An unbiased sensitivity analysis reveals important parameters controlling periodicity of circadian clock

To assess the importance of model parameters in kinetic models, sensitivity analysis is generally employed to provide key measures. However, it is quite often that no information is available for a significant number of parameters in biochemical models. Therefore, the results of sensitivity analysis that heavily rely on the accuracy of parameters are largely ambiguous. In this study, we propose a computational approach to determine the relative importance of parameters controlling the performance of the circadian clock in Drosophila. While previous attempts to sensitivity analysis largely depend on the knowledge of model parameters which are generally unknown, our study depicts a consistent picture of sensitivity assessment for a large number of parameters, even when the values of these parameters are not available in vivo. The resulting parametric sensitivity analysis suggests that PER/TIM negative loop is critical to maintain the stable periodicity of the circadian clock, which is consistent to the previously experimental and computational findings. Furthermore, our analysis generates a rich hypothesis of important parameters in the circadian clock that can be further tested experimentally. This approach can also be extended to assess the sensitivity of parameters in any biochemical system where a large number of parameters have unknown values. Biotechnol. Bioeng. 2010; 105: 250–259. © 2009 Wiley Periodicals, Inc.     

Quasi-multiparameter sensitivity measure for robustness analysis of complex biochemical networks

Robustness is the ability to resume reliable operation in the face of different types of perturbations. Analysis of how network structure achieves robustness enables one to understand and design cellular systems. It is typically true that all parameters simultaneously differ from their nominal values in vivo, but there have been few intelligible measures to estimate the robustness of a system's function to the uncertainty of all parameters.We propose a numerical and fast measure of a robust property to the uncertainty of all kinetic parameters, named quasi-multiparameter sensitivity (QMPS), which is defined as the sum of the squared magnitudes of single-parameter sensitivities. Despite its plain idea, it has hardly been employed in analysis of biological models. While QMPS is theoretically derived as a linear model, QMPS can be consistent with the expected variance simulated by the widely used Monte Carlo method in nonlinear biological models, when relatively small perturbations are given. To demonstrate the feasibility of QMPS, it is employed for numerical comparison to analyze the mechanism of how specific regulations generate robustness in typical biological models.QMPS characterizes the robustness much faster than the Monte Carlo method, thereby enabling the extensive search of a large parameter space to perform the numerical comparison between alternative or competing models. It provides a theoretical or quantitative insight to an understanding of how specific network structures are related to robustness. In circadian oscillators, a negative feedback loop with multiple phosphorylations is demonstrated to play a critical role in generating robust cycles to the uncertainty of multiple parameters.     

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

Background

Sensitivity and robustness are essential properties of circadian clock systems, enabling them to respond to the environment but resist noisy variations. These properties should be recapitulated in computational models of the circadian clock. Highly nonlinear kinetics and multiple loops are often incorporated into models to match experimental time-series data, but these also impact on model properties for clock models.

Methodology/Principal Findings

Here, we study the consequences of complicated structure and nonlinearity using simple Goodwin-type oscillators and the complex Arabidopsis circadian clock models. Sensitivity analysis of the simple oscillators implies that an interlocked multi-loop structure reinforces sensitivity/robustness properties, enhancing the response to external and internal variations. Furthermore, we found that reducing the degree of nonlinearity could sometimes enhance the robustness of models, implying that ad hoc incorporation of nonlinearity could be detrimental to a model''s perceived credibility.

Conclusion

The correct multi-loop structure and degree of nonlinearity are therefore critical in contributing to the desired properties of a model as well as its capacity to match experimental data.     

Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm

A mechanism for generating circadian rhythms has been of major interest in recent years. After the discovery of per and tim, a model with a simple feedback loop involving per and tim has been proposed. However, it is recognized that the simple feedback model cannot account for phenotypes generated by various mutants. A recent report by Glossop, Lyons & Hardin [Science286, 766 (1999)] on Drosophila suggests involvement of another feedback loop by dClk that is interlocked with per-tim feedback loop. In order to examine whether interlocked feedback loops can be a basic mechanism for circadian rhythms, a mathematical model was created and examined. Through extensive simulation and mathematical analysis, it was revealed that the interlocked feedback model accounts for the observations that are not explained by the simple feedback model. Moreover, the interlocked feedback model has robust properties in oscillations.     

果蝇昼夜节律的分子机制研究进展

果蝇由于遗传易操作性而成为一个研究昼夜节律分子机制的理想模式生物 . 到目前为止,通过遗传学和生物化学方法已经鉴定到 10 多个时钟基因 (clock genes) 和许多时钟相关基因,包括时钟输入基因和钟控基因 . 这些时钟基因以及它们的相应产物组成两个互相依赖的转录 / 翻译反馈环路,从而调节行为和生理的昼夜节律 . 果蝇这种核心钟的工作原理同样见于哺乳动物 .     

Network Features of the Mammalian Circadian Clock

The mammalian circadian clock is a cell-autonomous system that drives oscillations in behavior and physiology in anticipation of daily environmental change. To assess the robustness of a human molecular clock, we systematically depleted known clock components and observed that circadian oscillations are maintained over a wide range of disruptions. We developed a novel strategy termed Gene Dosage Network Analysis (GDNA) in which small interfering RNA (siRNA)-induced dose-dependent changes in gene expression were used to build gene association networks consistent with known biochemical constraints. The use of multiple doses powered the analysis to uncover several novel network features of the circadian clock, including proportional responses and signal propagation through interacting genetic modules. We also observed several examples where a gene is up-regulated following knockdown of its paralog, suggesting the clock network utilizes active compensatory mechanisms rather than simple redundancy to confer robustness and maintain function. We propose that these network features act in concert as a genetic buffering system to maintain clock function in the face of genetic and environmental perturbation.     

Stoichiometric relationship among clock proteins determines robustness of circadian rhythms

The mammalian circadian oscillator is primarily driven by an essential negative feedback loop comprising a positive component, the CLOCK-BMAL1 complex, and a negative component, the PER-CRY complex. Numerous studies suggest that feedback inhibition of CLOCK-BMAL1 is mediated by time-dependent physical interaction with its direct target gene products PER and CRY, suggesting that the ratio between the negative and positive complexes must be important for the molecular oscillator and rhythm generation. We explored this idea by altering expression of clock components in fibroblasts derived from Per2(Luc) and Per mutant mice, a cell system extensively used to study in vivo clock mechanisms. Our data demonstrate that the stoichiometric relationship between clock components is critical for the robustness of circadian rhythms and provide insights into the mechanistic organization of the negative feedback loop. Our findings may explain why certain mutant mice or cells are arrhythmic, whereas others are rhythmic, and suggest that robustness of circadian rhythms can be increased even in wild-type cells by modulating the stoichiometry.     

Robust entrainment of circadian oscillators requires specific phase response curves

The circadian clocks keeping time in many living organisms rely on self-sustained biochemical oscillations entrained by external cues, such as light, to the 24-h cycle induced by Earth's rotation. However, environmental cues are unreliable due to the variability of habitats, weather conditions, or cue-sensing mechanisms among individuals. A tempting hypothesis is that circadian clocks have evolved so as to be robust to fluctuations in the signal that entrains them. To support this hypothesis, we analyze the synchronization behavior of weakly and periodically forced oscillators in terms of their phase response curve (PRC), which measures phase changes induced by a perturbation applied at different times of the cycle. We establish a general relationship between the robustness of key entrainment properties, such as stability and oscillator phase, on the one hand, and the shape of the PRC as characterized by a specific curvature or the existence of a dead zone, on the other hand. The criteria obtained are applied to computational models of circadian clocks and account for the disparate robustness properties of various forcing schemes. Finally, the analysis of PRCs measured experimentally in several organisms strongly suggests a case of convergent evolution toward an optimal strategy for maintaining a clock that is accurate and robust to environmental fluctuations.     

‘Glocal’ Robustness Analysis and Model Discrimination for Circadian Oscillators

To characterize the behavior and robustness of cellular circuits with many unknown parameters is a major challenge for systems biology. Its difficulty rises exponentially with the number of circuit components. We here propose a novel analysis method to meet this challenge. Our method identifies the region of a high-dimensional parameter space where a circuit displays an experimentally observed behavior. It does so via a Monte Carlo approach guided by principal component analysis, in order to allow efficient sampling of this space. This ‘global’ analysis is then supplemented by a ‘local’ analysis, in which circuit robustness is determined for each of the thousands of parameter sets sampled in the global analysis. We apply this method to two prominent, recent models of the cyanobacterial circadian oscillator, an autocatalytic model, and a model centered on consecutive phosphorylation at two sites of the KaiC protein, a key circadian regulator. For these models, we find that the two-sites architecture is much more robust than the autocatalytic one, both globally and locally, based on five different quantifiers of robustness, including robustness to parameter perturbations and to molecular noise. Our ‘glocal’ combination of global and local analyses can also identify key causes of high or low robustness. In doing so, our approach helps to unravel the architectural origin of robust circuit behavior. Complementarily, identifying fragile aspects of system behavior can aid in designing perturbation experiments that may discriminate between competing mechanisms and different parameter sets.     

ELF4 is required for oscillatory properties of the circadian clock

Circadian clocks are required to coordinate metabolism and physiology with daily changes in the environment. Such clocks have several distinctive features, including a free-running rhythm of approximately 24 h and the ability to entrain to both light or temperature cycles (zeitgebers). We have previously characterized the EARLY FLOWERING4 (ELF4) locus of Arabidopsis (Arabidopsis thaliana) as being important for robust rhythms. Here, it is shown that ELF4 is necessary for at least two core clock functions: entrainment to an environmental cycle and rhythm sustainability under constant conditions. We show that elf4 demonstrates clock input defects in light responsiveness and in circadian gating. Rhythmicity in elf4 could be driven by an environmental cycle, but an increased sensitivity to light means the circadian system of elf4 plants does not entrain normally. Expression of putative core clock genes and outputs were characterized in various ELF4 backgrounds to establish the molecular network of action. ELF4 was found to be intimately associated with the CIRCADIAN CLOCK-ASSOCIATED1 (CCA1)/LONG ELONGATED HYPOCOTYL (LHY)-TIMING OF CAB EXPRESSION1 (TOC1) feedback loop because, under free run, ELF4 is required to regulate the expression of CCA1 and TOC1 and, further, elf4 is locked in the evening phase of this feedback loop. ELF4, therefore, can be considered a component of the central CCA1/LHY-TOC1 feedback loop in the plant circadian clock.     

Circadian systems and metabolism

Circadian systems direct many metabolic parameters and, at the same time, they appear to be exquisitely shielded from metabolic variations. Although the recent decade of circadian research has brought insights into how circadian periodicity may be generated at the molecular level, little is known about the relationship between this molecular feedback loop and metabolism both at the cellular and at the organismic level. In this theoretical paper, we conjecture about the interdependence between circadian rhythmicity and metabolism. A mathematical model based on the chemical reactions of photosynthesis demonstrates that metabolism as such may generate rhythmicity in the circadian range. Two additional models look at the possible function of feedback loops outside of the circadian oscillator. These feedback loops contribute to the robustness and sustainability of circadian oscillations and to compensation for long- and short-term metabolic variations. The specific circadian property of temperature compensation is put into the context of metabolism. As such, it represents a general compensatory mechanism that shields the clock from metabolic variations.     

Consistent robustness analysis (CRA) identifies biologically relevant properties of regulatory network models

Background

A number of studies have previously demonstrated that “goodness of fit” is insufficient in reliably classifying the credibility of a biological model. Robustness and/or sensitivity analysis is commonly employed as a secondary method for evaluating the suitability of a particular model. The results of such analyses invariably depend on the particular parameter set tested, yet many parameter values for biological models are uncertain.

Results

Here, we propose a novel robustness analysis that aims to determine the “common robustness” of the model with multiple, biologically plausible parameter sets, rather than the local robustness for a particular parameter set. Our method is applied to two published models of the Arabidopsis circadian clock (the one-loop [1] and two-loop [2] models). The results reinforce current findings suggesting the greater reliability of the two-loop model and pinpoint the crucial role of TOC1 in the circadian network.

Conclusions

Consistent Robustness Analysis can indicate both the relative plausibility of different models and also the critical components and processes controlling each model.