Robustness and period sensitivity analysis of minimal models for biochemical oscillators

Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms

Temperature compensation and robustness to biological noise are two key characteristics of the circadian clock. These features allow the circadian pacemaker to maintain a steady oscillation in a wide range of environmental conditions. The presence of a time-delayed negative feedback loop in the regulatory network generates autonomous circadian oscillations in eukaryotic systems. In comparison, the circadian clock of cyanobacteria is controlled by a strong positive feedback loop. Positive feedback loops with substrate depletion can also generate oscillations, inspiring other circadian clock models. What makes a circadian oscillatory network robust to extrinsic noise is unclear. We investigated four basic circadian oscillators with negative, positive, and combinations of positive and negative feedback loops to explore network features necessary for circadian clock resilience. We discovered that the negative feedback loop system performs the best in compensating temperature changes. We also show that a positive feedback loop can reduce extrinsic noise in periods of circadian oscillators, while intrinsic noise is reduced by negative feedback loops.

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.

Theoretical study on the regulation of circadian rhythms by RNA methylation

Messenger RNAs are often destabilized by methylation, suggesting that mRNA methylation alters mRNA and protein dynamics. This may indicate that the gene regulatory system is reflected by the metabolic system through mRNA methylation because methylation substrates are components of the metabolic system. Elucidating the mechanisms by which mRNA methylation regulates gene regulatory systems has posed considerable challenges due to the numerous targets of mRNA methylation. Recent studies have demonstrated that inhibition of mRNA N6-methyladenosine methylation elongates circadian periods. The aim of this study was to understand the mechanisms by which mRNA methylation regulates circadian rhythms. Using a detailed realistic model and a simple model, we demonstrated that period elongation of circadian rhythms by decreasing mRNA methylation can be achieved by two possibilities, i.e., decreasing mRNA methylation stabilizes nonoscillatory mRNAs such as Ck1δ and/or stabilizes oscillatory mRNAs of clock genes such as Per and Cry. In addition, we predicted that period elongation by stabilizing nonoscillatory mRNA (Ck1δ) should always be accompanied by the distortion of the circadian waveform. Finally, we discuss the validity of the two possible mechanisms on the regulation of circadian rhythms by mRNA methylation by quantifying waveform distortion of circadian gene activity data with or without mRNA methylation inhibitors.

Robustness, Entrainment, and Hybridization in Dissipative Molecular Networks, and the Origin of Life

How simple chemical reactions self-assembled into complex, robust networks at the origin of life is unknown. This general problem-self-assembly of dissipative molecular networks-is also important in understanding the growth of complexity from simplicity in molecular and biomolecular systems. Here, we describe how heterogeneity in the composition of a small network of oscillatory organic reactions can sustain (rather than stop) these oscillations, when homogeneity in their composition does not. Specifically, multiple reactants in an amide-forming network sustain oscillation when the environment (here, the space velocity) changes, while homogeneous networks-those with fewer reactants-do not. Remarkably, a mixture of two reactants of different structure-neither of which produces oscillations individually-oscillates when combined. These results demonstrate that molecular heterogeneity present in mixtures of reactants can promote rather than suppress complex behaviors.

A comparison of deterministic and stochastic approaches for sensitivity analysis in computational systems biology

With the recent rising application of mathematical models in the field of computational systems biology, the interest in sensitivity analysis methods had increased. The stochastic approach, based on chemical master equations, and the deterministic approach, based on ordinary differential equations (ODEs), are the two main approaches for analyzing mathematical models of biochemical systems. In this work, the performance of these approaches to compute sensitivity coefficients is explored in situations where stochastic and deterministic simulation can potentially provide different results (systems with unstable steady states, oscillators with population extinction and bistable systems). We consider two methods in the deterministic approach, namely the direct differential method and the finite difference method, and five methods in the stochastic approach, namely the Girsanov transformation, the independent random number method, the common random number method, the coupled finite difference method and the rejection-based finite difference method. The reviewed methods are compared in terms of sensitivity values and computational time to identify differences in outcome that can highlight conditions in which one approach performs better than the other.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

Conservation and Divergence of p53 Oscillation Dynamics across Species

The tumor-suppressing transcription factor p53 is highly conserved at the protein level and plays a key role in the DNA damage response. One important aspect of p53 regulation is its dynamics in response to DNA damage, which include oscillations. Here, we observe that, while the qualitative oscillatory nature of p53 dynamics is conserved across cell lines derived from human, monkey, dog, mouse, and rat, the oscillation period is variable. Specifically, rodent cells exhibit rapid p53 oscillations, whereas dog, monkey, and human cells show slower oscillations. Computational modeling and experiments identify stronger negative feedback between p53 and MDM2 as the driver of faster oscillations in rodents, suggesting that the period of oscillation is a network-level property. In total, our study shows that despite highly conserved signaling, the quantitative features of p53 oscillations can diverge across evolution. We caution that strong amino acid conservation of proteins and transcriptional network similarity do not necessarily imply conservation of time dynamics.

A novel interaction perturbation analysis reveals a comprehensive regulatory principle underlying various biochemical oscillators

Background

Biochemical oscillations play an important role in maintaining physiological and cellular homeostasis in biological systems. The frequency and amplitude of oscillations are regulated to properly adapt to environments by numerous interactions within biomolecular networks. Despite the advances in our understanding of biochemical oscillators, the relationship between the network structure of an oscillator and its regulatory function still remains unclear. To investigate such a relationship in a systematic way, we have developed a novel analysis method called interaction perturbation analysis that enables direct modulation of the strength of every interaction and evaluates its consequence on the regulatory function. We have applied this new method to the analysis of three representative types of oscillators.

Results

The results of interaction perturbation analysis showed different regulatory features according to the network structure of the oscillator: (1) both frequency and amplitude were seldom modulated in simple negative feedback oscillators; (2) frequency could be tuned in amplified negative feedback oscillators; (3) amplitude could be modulated in the incoherently amplified negative feedback oscillators. A further analysis of naturally-occurring biochemical oscillator models supported such different regulatory features according to their network structures.

Conclusions

Our results provide a clear evidence that different network structures have different regulatory features in modulating the oscillation frequency and amplitude. Our findings may help to elucidate the fundamental regulatory roles of network structures in biochemical oscillations.

Electronic supplementary material

The online version of this article (10.1186/s12918-017-0472-7) contains supplementary material, which is available to authorized users.

Classical Mechanics Approach Applied to Analysis of Genetic Oscillators

Biological oscillators present a fundamental part of several regulatory mechanisms that control the response of various biological systems. Several analytical approaches for their analysis have been reported recently. They are, however, limited to only specific oscillator topologies and/or to giving only qualitative answers, i.e., is the dynamics of an oscillator given the parameter space oscillatory or not. Here, we present a general analytical approach that can be applied to the analysis of biological oscillators. It relies on the projection of biological systems to classical mechanics systems. The approach is able to provide us with relatively accurate results in the meaning of type of behavior system reflects (i.e., oscillatory or not) and periods of potential oscillations without the necessity to conduct expensive numerical simulations. We demonstrate and verify the proposed approach on three different implementations of amplified negative feedback oscillator.

Feedback, Mass Conservation and Reaction Kinetics Impact the Robustness of Cellular Oscillations

Oscillations occur in a wide variety of cellular processes, for example in calcium and p53 signaling responses, in metabolic pathways or within gene-regulatory networks, e.g. the circadian system. Since it is of central importance to understand the influence of perturbations on the dynamics of these systems a number of experimental and theoretical studies have examined their robustness. The period of circadian oscillations has been found to be very robust and to provide reliable timing. For intracellular calcium oscillations the period has been shown to be very sensitive and to allow for frequency-encoded signaling. We here apply a comprehensive computational approach to study the robustness of period and amplitude of oscillatory systems. We employ different prototype oscillator models and a large number of parameter sets obtained by random sampling. This framework is used to examine the effect of three design principles on the sensitivities towards perturbations of the kinetic parameters. We find that a prototype oscillator with negative feedback has lower period sensitivities than a prototype oscillator relying on positive feedback, but on average higher amplitude sensitivities. For both oscillator types, the use of Michaelis-Menten instead of mass action kinetics in all degradation and conversion reactions leads to an increase in period as well as amplitude sensitivities. We observe moderate changes in sensitivities if replacing mass conversion reactions by purely regulatory reactions. These insights are validated for a set of established models of various cellular rhythms. Overall, our work highlights the importance of reaction kinetics and feedback type for the variability of period and amplitude and therefore for the establishment of predictive models.

Rhythmic behavior is omnipresent in biology and has many crucial functions. In cells the activation levels and abundances of signaling molecules such as NF-κB, p53, EGFR or calcium repeatedly increase and decrease in response to stimuli. Such a dynamic behavior can also be observed monitoring the concentrations of mRNAs and proteins in the circadian clock and the cell cycle. Period and amplitude which are the time span between peaks and the peak height, respectively, as well as their variabilities are important features of oscillations. The circadian period is very stable allowing for proper time keeping, whereas in calcium signaling the period is very variable encoding different stimulation strengths. Our goal is to examine the origin of differences in sensitivities of periods and amplitudes using a computational approach. We use prototype oscillators and demonstrate that they can be used to derive general principles that explain the degree of robustness in period and amplitude for a set of commonly used models of cellular oscillators. Our findings imply that the robustness of oscillating systems can be influenced by feedback type and kinetic properties to which special attention should be paid when designing mathematical models of cellular rhythms.