Dependence of the period on the rate of protein degradation in minimal models for circadian oscillations

The Goodwin Oscillator and its Legacy

In the 1960's Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin's work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin's findings.

Computer-Aided Whole-Cell Design: Taking a Holistic Approach by Integrating Synthetic With Systems Biology

Computer-aided design (CAD) for synthetic biology promises to accelerate the rational and robust engineering of biological systems. It requires both detailed and quantitative mathematical and experimental models of the processes to (re)design biology, and software and tools for genetic engineering and DNA assembly. Ultimately, the increased precision in the design phase will have a dramatic impact on the production of designer cells and organisms with bespoke functions and increased modularity. CAD strategies require quantitative models of cells that can capture multiscale processes and link genotypes to phenotypes. Here, we present a perspective on how whole-cell, multiscale models could transform design-build-test-learn cycles in synthetic biology. We show how these models could significantly aid in the design and learn phases while reducing experimental testing by presenting case studies spanning from genome minimization to cell-free systems. We also discuss several challenges for the realization of our vision. The possibility to describe and build whole-cells in silico offers an opportunity to develop increasingly automatized, precise and accessible CAD tools and strategies.

Theoretical study on the regulation of circadian rhythms by RNA methylation

Messenger RNAs are often destabilized by methylation, suggesting that mRNA methylation alters mRNA and protein dynamics. This may indicate that the gene regulatory system is reflected by the metabolic system through mRNA methylation because methylation substrates are components of the metabolic system. Elucidating the mechanisms by which mRNA methylation regulates gene regulatory systems has posed considerable challenges due to the numerous targets of mRNA methylation. Recent studies have demonstrated that inhibition of mRNA N6-methyladenosine methylation elongates circadian periods. The aim of this study was to understand the mechanisms by which mRNA methylation regulates circadian rhythms. Using a detailed realistic model and a simple model, we demonstrated that period elongation of circadian rhythms by decreasing mRNA methylation can be achieved by two possibilities, i.e., decreasing mRNA methylation stabilizes nonoscillatory mRNAs such as Ck1δ and/or stabilizes oscillatory mRNAs of clock genes such as Per and Cry. In addition, we predicted that period elongation by stabilizing nonoscillatory mRNA (Ck1δ) should always be accompanied by the distortion of the circadian waveform. Finally, we discuss the validity of the two possible mechanisms on the regulation of circadian rhythms by mRNA methylation by quantifying waveform distortion of circadian gene activity data with or without mRNA methylation inhibitors.

A Robust Model for Circadian Redox Oscillations

The circadian clock is an endogenous oscillator that controls daily rhythms in metabolism, physiology, and behavior. Although the timekeeping components differ among species, a common design principle is a transcription-translation negative feedback loop. However, it is becoming clear that other mechanisms can contribute to the generation of 24 h rhythms. Peroxiredoxins (Prxs) exhibit 24 h rhythms in their redox state in all kingdoms of life. In mammalian adrenal gland, heart and brown adipose tissue, such rhythms are generated as a result of an inactivating hyperoxidation reaction that is reduced by coordinated import of sulfiredoxin (Srx) into the mitochondria. However, a quantitative description of the Prx/Srx oscillating system is still missing. We investigate the basic principles that generate mitochondrial Prx/Srx rhythms using computational modeling. We observe that the previously described delay in mitochondrial Srx import, in combination with an appropriate separation of fast and slow reactions, is sufficient to generate robust self-sustained relaxation-like oscillations. We find that our conceptual model can be regarded as a series of three consecutive phases and two temporal switches, highlighting the importance of delayed negative feedback and switches in the generation of oscillations.

Time-resolved interaction proteomics of the GIGANTEA protein under diurnal cycles in Arabidopsis

The plant-specific protein GIGANTEA (GI) controls many developmental and physiological processes, mediating rhythmic post-translational regulation. GI physically binds several proteins implicated in the circadian clock, photoperiodic flowering, and abiotic stress responses. To understand GI's multifaceted function, we aimed to comprehensively and quantitatively identify potential interactors of GI in a time-specific manner, using proteomics on Arabidopsis plants expressing epitope-tagged GI. We detected previously identified (in)direct interactors of GI, as well as proteins implicated in protein folding, or degradation, and a previously uncharacterized transcription factor, CYCLING DOF FACTOR6 (CDF6). We verified CDF6's direct interaction with GI, and ZEITLUPE/FLAVIN-BINDING, KELCH REPEAT, F-BOX 1/LIGHT KELCH PROTEIN 2 proteins, and demonstrated its involvement in photoperiodic flowering. Extending interaction proteomics to time series provides a data resource of candidate protein targets for GI's post-translational control.

Design Principles of Phosphorylation-Dependent Timekeeping in Eukaryotic Circadian Clocks

The circadian clock in cyanobacteria employs a posttranslational oscillator composed of a sequential phosphorylation-dephosphorylation cycle of KaiC protein, in which the dynamics of protein structural changes driven by temperature-compensated KaiC's ATPase activity are critical for determining the period. On the other hand, circadian clocks in eukaryotes employ transcriptional feedback loops as a core mechanism. In this system, the dynamics of protein accumulation and degradation affect the circadian period. However, recent studies of eukaryotic circadian clocks reveal that the mechanism controlling the circadian period can be independent of the regulation of protein abundance. Instead, the circadian substrate is often phosphorylated at multiple sites at flexible protein regions to induce structural changes. The phosphorylation is catalyzed by kinases that induce sequential multisite phosphorylation such as casein kinase 1 (CK1) with temperature-compensated activity. We propose that the design principles of phosphorylation-dependent circadian-period determination in eukaryotes may share characteristics with the posttranslational oscillator in cyanobacteria.

Rigid Cooperation of Per1 and Per2 proteins

Period circadian clock (Per) genes Per1 and Per2 have essential roles in circadian oscillation. In this study, we identified a new role of Per1-Per2 cooperation, and its mechanism, using our new experimental methods. Under constant light conditions, the period length of Per1 and Per2 knockout mice depended on the copy number ratio of Per1:Per2. We then established a light-emitting diode-based lighting system that can generate any pattern of light intensity. Under gradually changing light in the absence of phase shift with different periods, both Per1^(−/−) and Per2^(−/−) mice were entrained to a broader range of period length than wild-type mice. To analyse Per1-Per2 cooperative roles at the cell culture level, we established a Per2 knockout-rescue system, which can detect period shortening in a familial advanced sleep phase syndrome (FASPS) mutant. Upon introduction of the Per1 coding region in this system, we saw period shortening. In conclusion, short period-associated protein Per1 and long period-associated Per2 cooperated to rigidly confine the circadian period to “circa” 24-h. These results suggest that the rigid circadian rhythm maintained through the cooperation of Per1-Per2 could negatively impact modern society, in which the use of artificial lighting is ubiquitous, and result in circadian disorders, including delirium.

Modelling the dual role of Per phosphorylation and its effect on the period and phase of the mammalian circadian clock

Circadian clocks are regulated at the post-translational level by a variety of processes among which protein phosphorylation plays a prominent, although complex, role. Thus, the phosphorylation of different sites on the clock protein PER by casein kinase I (CKI) can lead to opposite effects on the stability of the protein and on the period of circadian oscillations. Here the authors extend a computational model previously proposed for the mammalian circadian clock by incorporating two distinct phosphorylations of PER by CKI. On the basis of experimental observations the authors consider that phosphorylation at one site (denoted here PER-P1) enhances the rate of degradation of the protein and decreases the period, while phosphorylation at another site (PER-P2) stabilises the protein, enhances the transcription of the Per gene, and increases the period. The model also incorporates an additional phosphorylation of PER by the Glycogen Synthase Kinase 3 (GSK3). The authors show that the extended model incorporating the antagonistic effects of PER phosphorylations by CKI can account for observations pertaining to (i) the decrease in period in the Tau mutant, because of an increase in phosphorylation by CKI leading to PER-P1, and (ii) the familial advanced sleep phase syndrome (FASPS) in which the period is shortened and the phase of the oscillations is advanced when the rate of phosphorylation leading to PER-P2 is decreased. The model further accounts for the increase in period observed in the presence of CKI inhibitors that decrease the rate of phosphorylation leading to both PER-P1 and PER-P2. A similar increase in period results from inhibition of GSK3. [Includes supplementary material].

Robustness and period sensitivity analysis of minimal models for biochemical oscillators

Biological systems exhibit numerous oscillatory behaviors from calcium oscillations to circadian rhythms that recur daily. These autonomous oscillators contain complex feedbacks with nonlinear dynamics that enable spontaneous oscillations. The detailed nonlinear dynamics of such systems remains largely unknown. In this paper, we investigate robustness and dynamical differences of five minimal systems that may underlie fundamental molecular processes in biological oscillatory systems. Bifurcation analyses of these five models demonstrate an increase of oscillatory domains with a positive feedback mechanism that incorporates a reversible reaction, and dramatic changes in dynamics with small modifications in the wiring. Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology. In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type. We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.

Effects of different per translational kinetics on the dynamics of a core circadian clock model

Living beings display self-sustained daily rhythms in multiple biological processes, which persist in the absence of external cues since they are generated by endogenous circadian clocks. The period (per) gene is a central player within the core molecular mechanism for keeping circadian time in most animals. Recently, the modulation PER translation has been reported, both in mammals and flies, suggesting that translational regulation of clock components is important for the proper clock gene expression and molecular clock performance. Because translational regulation ultimately implies changes in the kinetics of translation and, therefore, in the circadian clock dynamics, we sought to study how and to what extent the molecular clock dynamics is affected by the kinetics of PER translation. With this objective, we used a minimal mathematical model of the molecular circadian clock to qualitatively characterize the dynamical changes derived from kinetically different PER translational mechanisms. We found that the emergence of self-sustained oscillations with characteristic period, amplitude, and phase lag (time delays) between per mRNA and protein expression depends on the kinetic parameters related to PER translation. Interestingly, under certain conditions, a PER translation mechanism with saturable kinetics introduces longer time delays than a mechanism ruled by a first-order kinetics. In addition, the kinetic laws of PER translation significantly changed the sensitivity of our model to parameters related to the synthesis and degradation of per mRNA and PER degradation. Lastly, we found a set of parameters, with realistic values, for which our model reproduces some experimental results reported recently for Drosophila melanogaster and we present some predictions derived from our analysis.

The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

Splitting strategy for simulating genetic regulatory networks

The splitting approach is developed for the numerical simulation of genetic regulatory networks with a stable steady-state structure. The numerical results of the simulation of a one-gene network, a two-gene network, and a p53-mdm2 network show that the new splitting methods constructed in this paper are remarkably more effective and more suitable for long-term computation with large steps than the traditional general-purpose Runge-Kutta methods. The new methods have no restriction on the choice of stepsize due to their infinitely large stability regions.

Oscillatory enzyme reactions and Michaelis-Menten kinetics

Oscillations occur in a number of enzymatic systems as a result of feedback regulation. How Michaelis-Menten kinetics influences oscillatory behavior in enzyme systems is investigated in models for oscillations in the activity of phosphofructokinase (PFK) in glycolysis and of cyclin-dependent kinases in the cell cycle. The model for the PFK reaction is based on a product-activated allosteric enzyme reaction coupled to enzymatic degradation of the reaction product. The Michaelian nature of the product decay term markedly influences the period, amplitude and waveform of the oscillations. Likewise, a model for oscillations of Cdc2 kinase in embryonic cell cycles based on Michaelis-Menten phosphorylation-dephosphorylation kinetics shows that the occurrence and amplitude of the oscillations strongly depend on the ultrasensitivity of the enzymatic cascade that controls the activity of the cyclin-dependent kinase.

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.

Roles of protein ubiquitination and degradation kinetics in biological oscillations

Protein ubiquitination and degradation play important roles in many biological functions and are associated with many human diseases. It is well known that for biochemical oscillations to occur, proper degradation rates of the participating proteins are needed. In most mathematical models of biochemical reactions, linear degradation kinetics has been used. However, the degradation kinetics in real systems may be nonlinear, and how nonlinear degradation kinetics affects biological oscillations are not well understood. In this study, we first develop a biochemical reaction model of protein ubiquitination and degradation and calculate the degradation rate against the concentration of the free substrate. We show that the protein degradation kinetics mainly follows the Michaelis-Menten formulation with a time delay caused by ubiquitination and deubiquitination. We then study analytically how the Michaelis-Menten degradation kinetics affects the instabilities that lead to oscillations using three generic oscillation models: 1) a positive feedback mediated oscillator; 2) a positive-plus-negative feedback mediated oscillator; and 3) a negative feedback mediated oscillator. In all three cases, nonlinear degradation kinetics promotes oscillations, especially for the negative feedback mediated oscillator, resulting in much larger oscillation amplitudes and slower frequencies than those observed with linear kinetics. However, the time delay due to protein ubiquitination and deubiquitination generally suppresses oscillations, reducing the amplitude and increasing the frequency of the oscillations. These theoretical analyses provide mechanistic insights into the effects of specific proteins in the ubiquitination-proteasome system on biological oscillations.

Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops

Feedback controls are central to cellular regulation. Negative-feedback mechanisms are well known to underline oscillatory dynamics. However, the presence of multiple negative-feedback mechanisms is common in oscillatory cellular systems, raising intriguing questions of how they cooperate to regulate oscillations. In this work, we studied the dynamical properties of a set of general biochemical motifs with dual, nested negative-feedback structures. We showed analytically and then confirmed numerically that, in these motifs, each negative-feedback loop exhibits distinctly different oscillation-controlling functions. The longer, outer feedback loop was found to promote oscillations, whereas the short, inner loop suppresses and can even eliminate oscillations. We found that the position of the inner loop within the coupled motifs affects its repression strength towards oscillatory dynamics. Bifurcation analysis indicated that emergence of oscillations may be a strict parametric requirement and thus evolutionarily tricky. Investigation of the quantitative features of oscillations (i.e. frequency, amplitude and mean value) revealed that coupling negative feedback provides robust tuning of the oscillation dynamics. Finally, we demonstrated that the mitogen-activated protein kinase (MAPK) cascades also display properties seen in the general nested feedback motifs. The findings and implications in this study provide novel understanding of biochemical negative-feedback regulation in a mixed wiring context.

Tuning the mammalian circadian clock: robust synergy of two loops

The circadian clock is accountable for the regulation of internal rhythms in most living organisms. It allows the anticipation of environmental changes during the day and a better adaptation of physiological processes. In mammals the main clock is located in the suprachiasmatic nucleus (SCN) and synchronizes secondary clocks throughout the body. Its molecular constituents form an intracellular network which dictates circadian time and regulates clock-controlled genes. These clock-controlled genes are involved in crucial biological processes including metabolism and cell cycle regulation. Its malfunction can lead to disruption of biological rhythms and cause severe damage to the organism. The detailed mechanisms that govern the circadian system are not yet completely understood. Mathematical models can be of great help to exploit the mechanism of the circadian circuitry. We built a mathematical model for the core clock system using available data on phases and amplitudes of clock components obtained from an extensive literature search. This model was used to answer complex questions for example: how does the degradation rate of Per affect the period of the system and what is the role of the ROR/Bmal/REV-ERB (RBR) loop? Our findings indicate that an increase in the RNA degradation rate of the clock gene Period (Per) can contribute to increase or decrease of the period--a consequence of a non-monotonic effect of Per transcript stability on the circadian period identified by our model. Furthermore, we provide theoretical evidence for a potential role of the RBR loop as an independent oscillator. We carried out overexpression experiments on members of the RBR loop which lead to loss of oscillations consistent with our predictions. These findings challenge the role of the RBR loop as a merely auxiliary loop and might change our view of the clock molecular circuitry and of the function of the nuclear receptors (REV-ERB and ROR) as a putative driving force of molecular oscillations.

Modeling circadian clocks: From equations to oscillations

Circadian rhythms are generated at the cellular level by a small but tightly regulated genetic network. In higher eukaryotes, interlocked transcriptional-translational feedback loops form the core of this network, which ensures the activation of the right genes (proteins) at the right time of the day. Understanding how such a complex molecular network can generate robust, self-sustained oscillations and accurately responds to signals from the environment (such as light and temperature) is greatly helped by mathematical modeling. In the present paper we review some mathematical models for circadian clocks, ranging from abstract, phenomenological models to the most detailed molecular models. We explain how the equations are derived, highlighting the challenges for the modelers, and how the models are analyzed. We show how to compute bifurcation diagrams, entrainment, and phase response curves. In the subsequent paper, we discuss, through a selection of examples, how modeling efforts have contributed to a better understanding of the dynamics of the circadian regulatory network.

Modeling oscillatory control in NF-κB, p53 and Wnt signaling

Oscillations are commonly observed in cellular behavior and span a wide range of timescales, from seconds in calcium signaling to 24 hours in circadian rhythms. In between lie oscillations with time periods of 1-5 hours seen in NF-κB, p53 and Wnt signaling, which play key roles in the immune system, cell growth/death and embryo development, respectively. In the first part of this article, we provide a brief overview of simple deterministic models of oscillations. In particular, we explain the mechanism of saturated degradation that has been used to model oscillations in the NF-κB, p53 and Wnt systems. The second part deals with the potential physiological role of oscillations. We use the simple models described earlier to explore whether oscillatory signals can encode more information than steady-state signals. We then discuss a few simple genetic circuits that could decode information stored in the average, amplitude or frequency of oscillations. The presence of frequency-detector circuit downstream of NF-κB or p53 would be a strong clue that oscillations are important for the physiological response of these signaling systems.