Abstract
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 timedelayed 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.
Introduction
Oscillations are everywhere, both in the physical and biological domains. Essential biological processes such as cell cycles^{1}, pacemaker cell’s response^{2,3,4}, circadian rhythms^{5,6,7}, calcium oscillations^{8}, transcription factor responses^{9,10,11,12,13}, hormone secretion^{14}, fertility cycles^{15} are only a few examples of biological oscillations^{16}. In all of these systems the emergence of oscillations results from a delayed negative feedback loop (NFB)^{17}. Based on a study by Novak et al.^{18}, nonlinear reaction kinetics, and opposing chemical processes with correct balanced timeframes are also required to create oscillations.
Among different types of biological oscillations, circadian oscillation is widely spread, as the internal clock of most organisms gets synchronized to the environmental day/night cycle changes. Nearly all living organisms maintain an internal clock with a freerunning period close to 24 h^{19}. Even in the presence of metabolic fluctuations, the circadian clock functions as a precise biological timekeeper^{20}. An examination of the nitrogenfixation property of a cyanobacterium Synechococcus sp. provided the first solid evidence for circadian rhythms in prokaryotes^{21}. Genetic studies in the fruit fly Drosophila melanogaster^{6} and the model filamentous fungus, Neurospora crassa^{22}, which were later extended to mammals^{23}, revealed that circadian oscillations are formed at the molecular level. The core clock genes in mammals (Per1, Per2, Per3, Cry1, Cry2, Clock, Bmal1, Reverbα, and Rorα) cause the rhythmic gene expression and govern the physiological features of circadian rhythms^{24,25}. The 2017 Nobel Prize in Physiology or Medicine was awarded to three scientists (Jeffrey Hall, Michael Rosbash, and Michael Young) who identified the working principles of circadian clocks^{26}. One of the most wellknown fundamental aspects of circadian period homeostasis is temperature compensation^{27,28}, which is the maintenance of relatively constant period at varying temperatures. Another unique feature of a circadian oscillator is its robustness against random fluctuations^{29,30}. The network motifs or detailed mechanisms that determine temperature compensation and robustness of circadian rhythms remain largely unknown.
Detailed mathematical models have been proposed to explain the behavior of the circadian oscillator. Some of the models contain a relatively small number of species and interactions to generate sustained oscillations. Goodwin developed the simplest model for limit cycle oscillations caused by negative feedback on gene expression^{31,32,33}. Further studies suggest that the presence of positive feedback (PFB) in a circadian clock controls the regulation of the peak concentration of proteins^{34}. Combination of a PFB with substrate depletion can also cause sustained oscillation^{35}. Later advances have revealed that the presence of PFB improves the robustness of circadian oscillation. PFB induces oscillations in NFB loops without requiring a high degree of cooperativity coefficient (a Hill coefficient of at least 8)^{36}. This Hill coefficient can also be reduced by introducing more variables in a NFB model. Based on the underlying mechanism, positive feedback may be divided into three categories: selfactivation, Michaelis–Menten degradation, and crossactivation^{37}. Ferrell et al. implied that a circadian oscillatory network with interlinked positive and negative feedback is more suited for creating oscillations with configurable frequency and constant amplitude^{1}.
Several earlier and recent research^{1,38,39,40} have contributed to an understanding of the molecular regulatory processes driving biological oscillations, as well as their beneficial qualities such as robustness, tunability, entrainment, and temperature compensation^{41}. The Arrhenius law^{42} states that reaction rates increase with temperature^{43,44}. As a result, it is expected that oscillation periods would shorten as temperatures rise^{45}. In contrast, the circadian clock is temperature compensated and the period is relatively independent of the temperature within a physiological range^{45,46,47}. The primary goal of this paper is to compare key features of basic circadian oscillatory networks in order to determine network motifs that are crucial for the robustness and temperature compensation of oscillation periods.
Results
We present four, greatly divergent, minimalistic models of circadian oscillations and compare them, to gain a better understanding of the key dynamical features of circadian rhythms. Based on the literature^{48,49}, most biological clocks^{50} rely on transcriptionaltranslational negative feedback loops (TTFL). In contrast, the cyanobacterial circadian rhythm^{24} relies on posttranslational changes of a single protein species, making the clock a posttranslational oscillator (PTO), which is controlled by a positive feedback system^{48,51}. This PTO is driven by three clock proteins, KaiA, KaiB, and KaiC, of Synechococcus elongatus. The cyanobacteria circadian clock also demonstrates temperature compensated oscillations in an in vitro system^{52}. In this experiment, KaiA, KaiB, KaiC, and adenosine triphosphate (ATP) were mixed together to generate sustained oscillation. In vivo, other proteins also bind to distinct forms of KaiC during the day or night. KaiC is a hexamer subunit, and it contains two phosphorylation sites (T432 and S431; abbreviated as T and S). In 24 h, the states change like – U (doubly unphosphorylated state) → T (S/pT, single phosphorylated state) → ST (pS/pT, doubly phosphorylated state) → S (pS/T, single phosphorylated state) → U (where p stands for the phosphorylated site) through phosphorylation and dephosphorylation process^{21}. During the daytime, KaiC autophosphorylates, and during the night KaiC dephosphorylates through the same active site. According to Rust et al.^{53}, the auto kinase activity of KaiC is increased due to the presence of KaiA, whereas the auto phosphatase activity is enhanced by KaiB^{54,55}. Kinetic and biochemical evidence show that one of these phospho forms limits KaiA activity via interacting with KaiB, providing the critical feedback that sustains circadian oscillations^{53}. We consider the complex model by Rust et al., which contains several positive feedback loops, and denote this network in the rest of the text as cyanoKaiABC (Fig. 1a).
We also study the ‘conservative Goodwin oscillator,’ a simple and wellstudied version of a twocomponent^{17} negative feedback loop motif (Fig. 1b), which is a simplified version of the TTFL^{56}, for this theoretical analysis. In this specific TwoVariableGoodwinNFB model, mRNA (variable X) is transcribed from a gene and then translated into protein (variable Y). The latter functions as a repressor, inhibiting the production of mRNA. To avoid negative variable concentration, we employed modified twovariable Goodwin model equations with Michaelis–Menten degradation kinetics^{17}.
In a recent study by Hernansaiz et al.^{57} introduced an alternative to Rust’s model^{53} that includes all mass action reactions but a mix of positive and negative feedback to expand the notion of positiveplusnegative feedback oscillators while keeping the kinetics simplified. A single molecule in this model passes through four different chemically modified states dependent on the number of modifications it has: none (OO), single (OP/PO), or double (PP). This core regulator interacts with a molecule that has two states (A and B). We termed the oscillatory network as combined positiveplusnegative feedback (cPNFB) (Fig. 1c).
For considering a positive feedback induced oscillator, we adopt Selkov’s substrate depletion system. We termed the network as SelkovPFB (Fig. 1d). Because of the presence of substrate inhibition and product activation reactions, this network depicts a simplified kinetic model of an open monosubstrate enzyme (phosphofructokinase) that creates selfoscillations in glycolysis^{35}. We include this in our analysis of circadian clock models to consider an oscillator that relies solely on positive feedback and lacks a direct negative feedback loop. Still, in this case we also see that while Y affects U positively, U affects Y negatively (by turning it to U), which leads to the expected opposite sign cross effects, necessary to induce oscillations.
In our investigation, we compared these four minimalistic circadian oscillatory networks. The dynamical mathematical equations of these networks, containing terms for the temperature dependence of each parameter, are given in the methods section. The Time course plot for these oscillatory networks at 298 K temperature is shown in Supplementary Fig. 1. The period of oscillation at 298 K for all the models is 24 h. In terms of regulatory network design, this will allow us to better understand the effects of the key elements of the circadian clock, such as temperature compensation and robustness.
Robustness analysis
The four investigated models require special parameter combinations to create oscillations with 24 h period (Supplementary Table 1). From a physiological perspective, it is crucial to maintain this period in a noisy chemical environment. We test how the periods of oscillations respond to changes in parameters in the four investigated models. Following earlier ideas on measuring the robustness of biological networks^{58,59}, we generated 1000 random parameter combinations from a lognormal distribution of all parameters of the models to measure the robustness of biological systems against the extrinsic fluctuations in parameters (randomly chosen from a lognormal distribution for a multiplicative factor with means equals 1 and standard deviation 0.0142 for all parameters, Supplementary Note 1 for details). We calculated the total parameter variation following earlier works^{58,59}, essentially, calculated the arithmetic mean (Supplementary Note 2A) of the reaction rates and displayed it against the periods of oscillations (Fig. 2). This helps to quantify the noise in individual oscillatory networks. The plot shows that the TwoVariableGoodwinNFB model has a wider distribution while the cPNFB model has a narrower distribution in terms of the period of oscillations.
For a better quantitative comparison of the extrinsic noise tolerance of the networks, we computed the percentage of coefficient of variation (% CV) of oscillation periods in a narrower range of parameter variations. The percentage of coefficient of variation (% CV) indicates that the TwoVariableGoodwinNFB model is the noisiest (% CV = 1.8571). In other words, the TwoVariableGoodwinNFB model is the least robust model while the combined positiveplusnegative feedback (cPNFB) model is the most robust. Furthermore, we extended the investigation of the period of oscillation with regards to total parameter fluctuation to seven different temperatures ranging from 283 to 313 K (Supplementary Fig. 2a). We chose a wide range of temperatures as our analysis does not focus on any particular organism. For instance, cyanobacteria can tolerate a large range of temperatures^{60,61,62}. In Supplementary Fig. 2b we present the relationship of the percentage of coefficient of variation (% CV) of the oscillation period changes to the temperature changes in the four different oscillatory networks (Fig. 1).
At all temperatures, we see that the cPNFB model has the smallest variation of periods, while the TwoVariableGoodwinNFB model has the largest change with respect to the periods of oscillations. This is true for parameter variations at all temperatures between 283 and 313 K. Only the robustness of the cyanoKaiABC model shows a major temperature dependence, while others seem to have a quite temperature independent robustness (Supplementary Fig. 2b).
The SelkovPFB model (with the least number of parameters—i.e., four) and the cyanoKaiABC model with 12 parameters show similar distributions despite having different numbers of parameters. We wondered how the number of parameters affect the noise canceling capabilities of models. Bayesian Information Criterion (BIC) is widely used to compare models with altering parameter numbers. It is mostly used to identify which model gives the best fit to an observation, while also considering the number of fitted parameters in the compared models. Thus, BIC is a method that penalizes more complex models by imposing a penalty dependent on the number of parameters evaluated in the model. Following this idea, we examine the correlation between BIC of matching the desired 24 h period of oscillations by the various models (Supplementary Note 2B) at different temperatures (Supplementary Fig. 3). We notice that the TwoVariableGoodwinNFB model is still the noisiest at most temperatures (shows the largest BIC), irrespective of its size difference from others. The small SelkovPFB and the cPNFB models show the lowest noise to parameter changes in this analysis. Thus, we conclude that the number of parameters has no significant impact on the robustness assessment, but it seems the presence of positive feedback in the network leads to lower noise (Supplementary Fig. 3). This is in contradiction with textbook claims on the role of positive and negative feedback in noise increase and decrease, respectively^{63}. Here we found the TwoVariableGoodwinNFB model as the least robust, despite it is known that negative feedback can serve as a noise reducing motif^{64}. In the next steps we will test if negative feedbacks can help temperature compensation of circadian clock models.
Temperature compensation analysis
When checking the temperature dependence of robustness (Supplementary Fig. 2), we can already notice that the models’ average period is changing with temperature changes, but the level of this change is different in the four models. In order to estimate the temperature compensation properties of these biological oscillators, we have measured their temperature coefficient (Q_{10}) values^{65,66}. In terms of period length changes, we simplify the computation as follows^{65}–
Because circadian oscillators should maintain their approximately 24 h period, they need to be temperature compensated. We plotted how the periods of oscillations respond to temperature changes in all four models (Fig. 3). We can observe that all of these models perform some temperature compensation, we still observe a great effect of temperature changes on the periods of oscillations.
Individual temperature compensated reactions
Instead of expecting that the period of oscillations is temperature compensated by a compensatory effect of reaction rate responses to temperature change^{67}, we can consider that an individual reaction may be temperature compensated, which have a major effect in compensating period changes^{27,44,68}. Following ideas by Hong et al.^{69} and observations on temperature insensitivities of key circadian clock reaction^{70,71}, we have checked what happens if we fix the rates of a single reaction while allowing all others to respond to temperature change? The goal of this research is to identify the reactions which could be responsible for temperature compensation in the various oscillatory networks. We explored how far the period of oscillations varies with temperature in a pure NFB (Supplementary Fig. 4b) and pure PFB motif (Supplementary Fig. 4d) when a single rate is constant, and we also checked the same in complex oscillatory networks with several combinations of positive and negative feedback loops (Supplementary Fig. 4a, Supplementary Fig. 4c) as well. Figure 4 depicts the best performing models, where a single rate was assumed to be temperature independent, while all others respond to temperature changes.
It can be concluded that the TwoVariableGoodwinNFB model outperforms the others (Fig. 4) when the α_{2} reaction rate (the rate at which the activator X triggers the synthesis of its inhibitor, Y, blue background on Fig. 1b) of the model is temperature resistant. The other models also improved on their Q_{10} values but are still quite sensitive to temperature changes. Interestingly Rust’s cyanobacterial circadian clock model, with rate \(k_{\rm{DS}}^0\) fixed (the rate at which the doubly phosphorylated state ‘ST’ transitions to the singly phosphorylated state ‘S’, blue background on Fig. 1a), show a temperature dependence curve with an increase in periods at larger temperatures.
If we consider temperature independence of two reaction rates, then we observe strong temperature compensation of the periods of the oscillators (Fig. 5). The TwoVariableGoodwinNFB (plotted in red color) and cPNFB (plotted in blue color) exhibit the predicted decrease in period for temperature increase (Fig. 5). On the other hand, the SelkovPFB network (plotted in yellow) contains only four parameters, and it seems, if we fix two of those, the other two could be quite well temperature compensated, but the period increases as a function of temperature. The most complicated model in our analysis is the cyanoKaiABC model (shown in green), which has a complex temperature response what shows a minimum around 24 h. The cyanoKaiABC model works well between 283 and 298 K temperatures, however, the period of oscillation begins to differ significantly from 24 h at 303 K.
Supplementary Figs. 5–8 show all possible combinations of two temperature compensated parameters for the TwoVariableGoodwinNFB, cyanoKaiABC, cPNFB, and SelkovPFB motifs, respectively. By comparing Figs. 4 and 5, we can deduce that TwoVariableGoodwinNFB with a single temperature compensated parameter (α_{2}) performs almost identically to the twoparameter fixed scenario (k_{1}α_{2}), while the other three networks show further improvements for the fixation of a second rate (these rates are labeled by blue and yellow on Fig. 1). It can be observed that the α_{2} rate of the TwoVariableGoodwinNFB network is directly controlling the negative feedback loop. The reaction rates, which should be temperature independent in the cyanoKaiABC and cPNFB models are also controllers of negative feedback loops. There is no negative feedback loop in the SelkovPFB, there the most crucial parameter seems to be the one that controls the synthesis of the substrate of the positive feedback loop. Still in all models we see some improvements on temperature compensation with the fixation of a second reaction rate. The reduced values of Q_{10} in Fig. 5 make the periods of oscillations better temperature compensated than shown in Fig. 4, where only one rate was fixed in each model.
Separating the effects of positive and negative feedbacks
The four above investigated networks have different kinetics and complexity (Fig. 1), in order to have a systematic approach to study the role of positive and negative feedbacks in the robustness and temperature compensation of circadian clock models, we have extended our analysis with a Selkovlike positive feedbackbased model with an additional negative feedback loop (Fig. 6a). In order to ensure the compared models have similar sizes and kinetics, we keep the size of the network the same and control the strength of a single rate for the positive (k_{3}) and the negative feedback loop (k_{5}) (parameter values are indicated in the Supplementary Table 1E, and rate constants at 298 K are displayed in the Supplementary Table 1E). The temperature compensation analysis of the combined model (Fig. 6b) suggests that strong NFB with weak PFB (small k_{3}large k_{5} case, with red line) leads to better temperature compensation (lowest Q_{10} values) than other combinations (Fig. 6b). This matches with our above findings (Fig. 3) that negative feedback improves temperature compensation. A robustness analysis (Fig. 6c) in the presence of extrinsic noise^{72} shows that robustness of networks has a strong temperature dependence. Combination of positive and negative feedbacks (blue) shows the least temperature dependence on noise, while strong PFB alone (large k_{3}small k_{5} case, with yellow line) is less robust. An interesting observation is that strong NFB (small k_{3}large k_{5} case, with red line) becomes less robust (Fig. 6c) at higher temperatures. This is probably because at higher temperatures the faster reaction rates reduce the delays of feedback loops, which is one of the key components for autonomous oscillations and period determination.
The observation that negative feedback might increase noise contradicts earlier findings, which showed noise reducing capabilities of negative feedback loops^{64}. Specifically, previous work indicated that negative feedback loops could reduce cellcell variation due to intrinsic noises^{64,72}. To investigate potential roles of negative and positive feedback loops with respect to noise, we performed stochastic simulations of the combined Selkovlike positivenegative feedback model (Fig. 6a). Stochastic simulations of this system at various low molecular abundance levels reveal that at 298 K, strong NFB with weak PFB (small k_{3}large k_{5} case) results in the most robust against intrinsic noise, caused by low molecular abundances (Fig. 7).
Discussion
The mechanisms for temperature compensation of biological oscillators remain an unsolved puzzle. We have shown here that robustness and temperature compensation of circadian oscillators come at the cost of one another. It is clear from Fig. 2 and Fig. 3 that TwoVariableGoodwinNFB is the least robust for parameter fluctuations (possesses a high % CV value for period of oscillations) while it is better temperature compensated (lower Q_{10} value) compared to the others. On the other end of the scales, the cPNFB circadian oscillatory network is the most robust and least temperature compensated (Fig. 2 and Supplementary Figs. 2, 3). These are two extreme examples of robustness and temperature compensation in the four models that we interrogated. Based on these results and on the systematic analysis of a combined positive plus negative feedback systems (Fig. 6) we can conclude that pure NFB cannot reduce noise in the system efficiently (TwoVariableGoodwinNFB), whereas the presence of a PFB in the network increases robustness of the system (at high temperatures) without reducing much on the temperature compensating capabilities of negative feedback loops in the presence of extrinsic noise (Fig. 6). In contrast, a NFB increases the robustness of the system in the presence of intrinsic noise (Fig. 7).
The cyanoKaiABC model of Rust et al.^{53} is a quite complex model, containing both positive and negative feedbacks and highly nonlinear kinetics. This model shows good temperature compensation when \(k_{\rm{DS}}^0\) and \(k_{\rm{SU}}^0\) are both fixed (Fig. 5), but it is not the most robust or best temperature compensated model even though it has the highest number of parameters (k = 12). Furthermore, we believe that the linear mass action kinetics in the cPNFB circadian oscillatory network might be the distinguishing factor for its least noisy behavior when compared to the other models, where nonlinear terms might exaggerate any noise in the system (“Methods” section). Mass action kinetics provides precision even at low input signal values, whereas nonlinear kinetics such as Hill or Michaelis–Menten lack precision due to overly complicated combinations and scaling factors of different reaction rates used to define the kinetics^{73}.
Minimal models of the plant circadian clock with various levels of complexity were also tested for temperature compensation by Avello et al.^{74}. Our systematic analysis revealed that negative feedback loops are needed for good temperature compensation and the combination of positive and negative feedback loops can lead to lower noise in circadian clocks. We also see that temperature independence of reactions, which are crucial for negative feedback loops can help the whole systems to drive temperature compensated oscillations. These results could explain why in higher eukaryotes the classical delayed negative feedback loopbased system drives circadian clock, while in the cyanobacteria in vitro circadian oscillator we see a positive feedbackbased KaiC system, where the temperature compensation is achieved through temperature independence of the key reactions in the system^{67}. These results also support earlier ideas that positive feedbacks can make circadian clock more robust^{1,69}.
Feedback loops can be constructed in many different ways on the molecular level^{75}. Feedbacks can drive activation or inhibition steps or affect both. Furthermore, kinetics of reactions could also have an effect of the precise dynamics of oscillators. We have investigated here a wide selection of such systems but systematically tested only a single combination model (Figs. 6 and 7). Thus, our analysis has its limitations and could be further expanded by studying effects of various kinetics, regulatory effects, and basal reaction rates.
Despite these limitations, the proposed model engineering and comparing strategies can be useful for the designs of synthetic circuits, which are desired to act robustly against environmental fluctuations, including temperature changes. This could be relevant to engineering technologies of cyanobacteria species for biomass production, involving lightharvesting^{76}. We believe that our study will further support this blooming research field.
Methods
Here we present the mathematical models (A–D) of the four oscillatory networks (Fig. 1) and an additional oscillatory network described in Fig. 6a (E):

A.
cyanoKaiABC network
$$U = {\rm{kaiC}}  T  D  S$$(2)Where, U = unphosphorylated form; T = threonine phosphorylated kaiC; D = double phosphorylated kaiC (= ST); and S = serine phosphorylated kaiC. The dynamical equations and parameter values (Supplementary Table 1A) for this model are taken from the original model of Rust et al.^{53}.
$${\rm{kaiA}} = A = {{{\mathrm{max}}}}(0,A  2 \cdot S)$$(3)$$k_{ij} = k_{ij}^0 + k_{ij}^A \cdot \frac{A}{{k_{\rm{half}} + A}}$$(4)Where, i,j ε {U,T,D,S}.
$$\frac{{dT}}{{dt}} = k_{UT} \cdot U + k_{DT} \cdot ST  k_{TU} \cdot T  k_{TD} \cdot T$$(5)$$\frac{{dST}}{{dt}} = k_{TD} \cdot T + k_{SD} \cdot ST  k_{DT} \cdot ST  k_{DS} \cdot ST$$(6)$$\frac{{dS}}{{dt}} = k_{US} \cdot U + k_{DS} \cdot ST  k_{SU} \cdot S  k_{SD} \cdot S$$(7) 
B.
TwoVariableGoodwinNFB network
The dynamical equations for this model are taken from Gonze et al.^{17}. The parameters for this network are changed in such a way (Supplementary Table 1B) that the period of oscillations is 24 hours at 298 K temperature.
(8)(9) 
C.
cPNFB Network
The dynamical equations for this model are taken from Hernansaiz et al.^{57}. The parameters for this network were changed in such a way that the period of oscillations (Supplementary Table 1C) is 24 h at 298 K temperature.
$$\frac{{dOO}}{{dt}} = k_{02} \cdot B \cdot PP + d_1 \cdot OO \cdot PO  k_{01} \cdot A \cdot OO  p_0 \cdot OO \cdot PP$$(10)$$\frac{{dPP}}{{dt}} = k_{01} \cdot A \cdot OO + p_3 \cdot OP \cdot PP  k_{02} \cdot B \cdot PP  d_2 \cdot OO \cdot PP$$(11)$$\frac{{dOP}}{{dt}} = p_0 \cdot OO \cdot PP  p_3 \cdot OP \cdot PP$$(12)$$\frac{{dPO}}{{dt}} = d_2 \cdot OO \cdot PP  d_1 \cdot OO \cdot PO$$(13)$$\frac{{dB}}{{dt}} = k_{03} \cdot A \cdot PP  k_{04} \cdot B \cdot OO$$(14)$$\frac{{dA}}{{dt}} = k_{04} \cdot B \cdot OO  k_{03} \cdot A \cdot PP$$(15) 
D.
SelkovPFB Network
The dynamical equations for this model are modified from the original paper by SEL’KOV^{35}. The parameters for this network were changed in such a way (Supplementary Table 1D) that the period of oscillations is 24 h at 298 K temperature.
$$\frac{{dY}}{{dt}} = k_1  k_2 \cdot Y  k_3 \cdot U^2 \cdot Y$$(16)$$\frac{{dU}}{{dt}} = k_2 \cdot Y + k_3 \cdot U^2 \cdot Y  k_4 \cdot U$$(17) 
E.
Selkovlike PFB based model with an additional NFB loop
The mathematical model of Selkovlike PFB based model with an additional NFB loop (Fig. 6a) is presented as:
Here, \(k_3\) is the inherent activatory autocatalytic positivefeedback influence on the species U in a typical Selkov model (Fig. 1d). U exhibits an additional NFB by imposing inhibition to the synthesis of species Y with the rate \(k_5\). Because of the additional NFB in this model, the total parameter number (Supplementary Table 2) is equal to 5.
The mathematical equation represents above can be expressed in terms of number of molecules by introducing a scaling factor (\(V_s\)) on the both sides of the equation. With the Gillespie approach^{77}, we utilized the following equation for the robustness analysis illustrated in Fig. 7:
Here, \(N_Y\) and \(N_U\) are the number of Y and U species present in the system.
The calculated reaction rates that apply to low k_{3} or k_{5} values are roughly onefifteenth of the larger k_{3} or k_{5} values. With a low k_{3} value and a large k_{5} value, the system will have a dominating NFB (strong NFB – weak PFB), whereas with a large k_{3} and a low k_{5}, the system will have a dominant PFB (weak NFB – strong PFB). When both the rates are large and equal, the system will display the combined effect of a PFB and a NFB (strong NFB – strong PFB).
The parameter values for the three cases shown in Fig. 6b, c and Fig. 7 are indicated in the Supplementary Table 1E.
Simulation settings
The dynamical equations discussed above have been simulated in Matlab (ver. R2021b) using the ODE45 solver tools. Each reaction rate represented in the ordinary differential equations above, has been rewritten in an Arrhenius equation form, representing the temperature dependence of reaction rates as:
where the preexponential factor for all rates is, A_{e} = 383.83 (arbitrary unit, ‘A.U.’)^{27}, and this is the parameter that is randomly chosen for the robustness analysis. The gas constant = R = 8.3144598 J K^{−1} mol^{−1} and T = temperature is measured in Kelvin.
Data availability
All data used for the simulations are presented in the Supplementary Information.
Code availability
The Matlab code of simulated models are available at GitHub: https://github.com/SuchanaChakravarty/SystematicanalysisofNFBPFBloopsforrobustnesstemperaturecompensationincircadianrhythm.git.
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Acknowledgements
This work was supported by the National Research, Development and Innovation Office of Hungary (K_20 134489) and (Thematic Excellence Programme  TKP2020NKA11). C.I.H. was supported by R01 DK117005.
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S.C. and A.C.N. both designed the research; performed the research; analyzed data; wrote the manuscript; and revised the manuscript. C.I.H. designed the research and revised the manuscript.
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Chakravarty, S., Hong, C.I. & CsikászNagy, A. Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms. npj Syst Biol Appl 9, 5 (2023). https://doi.org/10.1038/s41540023002687
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DOI: https://doi.org/10.1038/s41540023002687
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