Date Published: February 2, 2017
Publisher: Public Library of Science
Author(s): Lili Wu, Qi Ouyang, Hongli Wang, Gennady Cymbalyuk.
Nearly all living systems feature a temperature-independent oscillation period in circadian clocks. This ubiquitous property occurs at the system level and is rooted in the network architecture of the clock machinery. To investigate the mechanism of this prominent property of the circadian clock and provide general guidance for generating robust genetic oscillators with temperature-compensated oscillations, we theoretically explored the design principle and core network topologies preferred by oscillations with a temperature-independent period. By enumerating all topologies of genetic regulatory circuits with three genes, we obtained four network motifs, namely, a delayed negative feedback oscillator, repressilator, activator-inhibitor oscillator and substrate-depletion oscillator; hybrids of these motifs constitute the vast majority of target network topologies. These motifs are biased in their capacities for achieving oscillations and the temperature sensitivity of the period. The delayed negative feedback oscillator and repressilator are more robust for oscillations, whereas the activator-inhibitor and substrate-depletion oscillators are superior for maintaining a temperature-independent oscillation period. These results suggest that thermally robust oscillation can be more plausibly achieved by hybridizing these two categories of network motifs. Antagonistic balance and temperature insulation mechanisms for achieving temperature compensation are typically found in these topologies with temperature robustness. In the temperature insulation approach, the oscillation period relies on very few parameters, and these parameters are influenced only slightly by temperature. This approach prevents the temperature from affecting the oscillation period and generates circadian rhythms that are robust against environmental perturbations.
Robustness against environmental perturbations, particularly ambient temperature variations, is a key property of living systems. Thermal robustness has been reported recently in the signaling process of bacterial chemotaxis in E. coli  and in Notch signaling in the development of Drosophila . A prominent and intensively investigated example of thermal robustness is temperature compensation in circadian clocks; circadian clocks are ubiquitous in life forms from bacteria to humans [3–5]. Despite temperature changes, circadian clocks maintain endogenous and robust rhythmic activities with a period of approximately 24 hours in harmony with the environmental daily rhythm. A temperature-independent period and entrainment by zeitgebers are two fundamental qualities of circadian clocks. Over the past two decades, the molecular basis of circadian clocks, which is generally a network of transcription-translation feedback loops [3, 6, 7], has been delineated using model organisms [8–10]. Several explanations for the phenomena of temperature compensation have been proposed. A popular and mathematically natural mechanism is antagonistic balance [11–21], in which the temperature-independent period is achieved by a delicate balance that requires fine-tuning of parameters. To account for robustness to mutations in circadian clock genes and, consequently, changes in kinetic rate constants and activation energies, a switch-like mechanism has been proposed . Another scheme without the need for fine-tuning parameter values was proposed for systems with several reactions catalyzed by a common enzyme, in which the temperature compensation is based on an enzyme-limited mechanism [23, 24]. A recent notable explanation attributed compensation to an adaptation that buffers temperature changes [25, 26] via a temperature-insensitive core oscillator coupled to a specific adaptive temperature signaling pathway.
From the general condition for TCOs (Eq 1), the temperature-independent oscillation period involves two key factors, the control coefficient Ci and the activation energy Ei. The value of Ei is locally determined by the properties of the chemical reaction steps, which depend on the specific protein structures. Mutations can alter the chemical properties of proteins and thus the values of Ei, and appropriate protein mutations would lead to satisfaction of Eq 1. In this paper, we focused on the role of the other key factor, i.e., the control coefficient Ci in Eq 1. In contrast to the factor Ei, which is local, the control coefficient depends on the whole network topologies of the underlying biochemical interactions. This dependence is complex and difficult to resolve analytically. The main hypothesis of this work is that the network architecture plays an important role in TCOs. To test this hypothesis, networks for topologies preferred by TCOs were enumerated in the present study. Second, the temperature enters our models in the form of Arrhenius law. This is a simplification and assumption of the influence of temperature on gene expression. The real situation would be much more complex because gene transcription, mRNA processing, translation, protein stability, and protein-protein interactions  all depend strongly on temperature.