermore, mutations that were designed to specifically affect the FRQ-CK1 interaction resulted in impaired temperature compensation of the clock. Together, these results reveal the temperature-compensated FRQ-CK1 interaction, which results in temperature-compensated CK1-mediated FRQ and WC phosphorylation, as a main biochemical process that underlies the mechanism of circadian temperature compensation in Neurospora. IMPORTANCE Temperature compensation allows clocks to adapt to all seasons by having a relatively constant period length at different physiological temperatures, but the mechanism of temperature compensation is unclear. Stability of clock proteins was previously proposed to be a major factor that regulated temperature compensation. In this study, we showed that the interaction between CK1 and FRQ, but not FRQ stability, explains the circadian temperature compensation phenotypes in Neurospora. This study uncovered the key biochemical mechanism responsible for temperature compensation of the circadian clock and further established the mechanism for period length determination in Neurospora. Because the regulation of circadian clock proteins by CK1 and the formation of a stable clock complex with CK1 are highly conserved in eukaryotic clocks, a similar mechanism may also exist in animal clocks.

A mathematical model of circadian rhythms and dopamine

Background

The superchiasmatic nucleus (SCN) serves as the primary circadian (24hr) clock in mammals and is known to control important physiological functions such as the sleep-wake cycle, hormonal rhythms, and neurotransmitter regulation. Experimental results suggest that some of these functions reciprocally influence circadian rhythms, creating a highly complex network. Among the clock’s downstream products, orphan nuclear receptors REV-ERB and ROR are particularly interesting because they coordinately modulate the core clock circuitry. Recent experimental evidence shows that REV-ERB and ROR are not only crucial for lipid metabolism but are also involved in dopamine (DA) synthesis and degradation, which could have meaningful clinical implications for conditions such as Parkinson’s disease and mood disorders.

Methods

We create a mathematical model consisting of differential equations that express how the circadian variables are influenced by light, how REV-ERB and ROR feedback to the clock, and how REV-ERB, ROR, and BMAL1-CLOCK affect the dopaminergic system. The structure of the model is based on the findings of experimentalists.

Results

We compare our model predictions to experimental data on clock components in different light-dark conditions and in the presence of genetic perturbations. Our model results are consistent with experimental results on REV-ERB and ROR and allow us to predict the circadian variations in tyrosine hydroxylase and monoamine oxidase seen in experiments. By connecting our model to an extant model of dopamine synthesis, release, and reuptake, we are able to predict circadian oscillations in extracellular DA and homovanillic acid that correspond well with experimental observations.

Conclusions

The predictions of the mathematical model are consistent with a wide variety of experimental observations. Our calculations show that the mechanisms proposed by experimentalists by which REV-ERB, ROR, and BMAL1-CLOCK influence the DA system are sufficient to explain the circadian oscillations observed in dopaminergic variables. Our mathematical model can be used for further investigations of the effects of the mammalian circadian clock on the dopaminergic system. The model can also be used to predict how perturbations in the circadian clock disrupt the dopaminergic system and could potentially be used to find drug targets that ameliorate these disruptions.

Multi-scale modeling of the circadian modulation of learning and memory

We propose a multi-scale model to explain the time-of-day effects on learning and memory. We specifically model the circadian variation of hippocampus (HC) dependent long-term potentiation (LTP), depression (LTD), and the fear conditioning paradigm in amygdala. The model we built has both Goodwin type circadian gene regulatory network (GRN) and the conductance model of Morris-Lecar (ML) type to explain the spontaneous firing patterns (SFR) in suprachiasmatic nucleus (SCN). In the conductance model, we also include N-Methyl-D-aspartic acid receptor (NMDAR) to study the circadian dependent changes in LTP/LTD in hippocampus and include both NMDAR and α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR) dynamics to explain the circadian modulation of fear conditioning paradigm in memory acquisition, recall, and extinction as seen in amygdala. Our multi-scale model captures the essential dynamics seen in the experiments and strongly supports the circadian time-of-the-day effects on learning and memory.

Non-sinusoidal Waveform in Temperature-Compensated Circadian Oscillations

Time series of biological rhythms are of various shapes. Here, we investigated the waveforms of circadian rhythms in gene-protein dynamics using a newly developed, to our knowledge, index to quantify the degree of distortion from a sinusoidal waveform. In general, most biochemical reactions accelerate with increasing temperature, but the period of circadian rhythms remains relatively stable with temperature change, a phenomenon known as "temperature compensation." Despite extensive research, the mechanism underlying this remains unclear. To understand the mechanism, we used transcriptional-translational oscillator models for circadian rhythms in the fruit fly Drosophila and mammals. Given the assumption that reaction rates increase with temperature, mathematical analyses revealed that temperature compensation required waveforms that are more nonsinusoidal at higher temperatures. We then analyzed a post-translational oscillator (PTO) model of cyanobacteria circadian rhythms. Because the structure of the PTO is different from that of the transcriptional-translational oscillator, the condition for temperature compensation would be expected to differ. Unexpectedly, the computational analysis again showed that temperature compensation in the PTO model required a more nonsinusoidal waveform at higher temperatures. This finding held for both models even with a milder assumption that some reaction rates do not change with temperature, which is consistent with experimental evidence. Together, our theoretical analyses predict that the waveform of circadian gene-activity and/or protein phosphorylation rhythms would be more nonsinusoidal at higher temperatures, even when there are differences in the network structures.

Thermal adaptation and plasticity of the plant circadian clock

Contents Summary 1215 I. Introduction 1215 II. Molecular organization of the plant circadian clock 1216 III. Temperature compensation 1219 IV. Temperature regulation of circadian behaviors 1220 V. Thermal adaptation of the clock: evolutionary considerations 1223 VI. Light and temperature information for the clock function - synergic or individual? 1224 VII. Concluding remarks and future prospects 1225 Acknowledgements 1225 References 1225 SUMMARY: Plant growth and development is widely affected by diverse temperature conditions. Although studies have been focused mainly on the effects of stressful temperature extremes in recent decades, nonstressful ambient temperatures also influence an array of plant growth and morphogenic aspects, a process termed thermomorphogenesis. Notably, accumulating evidence indicates that both stressful and nonstressful temperatures modulate the functional process of the circadian clock, a molecular timer of biological rhythms in higher eukaryotes and photosynthetic prokaryotes. The circadian clock can sustain robust and precise timing over a range of physiological temperatures. Genes and molecular mechanisms governing the temperature compensation process have been explored in different plant species. In addition, a ZEITLUPE/HSP90-mediated protein quality control mechanism helps plants maintain the thermal stability of the clock under heat stress. The thermal adaptation capability and plasticity of the clock are of particular interest in view of the growing concern about global climate changes. Considering these circumstances in the field, we believe that it is timely to provide a provoking discussion on the current knowledge of temperature regulation of the clock function. The review also will discuss stimulating ideas on this topic along with ecosystem management and future agricultural innovation.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

Temperature-amplitude coupling for stable biological rhythms at different temperatures

Most biological processes accelerate with temperature, for example cell division. In contrast, the circadian rhythm period is robust to temperature fluctuation, termed temperature compensation. Temperature compensation is peculiar because a system-level property (i.e., the circadian period) is stable under varying temperature while individual components of the system (i.e., biochemical reactions) are usually temperature-sensitive. To understand the mechanism for period stability, we measured the time series of circadian clock transcripts in cultured C6 glioma cells. The amplitudes of Cry1 and Dbp circadian expression increased significantly with temperature. In contrast, other clock transcripts demonstrated no significant change in amplitude. To understand these experimental results, we analyzed mathematical models with different network topologies. It was found that the geometric mean amplitude of gene expression must increase to maintain a stable period with increasing temperatures and reaction speeds for all models studied. To investigate the generality of this temperature-amplitude coupling mechanism for period stability, we revisited data on the yeast metabolic cycle (YMC) period, which is also stable under temperature variation. We confirmed that the YMC amplitude increased at higher temperatures, suggesting temperature-amplitude coupling as a common mechanism shared by circadian and 4 h-metabolic rhythms.

Molecular Mechanisms Regulating Temperature Compensation of the Circadian Clock

An approximately 24-h biological timekeeping mechanism called the circadian clock is present in virtually all light-sensitive organisms from cyanobacteria to humans. The clock system regulates our sleep–wake cycle, feeding–fasting, hormonal secretion, body temperature, and many other physiological functions. Signals from the master circadian oscillator entrain peripheral clocks using a variety of neural and hormonal signals. Even centrally controlled internal temperature fluctuations can entrain the peripheral circadian clocks. But, unlike other chemical reactions, the output of the clock system remains nearly constant with fluctuations in ambient temperature, a phenomenon known as temperature compensation. In this brief review, we focus on recent advances in our understanding of the posttranslational modifications, especially a phosphoswitch mechanism controlling the stability of PER2 and its implications for the regulation of temperature compensation.

Robust network topologies for generating oscillations with temperature-independent periods

Nearly all living systems feature a temperature-independent oscillation period in circadian clocks. This ubiquitous property occurs at the system level and is rooted in the network architecture of the clock machinery. To investigate the mechanism of this prominent property of the circadian clock and provide general guidance for generating robust genetic oscillators with temperature-compensated oscillations, we theoretically explored the design principle and core network topologies preferred by oscillations with a temperature-independent period. By enumerating all topologies of genetic regulatory circuits with three genes, we obtained four network motifs, namely, a delayed negative feedback oscillator, repressilator, activator-inhibitor oscillator and substrate-depletion oscillator; hybrids of these motifs constitute the vast majority of target network topologies. These motifs are biased in their capacities for achieving oscillations and the temperature sensitivity of the period. The delayed negative feedback oscillator and repressilator are more robust for oscillations, whereas the activator-inhibitor and substrate-depletion oscillators are superior for maintaining a temperature-independent oscillation period. These results suggest that thermally robust oscillation can be more plausibly achieved by hybridizing these two categories of network motifs. Antagonistic balance and temperature insulation mechanisms for achieving temperature compensation are typically found in these topologies with temperature robustness. In the temperature insulation approach, the oscillation period relies on very few parameters, and these parameters are influenced only slightly by temperature. This approach prevents the temperature from affecting the oscillation period and generates circadian rhythms that are robust against environmental perturbations.

Temperature-Robust Neural Function from Activity-Dependent Ion Channel Regulation

Many species of cold-blooded animals experience substantial and rapid fluctuations in body temperature. Because biological processes are differentially temperature dependent, it is difficult to understand how physiological processes in such animals can be temperature robust [1-8]. Experiments have shown that core neural circuits, such as the pyloric circuit of the crab stomatogastric ganglion (STG), exhibit robust neural activity in spite of large (20°C) temperature fluctuations [3, 5, 7, 8]. This robustness is surprising because (1) each neuron has many different kinds of ion channels with different temperature dependencies (Q₁₀s) that interact in a highly nonlinear way to produce firing patterns and (2) across animals there is substantial variability in conductance densities that nonetheless produce almost identical firing properties. The high variability in conductance densities in these neurons [9, 10] appears to contradict the possibility that robustness is achieved through precise tuning of key temperature-dependent processes. In this paper, we develop a theoretical explanation for how temperature robustness can emerge from a simple regulatory control mechanism that is compatible with highly variable conductance densities [11-13]. The resulting model suggests a general mechanism for how nervous systems and excitable tissues can exploit degenerate relationships among temperature-sensitive processes to achieve robust function.

The effects of time-varying temperature on delays in genetic networks

Delays in gene networks result from the sequential nature of protein assembly. However, it is unclear how models of gene networks that use delays should be modified when considering time-dependent changes in temperature. This is important, as delay is often used in models of genetic oscillators that can be entrained by periodic fluctuations in temperature. Here, we analytically derive the time dependence of delay distributions in response to time-varying temperature changes. We find that the resulting time-varying delay is nonlinearly dependent on parameters of the time-varying temperature such as amplitude and frequency, therefore, applying an Arrhenius scaling may result in erroneous conclusions. We use these results to examine a model of a synthetic gene oscillator with temperature compensation. We show that temperature entrainment follows from the same mechanism that results in temperature compensation. Under a common Arrhenius scaling alone, the frequency of the oscillator is sensitive to changes in the mean temperature but robust to changes in the frequency of a periodically time-varying temperature. When a mechanism for temperature compensation is included in the model, however, we show that the oscillator is entrained by periodically varying temperature even when maintaining insensitivity to the mean temperature.

Temperature compensation and temperature sensation in the circadian clock

All known circadian clocks have an endogenous period that is remarkably insensitive to temperature, a property known as temperature compensation, while at the same time being readily entrained by a diurnal temperature oscillation. Although temperature compensation and entrainment are defining features of circadian clocks, their mechanisms remain poorly understood. Most models presume that multiple steps in the circadian cycle are temperature-dependent, thus facilitating temperature entrainment, but then insist that the effect of changes around the cycle sums to zero to enforce temperature compensation. An alternative theory proposes that the circadian oscillator evolved from an adaptive temperature sensor: a gene circuit that responds only to temperature changes. This theory implies that temperature changes should linearly rescale the amplitudes of clock component oscillations but leave phase relationships and shapes unchanged. We show using timeless luciferase reporter measurements and Western blots against TIMELESS protein that this prediction is satisfied by the Drosophila circadian clock. We also review evidence for pathways that couple temperature to the circadian clock, and show previously unidentified evidence for coupling between the Drosophila clock and the heat-shock pathway.

Effect of Multiple Clock Gene Ablations on the Circadian Period Length and Temperature Compensation in Mammalian Cells

Most organisms have cell-autonomous circadian clocks to coordinate their activity and physiology according to 24-h environmental changes. Despite recent progress in circadian studies, it is not fully understood how the period length and the robustness of mammalian circadian rhythms are determined. In this study, we established a series of mouse embryonic stem cell (ESC) lines with single or multiplex clock gene ablations using the CRISPR/Cas9-based genome editing method. ESC-based in vitro circadian clock formation assay shows that the CRISPR-mediated clock gene disruption not only reproduces the intrinsic circadian molecular rhythms of previously reported mice tissues and cells lacking clock genes but also reveals that complexed mutations, such as CKIδ(m/m):CKIε(+/m):Cry2(m/m) mutants, exhibit an additively lengthened circadian period. By using these mutant cells, we also investigated the relation between period length alteration and temperature compensation. Although CKIδ-deficient cells slightly affected the temperature insensitivity of period length, we demonstrated that the temperature compensation property is largely maintained in all mutants. These results show that the ESC-based assay system could offer a more systematic and comprehensive approach to the genotype-chronotype analysis of the intracellular circadian clockwork in mammals.

Heating and cooling the Drosophila melanogaster clock

Most biological phenomena are under control of a circuit known as the 'molecular circadian clock.' Over the past forty years of research in Drosophila melanogaster, studies have made significant advances in our understanding of the molecular timing mechanism of this circuit, which is determined by a core inhibitory feedback loop. While the timing mechanism of the molecular circadian clock is endogenous, it is well established that exogenous cues such as light and temperature modulate its timing. In the following article, we summarize our current understanding of how temperature interacts with the molecular circadian clock in adult Drosophila.

Robust concentration and frequency control in oscillatory homeostats

Homeostatic and adaptive control mechanisms are essential for keeping organisms structurally and functionally stable. Integral feedback is a control theoretic concept which has long been known to keep a controlled variable A robustly (i.e. perturbation-independent) at a given set-point A(set) by feeding the integrated error back into the process that generates A. The classical concept of homeostasis as robust regulation within narrow limits is often considered as unsatisfactory and even incompatible with many biological systems which show sustained oscillations, such as circadian rhythms and oscillatory calcium signaling. Nevertheless, there are many similarities between the biological processes which participate in oscillatory mechanisms and classical homeostatic (non-oscillatory) mechanisms. We have investigated whether biological oscillators can show robust homeostatic and adaptive behaviors, and this paper is an attempt to extend the homeostatic concept to include oscillatory conditions. Based on our previously published kinetic conditions on how to generate biochemical models with robust homeostasis we found two properties, which appear to be of general interest concerning oscillatory and homeostatic controlled biological systems. The first one is the ability of these oscillators ("oscillatory homeostats") to keep the average level of a controlled variable at a defined set-point by involving compensatory changes in frequency and/or amplitude. The second property is the ability to keep the period/frequency of the oscillator tuned within a certain well-defined range. In this paper we highlight mechanisms that lead to these two properties. The biological applications of these findings are discussed using three examples, the homeostatic aspects during oscillatory calcium and p53 signaling, and the involvement of circadian rhythms in homeostatic regulation.

Diversity in the dynamical behaviour of a compartmentalized programmable biochemical oscillator

In vitro compartmentalization of biochemical reaction networks is a crucial step towards engineering artificial cell-scale devices and systems. At this scale the dynamics of molecular systems becomes stochastic, which introduces several engineering challenges and opportunities. Here we study a programmable transcriptional oscillator system that is compartmentalized into microemulsion droplets with volumes between 33 fl and 16 pl. Simultaneous measurement of large populations of droplets reveals major variations in the amplitude, frequency and damping of the oscillations. Variability increases for smaller droplets and depends on the operating point of the oscillator. Rather than reflecting the stochastic kinetics of the chemical reaction network itself, the variability can be attributed to the statistical variation of reactant concentrations created during their partitioning into droplets. We anticipate that robustness to partitioning variability will be a critical challenge for engineering cell-scale systems, and that highly parallel time-series acquisition from microemulsion droplets will become a key tool for characterization of stochastic circuit function.

Engineered temperature compensation in a synthetic genetic clock

Synthetic biology promises to revolutionize biotechnology by providing the means to reengineer and reprogram cellular regulatory mechanisms. However, synthetic gene circuits are often unreliable, as changes to environmental conditions can fundamentally alter a circuit's behavior. One way to improve robustness is to use intrinsic properties of transcription factors within the circuit to buffer against intra- and extracellular variability. Here, we describe the design and construction of a synthetic gene oscillator in Escherichia coli that maintains a constant period over a range of temperatures. We started with a previously described synthetic dual-feedback oscillator with a temperature-dependent period. Computational modeling predicted and subsequent experiments confirmed that a single amino acid mutation to the core transcriptional repressor of the circuit results in temperature compensation. Specifically, we used a temperature-sensitive lactose repressor mutant that loses the ability to repress its target promoter at high temperatures. In the oscillator, this thermoinduction of the repressor leads to an increase in period at high temperatures that compensates for the decrease in period due to Arrhenius scaling of the reaction rates. The result is a transcriptional oscillator with a nearly constant period of 48 min for temperatures ranging from 30 °C to 41 °C. In contrast, in the absence of the mutation the period of the oscillator drops from 60 to 30 min over the same temperature range. This work demonstrates that synthetic gene circuits can be engineered to be robust to extracellular conditions through protein-level modifications.

Can temperate insects take the heat? A case study of the physiological and behavioural responses in a common ant, Iridomyrmex purpureus (Formicidae), with potential climate change

Insects in temperate regions are predicted to be at low risk of climate change relative to tropical species. However, these assumptions have generally been poorly examined in all regions, and such forecasting fails to account for microclimatic variation and behavioural optimisation. Here, we test how a population of the dominant ant species, Iridomyrmex purpureus, from temperate Australia responds to thermal stress. We show that ants regularly forage for short periods (minutes) at soil temperatures well above their upper thermal limits (upper lethal temperature = 45.8 ± 1.3°C; CT(max) = 46.1°C) determined over slightly longer periods (hours) and do not show any signs of a classic thermal performance curve in voluntary locomotion across soil surface temperatures of 18.6-57°C (equating to a body temperature of 24.5-43.1°C). Although ants were present all year round, and dynamically altered several aspects of their thermal biology to cope with low temperatures and seasonal variation, temperature-dependence of running speed remained invariant and ants were unable to elevate high temperature tolerance using plastic responses. Measurements of microclimate temperature were higher than ant body temperatures during the hottest part of the day, but exhibited a stronger relationship with each other than air temperatures from the closest weather station. Generally close associations of ant activity and performance with microclimatic conditions, possibly to maximise foraging times, suggest I. purpureus displays highly opportunistic thermal responses and readily adjusts behaviour to cope with high trail temperatures. Increasing frequency or duration of high temperatures is therefore likely to result in an immediate reduction in foraging efficiency. In summary, these results suggest that (1) soil-dwelling temperate insect populations may be at higher risks of thermal stress with increased frequency or duration of high temperatures resulting from climate change than previously thought, however, behavioural cues may be able to compensate to some extent; and (2) indices of climate change-related thermal stress, warming tolerance and thermal safety margin, are strongly influenced by the scale of climate metrics employed.

Generation of oscillating gene regulatory network motifs

Using an improved version of an evolutionary algorithm originally proposed by François and Hakim [Proc. Natl. Acad. Sci. USA 101, 580 (2004)], we generated small gene regulatory networks in which the concentration of a target protein oscillates in time. These networks may serve as candidates for oscillatory modules to be found in larger regulatory networks and protein interaction networks. The algorithm was run for 10(5) times to produce a large set of oscillating modules, which were systematically classified and analyzed. The robustness of the oscillations against variations of the kinetic rates was also determined, to filter out the least robust cases. Furthermore, we show that the set of evolved networks can serve as a database of models whose behavior can be compared to experimentally observed oscillations. The algorithm found three smallest (core) oscillators in which nonlinearities and number of components are minimal. Two of those are two-gene modules: the mixed feedback loop, already discussed in the literature, and an autorepressed gene coupled with a heterodimer. The third one is a single gene module which is competitively regulated by a monomer and a dimer. The evolutionary algorithm also generated larger oscillating networks, which are in part extensions of the three core modules and in part genuinely new modules. The latter includes oscillators which do not rely on feedback induced by transcription factors, but are purely of post-transcriptional type. Analysis of post-transcriptional mechanisms of oscillation may provide useful information for circadian clock research, as recent experiments showed that circadian rhythms are maintained even in the absence of transcription.

Molecular characterization of the transition to mid-life in Caenorhabditis elegans

We present an initial molecular characterization of a morphological transition between two early aging states. In previous work, an age score reflecting physiological age was developed using a machine classifier trained on images of worm populations at fixed chronological ages throughout their lifespan. The distribution of age scores identified three stable post-developmental states and transitions. The first transition occurs at day 5 post-hatching, where a significant percentage of the population exists in both state I and state II. The temperature dependence of the timing of this transition (Q 10 ~ 1.17) is too low to be explained by a stepwise process with an enzymatic or chemical rate-limiting step, potentially implicating a more complex mechanism. Individual animals at day 5 were sorted into state I and state II groups using the machine classifier and analyzed by microarray expression profiling. Despite being isogenic, grown for the same amount of time, and indistinguishable by eye, these two morphological states were confirmed to be molecularly distinct by hierarchical clustering and principal component analysis of the microarray results. These molecular differences suggest that pharynx morphology reflects the aging state of the whole organism. Our expression profiling yielded a gene set that showed significant overlap with those from three previous age-related studies and identified several genes not previously implicated in aging. A highly represented group of genes unique to this study is involved in targeted ubiquitin-mediated proteolysis, including Skp1-related (SKR), F-box-containing, and BTB motif adaptors.

Adaptive temperature compensation in circadian oscillations

A temperature independent period and temperature entrainment are two defining features of circadian oscillators. A default model of distributed temperature compensation satisfies these basic facts yet is not easily reconciled with other properties of circadian clocks, such as many mutants with altered but temperature compensated periods. The default model also suggests that the shape of the circadian limit cycle and the associated phase response curves (PRC) will vary since the average concentrations of clock proteins change with temperature. We propose an alternative class of models where the twin properties of a fixed period and entrainment are structural and arise from an underlying adaptive system that buffers temperature changes. These models are distinguished by a PRC whose shape is temperature independent and orbits whose extrema are temperature independent. They are readily evolved by local, hill climbing, optimization of gene networks for a common quality measure of biological clocks, phase anticipation. Interestingly a standard realization of the Goodwin model for temperature compensation displays properties of adaptive rather than distributed temperature compensation.

Generic temperature compensation of biological clocks by autonomous regulation of catalyst concentration

Circadian clocks--ubiquitous in life forms ranging from bacteria to multicellular organisms--often exhibit intrinsic temperature compensation; the period of circadian oscillators is maintained constant over a range of physiological temperatures, despite the expected Arrhenius form for the reaction coefficient. Observations have shown that the amplitude of the oscillation depends on the temperature but the period does not; this suggests that although not every reaction step is temperature independent, the total system comprising several reactions still exhibits compensation. Here we present a general mechanism for such temperature compensation. Consider a system with multiple activation energy barriers for reactions, with a common enzyme shared across several reaction steps. The steps with the highest activation energy rate-limit the cycle when the temperature is not high. If the total abundance of the enzyme is limited, the amount of free enzyme available to catalyze a specific reaction decreases as more substrates bind to the common enzyme. We show that this change in free enzyme abundance compensates for the Arrhenius-type temperature dependence of the reaction coefficient. Taking the example of circadian clocks with cyanobacterial proteins KaiABC, consisting of several phosphorylation sites, we show that this temperature compensation mechanism is indeed valid. Specifically, if the activation energy for phosphorylation is larger than that for dephosphorylation, competition for KaiA shared among the phosphorylation reactions leads to temperature compensation. Moreover, taking a simpler model, we demonstrate the generality of the proposed compensation mechanism, suggesting relevance not only to circadian clocks but to other (bio)chemical oscillators as well.

Modeling circadian clocks: Roles, advantages, and limitations

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

Circadian oscillators in eukaryotes

The biological clock, present in nearly all eukaryotes, has evolved such that organisms can adapt to our planet's rotation in order to anticipate the coming day or night as well as unfavorable seasons. As all modern high-precision chronometers, the biological clock uses oscillation as a timekeeping element. In this review, we describe briefly the discovery, historical development, and general properties of circadian oscillators. The issue of temperature compensation (TC) is discussed, and our present understanding of the underlying genetic and biochemical mechanisms in circadian oscillators are described with special emphasis on Neurospora crassa, mammals, and plants.

Studying adaptation and homeostatic behaviors of kinetic networks by using MATLAB

Organisms have the ability to counteract environmental perturbations and keep certain components within a cell homeostatically regulated. Closely related to homeostasis is the behavior of perfect adaptation where an organism responds to a step-wise perturbation by regulating some of its components, after a transient period, to their original pre-perturbation values. A particular interesting type of model relates to the so-called robust behavior where the homeostatic or perfect adaptation property is independent of the magnitude of the applied step-wise perturbation. It has been shown that this type of behavior is related to the control-theoretic concept of integral feedback (or integral control). Using downloadable MATLAB examples, we demonstrate how robust perfect adaptation sites can be identified in reaction kinetic networks by linearizing the system, applying the Laplace transform and inspecting the transfer function. We also show how the homeostatic set point in perfect adaptation is related to the presence of zero-order fluxes.

The functional interplay between protein kinase CK2 and CCA1 transcriptional activity is essential for clock temperature compensation in Arabidopsis

Circadian rhythms are daily biological oscillations driven by an endogenous mechanism known as circadian clock. The protein kinase CK2 is one of the few clock components that is evolutionary conserved among different taxonomic groups. CK2 regulates the stability and nuclear localization of essential clock proteins in mammals, fungi, and insects. Two CK2 regulatory subunits, CKB3 and CKB4, have been also linked with the Arabidopsis thaliana circadian system. However, the biological relevance and the precise mechanisms of CK2 function within the plant clockwork are not known. By using ChIP and Double-ChIP experiments together with in vivo luminescence assays at different temperatures, we were able to identify a temperature-dependent function for CK2 modulating circadian period length. Our study uncovers a previously unpredicted mechanism for CK2 antagonizing the key clock regulator CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1). CK2 activity does not alter protein accumulation or subcellular localization but interferes with CCA1 binding affinity to the promoters of the oscillator genes. High temperatures enhance the CCA1 binding activity, which is precisely counterbalanced by the CK2 opposing function. Altering this balance by over-expression, mutation, or pharmacological inhibition affects the temperature compensation profile, providing a mechanism by which plants regulate circadian period at changing temperatures. Therefore, our study establishes a new model demonstrating that two opposing and temperature-dependent activities (CCA1-CK2) are essential for clock temperature compensation in Arabidopsis.

The endo-siRNA pathway is essential for robust development of the Drosophila embryo

Background

Robustness to natural temperature fluctuations is critical to proper development in embryos and to cellular functions in adult organisms. However, mechanisms and pathways which govern temperature compensation remain largely unknown beyond circadian rhythms. Pathways which ensure robustness against temperature fluctuations may appear to be nonessential under favorable, uniform environmental conditions used in conventional laboratory experiments where there is little variation for which to compensate. The endo-siRNA pathway, which produces small double-stranded RNAs in Drosophila, appears to be nonessential for robust development of the embryo under ambient uniform temperature and to be necessary only for viral defense. Embryos lacking a functional endo-siRNA pathway develop into phenotypically normal adults. However, we hypothesized that small RNAs may regulate the embryo's response to temperature, as a ribonucleoprotein complex has been previously shown to mediate mammalian cell response to heat shock.

Principal Findings

Here, we show that the genes DICER-2 and ARGONAUTE2, which code for integral protein components of the endo-siRNA pathway, are essential for robust development and temperature compensation in the Drosophila embryo when exposed to temperature perturbations. The regulatory functions of DICER-2 and ARGONAUTE2 were uncovered by using microfluidics to expose developing Drosophila embryos to a temperature step, in which each half of the embryo develops at a different temperature through developmental cycle 14. Under this temperature perturbation, dicer-2 or argonaute2 embryos displayed abnormal segmentation. The abnormalities in segmentation are presumably due to the inability of the embryo to compensate for temperature-induced differences in rate of development and to coordinate developmental timing in the anterior and posterior halves. A deregulation of the length of nuclear division cycles 10–14 is also observed in dicer-2 embryos at high temperatures.

Conclusions

Results presented herein uncover a novel function of the endo-siRNA pathway in temperature compensation and cell cycle regulation, and we hypothesize that the endo-siRNA pathway may regulate the degradation of maternal cell cycle regulators. Endo-siRNAs may have a more general role buffering against environmental perturbations in other organisms.

The control of the controller: molecular mechanisms for robust perfect adaptation and temperature compensation

Organisms have the property to adapt to a changing environment and keep certain components within a cell regulated at the same level (homeostasis). "Perfect adaptation" describes an organism's response to an external stepwise perturbation by regulating some of its variables/components precisely to their original preperturbation values. Numerous examples of perfect adaptation/homeostasis have been found, as for example, in bacterial chemotaxis, photoreceptor responses, MAP kinase activities, or in metal-ion homeostasis. Two concepts have evolved to explain how perfect adaptation may be understood: In one approach (robust perfect adaptation), the adaptation is a network property, which is mostly, but not entirely, independent of rate constant values; in the other approach (nonrobust perfect adaptation), a fine-tuning of rate constant values is needed. Here we identify two classes of robust molecular homeostatic mechanisms, which compensate for environmental variations in a controlled variable's inflow or outflow fluxes, and allow for the presence of robust temperature compensation. These two classes of homeostatic mechanisms arise due to the fact that concentrations must have positive values. We show that the concept of integral control (or integral feedback), which leads to robust homeostasis, is associated with a control species that has to work under zero-order flux conditions and does not necessarily require the presence of a physico-chemical feedback structure. There are interesting links between the two identified classes of homeostatic mechanisms and molecular mechanisms found in mammalian iron and calcium homeostasis, indicating that homeostatic mechanisms may underlie similar molecular control structures.

A role for casein kinase 2 in the mechanism underlying circadian temperature compensation

Temperature compensation of circadian clocks is an unsolved problem with relevance to the general phenomenon of biological compensation. We identify casein kinase 2 (CK2) as a key regulator of temperature compensation of the Neurospora clock by determining that two long-standing clock mutants, chrono and period-3, displaying distinctive alterations in compensation encode the beta1 and alpha subunits of CK2, respectively. Reducing the dose of these subunits, particularly beta1, significantly alters temperature compensation without altering the enzyme's Q(10). By contrast, other kinases and phosphatases implicated in clock function do not play appreciable roles in temperature compensation. CK2 exerts its effects on the clock by directly phosphorylating FREQUENCY (FRQ), and this phosphorylation is compromised in CK2 hypomorphs. Finally, mutation of certain putative CK2 phosphosites on FRQ, shown to be phosphorylated in vivo, predictably alters temperature compensation profiles effectively phenocopying CK2 mutants.

Minimum criteria for DNA damage-induced phase advances in circadian rhythms

Robust oscillatory behaviors are common features of circadian and cell cycle rhythms. These cyclic processes, however, behave distinctively in terms of their periods and phases in response to external influences such as light, temperature, nutrients, etc. Nevertheless, several links have been found between these two oscillators. Cell division cycles gated by the circadian clock have been observed since the late 1950s. On the other hand, ionizing radiation (IR) treatments cause cells to undergo a DNA damage response, which leads to phase shifts (mostly advances) in circadian rhythms. Circadian gating of the cell cycle can be attributed to the cell cycle inhibitor kinase Wee1 (which is regulated by the heterodimeric circadian clock transcription factor, BMAL1/CLK), and possibly in conjunction with other cell cycle components that are known to be regulated by the circadian clock (i.e., c-Myc and cyclin D1). It has also been shown that DNA damage-induced activation of the cell cycle regulator, Chk2, leads to phosphorylation and destruction of a circadian clock component (i.e., PER1 in Mus or FRQ in Neurospora crassa). However, the molecular mechanism underlying how DNA damage causes predominantly phase advances in the circadian clock remains unknown. In order to address this question, we employ mathematical modeling to simulate different phase response curves (PRCs) from either dexamethasone (Dex) or IR treatment experiments. Dex is known to synchronize circadian rhythms in cell culture and may generate both phase advances and delays. We observe unique phase responses with minimum delays of the circadian clock upon DNA damage when two criteria are met: (1) existence of an autocatalytic positive feedback mechanism in addition to the time-delayed negative feedback loop in the clock system and (2) Chk2-dependent phosphorylation and degradation of PERs that are not bound to BMAL1/CLK.

Molecular components and mechanisms that connect cell cycle and circadian rhythms are important for the well-being of an organism. Cell cycle machinery regulates the progress of cell growth and division while the circadian rhythm network generates an ∼24 h time-keeping mechanism that regulates the daily processes of an organism (i.e. metabolism, bowel movements, body temperature, etc.). It is observed that cell divisions usually occur during a certain time window of a day, which indicated that there are circadian-gated cell divisions. Moreover, it's been shown that mice are more prone to develop cancer when certain clock genes are mutated resulting in an arrhythmic clock. Recently, a cell cycle checkpoint regulator, Chk2, was identified as a component that influences a core clock component and creates mostly phase advances (i.e., jet lags due to traveling east) in circadian rhythms upon DNA damage. This phase response with minimum delays is an unexpected result, and the molecular mechanism behind this phenomenon remains unknown. Our computational analyses of a mathematical model reveal two molecular criteria that account for the experimentally observed phase responses of the circadian clock upon DNA damage. These results demonstrate how circadian clock regulation by cell cycle checkpoint controllers provides another layer of complexity for efficient DNA damage responses.

Closing the circadian negative feedback loop: FRQ-dependent clearance of WC-1 from the nucleus

In Neurospora crassa, a transcription factor, WCC, activates the transcription of frq. FRQ forms homodimers as well as complexes with an RNA helicase, FRH, and the WCC, and translocates into the nucleus to inactivate the WCC, closing the time-delayed negative feedback loop. The detailed mechanism for closing this loop, however, remains incompletely understood. In particular within the nucleus, the low amount of FRQ compared with that of WC-1 creates a conundrum: How can the nuclear FRQ inactivate the larger amount of WCC? One possibility is that FRQ might function as a catalytic component in phosphorylation-dependent inhibition. However, in silico experiments reveal that stoichiometric noncatalytic binding and inhibition can generate a robust oscillator, even when nuclear FRQ levels are substantially lower than nuclear WCC, so long as there is FRQ-dependent clearance of WC-1 from the nucleus. Based on this model, we can predict and now demonstrate that WC-1 stability cycles, that WC-1 is stable in the absence of FRQ, and that physical binding between FRQ and WCC is essential for closure of the negative feedback loop. Moreover, and consistent with a noncatalytic clearance-based model for inhibition, appreciable amounts of the nuclear FRQ:WCC complex accumulate at some times of day, comprising as much as 10% of the nuclear WC-1.

A precise Bicoid gradient is nonessential during cycles 11-13 for precise patterning in the Drosophila blastoderm

BACKGROUND

During development, embryos decode maternal morphogen inputs into highly precise zygotic gene expression. The discovery of the morphogen Bicoid and its profound effect on developmental programming in the Drosophila embryo has been a cornerstone in understanding the decoding of maternal inputs. Bicoid has been described as a classical morphogen that forms a concentration gradient along the antero-posterior axis of the embryo by diffusion and initiates expression of target genes in a concentration-dependent manner in the syncytial blastoderm. Recent work has emphasized the stability of the Bicoid gradient as a function of egg length and the role of nuclear dynamics in maintaining the Bicoid gradient. Bicoid and nuclear dynamics were observed but not modulated under the ideal conditions used previously. Therefore, it has not been tested explicitly whether a temporally stable Bicoid gradient prior to cellularization is required for precise patterning.

PRINCIPAL FINDINGS

Here, we modulate both nuclear dynamics and the Bicoid gradient using laminar flows of different temperature in a microfluidic device to determine if stability of the Bicoid gradient prior to cellularization is essential for precise patterning. Dramatic motion of both cytoplasm and nuclei was observed prior to cellularization, and the Bicoid gradient was disrupted by nuclear motion and was highly abnormal as a function of egg length. Despite an abnormal Bicoid gradient during cycles 11-13, Even-skipped patterning in these embryos remained precise.

CONCLUSIONS

These results indicate that the stability of the Bicoid gradient as a function of egg length is nonessential during syncytial blastoderm stages. Further, presumably no gradient formed by simple diffusion on the scale of egg length could be responsible for the robust antero-posterior patterning observed, as severe cytoplasmic and nuclear motion would disrupt such a gradient. Additional mechanisms for how the embryo could sense its dimensions and interpret the Bicoid gradient are discussed.

Biological switches and clocks

To introduce this special issue on biological switches and clocks, we review the historical development of mathematical models of bistability and oscillations in chemical reaction networks. In the 1960s and 1970s, these models were limited to well-studied biochemical examples, such as glycolytic oscillations and cyclic AMP signalling. After the molecular genetics revolution of the 1980s, the field of molecular cell biology was thrown wide open to mathematical modellers. We review recent advances in modelling the gene-protein interaction networks that control circadian rhythms, cell cycle progression, signal processing and the design of synthetic gene networks.

Sloppiness, robustness, and evolvability in systems biology

The functioning of many biochemical networks is often robust-remarkably stable under changes in external conditions and internal reaction parameters. Much recent work on robustness and evolvability has focused on the structure of neutral spaces, in which system behavior remains invariant to mutations. Recently we have shown that the collective behavior of multiparameter models is most often sloppy: insensitive to changes except along a few 'stiff' combinations of parameters, with an enormous sloppy neutral subspace. Robustness is often assumed to be an emergent evolved property, but the sloppiness natural to biochemical networks offers an alternative nonadaptive explanation. Conversely, ideas developed to study evolvability in robust systems can be usefully extended to characterize sloppy systems.

Using chemistry and microfluidics to understand the spatial dynamics of complex biological networks

Understanding the spatial dynamics of biochemical networks is both fundamentally important for understanding life at the systems level and also has practical implications for medicine, engineering, biology, and chemistry. Studies at the level of individual reactions provide essential information about the function, interactions, and localization of individual molecular species and reactions in a network. However, analyzing the spatial dynamics of complex biochemical networks at this level is difficult. Biochemical networks are nonequilibrium systems containing dozens to hundreds of reactions with nonlinear and time-dependent interactions, and these interactions are influenced by diffusion, flow, and the relative values of state-dependent kinetic parameters. To achieve an overall understanding of the spatial dynamics of a network and the global mechanisms that drive its function, networks must be analyzed as a whole, where all of the components and influential parameters of a network are simultaneously considered. Here, we describe chemical concepts and microfluidic tools developed for network-level investigations of the spatial dynamics of these networks. Modular approaches can be used to simplify these networks by separating them into modules, and simple experimental or computational models can be created by replacing each module with a single reaction. Microfluidics can be used to implement these models as well as to analyze and perturb the complex network itself with spatial control on the micrometer scale. We also describe the application of these network-level approaches to elucidate the mechanisms governing the spatial dynamics of two networkshemostasis (blood clotting) and early patterning of the Drosophila embryo. To investigate the dynamics of the complex network of hemostasis, we simplified the network by using a modular mechanism and created a chemical model based on this mechanism by using microfluidics. Then, we used the mechanism and the model to predict the dynamics of initiation and propagation of blood clotting and tested these predictions with human blood plasma by using microfluidics. We discovered that both initiation and propagation of clotting are regulated by a threshold response to the concentration of activators of clotting, and that clotting is sensitive to the spatial localization of stimuli. To understand the dynamics of patterning of the Drosophila embryo, we used microfluidics to perturb the environment around a developing embryo and observe the effects of this perturbation on the expression of Hunchback, a protein whose localization is essential to proper development. We found that the mechanism that is responsible for Hunchback positioning is asymmetric, time-dependent, and more complex than previously proposed by studies of individual reactions. Overall, these approaches provide strategies for simplifying, modeling, and probing complex networks without sacrificing the functionality of the network. Such network-level strategies may be most useful for understanding systems with nonlinear interactions where spatial dynamics is essential for function. In addition, microfluidics provides an opportunity to investigate the mechanisms responsible for robust functioning of complex networks. By creating nonideal, stressful, and perturbed environments, microfluidic experiments could reveal the function of pathways thought to be nonessential under ideal conditions.

Simulating dark expressions and interactions of frq and wc-1 in the Neurospora circadian clock

Circadian rhythms are considered to play an essential part in the adaptation of organisms to their environments. The occurrence of circadian oscillations appears to be based on the presence of transcriptional-translational negative feedback loops. In Neurospora crassa, the protein FREQUENCY (FRQ) is part of such a negative feedback loop apparently by a direct interaction with its transcription factor WHITE COLLAR-1 (WC-1). Based on the observation that nuclear FRQ levels are significantly lower than nuclear WC-1 levels, it was suggested that FRQ would act more like a catalyst in inhibiting WC-1 rather than binding to WC-1 and making an inactive FRQ:WC-1 complex. Intrigued by this hypothesis, we constructed a model for the Neurospora circadian clock, which includes expression of the frq and the wc-1 genes and their possible interactions. The model suggests that even small amounts of nuclear FRQ-protein are capable of inhibiting frq transcription in a rhythmic manner by binding to WC-1 and promoting its degradation. Our model predicts the importance of a FRQ dependent degradation of WC-1 in closing the negative feedback loop. The model shows good agreement with experimental levels in nuclear and cytosolic FRQ and WC-1, their phase relationships, and several clock mutant phenotypes.

Computational analysis of mammalian cell division gated by a circadian clock: quantized cell cycles and cell size control

Cell cycle and circadian rhythms are conserved from cyanobacteria to humans with robust cyclic features. Recently, molecular links between these two cyclic processes have been discovered. Core clock transcription factors, Bmal1 and Clock (Clk), directly regulate Wee1 kinase, which inhibits entry into the mitosis. We investigate the effect of this connection on the timing of mammalian cell cycle processes with computational modeling tools. We connect a minimal model of circadian rhythms, which consists of transcription-translation feedback loops, with a modified mammalian cell cycle model from Novak and Tyson (2004). As we vary the mass doubling time (MDT) of the cell cycle, stochastic simulations reveal quantized cell cycles when the activity of Wee1 is influenced by clock components. The quantized cell cycles disappear in the absence of coupling or when the strength of this link is reduced. More intriguingly, our simulations indicate that the circadian clock triggers critical size control in the mammalian cell cycle. A periodic brake on the cell cycle progress via Wee1 enforces size control when the MDT is quite different from the circadian period. No size control is observed in the absence of coupling. The issue of size control in the mammalian system is debatable, whereas it is well established in yeast. It is possible that the size control is more readily observed in cell lines that contain circadian rhythms, since not all cell types have a circadian clock. This would be analogous to an ultradian clock intertwined with quantized cell cycles (and possibly cell size control) in yeast. We present the first coupled model between the mammalian cell cycle and circadian rhythms that reveals quantized cell cycles and cell size control influenced by the clock.

Temperature dependence and temperature compensation of kinetics of chemical oscillations; Belousov-Zhabotinskii reaction, glycolysis and circadian rhythms

Based on the distribution of activation energies around the experimental mean and averaging of rate constants we propose a theoretical scheme to examine the temperature dependence and temperature compensation of time periods of chemical oscillations. The critical finite width of the distribution is characteristic of endogeneous oscillations for compensating kinetics as observed in circadian oscillations, while the vanishing width corresponds to Arrhenius temperature dependent kinetics of non-endogeneous chemical oscillation in Belousov-Zhabotinskii reaction in a CSTR or glycolysis in cell-free yeast extracts. Our theoretical analysis is corroborated with experimental data.