The contributions of interlocking loops and extensive nonlinearity to the properties of circadian clock models

The Goodwin Oscillator and its Legacy

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

Comprehensive mapping of abiotic stress inputs into the soybean circadian clock

The plant circadian clock evolved to increase fitness by synchronizing physiological processes with environmental oscillations. Crop fitness was artificially selected through domestication and breeding, and the circadian clock was identified by both natural and artificial selections as a key to improved fitness. Despite progress in Arabidopsis, our understanding of the crop circadian clock is still limited, impeding its rational improvement for enhanced fitness. To unveil the interactions between the crop circadian clock and various environmental cues, we comprehensively mapped abiotic stress inputs to the soybean circadian clock using a 2-module discovery pipeline. Using the "molecular timetable" method, we computationally surveyed publicly available abiotic stress-related soybean transcriptomes to identify stresses that have strong impacts on the global rhythm. These findings were then experimentally confirmed using a multiplexed RNA sequencing technology. Specific clock components modulated by each stress were further identified. This comprehensive mapping uncovered inputs to the plant circadian clock such as alkaline stress. Moreover, short-term iron deficiency targeted different clock components in soybean and Arabidopsis and thus had opposite effects on the clocks of these 2 species. Comparing soybean varieties with different iron uptake efficiencies suggests that phase modulation might be a mechanism to alleviate iron deficiency symptoms in soybean. These unique responses in soybean demonstrate the need to directly study crop circadian clocks. Our discovery pipeline may serve as a broadly applicable tool to facilitate these explorations.

Deciphering the Dynamics of Interlocked Feedback Loops in a Model of the Mammalian Circadian Clock

Mathematical models of fundamental biological processes play an important role in consolidating theory and experiments, especially if they are systematically developed, thoroughly characterized, and well tested by experimental data. In this work, we report a detailed bifurcation analysis of a mathematical model of the mammalian circadian clock network developed by Relógio et al., noteworthy for its consistency with available data. Using one- and two-parameter bifurcation diagrams, we explore how oscillations in the model depend on the expression levels of its constituent genes and the activities of their encoded proteins. These bifurcation diagrams allow us to decipher the dynamics of interlocked feedback loops by parametric variation of genes and proteins in the model. Among other results, we find that REV-ERB, a member of a subfamily of orphan nuclear receptors, plays a critical role in the intertwined dynamics of Relógio's model. The bifurcation diagrams reported here can be used for predicting how the core clock network responds-in terms of period, amplitude and phases of oscillations-to different perturbations.

Modeling the interactions of sense and antisense Period transcripts in the mammalian circadian clock network

In recent years, it has become increasingly apparent that antisense transcription plays an important role in the regulation of gene expression. The circadian clock is no exception: an antisense transcript of the mammalian core-clock gene PERIOD2 (PER2), which we shall refer to as Per2AS RNA, oscillates with a circadian period and a nearly 12 h phase shift from the peak expression of Per2 mRNA. In this paper, we ask whether Per2AS plays a regulatory role in the mammalian circadian clock by studying in silico the potential effects of interactions between Per2 and Per2AS RNAs on circadian rhythms. Based on the antiphasic expression pattern, we consider two hypotheses about how Per2 and Per2AS mutually interfere with each other's expression. In our pre-transcriptional model, the transcription of Per2AS RNA from the non-coding strand represses the transcription of Per2 mRNA from the coding strand and vice versa. In our post-transcriptional model, Per2 and Per2AS transcripts form a double-stranded RNA duplex, which is rapidly degraded. To study these two possible mechanisms, we have added terms describing our alternative hypotheses to a published mathematical model of the molecular regulatory network of the mammalian circadian clock. Our pre-transcriptional model predicts that transcriptional interference between Per2 and Per2AS can generate alternative modes of circadian oscillations, which we characterize in terms of the amplitude and phase of oscillation of core clock genes. In our post-transcriptional model, Per2/Per2AS duplex formation dampens the circadian rhythm. In a model that combines pre- and post-transcriptional controls, the period, amplitude and phase of circadian proteins exhibit non-monotonic dependencies on the rate of expression of Per2AS. All three models provide potential explanations of the observed antiphasic, circadian oscillations of Per2 and Per2AS RNAs. They make discordant predictions that can be tested experimentally in order to distinguish among these alternative hypotheses.

Elements of biological oscillations in time and space

Oscillations in time and space are ubiquitous in nature and play critical roles in dynamic cellular processes. Although the molecular mechanisms underlying the generation of the dynamics are diverse, several distinct regulatory elements have been recognized as being critical in producing and modulating oscillatory dynamics. These include negative and positive feedback, time delay, nonlinearity in regulation, and random fluctuations ('noise'). Here we discuss the specific roles of these five elements in promoting or attenuating oscillatory dynamics, by drawing on insights from quantitative analyses of natural or synthetic biological networks.

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.

Using sign patterns to detect the possibility of periodicity in biological systems

Many models in the physical and life sciences, formulated as dynamical systems, exhibit a positive steady state, with its local qualitative behavior determined by the eigenvalues of its Jacobian matrix. Our interest lies in detecting if this steady state is linearly stable or if the system has periodic solutions arising from a Hopf bifurcation. We address this by considering the sign pattern of the Jacobian matrix and its set of allowed refined inertias. The refined inertia of a matrix, which is an extension of the classical matrix inertia, is a property of its eigenvalues. A Hopf bifurcation, leading to periodic solutions, may be possible if the sign pattern of the Jacobian matrix allows a specific set of refined inertias. For most systems, we also need to consider magnitude restrictions on the entries of the Jacobian matrix that are a consequence of the particular biological model. The usefulness of sign pattern analysis to detect linear stability or the possibility of periodicity is illustrated with several biological examples, including metabolic-genetic circuits, biochemical reaction networks, predator-prey and competition systems.

Network representations and methods for the analysis of chemical and biochemical pathways

Systems biologists increasingly use network representations to investigate biochemical pathways and their dynamic behaviours. In this critical review, we discuss four commonly used network representations of chemical and biochemical pathways. We illustrate how some of these representations reduce network complexity but result in the ambiguous representation of biochemical pathways. We also examine the current theoretical approaches available to investigate the dynamic behaviour of chemical and biochemical networks. Finally, we describe how the critical chemical and biochemical pathways responsible for emergent dynamic behaviour can be identified using network mining and functional mapping approaches.

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.

Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction-diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.

(Re)inventing the circadian feedback loop

For 20 years, researchers have thought that circadian clocks are defined by feedback loops of transcription and translation. The rediscovery of posttranslational circadian oscillators in diverse organisms forces us to rethink this paradigm. Meanwhile, the original "basic" feedback loops of canonical circadian clocks have swelled to include dozens of additional proteins acting in interlocked loops. We review several self-sustained clock mechanisms and propose that minimum requirements for diurnal timekeeping might be simpler than those of actual free-running circadian oscillators. Thus, complex mechanisms of circadian timekeeping might have evolved from random connections between unrelated feedback loops with independent but limited time-telling capability.

Model Misinterpretation within Biology: Phenotypes, Statistics, Networks, and Inference

Models of myriad forms are rapidly becoming central to biology. These range from statistical models that are fundamental to the interpretation of experimental results to ordinary differential equation models that attempt to describe the results in a mechanistic format. Models will be more and more essential to biologists but this growing importance requires all model users to become more sophisticated about what is in a model and how that limits the usability of the model. This review attempts to relay the potential pitfalls that can lie within a model.

Effective models of periodically driven networks

Circadian rhythms are governed by a highly coupled, complex network of genes. Due to feedback within the network, any modification of the system's state requires coherent changes in several nodes. A model of the underlying network is necessary to compute these modifications. We use an effective modeling approach for this task. Rather than inferred biochemical interactions, our method utilizes microarray data from a group of mutants for its construction. With simulated data, we develop an effective model for a circadian network in a peripheral tissue, subject to driving by the suprachiasmatic nucleus, the mammalian pacemaker. The effective network can predict time-dependent gene expression levels in other mutants.

Efficient characterization of high-dimensional parameter spaces for systems biology

Background

A biological system's robustness to mutations and its evolution are influenced by the structure of its viable space, the region of its space of biochemical parameters where it can exert its function. In systems with a large number of biochemical parameters, viable regions with potentially complex geometries fill a tiny fraction of the whole parameter space. This hampers explorations of the viable space based on "brute force" or Gaussian sampling.

Results

We here propose a novel algorithm to characterize viable spaces efficiently. The algorithm combines global and local explorations of a parameter space. The global exploration involves an out-of-equilibrium adaptive Metropolis Monte Carlo method aimed at identifying poorly connected viable regions. The local exploration then samples these regions in detail by a method we call multiple ellipsoid-based sampling. Our algorithm explores efficiently nonconvex and poorly connected viable regions of different test-problems. Most importantly, its computational effort scales linearly with the number of dimensions, in contrast to "brute force" sampling that shows an exponential dependence on the number of dimensions. We also apply this algorithm to a simplified model of a biochemical oscillator with positive and negative feedback loops. A detailed characterization of the model's viable space captures well known structural properties of circadian oscillators. Concretely, we find that model topologies with an essential negative feedback loop and a nonessential positive feedback loop provide the most robust fixed period oscillations. Moreover, the connectedness of the model's viable space suggests that biochemical oscillators with varying topologies can evolve from one another.

Conclusions

Our algorithm permits an efficient analysis of high-dimensional, nonconvex, and poorly connected viable spaces characteristic of complex biological circuitry. It allows a systematic use of robustness as a tool for model discrimination.