Robustness from flexibility in the fungal circadian clock

Alternative polyadenylation factor CPSF6 regulates temperature compensation of the mammalian circadian clock

A defining property of circadian clocks is temperature compensation, characterized by the resilience of their near 24-hour free-running periods against changes in environmental temperature within the physiological range. While temperature compensation is evolutionary conserved across different taxa of life and has been studied within many model organisms, its molecular underpinnings remain elusive. Posttranscriptional regulations such as temperature-sensitive alternative splicing or phosphorylation have been described as underlying reactions. Here, we show that knockdown of cleavage and polyadenylation specificity factor subunit 6 (CPSF6), a key regulator of 3'-end cleavage and polyadenylation, significantly alters circadian temperature compensation in human U-2 OS cells. We apply a combination of 3'-end-RNA-seq and mass spectrometry-based proteomics to globally quantify changes in 3' UTR length as well as gene and protein expression between wild-type and CPSF6 knockdown cells and their dependency on temperature. Since changes in temperature compensation behavior should be reflected in alterations of temperature responses within one or all of the 3 regulatory layers, we statistically assess differential responses upon changes in ambient temperature between wild-type and CPSF6 knockdown cells. By this means, we reveal candidate genes underlying circadian temperature compensation, including eukaryotic translation initiation factor 2 subunit 1 (EIF2S1).

BDEtools: A MATLAB Package for Boolean Delay Equation Modeling

Boolean Delay Equations (BDEs) can simulate surprisingly complex behavior, despite their relative simplicity. In addition to steady-state dynamics, BDEs can also generate periodic and quasiperiodic oscillations, m:n frequency locking, and even chaos. Further, the enumerability of Boolean update functions and their compact parametrization means that BDEs can be leveraged to generate low-level descriptions of biological networks, from which more detailed formulations (e.g., differential equation models) can be constructed. However, although several studies have demonstrated the utility of BDE modeling in computational biology, a current barrier to the wider adoption of the BDE approach is the absence of freely available simulation software. In this work, we present BDEtools-an open-source MATLAB package for numerically solving BDE models. After giving a brief introduction to BDE modeling, we describe the package's solver algorithms, together with several utility functions that can be used to provide solver inputs and to process solver outputs. We also demonstrate the functionality of BDEtools by illustrating its application to an established model of a gene regulatory network that controls circadian rhythms. BDEtools makes it straightforward for researchers to quickly build reliable BDE models of biological networks. We hope that its ease of use and free availability will encourage more researchers to explore BDE formulations of their systems of interest. Through the continued use of BDEs by the computational biology community, we will, no doubt, discover their potential applicability to a broader class of biological networks.

It's about time: Analysing simplifying assumptions for modelling multi-step pathways in systems biology

Thoughtful use of simplifying assumptions is crucial to make systems biology models tractable while still representative of the underlying biology. A useful simplification can elucidate the core dynamics of a system. A poorly chosen assumption can, however, either render a model too complicated for making conclusions or it can prevent an otherwise accurate model from describing experimentally observed dynamics. Here, we perform a computational investigation of sequential multi-step pathway models that contain fewer pathway steps than the system they are designed to emulate. We demonstrate when such models will fail to reproduce data and how detrimental truncation of a pathway leads to detectable signatures in model dynamics and its optimised parameters. An alternative assumption is suggested for simplifying such pathways. Rather than assuming a truncated number of pathway steps, we propose to use the assumption that the rates of information propagation along the pathway is homogeneous and, instead, letting the length of the pathway be a free parameter. We first focus on linear pathways that are sequential and have first-order kinetics, and we show how this assumption results in a three-parameter model that consistently outperforms its truncated rival and a delay differential equation alternative in recapitulating observed dynamics. We then show how the proposed assumption allows for similarly terse and effective models of non-linear pathways. Our results provide a foundation for well-informed decision making during model simplifications.

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.

A simplified modelling framework facilitates more complex representations of plant circadian clocks

The circadian clock orchestrates biological processes so that they occur at specific times of the day, thereby facilitating adaptation to diurnal and seasonal environmental changes. In plants, mathematical modelling has been comprehensively integrated with experimental studies to gain a better mechanistic understanding of the complex genetic regulatory network comprising the clock. However, with an increasing number of circadian genes being discovered, there is a pressing need for methods facilitating the expansion of computational models to incorporate these newly-discovered components. Conventionally, plant clock models have comprised differential equation systems based on Michaelis-Menten kinetics. However, the difficulties associated with modifying interactions using this approach-and the concomitant problem of robustly identifying regulation types-has contributed to a complexity bottleneck, with quantitative fits to experimental data rapidly becoming computationally intractable for models possessing more than ≈50 parameters. Here, we address these issues by constructing the first plant clock models based on the S-System formalism originally developed by Savageau for analysing biochemical networks. We show that despite its relative simplicity, this approach yields clock models with comparable accuracy to the conventional Michaelis-Menten formalism. The S-System formulation also confers several key advantages in terms of model construction and expansion. In particular, it simplifies the inclusion of new interactions, whilst also facilitating the modification of regulation types, thereby making it well-suited to network inference. Furthermore, S-System models mitigate the issue of parameter identifiability. Finally, by applying linear systems theory to the models considered, we provide some justification for the increased use of aggregated protein equations in recent plant clock modelling, replacing the separate cytoplasmic/nuclear protein compartments that were characteristic of the earlier models. We conclude that as well as providing a simplified framework for model development, the S-System formalism also possesses significant potential as a robust modelling method for designing synthetic gene circuits.

Weak coupling between intracellular feedback loops explains dissociation of clock gene dynamics

Circadian rhythms are generated by interlocked transcriptional-translational negative feedback loops (TTFLs), the molecular process implemented within a cell. The contributions, weighting and balancing between the multiple feedback loops remain debated. Dissociated, free-running dynamics in the expression of distinct clock genes has been described in recent experimental studies that applied various perturbations such as slice preparations, light pulses, jet-lag, and culture medium exchange. In this paper, we provide evidence that this "presumably transient" dissociation of circadian gene expression oscillations may occur at the single-cell level. Conceptual and detailed mechanistic mathematical modeling suggests that such dissociation is due to a weak interaction between multiple feedback loops present within a single cell. The dissociable loops provide insights into underlying mechanisms and general design principles of the molecular circadian clock.

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.

An incoherent feed-forward loop switches the Arabidopsis clock rapidly between two hysteretic states

In higher plants (e.g., Arabidopsis thaliana), the core structure of the circadian clock is mostly governed by a repression process with very few direct activators. With a series of simplified models, we studied the underlying mechanism and found that the Arabidopsis clock consists of type-2 incoherent feed-forward loops (IFFLs), one of them creating a pulse-like expression in PRR9/7. The double-negative feedback loop between CCA1/LHY and PRR5/TOC1 generates a bistable, hysteretic behavior in the Arabidopsis circadian clock. We found that the IFFL involving PRR9/7 breaks the bistability and moves the system forward with a rapid pulse in the daytime, and the evening complex (EC) breaks it in the evening. With this illustration, we can intuitively explain the behavior of the clock under mutant conditions. Thus, our results provide new insights into the underlying network structures of the Arabidopsis core oscillator.

A transcription network of interlocking positive feedback loops maintains intracellular iron balance in archaea

Iron is required for key metabolic processes but is toxic in excess. This circumstance forces organisms across the tree of life to tightly regulate iron homeostasis. In hypersaline lakes dominated by archaeal species, iron levels are extremely low and subject to environmental change; however, mechanisms regulating iron homeostasis in archaea remain unclear. In previous work, we demonstrated that two transcription factors (TFs), Idr1 and Idr2, collaboratively regulate aspects of iron homeostasis in the model species Halobacterium salinarum. Here we show that Idr1 and Idr2 are part of an extended regulatory network of four TFs of the bacterial DtxR family that maintains intracellular iron balance. We demonstrate that each TF directly regulates at least one of the other DtxR TFs at the level of transcription. Dynamical modeling revealed interlocking positive feedback loop architecture, which exhibits bistable or oscillatory network dynamics depending on iron availability. TF knockout mutant phenotypes are consistent with model predictions. Together, our results support that this network regulates iron homeostasis despite variation in extracellular iron levels, consistent with dynamical properties of interlocking feedback architecture in eukaryotes. These results suggest that archaea use bacterial-type TFs in a eukaryotic regulatory network topology to adapt to harsh environments.

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.

Robustness, flexibility, and sensitivity in a multifunctional motor control model

Motor systems must adapt to perturbations and changing conditions both within and outside the body. We refer to the ability of a system to maintain performance despite perturbations as "robustness," and the ability of a system to deploy alternative strategies that improve fitness as "flexibility." Different classes of pattern-generating circuits yield dynamics with differential sensitivities to perturbations and parameter variation. Depending on the task and the type of perturbation, high sensitivity can either facilitate or hinder robustness and flexibility. Here we explore the role of multiple coexisting oscillatory modes and sensory feedback in allowing multiphasic motor pattern generation to be both robust and flexible. As a concrete example, we focus on a nominal neuromechanical model of triphasic motor patterns in the feeding apparatus of the marine mollusk Aplysia californica. We find that the model can operate within two distinct oscillatory modes and that the system exhibits bistability between the two. In the "heteroclinic mode," higher sensitivity makes the system more robust to changing mechanical loads, but less robust to internal parameter variations. In the "limit cycle mode," lower sensitivity makes the system more robust to changes in internal parameter values, but less robust to changes in mechanical load. Finally, we show that overall performance on a variable feeding task is improved when the system can flexibly transition between oscillatory modes in response to the changing demands of the task. Thus, our results suggest that the interplay of sensory feedback and multiple oscillatory modes can allow motor systems to be both robust and flexible in a variable environment.

Evolution of robust circadian clocks in Drosophila melanogaster populations reared in constant dark for over 330 generations

Robustness is considered to be an important feature of biological systems which may evolve when the functionality of a trait is associated with higher fitness across multiple environmental conditions. Thus, the ability to maintain stable biological phenotypes across environments is thought to be of adaptive value. Previously, we have reported higher intrinsic activity levels (activity levels of free-running rhythm in constant darkness) and power of rhythm (as assessed by amplitude of the periodogram) in Drosophila melanogaster populations (stocks) reared in constant darkness (DD stocks) as compared to those reared in constant light (LL stocks) and 12:12-h light-dark cycles (LD stocks) for over 19 years (∼330 generations). In the current study, we intended to examine whether the enhanced levels of activity observed in DD stocks persist under various environments such as photoperiods, ambient temperatures, non-24-h light-dark (LD) cycles, and semi-natural conditions (SN). We found that DD stocks largely retain their phenotype of enhanced activity levels across most of the above-mentioned environments suggesting the evolution of robust circadian clocks in DD stocks. Furthermore, we compared the peak activity levels of the three stocks across different environmental conditions relative to their peaks in constant darkness and found that the change in peak activity levels upon entrainment was not significantly different across the three stocks for any of the examined environmental conditions. This suggests that the enhancement of activity levels in DD stocks is not due to differential sensitivity to environment. Thus, these results suggest that rearing in constant darkness (DD) leads to evolution of robust circadian clocks suggesting a possible adaptive value of possessing such rhythms under constant dark environments.

Nonlinear Dynamics in Gene Regulation Promote Robustness and Evolvability of Gene Expression Levels

Cellular phenotypes underpinned by regulatory networks need to respond to evolutionary pressures to allow adaptation, but at the same time be robust to perturbations. This creates a conflict in which mutations affecting regulatory networks must both generate variance but also be tolerated at the phenotype level. Here, we perform mathematical analyses and simulations of regulatory networks to better understand the potential trade-off between robustness and evolvability. Examining the phenotypic effects of mutations, we find an inverse correlation between robustness and evolvability that breaks only with nonlinearity in the network dynamics, through the creation of regions presenting sudden changes in phenotype with small changes in genotype. For genotypes embedding low levels of nonlinearity, robustness and evolvability correlate negatively and almost perfectly. By contrast, genotypes embedding nonlinear dynamics allow expression levels to be robust to small perturbations, while generating high diversity (evolvability) under larger perturbations. Thus, nonlinearity breaks the robustness-evolvability trade-off in gene expression levels by allowing disparate responses to different mutations. Using analytical derivations of robustness and system sensitivity, we show that these findings extend to a large class of gene regulatory network architectures and also hold for experimentally observed parameter regimes. Further, the effect of nonlinearity on the robustness-evolvability trade-off is ensured as long as key parameters of the system display specific relations irrespective of their absolute values. We find that within this parameter regime genotypes display low and noisy expression levels. Examining the phenotypic effects of mutations, we find an inverse correlation between robustness and evolvability that breaks only with nonlinearity in the network dynamics. Our results provide a possible solution to the robustness-evolvability trade-off, suggest an explanation for the ubiquity of nonlinear dynamics in gene expression networks, and generate useful guidelines for the design of synthetic gene circuits.

Reciprocity Between Robustness of Period and Plasticity of Phase in Biological Clocks

Circadian clocks exhibit the robustness of period and plasticity of phase against environmental changes such as temperature and nutrient conditions. Thus far, however, it is unclear how both are simultaneously achieved. By investigating distinct models of circadian clocks, we demonstrate reciprocity between robustness and plasticity: higher robustness in the period implies higher plasticity in the phase, where changes in period and in phase follow a linear relationship with a negative coefficient. The robustness of period is achieved by the adaptation on the limit cycle via a concentration change of a buffer molecule, whose temporal change leads to a phase shift following a shift of the limit-cycle orbit in phase space. Generality of reciprocity in clocks with the adaptation mechanism is confirmed with theoretical analysis of simple models, while biological significance is discussed.

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.

Light and circadian regulation of clock components aids flexible responses to environmental signals

The circadian clock measures time across a 24 h period, increasing fitness by phasing biological processes to the most appropriate time of day. The interlocking feedback loop mechanism of the clock is conserved across species; however, the number of loops varies. Mathematical and computational analyses have suggested that loop complexity affects the overall flexibility of the oscillator, including its responses to entrainment signals. We used a discriminating experimental assay, at the transition between different photoperiods, in order to test this proposal in a minimal circadian network (in Ostreococcus tauri) and a more complex network (in Arabidopsis thaliana). Transcriptional and translational reporters in O. tauri primarily tracked dawn or dusk, whereas in A. thaliana, a wider range of responses were observed, consistent with its more flexible clock. Model analysis supported the requirement for this diversity of responses among the components of the more complex network. However, these and earlier data showed that the O. tauri network retains surprising flexibility, despite its simple circuit. We found that models constructed from experimental data can show flexibility either from multiple loops and/or from multiple light inputs. Our results suggest that O. tauri has adopted the latter strategy, possibly as a consequence of genomic reduction.

Mathematical models light up plant signaling

Plants respond to changes in the environment by triggering a suite of regulatory networks that control and synchronize molecular signaling in different tissues, organs, and the whole plant. Molecular studies through genetic and environmental perturbations, particularly in the model plant Arabidopsis thaliana, have revealed many of the mechanisms by which these responses are actuated. In recent years, mathematical modeling has become a complementary tool to the experimental approach that has furthered our understanding of biological mechanisms. In this review, we present modeling examples encompassing a range of different biological processes, in particular those regulated by light. Current issues and future directions in the modeling of plant systems are discussed.

A positive feedback loop links circadian clock factor CLOCK-BMAL1 to the basic transcriptional machinery

Circadian clocks in mammals are built on a negative feedback loop in which the heterodimeric transcription factor circadian locomotor output cycles kaput (CLOCK)-brain, muscle Arnt-like 1 (BMAL1) drives the expression of its own inhibitors, the PERIOD and CRYPTOCHROME proteins. Reactivation of CLOCK-BMAL1 occurs at a specific time several hours after PERIOD and CRYPTOCHROME protein turnover, but the mechanism underlying this process is unknown. We found that mouse BMAL1 complexes include TRAP150 (thyroid hormone receptor-associated protein-150; also known as THRAP3). TRAP150 is a selective coactivator for CLOCK-BMAL1, which oscillates under CLOCK-BMAL1 transcriptional control. TRAP150 promotes CLOCK-BMAL1 binding to target genes and links CLOCK-BMAL1 to the transcriptional machinery at target-gene promoters. Depletion of TRAP150 caused low-amplitude, long-period rhythms, identifying it as a positive clock element. The activity of TRAP150 defines a positive feedback loop within the clock and provides a potential mechanism for timing the reactivation of circadian transcription.

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

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

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

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

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

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

Evolutionary systems biology: what it is and why it matters

Evolutionary systems biology (ESB) is a rapidly growing integrative approach that has the core aim of generating mechanistic and evolutionary understanding of genotype-phenotype relationships at multiple levels. ESB's more specific objectives include extending knowledge gained from model organisms to non-model organisms, predicting the effects of mutations, and defining the core network structures and dynamics that have evolved to cause particular intracellular and intercellular responses. By combining mathematical, molecular, and cellular approaches to evolution, ESB adds new insights and methods to the modern evolutionary synthesis, and offers ways in which to enhance its explanatory and predictive capacities. This combination of prediction and explanation marks ESB out as a research manifesto that goes further than its two contributing fields. Here, we summarize ESB via an analysis of characteristic research examples and exploratory questions, while also making a case for why these integrative efforts are worth pursuing.

Robustness of circadian timing systems evolves in the fruit fly Drosophila melanogaster as a correlated response to selection for adult emergence in a narrow window of time

Robustness is a fundamental property of biological timing systems that is likely to ensure their efficient functioning under a wide range of environmental conditions. Here we report the findings of our study aimed at examining robustness of circadian clocks in fruit fly Drosophila melanogaster populations selected to emerge as adults within a narrow window of time. Previously, we have reported that such flies display enhanced synchrony, accuracy, and precision in their adult emergence and activity/rest rhythms. Since it is expected that accurate and precise circadian clocks may confer enhanced stability in circadian time-keeping, we decided to examine robustness in circadian rhythms of flies from the selected populations by subjecting them to a variety of environmental conditions comprising of a range of photoperiods, light intensities, ambient temperatures, and constant darkness. The results revealed that adult emergence and activity/rest rhythms of flies from the selected stocks were more robust than controls, as they displayed enhanced stability under a wide variety of environmental conditions. These results suggest that selection for adult emergence within a narrow window of time results in the evolution of robustness in circadian timing systems of the fruit fly D. melanogaster.

Non-transcriptional oscillators in circadian timekeeping

Circadian clocks have evolved as an adaptation to life on a rotating planet, and orchestrate rhythmic changes in physiology to match the time of day. For decades, cellular circadian rhythms were considered to solely result from feedback between the products of rhythmically expressed genes. These transcriptional/translational feedback loops (TTFLs) have been ubiquitously studied, and explain the majority of circadian outputs. In recent years, however, non-transcriptional processes were shown to be major contributors to circadian rhythmicity. These key findings have profound implications on our understanding of the evolution and mechanistic basis of cellular circadian timekeeping. This review summarises and discusses these results and the experimental and theoretical evidence of a possible relation between non-transcriptional oscillator (NTO) mechanisms and TTFL oscillations.

Period coding of Bmal1 oscillators in the suprachiasmatic nucleus

Circadian oscillators in the suprachiasmatic nucleus (SCN) collectively orchestrate 24 h rhythms in the body while also coding for seasonal rhythms. Although synchronization is required among SCN oscillators to provide robustness for regular timekeeping (Herzog et al., 2004), heterogeneity of period and phase distributions is needed to accommodate seasonal variations in light duration (Pittendrigh and Daan, 1976b). In the mouse SCN, the heterogeneous phase distribution has been recently found in the cycling of clock genes Period 1 and Period 2 (Per1, Per2) and has been shown to reorganize by relative day lengths (Inagaki et al., 2007). However, it is not yet clearly understood what underlies the spatial patterning of Per1 and Per2 expression (Yamaguchi et al., 2003; Foley et al., 2011) and its plasticity. We found that the period of the oscillation in Bmal1 expression, a positive-feedback component of the circadian clock, preserves the behavioral circadian period under culture and drives clustered oscillations in the mouse SCN. Pharmacological and physical isolations of SCN subregions indicate that the period of Bmal1 oscillation is subregion specific and is preserved during culture. Together with computer simulations, we show that either the intercellular coupling does not strongly influence the Bmal1 oscillation or the nature of the coupling is more complex than previously assumed. Furthermore, we have found that the region-specific periods are modulated by the light conditions that an animal is exposed to. Based on these, we suggest that the period forms the basis of seasonal coding in the SCN.

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.

Evolutionary principles underlying structure and response dynamics of cellular networks

The network view in systems biology, in conjunction with the continuing development of experimental technologies, is providing us with the key structural and dynamical features of both cell-wide and pathway-level regulatory, signaling and metabolic systems. These include for example modularity and presence of hub proteins at the structural level and ultrasensitivity and feedback control at the level of dynamics. The uncovering of such features, and the seeming commonality of some of them, makes many systems biologists believe that these could represent design principles that underpin cellular systems across organisms. Here, we argue that such claims on any observed feature requires an understanding of how it has emerged in evolution and how it can shape subsequent evolution. We review recent and past studies that aim to achieve such evolutionary understanding for observed features of cellular networks. We argue that this evolutionary framework could lead to deciphering evolutionary origin and relevance of proposed design principles, thereby allowing to predict their presence or absence in an organism based on its environment and biochemistry and their effect on its future evolution.

Validation and invalidation of systems biology models using robustness analysis

Robustness, the ability of a system to function correctly in the presence of both internal and external uncertainty, has emerged as a key organising principle in many biological systems. Biological robustness has thus become a major focus of research in Systems Biology, particularly on the engineering-biology interface, since the concept of robustness was first rigorously defined in the context of engineering control systems. This review focuses on one particularly important aspect of robustness in Systems Biology, that is, the use of robustness analysis methods for the validation or invalidation of models of biological systems. With the explosive growth in quantitative modelling brought about by Systems Biology, the problem of validating, invalidating and discriminating between competing models of a biological system has become an increasingly important one. In this review, the authors provide a comprehensive overview of the tools and methods that are available for this task, and illustrate the wide range of biological systems to which this approach has been successfully applied.

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.

Quantitative analysis of regulatory flexibility under changing environmental conditions

The circadian clock controls 24-h rhythms in many biological processes, allowing appropriate timing of biological rhythms relative to dawn and dusk. Known clock circuits include multiple, interlocked feedback loops. Theory suggested that multiple loops contribute the flexibility for molecular rhythms to track multiple phases of the external cycle. Clear dawn- and dusk-tracking rhythms illustrate the flexibility of timing in Ipomoea nil. Molecular clock components in Arabidopsis thaliana showed complex, photoperiod-dependent regulation, which was analysed by comparison with three contrasting models. A simple, quantitative measure, Dusk Sensitivity, was introduced to compare the behaviour of clock models with varying loop complexity. Evening-expressed clock genes showed photoperiod-dependent dusk sensitivity, as predicted by the three-loop model, whereas the one- and two-loop models tracked dawn and dusk, respectively. Output genes for starch degradation achieved dusk-tracking expression through light regulation, rather than a dusk-tracking rhythm. Model analysis predicted which biochemical processes could be manipulated to extend dusk tracking. Our results reveal how an operating principle of biological regulators applies specifically to the plant circadian clock.

LIGHT-REGULATED WD1 and PSEUDO-RESPONSE REGULATOR9 form a positive feedback regulatory loop in the Arabidopsis circadian clock

In Arabidopsis thaliana, central circadian clock genes constitute several feedback loops. These interlocking loops generate an ~24-h oscillation that enables plants to anticipate the daily diurnal environment. The identification of additional clock proteins can help dissect the complex nature of the circadian clock. Previously, LIGHT-REGULATED WD1 (LWD1) and LWD2 were identified as two clock proteins regulating circadian period length and photoperiodic flowering. Here, we systematically studied the function of LWD1/2 in the Arabidopsis circadian clock. Analysis of the lwd1 lwd2 double mutant revealed that LWD1/2 plays dual functions in the light input pathway and the regulation of the central oscillator. Promoter:luciferase fusion studies showed that activities of LWD1/2 promoters are rhythmic and depend on functional PSEUDO-RESPONSE REGULATOR9 (PRR9) and PRR7. LWD1/2 is also needed for the expression of PRR9, PRR7, and PRR5. LWD1 is preferentially localized within the nucleus and associates with promoters of PRR9, PRR5, and TOC1 in vivo. Our results support the existence of a positive feedback loop within the Arabidopsis circadian clock. Further mechanistic studies of this positive feedback loop and its regulatory effects on the other clock components will further elucidate the complex nature of the Arabidopsis circadian clock.

Analysis of signal propagation in an elastic-tube flow model

We combine linear and nonlinear signal analysis techniques to investigate the transmission of pressure signals along a one-dimensional model of fluid flow in an elastic tube. We derive a simple, generally applicable measure for the robustness of a simulated vessel against in vivo pressure fluctuations, based on quantifying the degree of synchronization between proximal and distal pressure pulses. The practical use of this measure will be in its application to simulated pulses generated in response to a stochastic forcing term mimicking biological variations of root pressure in arterial blood flow.

Computational modeling of biological pathways by executable biology

"In silico" experiments (i.e., computer simulation) constitute an aid to traditional biological research, by allowing biologists to execute efficient simulations taking into consideration the data obtained in wet experiments and to generate new hypotheses, which can be later verified in additional wet experiments. In addition to being much cheaper and faster than wet experiments, computer simulation has other advantages: it allows us to run experiments in which several species can be monitored at the same time, to explore quickly various conditions by varying species and parameters in different runs, and in some cases to observe the behavior of the system at a greater level of detail than the one permitted by experimental techniques. In the past few years there has been a considerable effort in the computer science community to develop computational languages and software tools for modeling and analysing biochemical systems. Among the challenges which must be addressed in this context, there are: the definition of languages powerful enough to express all the relevant features of biochemical systems, the development of efficient algorithms to analyze models and interpret the results, and the implementation of modeling platforms which are usable by nonprogrammers. In this chapter, we focus on the use of computational modeling to the analysis of biochemical systems. Computational modeling, in conjunction with the use of formal intuitive modeling languages, enables biologists to define models using a notation very similar to the informal descriptions they commonly use, but formal and, hence, automatically executable. We describe the main features of the existing textual computational languages and the tool support available for model development and analysis.