A design principle for a posttranslational biochemical oscillator

Synthesis of causal and surrogate models by non-equilibrium thermodynamics in biological systems

We developed a model to represent the time evolution phenomena of life through physics constraints. To do this, we took into account that living organisms are open systems that exchange messages through intracellular communication, intercellular communication and sensory systems, and introduced the concept of a message force field. As a result, we showed that the maximum entropy generation principle is valid in time evolution. Then, in order to explain life phenomena based on this principle, we modelled the living system as a nonlinear oscillator coupled by a message and derived the governing equations. The governing equations consist of two laws: one states that the systems are synchronized when the variation of the natural frequencies between them is small or the coupling strength through the message is sufficiently large, and the other states that the synchronization is broken by the proliferation of biological systems. Next, to simulate the phenomena using data obtained from observations of the temporal evolution of life, we developed an inference model that combines physics constraints and a discrete surrogate model using category theory, and simulated the phenomenon of early embryogenesis using this inference model. The results show that symmetry creation and breaking based on message force fields can be widely used to model life phenomena.

Network regulation meets substrate modification chemistry

Biochemical networks are at the heart of cellular information processing. These networks contain distinct facets: (i) processing of information from the environment via cascades/pathways along with network regulation and (ii) modification of substrates in different ways, to confer protein functionality, stability and processing. While many studies focus on these factors individually, how they interact and the consequences for cellular systems behaviour are poorly understood. We develop a systems framework for this purpose by examining the interplay of network regulation (canonical feedback and feed-forward circuits) and multisite modification, as an exemplar of substrate modification. Using computational, analytical and semi-analytical approaches, we reveal distinct and unexpected ways in which the substrate modification and network levels combine and the emergent behaviour arising therefrom. This has important consequences for dissecting the behaviour of specific signalling networks, tracing the origins of systems behaviour, inference of networks from data, robustness/evolvability and multi-level engineering of biomolecular networks. Overall, we repeatedly demonstrate how focusing on only one level (say network regulation) can lead to profoundly misleading conclusions about all these aspects, and reveal a number of important consequences for experimental/theoretical/data-driven interrogations of cellular signalling systems.

Circadian clock genes as promising therapeutic targets for bone loss

The circadian clock regulates many key physiological processes such as the sleep-wake cycle, hormone release, cardiovascular health, glucose metabolism and body temperature. Recent evidence has suggested a critical role of the circadian system in controlling bone metabolism. Here we review the connection between bone metabolism and the biological clock, and the roles of these mechanisms in bone loss. We also analyze the regulatory effects of clock-related genes on signaling pathways and transcription factors in osteoblasts and osteoclasts. Additionally, osteocytes and endothelial cells (ECs) regulated by the circadian clock are also discussed in our review. Furthermore, we also summarize the regulation of circadian clock genes by some novel modulators, which provides us with a new insight into a potential strategy to prevent and treat bone diseases such as osteoporosis by targeting circadian genes.

Dynamics and Sensitivity of Signaling Pathways

Purpose of Review

Signaling pathways serve to communicate information about extracellular conditions into the cell, to both the nucleus and cytoplasmic processes to control cell responses. Genetic mutations in signaling network components are frequently associated with cancer and can result in cells acquiring an ability to divide and grow uncontrollably. Because signaling pathways play such a significant role in cancer initiation and advancement, their constituent proteins are attractive therapeutic targets. In this review, we discuss how signaling pathway modeling can assist with identifying effective drugs for treating diseases, such as cancer. An achievement that would facilitate the use of such models is their ability to identify controlling biochemical parameters in signaling pathways, such as molecular abundances and chemical reaction rates, because this would help determine effective points of attack by therapeutics.

Recent Findings

We summarize the current state of understanding the sensitivity of phosphorylation cycles with and without sequestration. We also describe some basic properties of regulatory motifs including feedback and feedforward regulation.

Summary

Although much recent work has focused on understanding the dynamics and particularly the sensitivity of signaling networks in eukaryotic systems, there is still an urgent need to build more scalable models of signaling networks that can appropriately represent their complexity across different cell types and tumors.

Elucidation of master allostery essential for circadian clock oscillation in cyanobacteria

Spatiotemporal allostery is the source of complex but ordered biological phenomena. To identify the structural basis for allostery that drives the cyanobacterial circadian clock, we crystallized the clock protein KaiC in four distinct states, which cover a whole cycle of phosphor-transfer events at Ser⁴³¹ and Thr⁴³². The minimal set of allosteric events required for oscillatory nature is a bidirectional coupling between the coil-to-helix transition of the Ser⁴³¹-dependent phospho-switch in the C-terminal domain of KaiC and adenosine 5'-diphosphate release from its N-terminal domain during adenosine triphosphatase cycle. An engineered KaiC protein oscillator consisting of a minimal set of the identified master allosteric events exhibited a monophosphorylation cycle of Ser⁴³¹ with a temperature-compensated circadian period, providing design principles for simple posttranslational biochemical circadian oscillators.

A design principle of spindle oscillations in mammalian sleep

Neural oscillations are mainly regulated by molecular mechanisms and network connectivity of neurons. Large-scale simulations of neuronal networks have driven the population-level understanding of neural oscillations. However, cell-intrinsic mechanisms, especially a design principle, of neural oscillations remain largely elusive. Herein, we developed a minimal, Hodgkin-Huxley-type model of groups of neurons to investigate molecular mechanisms underlying spindle oscillation, which is synchronized oscillatory activity predominantly observed during mammalian sleep. We discovered that slowly inactivating potassium channels played an essential role in characterizing the firing pattern. The detailed analysis of the minimal model revealed that leak sodium and potassium channels, which controlled passive properties of the fast variable (i.e., membrane potential), competitively regulated the base value and time constant of the slow variable (i.e., cytosolic calcium concentration). Consequently, we propose a theoretical design principle of spindle oscillations that may explain intracellular mechanisms behind the flexible control over oscillation density and calcium setpoint.

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A minimal, Hodgkin-Huxley-type model of spindle oscillations is developed

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The property of delayed rectifier K⁺ channels characterizes spindle oscillations

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The combination of bifurcations specifies spindle oscillations

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Spindle oscillations are controlled by the balance of inward and outward currents

Chemical engines: driving systems away from equilibrium through catalyst reaction cycles

Biological systems exhibit a range of complex functions at the micro- and nanoscales under non-equilibrium conditions (for example, transportation and motility, temporal control, information processing and so on). Chemists also employ out-of-equilibrium systems, for example in kinetic selection during catalysis, self-replication, dissipative self-assembly and synthetic molecular machinery, and in the form of chemical oscillators. Key to non-equilibrium behaviour are the mechanisms through which systems are able to extract energy from the chemical reactants ('fuel') that drive such processes. In this Perspective we relate different examples of such powering mechanisms using a common conceptual framework. We discuss how reaction cycles can be coupled to other dynamic processes through positive (acceleration) or negative (inhibition) catalysis to provide the thermodynamic impetus for diverse non-equilibrium behaviour, in effect acting as a 'chemical engine'. We explore the way in which the energy released from reaction cycles is harnessed through kinetic selection in a series of what have sometimes been considered somewhat disparate fields (systems chemistry, molecular machinery, dissipative assembly and chemical oscillators), highlight common mechanistic principles and the potential for the synchronization of chemical reaction cycles, and identify future challenges for the invention and application of non-equilibrium systems. Explicit recognition of the use of fuelling reactions to power structural change in catalysts may stimulate the investigation of known catalytic cycles as potential elements for chemical engines, a currently unexplored area of catalysis research.

Symmetry breaking meets multisite modification

Multisite modification is a basic way of conferring functionality to proteins and a key component of post-translational modification networks. Additional interest in multisite modification stems from its capability of acting as complex information processors. In this paper, we connect two seemingly disparate themes: symmetry and multisite modification. We examine different classes of random modification networks of substrates involving separate or common enzymes. We demonstrate that under different instances of symmetry of the modification network (invoked explicitly or implicitly and discussed in the literature), the biochemistry of multisite modification can lead to the symmetry being broken. This is shown computationally and consolidated analytically, revealing parameter regions where this can (and in fact does) happen, and characteristics of the symmetry-broken state. We discuss the relevance of these results in situations where exact symmetry is not present. Overall, through our study we show how symmetry breaking (i) can confer new capabilities to protein networks, including concentration robustness of different combinations of species (in conjunction with multiple steady states); (ii) could have been the basis for ordering of multisite modification, which is widely observed in cells; (iii) can significantly impact information processing in multisite modification and in cell signalling networks/pathways where multisite modification is present; and (iv) can be a fruitful new angle for engineering in synthetic biology and chemistry. All in all, the emerging conceptual synthesis provides a new vantage point for the elucidation and the engineering of molecular systems at the junction of chemical and biological systems.

Proteins help our cells perform the chemical reactions necessary for life. Once proteins are made, they can also be modified in different ways. This can simply change their activity, or otherwise make them better suited for their specific jobs within the cell. Biological ‘catalysts’ called enzymes carry out protein modifications by reversibly adding (or removing) chemical groups, such as phosphate groups.

‘Multisite modifications’ occur when a protein has two or more modifications in different areas, which can be added randomly or in a specific sequence. The combination of all the modifications attached to a protein acts like a chemical barcode and confers a specific function to the protein.

Modification networks add levels of complexity above individual proteins. These encompass not only the proteins in a cell or tissue, but also the different enzymes that can modify them, and how they all interact with each other. Although our knowledge of these networks is substantial, basic aspects, such as how the ordering of multisite modification systems emerges, is still not well understood.

Using a simple set of multisite modifications, Ramesh and Krishnan set out to study the potential mechanisms allowing the creation of order in this context. Symmetry is a pervasive theme across the sciences. In biology, symmetry and how it may be broken, is important to understand, for example, how organism develop. Ramesh and Krishnan used the perspective of symmetry in protein networks to uncover the origins of ordering.

First, mathematical models of simple modification networks were created based on their basic descriptions. This system centred on proteins that could have phosphate modifications at two possible sites. The network was ‘symmetric’, meaning that the rate of different sets of chemical reactions was identical, as were the amounts of all the enzymes involved.

Dissecting the simulated network using a variety of mathematical approaches showed that its initial symmetry could break, giving rise to sets of ordered multisite modifications. Breaking symmetry did not require any additional features or factors; the basic chemical ‘ingredients’ of protein modification were all that was needed. The prism of symmetry also revealed other aspects of these multisite modification networks, such as robustness and oscillations.

This study sheds new light on the mechanism behind ordering of protein modifications. In the future, Ramesh and Krishnan hope that this approach can be applied to the study of not just proteins but also a wider range of biochemical networks.

CRYPTOCHROMES confer robustness, not rhythmicity, to circadian timekeeping

Circadian rhythms are a pervasive property of mammalian cells, tissues and behaviour, ensuring physiological adaptation to solar time. Models of cellular timekeeping revolve around transcriptional feedback repression, whereby CLOCK and BMAL1 activate the expression of PERIOD (PER) and CRYPTOCHROME (CRY), which in turn repress CLOCK/BMAL1 activity. CRY proteins are therefore considered essential components of the cellular clock mechanism, supported by behavioural arrhythmicity of CRY-deficient (CKO) mice under constant conditions. Challenging this interpretation, we find locomotor rhythms in adult CKO mice under specific environmental conditions and circadian rhythms in cellular PER2 levels when CRY is absent. CRY-less oscillations are variable in their expression and have shorter periods than wild-type controls. Importantly, we find classic circadian hallmarks such as temperature compensation and period determination by CK1δ/ε activity to be maintained. In the absence of CRY-mediated feedback repression and rhythmic Per2 transcription, PER2 protein rhythms are sustained for several cycles, accompanied by circadian variation in protein stability. We suggest that, whereas circadian transcriptional feedback imparts robustness and functionality onto biological clocks, the core timekeeping mechanism is post-translational.

A design principle for posttranslational chaotic oscillators

Chaos behavior has been observed in various cellular and molecular processes. Here, we modeled reversible phosphorylation dynamics to elucidate a design principle for autonomous chaos generation that may arise from generic enzymatic reactions. A comprehensive parameter search demonstrated that the reaction system composed of a set of kinases and phosphatases and two substrates with two modification sites exhibits chaos behavior. All reactions are described according to the Michaelis-Menten reaction scheme without exotic functions being applied to enzymes and substrates. Clustering analysis of parameter sets that can generate chaos behavior revealed the existence of motif structures. These chaos motifs allow the two-substrate species to interact via enzyme availability and constrain the two substrates' dynamic changes in phosphorylation status so that they occur at different timescales. This chaos motif structure is found in several enzymatic reactions, suggesting that chaos behavior may underlie cellular autonomy in a variety of biochemical systems.

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Two substrates with reversible two-site phosphorylation can exhibit chaos behavior

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The chaos does not require autocatalysis or allosteric regulation of enzymes

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The chaos is a result of the coupling of two substrates via enzyme availability

Simple Kinetic Models in Molecular Chronobiology

Circadian rhythms are constituted by a complex dynamical system with intertwined feedback loops, molecular switches, and self-sustained oscillations. Mathematical modeling supports understanding available heterogeneous kinetic data, highlights basic mechanisms, and can guide experimental research. Here, we introduce the basic steps from a biological question to simple models providing insight into gene-regulatory mechanisms. We illustrate the general approach by three examples: modeling decay processes, clock-controlled genes, and self-sustained oscillations.

Self-assembly-based posttranslational protein oscillators

Recent advances in synthetic posttranslational protein circuits are substantially impacting the landscape of cellular engineering and offer several advantages compared to traditional gene circuits. However, engineering dynamic phenomena such as oscillations in protein-level circuits remains an outstanding challenge. Few examples of biological posttranslational oscillators are known, necessitating theoretical progress to determine realizable oscillators. We construct mathematical models for two posttranslational oscillators, using few components that interact only through reversible binding and phosphorylation/dephosphorylation reactions. Our designed oscillators rely on the self-assembly of two protein species into multimeric functional enzymes that respectively inhibit and enhance this self-assembly. We limit our analysis to within experimental constraints, finding (i) significant portions of the restricted parameter space yielding oscillations and (ii) that oscillation periods can be tuned by several orders of magnitude using recent advances in computational protein design. Our work paves the way for the rational design and realization of protein-based dynamic systems.

Exploring cyclic networks of multisite modification reveals origins of information processing characteristics

Multisite phosphorylation (and generally multisite modification) is a basic way of encoding substrate function and circuits/networks of post-translational modifications (PTM) are ubiquitous in cell signalling. The information processing characteristics of PTM systems are a focal point of broad interest. The ordering of modifications is a key aspect of multisite modification, and a broad synthesis of the impact of ordering of modifications is still missing. We focus on a basic class of multisite modification circuits: the cyclic mechanism, which corresponds to the same ordering of phosphorylation and dephosphorylation, and examine multiple variants involving common/separate kinases and common/separate phosphatases. This is of interest both because it is encountered in concrete cellular contexts, and because it serves as a bridge between ordered (sequential) mechanisms (representing one type of ordering) and random mechanisms (which have no ordering). We show that bistability and biphasic dose response curves of the maximally modified phosphoform are ruled out for basic structural reasons independent of parameters, while oscillations can result with even just one shared enzyme. We then examine the effect of relaxing some basic assumptions about the ordering of modification. We show computationally and analytically how bistability, biphasic responses and oscillations can be generated by minimal augmentations to the cyclic mechanism even when these augmentations involved reactions operating in the unsaturated limit. All in all, using this approach we demonstrate (1) how the cyclic mechanism (with single augmentations) represents a modification circuit using minimal ingredients (in terms of shared enzymes and sequestration of enzymes) to generate bistability and oscillations, when compared to other mechanisms, (2) new design principles for rationally designing PTM systems for a variety of behaviour, (3) a basis and a necessary step for understanding the origins and robustness of behaviour observed in basic multisite modification systems.

Asymmetry between Activators and Deactivators in Functional Protein Networks

Are “turn-on” and “turn-off” functions in protein-protein interaction networks exact opposites of each other? To answer this question, we implement a minimal model for the evolution of functional protein-interaction networks using a sequence-based mutational algorithm, and apply the model to study neutral drift in networks that yield oscillatory dynamics. We study the roles of activators and deactivators, two core components of oscillatory protein interaction networks, and find a striking asymmetry in the roles of activating and deactivating proteins, where activating proteins tend to be synergistic and deactivating proteins tend to be competitive.

Bayesian modeling reveals metabolite-dependent ultrasensitivity in the cyanobacterial circadian clock

Mathematical models can enable a predictive understanding of mechanism in cell biology by quantitatively describing complex networks of interactions, but such models are often poorly constrained by available data. Owing to its relative biochemical simplicity, the core circadian oscillator in Synechococcus elongatus has become a prototypical system for studying how collective dynamics emerge from molecular interactions. The oscillator consists of only three proteins, KaiA, KaiB, and KaiC, and near-24-h cycles of KaiC phosphorylation can be reconstituted in vitro. Here, we formulate a molecularly detailed but mechanistically naive model of the KaiA-KaiC subsystem and fit it directly to experimental data within a Bayesian parameter estimation framework. Analysis of the fits consistently reveals an ultrasensitive response for KaiC phosphorylation as a function of KaiA concentration, which we confirm experimentally. This ultrasensitivity primarily results from the differential affinity of KaiA for competing nucleotide-bound states of KaiC. We argue that the ultrasensitive stimulus-response relation likely plays an important role in metabolic compensation by suppressing premature phosphorylation at nighttime.

Exploring the intrinsic behaviour of multisite phosphorylation systems as part of signalling pathways

Multisite phosphorylation is a basic way of chemically encoding substrate function and a recurring feature of cell signalling pathways. A number of studies have explored information processing characteristics of multisite phosphorylation, through studies of the intrinsic kinetics. Many of these studies focus on the module in isolation. In this paper, we build a bridge to connect the behaviour of multisite modification in isolation to that as part of pathways. We study the effect of activation of the enzymes (which are basic ways in which the module may be regulated), as well the effects of the modified substrates being involved in further modifications or exiting reaction compartments. We find that these effects can induce multiple kinds of transitions, including to behaviour not seen intrinsically in the multisite modification module. We then build on these insights to investigate how these multisite modification systems can be tuned by enzyme activation to realize a range of information processing outcomes for the design of synthetic phosphorylation circuits. Connecting the complexity of multisite modification kinetics, with the pathways in which they are embedded, serves as a basis for teasing out many aspects of their interaction, providing insights of relevance in systems biology, synthetic biology/chemistry and chemical information processing.

Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions

Posttranslational modification of proteins is important for signal transduction, and hence significant effort has gone toward understanding how posttranslational modification networks process information. This involves, on the theory side, analyzing the dynamical systems arising from such networks. Which networks are, for instance, bistable? Which networks admit sustained oscillations? Which parameter values enable such behaviors? In this Biophysical Perspective, we highlight recent progress in this area and point out some important future directions. Along the way, we summarize several techniques for analyzing general networks, such as eliminating variables to obtain steady-state parameterizations, and harnessing results on how incorporating intermediates affects dynamics.

Computational modelling unravels the precise clockwork of cyanobacteria

Precisely timing the regulation of gene expression by anticipating recurring environmental changes is a fundamental part of global gene regulation. Circadian clocks are one form of this regulation, which is found in both eukaryotes and prokaryotes, providing a fitness advantage for these organisms. Whereas many different eukaryotic groups harbour circadian clocks, cyanobacteria are the only known oxygenic phototrophic prokaryotes to regulate large parts of their genes in a circadian fashion. A decade of intensive research on the mechanisms and functionality using computational and mathematical approaches in addition to the detailed biochemical and biophysical understanding make this the best understood circadian clock. Here, we summarize the findings and insights into various parts of the cyanobacterial circadian clock made by mathematical modelling. These findings have implications for eukaryotic circadian research as well as synthetic biology harnessing the power and efficiency of global gene regulation.

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.

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.

The Design Space of the Embryonic Cell Cycle Oscillator

One of the main tasks in the analysis of models of biomolecular networks is to characterize the domain of the parameter space that corresponds to a specific behavior. Given the large number of parameters in most models, this is no trivial task. We use a model of the embryonic cell cycle to illustrate the approaches that can be used to characterize the domain of parameter space corresponding to limit cycle oscillations, a regime that coordinates periodic entry into and exit from mitosis. Our approach relies on geometric construction of bifurcation sets, numerical continuation, and random sampling of parameters. We delineate the multidimensional oscillatory domain and use it to quantify the robustness of periodic trajectories. Although some of our techniques explore the specific features of the chosen system, the general approach can be extended to other models of the cell cycle engine and other biomolecular networks.

Systems Chronotherapeutics

Chronotherapeutics aim at treating illnesses according to the endogenous biologic rhythms, which moderate xenobiotic metabolism and cellular drug response. The molecular clocks present in individual cells involve approximately fifteen clock genes interconnected in regulatory feedback loops. They are coordinated by the suprachiasmatic nuclei, a hypothalamic pacemaker, which also adjusts the circadian rhythms to environmental cycles. As a result, many mechanisms of diseases and drug effects are controlled by the circadian timing system. Thus, the tolerability of nearly 500 medications varies by up to fivefold according to circadian scheduling, both in experimental models and/or patients. Moreover, treatment itself disrupted, maintained, or improved the circadian timing system as a function of drug timing. Improved patient outcomes on circadian-based treatments (chronotherapy) have been demonstrated in randomized clinical trials, especially for cancer and inflammatory diseases. However, recent technological advances have highlighted large interpatient differences in circadian functions resulting in significant variability in chronotherapy response. Such findings advocate for the advancement of personalized chronotherapeutics through interdisciplinary systems approaches. Thus, the combination of mathematical, statistical, technological, experimental, and clinical expertise is now shaping the development of dedicated devices and diagnostic and delivery algorithms enabling treatment individualization. In particular, multiscale systems chronopharmacology approaches currently combine mathematical modeling based on cellular and whole-body physiology to preclinical and clinical investigations toward the design of patient-tailored chronotherapies. We review recent systems research works aiming to the individualization of disease treatment, with emphasis on both cancer management and circadian timing system–resetting strategies for improving chronic disease control and patient outcomes.

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.

Feedback Loops of the Mammalian Circadian Clock Constitute Repressilator

Mammals evolved an endogenous timing system to coordinate their physiology and behaviour to the 24h period of the solar day. While it is well accepted that circadian rhythms are generated by intracellular transcriptional feedback loops, it is still debated which network motifs are necessary and sufficient for generating self-sustained oscillations. Here, we systematically explore a data-based circadian oscillator model with multiple negative and positive feedback loops and identify a series of three subsequent inhibitions known as "repressilator" as a core element of the mammalian circadian oscillator. The central role of the repressilator motif is consistent with time-resolved ChIP-seq experiments of circadian clock transcription factors and loss of rhythmicity in core clock gene knockouts.

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.

Long-term dynamics of multisite phosphorylation

A systematic framework for exploring the long-term dynamics of a reaction network is applied to a minimal model of ERK regulation that distinguishes both monophosphorylated forms and allows for nonzero enzyme processivity. Bistability and oscillations can be observed at high levels of processivity.

Multisite phosphorylation cycles are ubiquitous in cell regulation systems and are studied at multiple levels of complexity, from molecules to organisms, with the ultimate goal of establishing predictive understanding of the effects of genetic and pharmacological perturbations of protein phosphorylation in vivo. Achieving this goal is essentially impossible without mathematical models, which provide a systematic framework for exploring dynamic interactions of multiple network components. Most of the models studied to date do not discriminate between the distinct partially phosphorylated forms and focus on two limiting reaction regimes, distributive and processive, which differ in the number of enzyme–substrate binding events needed for complete phosphorylation or dephosphorylation. Here we use a minimal model of extracellular signal-related kinase regulation to explore the dynamics of a reaction network that includes all essential phosphorylation forms and arbitrary levels of reaction processivity. In addition to bistability, which has been studied extensively in distributive mechanisms, this network can generate periodic oscillations. Both bistability and oscillations can be realized at high levels of reaction processivity. Our work provides a general framework for systematic analysis of dynamics in multisite phosphorylation systems.

Using computational models to discover and understand mechanisms

Areas of biology such as cell and molecular biology have been dominated by research directed at constructing mechanistic explanations that identify parts and operations that when organized appropriately are responsible for the various phenomena they investigate. Increasingly the mechanisms hypothesized involve non-sequential organization of non-linear operations and so exceed the ability of researchers to mentally rehearse their behavior. Accordingly, scientists rely on tools of computational modeling and dynamical systems theory in advancing dynamic mechanistic explanations. Using circadian rhythm research as an exemplar, this paper explores the variety of roles computational modeling is playing. They serve not just to determine whether the mechanism will produce the desired behavior, but in the discovery process of hypothesizing mechanisms and in understanding why proposed mechanisms behave as they do.

An Analytical Framework for Studying Small-Number Effects in Catalytic Reaction Networks: A Probability Generating Function Approach to Chemical Master Equations

Cell activities primarily depend on chemical reactions, especially those mediated by enzymes, and this has led to these activities being modeled as catalytic reaction networks. Although deterministic ordinary differential equations of concentrations (rate equations) have been widely used for modeling purposes in the field of systems biology, it has been pointed out that these catalytic reaction networks may behave in a way that is qualitatively different from such deterministic representation when the number of molecules for certain chemical species in the system is small. Apart from this, representing these phenomena by simple binary (on/off) systems that omit the quantities would also not be feasible. As recent experiments have revealed the existence of rare chemical species in cells, the importance of being able to model potential small-number phenomena is being recognized. However, most preceding studies were based on numerical simulations, and theoretical frameworks to analyze these phenomena have not been sufficiently developed. Motivated by the small-number issue, this work aimed to develop an analytical framework for the chemical master equation describing the distributional behavior of catalytic reaction networks. For simplicity, we considered networks consisting of two-body catalytic reactions. We used the probability generating function method to obtain the steady-state solutions of the chemical master equation without specifying the parameters. We obtained the time evolution equations of the first- and second-order moments of concentrations, and the steady-state analytical solution of the chemical master equation under certain conditions. These results led to the rank conservation law, the connecting state to the winner-takes-all state, and analysis of 2-molecules M-species systems. A possible interpretation of the theoretical conclusion for actual biochemical pathways is also discussed.

Mixed mechanisms of multi-site phosphorylation

Multi-site phosphorylation is ubiquitous in cell biology and has been widely studied experimentally and theoretically. The underlying chemical modification mechanisms are typically assumed to be distributive or processive. In this paper, we study the behaviour of mixed mechanisms that can arise either because phosphorylation and dephosphorylation involve different mechanisms or because phosphorylation and/or dephosphorylation can occur through a combination of mechanisms. We examine a hierarchy of models to assess chemical information processing through different mixed mechanisms, using simulations, bifurcation analysis and analytical work. We demonstrate how mixed mechanisms can show important and unintuitive differences from pure distributive and processive mechanisms, in some cases resulting in monostable behaviour with simple dose-response behaviour, while in other cases generating new behaviour-like oscillations. Our results also suggest patterns of information processing that are relevant as the number of modification sites increases. Overall, our work creates a framework to examine information processing arising from complexities of multi-site modification mechanisms and their impact on signal transduction.

PER2 Differentially Regulates Clock Phosphorylation versus Transcription by Reciprocal Switching of CK1ε Activity

Casein kinase 1ε (CK1ε) performs key phosphorylation reactions in the circadian clock mechanism that determine period. We show that the central clock protein PERIOD2 (PER2) not only acts as a transcriptional repressor but also inhibits the autoinactivation of CK1ε, thereby promoting CK1ε activity. Moreover, PER2 reciprocally regulates CK1ε's ability to phosphorylate other substrates. On output pathway substrates (e.g., P53), PER2 inhibits the activity of CK1ε. However, in the case of central clock proteins (e.g., CRYPTOCHROME2), PER2 stimulates the CK1ε-mediated phosphorylation of CRY2. CK1ε activity is temperature compensated on the core clock substrate CRY2 but not on output substrates, for example, the physiological output protein substrate P53 and its nonphysiological correlate, bovine serum albumin (BSA). These results indicate heretofore unrecognized pivotal roles of PER2; it not only regulates the central transcription/translation feedback loop but also differentially controls kinase activity CK1ε in its phosphorylation of central clock (e.g., CRY2) versus output (e.g., P53) substrates.

Exponentially Fitted Two-Derivative Runge-Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems

Oscillation is one of the most important phenomena in the chemical reaction systems in living cells. The general purpose simulation algorithms fail to take into account this special character and produce unsatisfying results. In order to enhance the accuracy of the integrator, the second-order derivative is incorporated in the scheme. The oscillatory feature of the solution is captured by the integrators with an exponential fitting property. Three practical exponentially fitted TDRK (EFTDRK) methods are derived. To test the effectiveness of the new EFTDRK methods, the two-gene system with cross-regulation and the circadian oscillation of the period protein in Drosophila are simulated. Each EFTDRK method has the best fitting frequency which minimizes the global error. The numerical results show that the new EFTDRK methods are more accurate and more efficient than their prototype TDRK methods or RK methods of the same order and the traditional exponentially fitted RK method in the literature.

A modular approach to self-oscillating polymer systems driven by the Belousov–Zhabotinsky reaction

This review explores the principle, modular construction, integral control and engineering aspects of self-oscillating polymer systems driven by the Belousov–Zhabotinsky reaction.

Analytical study of robustness of a negative feedback oscillator by multiparameter sensitivity

Background

One of the distinctive features of biological oscillators such as circadian clocks and cell cycles is robustness which is the ability to resume reliable operation in the face of different types of perturbations. In the previous study, we proposed multiparameter sensitivity (MPS) as an intelligible measure for robustness to fluctuations in kinetic parameters. Analytical solutions directly connect the mechanisms and kinetic parameters to dynamic properties such as period, amplitude and their associated MPSs. Although negative feedback loops are known as common structures to biological oscillators, the analytical solutions have not been presented for a general model of negative feedback oscillators.

Results

We present the analytical expressions for the period, amplitude and their associated MPSs for a general model of negative feedback oscillators. The analytical solutions are validated by comparing them with numerical solutions. The analytical solutions explicitly show how the dynamic properties depend on the kinetic parameters. The ratio of a threshold to the amplitude has a strong impact on the period MPS. As the ratio approaches to one, the MPS increases, indicating that the period becomes more sensitive to changes in kinetic parameters. We present the first mathematical proof that the distributed time-delay mechanism contributes to making the oscillation period robust to parameter fluctuations. The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).

Conclusions

Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators. This study strongly supports that the hypothesis that phosphorylations of clock proteins contribute to the robustness of circadian rhythms. The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

Homeostasis of the period of post-translational biochemical oscillators

Generally, circadian clocks or biological oscillations are resistant to external conditions such as temperature and nutrient concentration. We propose that enzyme-limited competition provides a general mechanism of homeostasis of the period of post-translational oscillators based on protein modifications, and demonstrate it by nutrient compensation in a theoretical model of cyanobacterial circadian clock. The rate change by nutrient concentration is counterbalanced by the amount of available free enzyme, which occurs because of the competition among the various substrates for the limited enzyme. The temperature and nutrient compensation are determined by the postulate that the catalytic modification reactions are rate limiting.

Molecular mechanisms that regulate the coupled period of the mammalian circadian clock

In mammals, most cells in the brain and peripheral tissues generate circadian (∼24 h) rhythms autonomously. These self-sustained rhythms are coordinated and entrained by a master circadian clock in the suprachiasmatic nucleus (SCN). Within the SCN, the individual rhythms of each neuron are synchronized through intercellular signaling. One important feature of SCN is that the synchronized period is close to the population mean of cells' intrinsic periods. In this way, the synchronized period of the SCN stays close to the periods of cells in peripheral tissues. This is important because the SCN must entrain cells throughout the body. However, the mechanism that drives the period of the coupled SCN cells to the population mean is not known. We use mathematical modeling and analysis to show that the mechanism of transcription repression in the intracellular feedback loop plays a pivotal role in regulating the coupled period. Specifically, we use phase response curve analysis to show that the coupled period within the SCN stays near the population mean if transcriptional repression occurs via protein sequestration. In contrast, the coupled period is far from the mean if repression occurs through highly nonlinear Hill-type regulation (e.g., oligomer- or phosphorylation-based repression), as widely assumed in previous mathematical models. Furthermore, we find that the timescale of intercellular coupling needs to be fast compared to that of intracellular feedback to maintain the mean period. These findings reveal the important relationship between the intracellular transcriptional feedback loop and intercellular coupling. This relationship explains why transcriptional repression appears to occur via protein sequestration in multicellular organisms, mammals, and Drosophila, in contrast with the phosphorylation-based repression in unicellular organisms and syncytia. That is, transition to protein sequestration is essential for synchronizing multiple cells with a period close to the population mean (∼24 h).

Metabolic and nontranscriptional circadian clocks: eukaryotes

Circadian clocks are cellular timekeeping mechanisms that coordinate behavior and physiology around the 24-h day in most living organisms. Misalignment of an organism's clock with its environment is associated with long-term adverse fitness consequences, as exemplified by the link between circadian disruption and various age-related diseases in humans. Current eukaryotic models of the circadian oscillator rely on transcription/translation feedback loop mechanisms, supplemented with accessory cytosolic loops that connect them to cellular physiology. However, mounting evidence is questioning the absolute necessity of transcription-based oscillators for circadian rhythmicity, supported by the recent discovery of oxidation-reduction cycles of peroxiredoxin proteins, which persist even in the absence of transcription. A more fundamental mechanism based on metabolic cycles could thus underlie circadian transcriptional and cytosolic rhythms, thereby promoting circadian oscillations to integral properties of cellular metabolism.

The Goodwin model: behind the Hill function

The Goodwin model is a 3-variable model demonstrating the emergence of oscillations in a delayed negative feedback-based system at the molecular level. This prototypical model and its variants have been commonly used to model circadian and other genetic oscillators in biology. The only source of non-linearity in this model is a Hill function, characterizing the repression process. It was mathematically shown that to obtain limit-cycle oscillations, the Hill coefficient must be larger than 8, a value often considered unrealistic. It is indeed difficult to explain such a high coefficient with simple cooperative dynamics. We present here molecular models of the standard Goodwin model, based on single or multisite phosphorylation/dephosphorylation processes of a transcription factor, which have been previously shown to generate switch-like responses. We show that when the phosphorylation/dephosphorylation processes are fast enough, the limit-cycle obtained with a multisite phosphorylation-based mechanism is in very good quantitative agreement with the oscillations observed in the Goodwin model. Conditions in which the detailed mechanism is well approximated by the Goodwin model are given. A variant of the Goodwin model which displays sharp thresholds and relaxation oscillations is also explained by a double phosphorylation/dephosphorylation-based mechanism through a bistable behavior. These results not only provide rational support for the Goodwin model but also highlight the crucial role of the speed of post-translational processes, whose response curve are usually established at a steady state, in biochemical oscillators.