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

The circadian clock keeps its 24h period at a wide range of temperatures in the noisy environment of cells. Various models have been proposed to understand the behaviour of the controlling network of this periodic cycle. These models were compared here to identify which feedback loops are important for such robustness of the circadian clock.
Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms

What is circadian rhythm? Oscillations play a significant part in the physiological processes that occur within any living body. The circadian rhythm is one such example of an essential biological oscillator. Due to the presence of circadian oscillation among most organisms, the inherent biological clock is synced with the external diurnal clock. Nearly all living organisms have an internal biological clock that repeats every 24 hours. The body's internal clock controls a variety of physiological cycles, including the sleep-wake cycle, hormone cycle, metabolic cycle, and many more. Oscillation can be produced when there is a delayed negative feedback loop present in the system.

The circadian clock is made up of certain sets of proteins that are present in cells and tissues. Circadian oscillation takes place at the molecular level. The suprachiasmatic nucleus (SCN) in the hypothalamus is where mammals' core circadian clock resides. Clock, Bmal1, Per1, Per2, Per3, Cry1, Cry2, Rev-erbα, and Rorα are the essential clock genes that regulate circadian rhythm in mammals. One of the unique properties of the circadian clock is temperature compensation. In this scenario, the period of the clock is not affected by the temperature, but it still has the capacity to synchronize itself to the various temperature cycles. The other remarkable property of a circadian oscillator is its robustness towards cellular noise. The circadian oscillators are built differently for different organisms. For example, in eukaryotic systems, circadian oscillations are caused by a delayed negative feedback loop, while in cyanobacteria, the circadian clock is governed by a positive feedback loop.

Why are we interested in circadian oscillators? Through this study, we aim to understand more about how network design impacts behavioral changes in the circadian oscillator, particularly in terms of temperature compensation and robustness against biological noises. In order to understand the network properties required for the robustness and temperature compensation of the circadian clock, we studied four fundamental circadian clocks containing negative, positive, or a combination of both positive and negative feedback mechanisms.

We have looked at the circadian clock model of S. elongatus, which is made up of three clock proteins: KaiA, KaiB, and KaiC. We explored this complex network by Rust et al.1, which is comprised of numerous positive feedback chains, and refer to it as the cyano-KaiABC model (Fig. 1a) throughout the text. The "conservative Goodwin oscillator"2, a basic two-component negative-feedback loop motif, is also explored. In this text, we refer to this network as the Two-Variable-Goodwin-NFB model (Fig. 1b). Hernansaiz et al.3 have proposed an alternative to Rust's network that incorporates all mass action processes and makes them simpler. This network is composed of both positive and negative feedback loops. We have termed the network as combined positive-plus-negative feedback (cPNFB) loop model (Fig. 1c). We chose Selkov's substrate depletion4 system to think about an oscillator with positive feedback. We named the model as Selkov-PFB (Fig. 1d).

Fig. 1 Oscillatory networks consist of positive and negative feedback loops.

Cyano-KaiABC (a); Two-Variable-Goodwin-NFB (b); cPNFB (c); and Selkov-PFB (d) are shown in a schematic figure. A dual arrow pointing both ways represents the reversible reactions. The solid arrows show the direct processes (synthesis/degradation, phosphorylation/dephosphorylation), whereas the dashed arrows reflect the indirect regulatory reactions (activation (pointed arrowhead) or inhibition (blunt arrowhead)).

Findings and discussions: We found that pure negative feedback (Two-Variable-Goodwin-NFB) is best for temperature compensation among these networks (Fig. 2). Q10 measures how temperature influences any biological process and a lower Q10 indicates a greater degree of temperature compensation. 

Fig. 2 Temperature-dependent oscillation periods.

The oscillation period decreases as temperature increases in all four networks. The legend displays the Q10 values estimated for each model between 293K and 303K temperature.

We have also identified the individual reactions that are temperature compensated. The two-variable Goodwin-NFB model outperforms others in both the single (Fig. 3a) and two (Fig. 3b) temperature-independent reaction scenarios. Whether it’s α2 (Two-Variable-Goodwin-NFB) or  (cyano-KaiABC) reaction, these are the direct or indirect controllers of the negative feedback. All these primary and secondary temperature-independent reactions are highlighted with a shaded blue or yellow color in Fig. 1.

Fig. 3 Temperature compensated reactions.

The figure shows how much the oscillation period varies in all four oscillatory network motifs when the rate of a single reaction (a) or two reactions (b) is constant, but all others are permitted to respond to temperature variations. Q10 values for the corresponding reactions are displayed in the legend.

We further extended our research in the robustness analysis of these four networks against extrinsic noise. We found that Two-Variable-Goodwin-NFB model is least robust (having highest percentage co-efficient of variation ‘%CV’) whereas cPNFB model is the most robust network (having lowest %CV) (Fig. 4). It seems that having positive feedback in the system reduces extrinsic noise, which a negative feedback loop alone cannot do.

Fig. 4 Robustness analysis of four distinct oscillatory networks.

At 298K, the graph shows how the period of oscillations varies in response to total parameter variation (how far a randomly picked parameter deviates from the baseline value). The percentage of co-efficient of variation (% CV) for each of the 200 sampled parameter sets for each model between 0.005 and 0.015 for the total parameter variation is shown with the corresponding color.

Finally our study has been able to identify that a combination of positive and negative feedback loops can reduce noise better in circadian clocks and that negative feedback loops are necessary for better temperature compensation. We further demonstrated that reactions driven by negative feedback loops generate temperature compensated oscillations.


  1. Rust, M. J., Markson, J. S., Lane, W. S., Fisher, D. S. & O’Shea, E. K. Ordered phosphorylation governs oscillation of a three-protein circadian clock. Science 318, 809–812 (2007).
  2. Gonze, D. & Ruoff, P. The Goodwin Oscillator and its Legacy. Acta Biotheor. 69, 857–874 (2021).
  3. Hernansaiz-Ballesteros, R. D., Cardelli, L. & Csikász-Nagy, A. Single molecules can operate as primitive biological sensors, switches and oscillators. BMC Syst. Biol. 12, 70 (2018).
  4. SEL’KOV, E. E. Self-Oscillations in Glycolysis 1. A Simple Kinetic Model. Eur. J. Biochem. 4, 79–86 (1968).

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