Modeling circadian oscillations with interlocking positive and negative feedback loops

Computational modeling establishes mechanotransduction as a potent modulator of the mammalian circadian clock

Mechanotransduction, which is the integration of mechanical signals from the external environment of a cell to changes in intracellular signaling, governs many cellular functions. Recent studies have shown that the mechanical state of the cell is also coupled to the cellular circadian clock. To investigate possible interactions between circadian rhythms and cellular mechanotransduction, we have developed a computational model that integrates the two pathways. We postulated that translocation of the transcriptional regulators MRTF (herein referring to both MRTF-A and MRTF-B), YAP and TAZ (also known as YAP1 and WWTR1, respectively; collectively denoted YAP/TAZ) into the nucleus leads to altered expression of circadian proteins. Simulations from our model predict that lower levels of cytoskeletal activity are associated with longer circadian oscillation periods and higher oscillation amplitudes, which is consistent with recent experimental observations. Furthermore, accumulation of YAP/TAZ and MRTF in the nucleus causes circadian oscillations to decay in our model. These effects hold both at the single-cell level and within a population-level framework. Finally, we investigated the effects of mutations in YAP or lamin A, the latter of which result in a class of diseases known as laminopathies. In silico, oscillations in circadian proteins are substantially weaker in populations of cells with mutations in YAP or lamin A, suggesting that defects in mechanotransduction can disrupt the circadian clock in certain disease states; however, reducing substrate stiffness in the model restores normal oscillatory behavior, suggesting a possible compensatory mechanism. Thus, our study identifies that mechanotransduction could be a potent modulatory cue for cellular clocks and that this crosstalk can be leveraged to rescue the circadian clock in disease states.

A topological mechanism for robust and efficient global oscillations in biological networks

Long and stable timescales are often observed in complex biochemical networks, such as in emergent oscillations. How these robust dynamics persist remains unclear, given the many stochastic reactions and shorter time scales demonstrated by underlying components. We propose a topological model that produces long oscillations around the network boundary, reducing the system dynamics to a lower-dimensional current in a robust manner. Using this to model KaiC, which regulates the circadian rhythm in cyanobacteria, we compare the coherence of oscillations to that in other KaiC models. Our topological model localizes currents on the system edge, with an efficient regime of simultaneously increased precision and decreased cost. Further, we introduce a new predictor of coherence from the analysis of spectral gaps, and show that our model saturates a global thermodynamic bound. Our work presents a new mechanism and parsimonious description for robust emergent oscillations in complex biological networks.

miR-277 regulates the phase of circadian activity-rest rhythm in Drosophila melanogaster

Circadian clocks temporally organize behaviour and physiology of organisms with a rhythmicity of about 24 h. In Drosophila, the circadian clock is composed of mainly four clock genes: period (per), timeless (tim), Clock (Clk) and cycle (cyc) which constitutes the transcription-translation feedback loop. The circadian clock is further regulated via post-transcriptional and post-translational mechanisms among which microRNAs (miRNAs) are well known post-transcriptional regulatory molecules. Here, we identified and characterized the role of miRNA-277 (miR-277) expressed in the clock neurons in regulating the circadian rhythm. Downregulation of miR-277 in the pacemaker neurons expressing circadian neuropeptide, pigment dispersing factor (PDF) advanced the phase of the morning activity peak under 12 h light: 12 h dark cycles (LD) at lower light intensities and these flies exhibited less robust rhythms compared to the controls under constant darkness. In addition, downregulation of miR-277 in the PDF expressing neurons abolished the Clk gene transcript oscillation under LD. Our study points to the potential role of miR-277 in fine tuning the Clk expression and in maintaining the phase of the circadian rhythm in Drosophila.

Dynamic properties of a self-replicating peptide network with inhibition

In this paper, we report an open system consisting of three self-replicating peptides, in which peptide 1 inhibits the duplex template of peptide 2, peptide 2 inhibits duplex 3, and peptide 3 inhibits duplex 1 to complete the negative feedback loop. This interacting chemical network yields oscillations in the concentrations of all species over time and establishes a possible mechanism for pre-biotic chemical systems organization. The first focus of our analysis is the effect of altering rates of duplex formation and inhibition on oscillations. We then examine the autocatalytic rate constant in the symmetric and asymmetric cases.

Long-range connections are crucial for synchronization transition in a computational model of Drosophila brain dynamics

The synchronization transition type has been the focus of attention in recent years because it is associated with many functional characteristics of the brain. In this paper, the synchronization transition in neural networks with sleep-related biological drives in Drosophila is investigated. An electrical synaptic neural network is established to research the difference between the synchronization transition of the network during sleep and wake, in which neurons regularly spike during sleep and chaotically spike during wake. The synchronization transition curves are calculated mainly using the global instantaneous order parameters S. The underlying mechanisms and types of synchronization transition during sleep are different from those during wake. During sleep, regardless of the network structure, a frustrated (discontinuous) transition can be observed. Moreover, the phenomenon of quasi periodic partial synchronization is observed in ring-shaped regular network with and without random long-range connections. As the network becomes dense, the synchronization of the network only needs to slightly increase the coupling strength g. While during wake, the synchronization transition of the neural network is very dependent on the network structure, and three mechanisms of synchronization transition have emerged: discontinuous synchronization (explosive synchronization and frustrated synchronization), and continuous synchronization. The random long-range connections is the main topological factor that plays an important role in the resulting synchronization transition. Furthermore, similarities and differences are found by comparing synchronization transition research for the Hodgkin-Huxley neural network in the beta-band and gammma-band, which can further improve the synchronization phase transition research of biologically motivated neural networks. A complete research framework can also be used to study coupled nervous systems, which can be extended to general coupled dynamic systems.

WNK3-PER1 interactions regulate the circadian rhythm in the suprachiasmatic nucleus in rats

PER1 is a core component of the internal time-keeping system. In the suprachiasmatic nucleus, it serves as the primary circadian pacemaker in mammalian brains. PER1 functions with other clock components to generate a feedback loop involving the transcriptional repression of gene expression to produce a circadian rhythm with an approximately 24-hour cycle. Post-transcriptional modifications (PTMs) are a basic regulatory mechanism that both perpetuate self-sustained oscillations and interpret metabolic input into circadian physiology by affecting factors such as protein stability, interactions, localization, and activity. Here we examined whether the serine/threonine protein kinase WNK3, which is expressed in a circadian rhythm, can interact and colocalize with PER1 in the SCN. In rats, WNK3 knockdown in the SCN is associated with altered sleep patterns. Moreover, WNK3 can phosphorylate PER1 to promote its degradation and is associated with circadian oscillations when PER1 is expressed in vitro.

Optimization with delay-induced bifurcations

Optimization is finding the best solution, which mathematically amounts to locating the global minimum of some cost function. Optimization is traditionally automated with digital or quantum computers, each having their limitations and none guaranteeing an optimal solution. Here, we conceive a principle behind optimization based on delay-induced bifurcations, which is potentially implementable in non-quantum analog devices. Often, optimization techniques are interpreted via a particle moving in multi-well energy landscape and to prevent confinement to a non-global minima they should incorporate mechanisms to overcome barriers between the minima. Particularly, simulated annealing digitally emulates pushing a fictitious particle over a barrier by random noise, whereas quantum computers utilize tunneling through barriers. In our principle, the barriers are effectively destroyed by delay-induced bifurcations. Although bifurcation scenarios in nonlinear delay-differential equations can be very complex and are notoriously difficult to predict, we hypothesize, verify, and utilize the finding that they become considerably more predictable in dynamical systems, where the right-hand side depends only on the delayed variable and represents a gradient of some potential energy function. By tuning the delay introduced into the gradient descent setting, thanks to global bifurcations destroying local attractors, one could force the system to spontaneously wander around all minima. This would be similar to noise-induced behavior in simulated annealing but achieved deterministically. Ideally, a slow increase and then decrease of the delay should automatically push the system toward the global minimum. We explore the possibility of this scenario and formulate some prerequisites.

Mathematical Modelling of p53 Signalling during DNA Damage Response: A Survey

No gene has garnered more interest than p53 since its discovery over 40 years ago. In the last two decades, thanks to seminal work from Uri Alon and Ghalit Lahav, p53 has defined a truly synergistic topic in the field of mathematical biology, with a rich body of research connecting mathematic endeavour with experimental design and data. In this review we survey and distill the extensive literature of mathematical models of p53. Specifically, we focus on models which seek to reproduce the oscillatory dynamics of p53 in response to DNA damage. We review the standard modelling approaches used in the field categorising them into three types: time delay models, spatial models and coupled negative-positive feedback models, providing sample model equations and simulation results which show clear oscillatory dynamics. We discuss the interplay between mathematics and biology and show how one informs the other; the deep connections between the two disciplines has helped to develop our understanding of this complex gene and paint a picture of its dynamical response. Although yet more is to be elucidated, we offer the current state-of-the-art understanding of p53 response to DNA damage.

Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization

Nonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze and predict because delay-induced effects strongly depend on the form of the nonlinearities involved and on the exact way the delay enters the system. We consider a special class of nonlinear systems with delay obtained by taking a gradient dynamical system with a two-well "potential" function and replacing the argument of the right-hand side function with its delayed version. This choice of the system is motivated by the relative ease of its graphical interpretation and by its relevance to a recent approach to use delay in finding the global minimum of a multi-well function. Here, the simplest type of such systems is explored for which we hypothesize and verify the possibility to qualitatively predict the delay-induced effects, such as a chain of homoclinic bifurcations one by one eliminating local attractors and enabling the phase trajectory to spontaneously visit vicinities of all local minima. The key phenomenon here is delay-induced reorganization of manifolds, which cease to serve as barriers between the local minima after homoclinic bifurcations. Despite the general scenario being quite universal in two-well potentials, the homoclinic bifurcation comes in various versions depending on the fine features of the potential. Our results are a pre-requisite for understanding general highly nonlinear multistable systems with delay. They also reveal the mechanisms behind the possible role of delay in optimization.

Collective Dynamics of Neural Networks With Sleep-Related Biological Drives in Drosophila

The collective electrophysiological dynamics of the brain as a result of sleep-related biological drives in Drosophila are investigated in this paper. Based on the Huber-Braun thermoreceptor model, the conductance-based neurons model is extended to a coupled neural network to analyze the local field potential (LFP). The LFP is calculated by using two different metrics: the mean value and the distance-dependent LFP. The distribution of neurons around the electrodes is assumed to have a circular or grid distribution on a two-dimensional plane. Regardless of which method is used, qualitatively similar results are obtained that are roughly consistent with the experimental data. During wake, the LFP has an irregular or a regular spike. However, the LFP becomes regular bursting during sleep. To further analyze the results, wavelet analysis and raster plots are used to examine how the LFP frequencies changed. The synchronization of neurons under different network structures is also studied. The results demonstrate that there are obvious oscillations at approximately 8 Hz during sleep that are absent during wake. Different time series of the LFP can be obtained under different network structures and the density of the network will also affect the magnitude of the potential. As the number of coupled neurons increases, the neural network becomes easier to synchronize, but the sleep and wake time described by the LFP spectrogram do not change. Moreover, the parameters that affect the durations of sleep and wake are analyzed.

Coupling protocol of interlocked feedback oscillators in circadian clocks

Circadian rhythms (approx. 24 h) show the robustness of key oscillatory features such as phase, period and amplitude against external and internal variations. The robustness of Drosophila circadian clocks can be generated by interlocked transcriptional-translational feedback loops, where two negative feedback loops are coupled through mutual activations. The mechanisms by which such coupling protocols have survived out of many possible protocols remain to be revealed. To address this question, we investigated two distinct coupling protocols: activator-coupled oscillators (ACO) and repressor-coupled oscillators (RCO). We focused on the two coupling parameters: coupling dissociation constant and coupling time-delay. Interestingly, the ACO was able to produce anti-phase or morning-evening cycles, whereas the RCO produced in-phase ones. Deterministic and stochastic analyses demonstrated that the anti-phase ACO provided greater fluctuations in amplitude not only with respect to changes in coupling parameters but also to random parameter perturbations than the in-phase RCO. Moreover, the ACO deteriorated the entrainability to the day-night master clock, whereas the RCO produced high entrainability. Considering that the real, interlocked feedback loops have evolved as the ACO, instead of the RCO, we first proposed a hypothesis that the morning-evening or anti-phase cycle is more essential for Drosophila than achieving robustness and entrainability.

Delay stability of reaction systems

Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.

Towards the revival of oscillation from complete cessation in stochastic systems for application in molecular biology

Delay and noise are inevitable in complex systems that are common in biochemical networks. The system is often disturbed at various states irrespective of the size (small or large) of delay and noise. Therefore, it is of interest to describe the significance of delay and noise in stochastic Willamowski-Rossler chemical oscillator model using a delay stochastic (having random probability distribution) simulation algorithm. Oscillating dynamics moves to stable fixed point when delay at a fixed magnitude of noise drives the system from oscillating state to stochastic amplitude death state (complete cessation). However, the amplitude death state is induced to a revived oscillating state in stochastic system (which is far from equilibrium state) for noise with a fixed value of delay. Thus, significantly large and small noise induces the dynamics of the system to amplitude death state. Hence, we describe the interplay of delay and noise in stochastic systems for the proper and efficient functioning of the complex system that are frequent in biological networks.

Genotype Components as Predictors of Phenotype in Model Gene Regulatory Networks

Models of gene regulatory networks (GRN) have proven useful for understanding many aspects of the highly complex behavior of biological control networks. Randomly generated non-Boolean networks were used in experimental simulations to generate data on dynamic phenotypes as a function of several genotypic parameters. We found that predictive relationships between some phenotypes and quantitative genotypic parameters such as number of network genes, interaction density, and initial condition could be derived depending on the strength of the topological (positional) genotype on specific phenotypes. We quantitated the strength of the topological genotype effect (TGE) on a number of phenotypes in multi-gene networks. For phenotypes with a low influence of topological genotype, derived and empirical relationships using quantitative genotype parameters were accurate in phenotypic outcomes. We found a number of dynamic network properties, including oscillation behaviors, that were largely dependent on genotype topology, and for which no such general quantitative relationships were determinable. It remains to be determined if these results are applicable to biological gene regulatory networks.

Global stability analysis of fractional-order gene regulatory networks with time delay

Fractional-order gene regulatory networks with time delay (DFGRNs) have proven that they are more suitable to model gene regulation mechanism than integer-order. In this paper, a novel DFGRN is proposed. The existence and uniqueness of the equilibrium point for the DFGRN are proved under certain conditions. On this basis, the conditions on the global asymptotic stability are established by using the Lyapunov method and comparison theorem for the DFGRN, and the stability conditions are dependent on the fractional-order [Formula: see text]. Finally, numerical simulations show that the obtained results are reasonable.

Emergent dynamics of coordinated cells with time delays in a tissue

In this article, we investigate the emergence of tissue dynamics with time delays of diffusion. Such emergent dynamics, describing the tissue homeostasis, usually correspond to particular tissue functions, which are attracting a tremendous amount of attention from both communities of mathematical modeling and systems biology. Specifically, in addition to the within-cell genome dynamics and the diffusion among the cells, we consider several types of time delays of diffusion present in the coordinated cells. We establish several generalized versions of the "monotonicity condition" (MC), whose traditional version [I. Rajapakse and S. Smale, Proc. Natl. Acad. Sci. U.S.A. 114, 1462-1467 (2017)] guaranteed the stability of the equilibrium in a system of coordinated cells without time delay. Indeed, we find that one generalized MC we establish still guarantees the stability of the time-delayed system's equilibrium, which corresponds to a formation of tissue functions depending primarily on individual genome dynamics but less on interacting structures and time delays of diffusion. We also find that, when the generalized MC is further relaxed, the system is able to sustain periodic oscillations, whose periods are verified to have delicate dependence with the selected time delays. These produced oscillations usually represent realistic behaviors of "alive" cells. We use several representative examples to demonstrate the usefulness of the established analytical conditions to the understanding of the emergent dynamics observed in computational models and in real systems as well.

Mathematical modeling of circadian rhythms

Circadian rhythms are endogenous ~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Reducing the complexity of mathematical models for the plant circadian clock by distributed delays

A major bottleneck in the modelling of biological networks is the parameter explosion problem - the exponential increase in the number of parameters that need to be optimised to data as the size of the model increases. Here, we address this problem in the context of the plant circadian clock by applying the method of distributed delays. We show that using this approach, the system architecture can be simplified efficiently - reducing the number of parameters - whilst still preserving the core mechanistic dynamics of the gene regulatory network. Compared to models with discrete time-delays, which are governed by functional differential equations, the distributed delay models can be converted into sets of equivalent ordinary differential equations, enabling the use of standard methods for numerical integration, and for stability and bifurcation analyses. We demonstrate the efficiency of our modelling approach by applying it to three exemplar mathematical models of the Arabidopsis circadian clock of varying complexity, obtaining significant reductions in complexity in each case. Moreover, we revise one of the most up-to-date Arabidopsis models, updating the regulation of the PRR9 and PRR7 genes by LHY in accordance with recent experimental data. The revised model more accurately reproduces the LHY-induction experiments of core clock genes, compared with the original model. Our work thus shows that the method of distributed delays facilitates the optimisation and reformulation of genetic network models.

Computational and experimental insights into the circadian effects of SIRT1

The circadian clock orchestrates 24-h rhythms in physiology in most living organisms. At the molecular level, the dogma is that circadian oscillations are based on a negative transcriptional feedback loop. Recent studies found the NAD+-dependent histone deacetylase, SIRT1, directly regulates acetylation status of clock components and influences circadian amplitude in cells. While Nakahata et al. [Nakahata Y, Kaluzova M (2008) Cell 134:329-340] reported that loss of SIRT1 increases amplitude through BMAL1 acetylation, Asher et al. [Asher G, Gatfield D (2008) Cell 134:317-328] reported that loss of SIRT1 decreases amplitude through an increase in acetylated PER2. To address this SIRT1 paradox, we developed a circadian enzymatic model. Predictions from this model and experimental validation strongly align with the findings of Asher et al., with PER2 as the primary target of SIRT1. Further, the model suggested SIRT1 influences BMAL1 expression through actions on PGC1α. We validated this finding experimentally. Thus, our computational and experimental approaches suggest SIRT1 positively regulates clock function through actions on PER2 and PGC1α.

Period-amplitude co-variation in biomolecular oscillators

The period and amplitude of biomolecular oscillators are functionally important properties in multiple contexts. For a biomolecular oscillator, the overall constraints in how tuning of amplitude affects period, and vice versa, are generally unclear. Here, the authors investigate this co-variation of the period and amplitude in mathematical models of biomolecular oscillators using both simulations and analytical approximations. The authors computed the amplitude-period co-variation of 11 benchmark biomolecular oscillators as their parameters were individually varied around a nominal value, classifying the various co-variation patterns such as a simultaneous increase/decrease in period and amplitude. Next, the authors repeated the classification using a power norm-based amplitude metric, to account for the amplitudes of the many biomolecular species that may be part of the oscillations, finding largely similar trends. Finally, the authors calculate 'scaling laws' of period-amplitude co-variation for a subset of these benchmark oscillators finding that as the approximated period increases, the upper bound of the amplitude increases, or reaches a constant value. Based on these results, the authors discuss the effect of different parameters on the type of period-amplitude co-variation as well as the difficulty in achieving an oscillation with large amplitude and small period.

RACIPE: a computational tool for modeling gene regulatory circuits using randomization

Background

One of the major challenges in traditional mathematical modeling of gene regulatory circuits is the insufficient knowledge of kinetic parameters. These parameters are often inferred from existing experimental data and/or educated guesses, which can be time-consuming and error-prone, especially for large networks.

Results

We present a user-friendly computational tool for the community to use our newly developed method named random circuit perturbation (RACIPE), to explore the robust dynamical features of gene regulatory circuits without the requirement of detailed kinetic parameters. Taking the network topology as the only input, RACIPE generates an ensemble of circuit models with distinct randomized parameters and uniquely identifies robust dynamical properties by statistical analysis. Here, we discuss the implementation of the software and the statistical analysis methods of RACIPE-generated data to identify robust gene expression patterns and the functions of genes and regulatory links. Finally, we apply the tool on coupled toggle-switch circuits and a published circuit of B-lymphopoiesis.

Conclusions

We expect our new computational tool to contribute to a more comprehensive and unbiased understanding of mechanisms underlying gene regulatory networks. RACIPE is a free open source software distributed under (Apache 2.0) license and can be downloaded from GitHub (https://github.com/simonhb1990/RACIPE-1.0).

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0594-6) contains supplementary material, which is available to authorized users.

Minimum Action Path Theory Reveals the Details of Stochastic Transitions Out of Oscillatory States

Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Transitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending nonlocally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.

Aging transition in systems of oscillators with global distributed-delay coupling

We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-[Formula: see text]B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-[Formula: see text]B) and through the use of high-performance computing be capable of simulating up to [Formula: see text] cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

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.

The Neurospora Circadian System

The eukaryotic filamentous fungus Neurospora crassa has proven to be a durable and dependable model system for the analysis of the cellular and molecular bases of circadian rhythms. Pioneering genetic analyses identified clock genes, and beginning with the cloning of frequency ( frq), work over the past 2 decades has revealed the molecular basis of a core circadian clock feedback loop that has illuminated our understanding of circadian oscillators in microbes, plants, and animals. In this transcription/translation-based feedback loop, a heterodimer of the White Collar-1 (WC-1) and WC-2 proteins acts both as the circadian photoreceptor and, in the dark, as a transcription factor that promotes the expression of the frq gene. FRQ dimerizes and feeds back to block the activity of its activators (making a negative feedback loop), as well as feeding forward to promote the synthesis of its activator, WC-1. Phosphorylation of FRQ by several kinases leads to its ubiquitination and turnover, releasing the WC-1/WC-2 dimer to reactivate frq expression and restart the circadian cycle. Light resetting of the clock can be understood through the rapid light induction of frq expression and temperature resetting through the influence of elevated temperaturesin driving higher levels of FRQ. Several FRQ- and WC-independent, noncircadian FRQ-less oscillators (FLOs) have been described, each of which appears to regulate aspects of Neurospora growth or development. Overall, the FRQ/white collar complex feedback loop appears to coordinate the circadian system through its activity to regulate downstream-target clock-controlled genes, either directly or via regulation of driven FLOs.

Multiscale modeling of tumor growth induced by circadian rhythm disruption in epithelial tissue

We propose a multiscale chemo-mechanical model of cancer tumor development in epithelial tissue. The model is based on the transformation of normal cells into a cancerous state triggered by a local failure of spatial synchronization of the circadian rhythm. The model includes mechanical interactions and a chemical signal exchange between neighboring cells, as well as a division of cells and intercalation that allows for modification of the respective parameters following transformation into the cancerous state. The numerical simulations reproduce different dephasing patterns--spiral waves and quasistationary clustering, with the latter being conducive to cancer formation. Modification of mechanical properties reproduces a distinct behavior of invasive and localized carcinoma.

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.

Analysis of Feedback Mechanisms with Unknown Delay Using Sparse Multivariate Autoregressive Method

This paper discusses the study of two interacting processes in which a feedback mechanism exists between the processes. The study was motivated by problems such as the circadian oscillation of gene expression where two interacting protein transcriptions form both negative and positive feedback loops with long delays to equilibrium. Traditionally, data of this type could be examined using autoregressive analysis. However, in circadian oscillation the order of an autoregressive model cannot be determined a priori. We propose a sparse multivariate autoregressive method that incorporates mixed linear effects into regression analysis, and uses a forward-backward greedy search algorithm to select non-zero entries in the regression coefficients, the number of which is constrained not to exceed a pre-specified number. A small simulation study provides preliminary evidence of the validity of the method. Besides the circadian oscillation example, an additional example of blood pressure variations using data from an intervention study is used to illustrate the method and the interpretation of the results obtained from the sparse matrix method. These applications demonstrate how sparse representation can be used for handling high dimensional variables that feature dynamic, reciprocal relationships.

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

Rhythmic behavior is essential for plants; for example, daily (circadian) rhythms control photosynthesis and seasonal rhythms regulate their life cycle. The core of the circadian clock is a genetic network that coordinates the expression of specific clock genes in a circadian rhythm reflecting the 24-h day/night cycle. Circadian clocks exhibit stochastic noise due to the low copy numbers of clock genes and the consequent cell-to-cell variation: this intrinsic noise plays a major role in circadian clocks by inducing more robust oscillatory behavior. Another source of noise is the environment, which causes variation in temperature and light intensity: this extrinsic noise is part of the requirement for the structural complexity of clock networks. Advances in experimental techniques now permit single-cell measurements and the development of single-cell models. Here we present some modeling studies showing the importance of considering both types of noise in understanding how plants adapt to regular and irregular light variations. Stochastic models have proven useful for understanding the effect of regular variations. By contrast, the impact of irregular variations and the interaction of different noise sources are less well studied.

Brain clock driven by neuropeptides and second messengers

The master circadian pacemaker in mammals is localized in a small portion of the brain called the suprachiasmatic nucleus (SCN). It is unclear how the SCN produces circadian rhythms. A common interpretation is that the SCN produces oscillations through the coupling of genetic oscillators in the neurons. The coupling is effected by a network of neuropeptides and second messengers. This network is crucial for the correct function of the SCN. However, models that study a possible oscillatory behavior of the network itself have received little attention. Here we propose and analyze a model to examine this oscillatory potential. We show that an intercellular oscillator emerges in the SCN as a result of the neuropeptide and second messenger dynamics. We find that this intercellular clock can produce circadian rhythms by itself with and without genetic clocks. We also found that the model is robust to perturbation of parameters and can be entrained by light-dark cycles.

Improved understanding of gene expression regulation using systems biology

This article reviews the current state of systems biology approaches, including the experimental tools used to generate 'omic' data and computational frameworks to interpret this data. Through illustrative examples, systems biology approaches to understand gene expression and gene expression regulation are discussed. Some of the challenges facing this field and the future opportunities in the systems biology era are highlighted.

From molecules to behavior and the clinic: Integration in chronobiology

Chronobiology, especially the study of circadian rhythms, provides a model scientific field in which philosophers can study how investigators from a variety of disciplines working at different levels of organization are each contributing to a multi-level account of the responsible mechanism. I focus on how the framework of mechanistic explanation integrates research designed to decompose the mechanism with efforts directed at recomposition that relies especially on computation models. I also examine how recently the integration has extended beyond basic research to the processes through which the disruption of circadian rhythms contributes to disease, including various forms of cancer. Understanding these linkages has been facilitated by discoveries about how circadian mechanisms interact with mechanisms involved in other physiological processes, including the cell cycle and the immune system.

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.

The interplay of cis-regulatory elements rules circadian rhythms in mouse liver

The mammalian circadian clock is driven by cell-autonomous transcriptional feedback loops that involve E-boxes, D-boxes, and ROR-elements. In peripheral organs, circadian rhythms are additionally affected by systemic factors. We show that intrinsic combinatorial gene regulation governs the liver clock. With a temporal resolution of 2 h, we measured the expression of 21 clock genes in mouse liver under constant darkness and equinoctial light-dark cycles. Based on these data and known transcription factor binding sites, we develop a six-variable gene regulatory network. The transcriptional feedback loops are represented by equations with time-delayed variables, which substantially simplifies modelling of intermediate protein dynamics. Our model accurately reproduces measured phases, amplitudes, and waveforms of clock genes. Analysis of the network reveals properties of the clock: overcritical delays generate oscillations; synergy of inhibition and activation enhances amplitudes; and combinatorial modulation of transcription controls the phases. The agreement of measurements and simulations suggests that the intrinsic gene regulatory network primarily determines the circadian clock in liver, whereas systemic cues such as light-dark cycles serve to fine-tune the rhythms.

Stochastic gene expression modeling with Hill function for switch-like gene responses

Gene expression models play a key role to understand the mechanisms of gene regulation whose aspects are grade and switch-like responses. Though many stochastic approaches attempt to explain the gene expression mechanisms, the Gillespie algorithm which is commonly used to simulate the stochastic models requires additional gene cascade to explain the switch-like behaviors of gene responses. In this study, we propose a stochastic gene expression model describing the switch-like behaviors of a gene by employing Hill functions to the conventional Gillespie algorithm. We assume eight processes of gene expression and their biologically appropriate reaction rates are estimated based on published literatures. We observed that the state of the system of the toggled switch model is rarely changed since the Hill function prevents the activation of involved proteins when their concentrations stay below a criterion. In ScbA-ScbR system, which can control the antibiotic metabolite production of microorganisms, our modified Gillespie algorithm successfully describes the switch-like behaviors of gene responses and oscillatory expressions which are consistent with the published experimental study.

Digital clocks: simple Boolean models can quantitatively describe circadian systems

The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day–night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock's oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.

A symmetric dual feedback system provides a robust and entrainable oscillator

Many organisms have evolved molecular clocks to anticipate daily changes in their environment. The molecular mechanisms by which the circadian clock network produces sustained cycles have extensively been studied and transcriptional-translational feedback loops are common structures to many organisms. Although a simple or single feedback loop is sufficient for sustained oscillations, circadian clocks implement multiple, complicated feedback loops. In general, different types of feedback loops are suggested to affect the robustness and entrainment of circadian rhythms.

To reveal the mechanism by which such a complex feedback system evolves, we quantify the robustness and light entrainment of four competing models: the single, semi-dual, dual, and redundant feedback models. To extract the global properties of those models, all plausible kinetic parameter sets that generate circadian oscillations are searched to characterize their oscillatory features. To efficiently perform such analyses, we used the two-phase search (TPS) method as a fast and non-biased search method and quasi-multiparameter sensitivity (QMPS) as a fast and exact measure of robustness to uncertainty of all kinetic parameters.

So far the redundant feedback model has been regarded as the most robust oscillator, but our extensive analysis corrects or overcomes this hypothesis. The dual feedback model, which is employed in biology, provides the most robust oscillator to multiple parameter perturbations within a cell and most readily entrains to a wide range of light-dark cycles. The kinetic symmetry between the dual loops and their coupling via a protein complex are found critically responsible for robust and entrainable oscillations. We first demonstrate how the dual feedback architecture with kinetic symmetry evolves out of many competing feedback systems.

Modeling circadian clocks: Roles, advantages, and limitations

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

Robustness of circadian rhythms in the presence of molecular fluctuations: an investigation based on a mechanistic, statistical theory and a simulation algorithm

After a very brief introduction to a mechanistic and statistical theory of molecular fluctuations in chemical reactions developed by Joel Keizer, we explore the robustness of a circadian rhythm model by using the theory and the exact stochastic simulation (ESS). The comparative study shows that the theory reflects the effects of the dynamics of the model on the robustness more than ESS does. Even though the theory is a macroscopic one, the robustness of the model compares well with that computed from the ESS when the system size is larger than 50. The robustness increases nonlinearly with the system size and it reaches an asymptotic value at higher system sizes. As we can expect from the dynamics of the system, the robustness is minimum near the bifurcation point and as the most sensitive parameter increases away from the bifurcation point the robustness according to the theory as well as the ESS increases and then reaches to a steady value.

Modeling light adaptation in circadian clock: prediction of the response that stabilizes entrainment

Periods of biological clocks are close to but often different from the rotation period of the earth. Thus, the clocks of organisms must be adjusted to synchronize with day-night cycles. The primary signal that adjusts the clocks is light. In Neurospora, light transiently up-regulates the expression of specific clock genes. This molecular response to light is called light adaptation. Does light adaptation occur in other organisms? Using published experimental data, we first estimated the time course of the up-regulation rate of gene expression by light. Intriguingly, the estimated up-regulation rate was transient during light period in mice as well as Neurospora. Next, we constructed a computational model to consider how light adaptation had an effect on the entrainment of circadian oscillation to 24-h light-dark cycles. We found that cellular oscillations are more likely to be destabilized without light adaption especially when light intensity is very high. From the present results, we predict that the instability of circadian oscillations under 24-h light-dark cycles can be experimentally observed if light adaptation is altered. We conclude that the functional consequence of light adaptation is to increase the adjustability to 24-h light-dark cycles and then adapt to fluctuating environments in nature.

Internal noise enhanced oscillation in a delayed circadian pacemaker

The effect of internal noise in a delayed circadian oscillator is studied by using both chemical Langevin equations and stochastic normal form theory. It is found that internal noise can induce circadian oscillation even if the delay time τ is below the deterministic Hopf bifurcation τ(h). We use signal-to-noise ratio (SNR) to quantitatively characterize the performance of such noise induced oscillations and a threshold value of SNR is introduced to define the so-called effective oscillation. Interestingly, the τ-range for effective stochastic oscillation, denoted as Δτ(EO), shows a bell-shaped dependence on the intensity of internal noise which is inversely proportional to the system size. We have also investigated how the rates of synthesis and degradation of the clock protein influence the SNR and thus Δτ(EO). The decay rate K(d) could significantly affect Δτ(EO), while varying the gene expression rate K(e) has no obvious effect if K(e) is not too small. Stochastic normal form analysis and numerical simulations are in good consistency with each other. This work provides us comprehensive understandings of how internal noise and time delay work cooperatively to influence the dynamics of circadian oscillations.

Multiple light inputs to a simple clock circuit allow complex biological rhythms

Circadian clocks are biological timekeepers that allow living cells to time their activity in anticipation of predictable environmental changes. Detailed understanding of the circadian network of higher plants, such as Arabidopsis thaliana, is hampered by the high number of partially redundant genes. However, the picoeukaryotic alga Ostreococcus tauri, which was recently shown to possess a small number of non-redundant clock genes, presents an attractive alternative target for detailed modelling of circadian clocks in the green lineage. Based on extensive time-series data from in vivo reporter gene assays, we developed a model of the Ostreococcus clock as a feedback loop between the genes TOC1 and CCA1. The model reproduces the dynamics of the transcriptional and translational reporters over a range of photoperiods. Surprisingly, the model is also able to predict the transient behaviour of the clock when the light conditions are altered. Despite the apparent simplicity of the clock circuit, it displays considerable complexity in its response to changing light conditions. Systematic screening of the effects of altered day length revealed a complex relationship between phase and photoperiod, which is also captured by the model. The complex light response is shown to stem from circadian gating of light-dependent mechanisms. This study provides insights into the contributions of light inputs to the Ostreococcus clock. The model suggests that a high number of light-dependent reactions are important for flexible timing in a circadian clock with only one feedback loop.