Comprehensive modelling of the Neurospora circadian clock and its temperature compensation

Comparative mathematical modeling reveals the differential effects of high-fat diet and ketogenic diet on the PI3K-Akt signaling pathway in heart

BACKGROUND

Obesity is a global health concern associated with increased risk of diseases like cardiovascular conditions including ischemic heart disease, a leading cause of mortality. The ketogenic diet (KD) has potential therapeutic applications in managing obesity and related disorders. However, the intricate effects of KD on diverse physiological conditions remain incompletely understood. The PI3K-Akt signaling pathway is critical for heart health, and its dysregulation implicates numerous cardiac diseases.

METHODS

We developed comprehensive mathematical models of the PI3K-Akt signaling pathway under high-fat diet (HFD) and KD conditions to elucidate their differential impacts and quantify apoptosis. Simulations and sensitivity analysis were performed.

RESULTS

Simulations demonstrate that KD can reduce the activation of key molecules like Erk and Trp53 to mitigate apoptosis compared to HFD. Findings align with experimental data, highlighting the potential cardiac benefits of KD. Sensitivity analysis identifies regulators like Trp53 and Bcl2l1 that critically influence apoptosis under HFD.

CONCLUSIONS

Mathematical modeling provides quantitative insights into the contrasting effects of HFD and KD on cardiac PI3K-Akt signaling and apoptosis. Findings have implications for precision nutrition and developing novel therapeutic strategies to address obesity-related cardiovascular diseases.

Circadian regulation of metabolism across photosynthetic organisms

Circadian regulation produces a biological measure of time within cells. The daily cycle in the availability of light for photosynthesis causes dramatic changes in biochemical processes in photosynthetic organisms, with the circadian clock having crucial roles in adaptation to these fluctuating conditions. Correct alignment between the circadian clock and environmental day-night cycles maximizes plant productivity through its regulation of metabolism. Therefore, the processes that integrate circadian regulation with metabolism are key to understanding how the circadian clock contributes to plant productivity. This forms an important part of exploiting knowledge of circadian regulation to enhance sustainable crop production. Here, we examine the roles of circadian regulation in metabolic processes in source and sink organ structures of Arabidopsis. We also evaluate possible roles for circadian regulation in root exudation processes that deposit carbon into the soil, and the nature of the rhythmic interactions between plants and their associated microbial communities. Finally, we examine shared and differing aspects of the circadian regulation of metabolism between Arabidopsis and other model photosynthetic organisms, and between circadian control of metabolism in photosynthetic and non-photosynthetic organisms. This synthesis identifies a variety of future research topics, including a focus on metabolic processes that underlie biotic interactions within ecosystems.

Data-driven modelling captures dynamics of the circadian clock of Neurospora crassa

Eukaryotic circadian clocks are based on self-sustaining, cell-autonomous oscillatory feedback loops that can synchronize with the environment via recurrent stimuli (zeitgebers) such as light. The components of biological clocks and their network interactions are becoming increasingly known, calling for a quantitative understanding of their role for clock function. However, the development of data-driven mathematical clock models has remained limited by the lack of sufficiently accurate data. Here we present a comprehensive model of the circadian clock of Neurospora crassa that describe free-running oscillations in constant darkness and entrainment in light-dark cycles. To parameterize the model, we measured high-resolution time courses of luciferase reporters of morning and evening specific clock genes in WT and a mutant strain. Fitting the model to such comprehensive data allowed estimating parameters governing circadian phase, period length and amplitude, and the response of genes to light cues. Our model suggests that functional maturation of the core clock protein Frequency causes a delay in negative feedback that is critical for generating circadian rhythms.

Circadian rhythms are endogenous autonomous clocks that emancipate daily rhythms in physiology and behavior. Lately, a large body of research has contributed to our understanding of clocks’ genetic and mechanistic basis across kingdoms of life, i.e., mammals, fungi, plants, and bacteria. Several mathematical models have made key contributions to our current understanding of the design principles of the Neurospora crassa circadian clock and conditions for self-sustained oscillations. However, previous models uncovered and described the principle properties of the clock in generic manner due to a lack of experimental data. In this study, we developed a mathematical model based on systems of differential equations to describe the core clock components and estimated model parameters from luciferase data that capture experimental observations. We demonstrate the model predictive control simulation emphasizing the importance of functional maturation of the core clock protein Frequency in generating circadian rhythms.

A Unified Model for Entrainment by Circadian Clocks: Dynamic Circadian Integrated Response Characteristic (dCiRC)

Circadian rhythms are ubiquitous and are observed in all biological kingdoms. In nature, their primary characteristic or phenotype is the phase of entrainment. There are two main hypotheses related to how circadian clocks entrain, parametric and non-parametric models. The parametric model focuses on the gradual changes of the clock parameters in response to the changing ambient condition, whereas the non-parametric model focuses on the instantaneous change of the phase of the clock in response to the zeitgeber. There are ample empirical data supporting both models. However, only recently has a unifying model been proposed, the circadian integrated response characteristic (CiRC). In the current study, we developed a system of ordinary differential equations, dynamic CiRC (dCiRC), that describes parameters of circadian rhythms and predicts the phase of entrainment in zeitgeber cycles. dCiRC mathematically extracts the underlying information of velocity changes of the internal clock that reflects the parametric model and the phase shift trajectory that reflects the non-parametric model from phase data under entraining conditions. As a proof of concept, we measured clock parameters of 26 Neurospora crassa ecotypes in both cycling and constant conditions using dCiRC. Our data showed that the morning light shortens the period of the clock while the afternoon light lengthens it. We also found that individual ecotypes have different strategies of integrating light effects to accomplish the optimal phase of entrainment, a model feature that is consistent with our knowledge of how circadian clocks are organized and encoded. The unified model dCiRC will provide new insights into how circadian clocks function under different zeitgeber conditions. We suggest that this type of model may be useful in the advent of chronotherapies.

Principles underlying the complex dynamics of temperature entrainment by a circadian clock

Autonomously oscillating circadian clocks resonate with daily environmental (zeitgeber) rhythms to organize physiology around the solar day. Although entrainment properties and mechanisms have been studied widely and in great detail for light-dark cycles, entrainment to daily temperature rhythms remains poorly understood despite that they are potent zeitgebers. Here we investigate the entrainment of the chronobiological model organism Neurospora crassa, subject to thermocycles of different periods and fractions of warm versus cold phases, mimicking seasonal variations. Depending on the properties of these thermocycles, regularly entrained rhythms, period-doubling (frequency demultiplication) but also irregular aperiodic behavior occurs. We demonstrate that the complex nonlinear phenomena of experimentally observed entrainment dynamics can be understood by molecular mathematical modeling.

Circadian rhythm shows potential for mRNA efficiency and self-organized division of labor in multinucleate cells

Multinucleate cells occur in every biosphere and across the kingdoms of life, including in the human body as muscle cells and bone-forming cells. Data from filamentous fungi suggest that, even when bathed in a common cytoplasm, nuclei are capable of autonomous behaviors, including division. How does this potential for autonomy affect the organization of cellular processes between nuclei? Here we analyze a simplified model of circadian rhythm, a form of cellular oscillator, in a mathematical model of the filamentous fungus Neurospora crassa. Our results highlight a potential role played by mRNA-protein phase separation to keep mRNAs close to the nuclei from which they originate, while allowing proteins to diffuse freely between nuclei. Our modeling shows that syncytism allows for extreme mRNA efficiency-we demonstrate assembly of a robust oscillator with a transcription rate a thousand-fold less than in comparable uninucleate cells. We also show self-organized division of the labor of mRNA production, with one nucleus in a two-nucleus syncytium producing at least twice as many mRNAs as the other in 30% of cycles. This division can occur spontaneously, but division of labor can also be controlled by regulating the amount of cytoplasmic volume available to each nucleus. Taken together, our results show the intriguing richness and potential for emergent organization among nuclei in multinucleate cells. They also highlight the role of previously studied mechanisms of cellular organization, including nuclear space control and localization of mRNAs through RNA-protein phase separation, in regulating nuclear coordination.

ECHO: an application for detection and analysis of oscillators identifies metabolic regulation on genome-wide circadian output

MOTIVATION

Time courses utilizing genome scale data are a common approach to identifying the biological pathways that are controlled by the circadian clock, an important regulator of organismal fitness. However, the methods used to detect circadian oscillations in these datasets are not able to accommodate changes in the amplitude of the oscillations over time, leading to an underestimation of the impact of the clock on biological systems.

RESULTS

We have created a program to efficaciously identify oscillations in large-scale datasets, called the Extended Circadian Harmonic Oscillator application, or ECHO. ECHO utilizes an extended solution of the fixed amplitude oscillator that incorporates the amplitude change coefficient. Employing synthetic datasets, we determined that ECHO outperforms existing methods in detecting rhythms with decreasing oscillation amplitudes and in recovering phase shift. Rhythms with changing amplitudes identified from published biological datasets revealed distinct functions from those oscillations that were harmonic, suggesting purposeful biologic regulation to create this subtype of circadian rhythms.

AVAILABILITY AND IMPLEMENTATION

ECHO's full interface is available at https://github.com/delosh653/ECHO. An R package for this functionality, echo.find, can be downloaded at https://CRAN.R-project.org/package=echo.find.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

FRQ-CK1 interaction determines the period of circadian rhythms in Neurospora

Circadian clock mechanisms have been extensively investigated but the main rate-limiting step that determines circadian period remains unclear. Formation of a stable complex between clock proteins and CK1 is a conserved feature in eukaryotic circadian mechanisms. Here we show that the FRQ-CK1 interaction, but not FRQ stability, correlates with circadian period in Neurospora circadian clock mutants. Mutations that specifically affect the FRQ-CK1 interaction lead to severe alterations in circadian period. The FRQ-CK1 interaction has two roles in the circadian negative feedback loop. First, it determines the FRQ phosphorylation profile, which regulates FRQ stability and also feeds back to either promote or reduce the interaction itself. Second, it determines the efficiency of circadian negative feedback process by mediating FRQ-dependent WC phosphorylation. Our conclusions are further supported by mathematical modeling and in silico experiments. Together, these results suggest that the FRQ-CK1 interaction is a major rate-limiting step in circadian period determination.

Circadian clocks control daily rhythms of molecular and physiological activities. Here, the authors show that the interaction between proteins FRQ and CK1, rather than FRQ stability, is a major rate-limiting step in circadian period determination in the model fungus Neurospora.

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.

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.

Modeling Reveals a Key Mechanism for Light-Dependent Phase Shifts of Neurospora Circadian Rhythms

Light shifts and synchronizes the phase of the circadian clock to daily environments, which is critical for maintaining the daily activities of an organism. It has been proposed that such light-dependent phase shifts are triggered by light-induced upregulation of a negative element of the core circadian clock (i.e., frq, Per1/2) in many organisms, including fungi. However, we find, using systematic mathematical modeling of the Neurospora crassa circadian clock, that the upregulation of the frq gene expression alone is unable to reproduce the observed light-dependent phase responses. Indeed, we find that the depression of the transcriptional activator white-collar-1, previously shown to be promoted by FRQ and VVD, is a key molecular mechanism for accurately simulating light-induced phase response curves for wild-type and mutant strains of Neurospora. Our findings elucidate specific molecular pathways that can be utilized to control phase resetting of circadian rhythms.

Circadian Rhythms in Neurospora Exhibit Biologically Relevant Driven and Damped Harmonic Oscillations

Circadian rhythms are endogenous cycles of approximately 24 hours reinforced by external cues such as light. These cycles are typically modeled as harmonic oscillators with fixed amplitude peaks. Using experimental data measuring global gene transcription in Neurospora crassa over 48 hours in the dark (i.e. with external queues removed), we demonstrate that many circadian genes frequently exhibit either damped harmonic oscillations, in which the peak amplitudes decrease each day, or driven harmonic oscillations, in which the peak amplitudes increase each day. By fitting extended harmonic oscillator models which include a damping ratio coefficient, we detected additional circadian genes that were not identified by the current standard tools that use fixed amplitude waves as reference, e.g. JTK_CYCLE. Functional Catalogue analysis confirms that our identified damped or driven genes exhibit distinct biological functions. The application of extended damped/driven harmonic oscillator models thus can elucidate, not only previously unidentified circadian genes, but also characterize gene subsets with expression patterns of biological relevance. Thus, expanded harmonic oscillators provide a powerful new tool for circadian system biology.

Systems Biology-Derived Discoveries of Intrinsic Clocks

A systems approach to studying biology uses a variety of mathematical, computational, and engineering tools to holistically understand and model properties of cells, tissues, and organisms. Building from early biochemical, genetic, and physiological studies, systems biology became established through the development of genome-wide methods, high-throughput procedures, modern computational processing power, and bioinformatics. Here, we highlight a variety of systems approaches to the study of biological rhythms that occur with a 24-h period-circadian rhythms. We review how systems methods have helped to elucidate complex behaviors of the circadian clock including temperature compensation, rhythmicity, and robustness. Finally, we explain the contribution of systems biology to the transcription-translation feedback loop and posttranslational oscillator models of circadian rhythms and describe new technologies and "-omics" approaches to understand circadian timekeeping and neurophysiology.

Asynchronous τ-leaping

Stochastic simulation of cell signaling pathways and genetic regulatory networks has contributed to the understanding of cell function; however, investigation of larger, more complicated systems requires computationally efficient algorithms. τ-leaping methods, which improve efficiency when some molecules have high copy numbers, either use a fixed leap size, which does not adapt to changing state, or recalculate leap size at a heavy computational cost. We present a hybrid simulation method for reaction-diffusion systems which combines exact stochastic simulation and τ-leaping in a dynamic way. Putative times of events are stored in a priority queue, which reduces the cost of each step of the simulation. For every reaction and diffusion channel at each step of the simulation the more efficient of an exact stochastic event or a τ-leap is chosen. This new approach removes the inherent trade-off between speed and accuracy in stiff systems which was present in all τ-leaping methods by allowing each reaction channel to proceed at its own pace. Both directions of reversible reactions and diffusion are combined in a single event, allowing bigger leaps to be taken. This improves efficiency for systems near equilibrium where forward and backward events are approximately equally frequent. Comparison with existing algorithms and behaviour for five test cases of varying complexity shows that the new method is almost as accurate as exact stochastic simulation, scales well for large systems, and for various problems can be significantly faster than τ-leaping.

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.

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.

The Neurospora photoreceptor VIVID exerts negative and positive control on light sensing to achieve adaptation

Light adaptation in Neurospora is mediated by the photoreceptor VIVID, which exerts both a negative and positive effect on light sensing. These apparently paradoxical roles of VIVID are explained by the dynamics of a network motif that utilizes futile cycling.

The fungus Neurospora detects relative changes in light intensity by adapting to the ambient light level and remaining responsive to increases in light intensity.

Both the downregulation of the acute light response and maintained responsiveness are mediated by the photoreceptor VIVID (VVD).

Data-based mathematical modeling shows that this paradoxical function of VVD can be realized by a futile-cycle network motif that turns feedback inhibition into sensory adaptation.

The light response in Neurospora is mediated by the photoreceptor and circadian transcription factor White Collar Complex (WCC). The expression rate of the WCC target genes adapts in daylight and remains refractory to moonlight, despite the extraordinary light sensitivity of the WCC. To explain this photoadaptation, feedback inhibition by the WCC interaction partner VIVID (VVD) has been invoked. Here we show through data-driven mathematical modeling that VVD allows Neurospora to detect relative changes in light intensity. To achieve this behavior, VVD acts as an inhibitor of WCC-driven gene expression and, at the same time, as a positive regulator that maintains the responsiveness of the photosystem. Our data indicate that this paradoxical function is realized by a futile cycle that involves the light-induced sequestration of active WCC by VVD and the replenishment of the activatable WCC pool through the decay of the photoactivated state. Our quantitative study uncovers a novel network motif for achieving sensory adaptation and defines a core input module of the circadian clock in Neurospora.

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.

Exploring the genetic control of glycolytic oscillations in Saccharomyces cerevisiae

BACKGROUND

A well known example of oscillatory phenomena is the transient oscillations of glycolytic intermediates in Saccharomyces cerevisiae, their regulation being predominantly investigated by mathematical modeling. To our knowledge there has not been a genetic approach to elucidate the regulatory role of the different enzymes of the glycolytic pathway.

RESULTS

We report that the laboratory strain BY4743 could also be used to investigate this oscillatory phenomenon, which traditionally has been studied using S. cerevisiae X2180. This has enabled us to employ existing isogenic deletion mutants and dissect the roles of isoforms, or subunits of key glycolytic enzymes in glycolytic oscillations. We demonstrate that deletion of TDH3 but not TDH2 and TDH1 (encoding glyceraldehyde-3-phosphate dehydrogenase: GAPDH) abolishes NADH oscillations. While deletion of each of the hexokinase (HK) encoding genes (HXK1 and HXK2) leads to oscillations that are longer lasting with lower amplitude, the effect of HXK2 deletion on the duration of the oscillations is stronger than that of HXK1. Most importantly our results show that the presence of beta (Pfk2) but not that of alpha subunits (Pfk1) of the hetero-octameric enzyme phosphofructokinase (PFK) is necessary to achieve these oscillations. Furthermore, we report that the cAMP-mediated PKA pathway (via some of its components responsible for feedback down-regulation) modulates the activity of glycoytic enzymes thus affecting oscillations. Deletion of both PDE2 (encoding a high affinity cAMP-phosphodiesterase) and IRA2 (encoding a GTPase activating protein- Ras-GAP, responsible for inactivating Ras-GTP) abolished glycolytic oscillations.

CONCLUSIONS

The genetic approach to characterising the glycolytic oscillations in yeast has demonstrated differential roles of the two types of subunits of PFK, and the isoforms of GAPDH and HK. Furthermore, it has shown that PDE2 and IRA2, encoding components of the cAMP pathway responsible for negative feedback regulation of PKA, are required for glycolytic oscillations, suggesting an enticing link between these cAMP pathway components and the glycolysis pathway enzymes shown to have the greatest role in glycolytic oscillation. This study suggests that a systematic genetic approach combined with mathematical modelling can advance the study of oscillatory phenomena.