The circadian clock controls many physiological processes in higher plants and causes a large fraction of the genome to be expressed with a 24h rhythm. The transcripts encoding the RNA-binding proteins AtGRP7 (Arabidopsis thaliana Glycine Rich Protein 7) and AtGRP8 oscillate with evening peaks. The circadian clock components CCA1 and LHY negatively affect AtGRP7 expression at the level of transcription. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate post-transcriptionally: high protein levels promote the generation of an alternative splice form that is rapidly degraded. This clock-regulated feedback loop has been proposed to act as a molecular slave oscillator in clock output. While mathematical models describing the circadian core oscillator in Arabidopsis thaliana were introduced recently, we propose here the first model of a circadian slave oscillator. We define the slave oscillator in terms of ordinary differential equations and identify the model's parameters by an optimization procedure based on experimental results. The model successfully reproduces the pertinent experimental findings such as waveforms, phases, and half-lives of the time-dependent concentrations. Furthermore, we obtain insights into possible mechanisms underlying the observed experimental dynamics: the negative auto-regulation and reciprocal cross-regulation via alternative splicing could be responsible for the sharply peaking waveforms of the AtGRP7 and AtGRP8 mRNA. Moreover, our results suggest that the AtGRP8 transcript oscillations are subordinated to those of AtGRP7 due to a higher impact of AtGRP7 protein on alternative splicing of its own and of the AtGRP8 pre-mRNA compared to the impact of AtGRP8 protein. Importantly, a bifurcation analysis provides theoretical evidence that the slave oscillator could be a toggle switch, arising from the reciprocal cross-regulation at the post-transcriptional level. In view of this, transcriptional repression of AtGRP7 and AtGRP8 by LHY and CCA1 induces oscillations of the toggle switch, leading to the observed high-amplitude oscillations of AtGRP7 mRNA.
The circadian clock organizes the day in the life of a plant by causing 24h rhythms in gene expression. For example, the core clockwork of the model plant Arabidopsis thaliana causes the transcripts encoding the RNA-binding proteins AtGRP7 and AtGRP8 to undergo high amplitude oscillations with a peak at the end of the day. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate their own expression by causing alternative splicing of their pre-mRNAs, followed by rapid degradation of the alternatively spliced transcripts. This has led to the suggestion that they represent molecular slave oscillators downstream of the core clock. Using a mathematical model we obtain insights into possible mechanisms underlying the experimentally observed dynamics, e.g. a higher impact of AtGRP7 protein compared to the impact of AtGRP8 protein on the alternative splicing explains the experimentally observed phases of their transcript. Previously, components that reciprocally repress their own transcription (double negative loops) have been shown to potentially act as a toggle switch between two states. We provide theoretical evidence that the slave oscillator could be a bistable toggle switch as well, operating at the post-transcriptional level.
Citation: Schmal C, Reimann P, Staiger D (2013) A Circadian Clock-Regulated Toggle Switch Explains AtGRP7 and AtGRP8 Oscillations in Arabidopsis thaliana. PLoS Comput Biol 9(3): e1002986. doi:10.1371/journal.pcbi.1002986
Editor: Christopher V. Rao, University of Illinois at Urbana-Champaign, United States of America
Received: November 10, 2012; Accepted: January 29, 2013; Published: March 28, 2013
Copyright: © 2013 Schmal et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: CS is a fellow of the International Graduate Program “Bioinformatics of Signaling Networks” funded by Bielefeld University. This work was supported by the DFG through grants STA 653/2 (DS) and SFB613 (DS and PR). We acknowledge the support for the publication fee by the DFG and the Open Access Publication Funds of Bielefeld University. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Circadian clocks are endogenous timekeepers that can be found among all taxa of life –. They are able to generate stable oscillations with a period of approximately 24h that persist even under constant (free-running) conditions, i.e. in the absence of any rhythmic environmental influences that impact the clock. Entrainment by environmental signals such as light and temperature can synchronize the clock to the period of the Earth's rotation. Such a clockwork may confer a higher fitness to an organism as it allows to anticipate daily cycles of light and temperature in a spinning world , .
Circadian clocks are usually described as molecular networks including (interlocked) transcriptional - translational feedback loops . In the higher plants model organism Arabidopsis thaliana an interplay of experiments and mathematical modeling shaped the current view on the circadian clock's network –. Locke et al. first modeled the structure of the circadian clock as a “simple” two-gene negative feedback loop , where the two partially redundant MYB transcription factors LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1) (combined to one variable LHY/CCA1) inhibit the transcription of their activator TIMING OF CAB EXPRESSION 1 (TOC1). However, in silico and experimental mutant analysis revealed inconsistencies between the model and data , . The assumed circadian clock architecture was therefore extended in successive steps – from this simple design to the idea of a clockwork that has a repressilator-like architecture at its core . In this recent picture a “morning loop” consists of the morning-expressed genes LHY/CCA1 that activate the transcription of the PSEUDO RESPONSE REGULATORS 9, 7 and 5 (PRR9, PRR7 and PRR5) which in turn inhibit the transcription of LHY/CCA1. Furthermore, LHY/CCA1 is assumed to repress the transcription of the “evening loop” genes EARLY FLOWERING 3 (ELF3) and 4 (ELF4), LUX ARRHYTHMO (LUX), GIGANTEA (GI), and TOC1, respectively. ELF3, ELF4 and LUX form a protein complex (evening complex, EC) that inhibits the transcription of PRR9, thereby connecting the evening loop with the morning loop, which closes the feedback loop circuitry .
The circadian clock affects many physiological processes in Arabidopsis thaliana, including the oscillation of free cytosolic calcium , stomatal opening, cotyledon and leaf movement , and even enables the plant to measure day-length, track seasons and thereby triggers the onset of flowering . Underlying these physiological rhythms is a widespread control of gene expression by the circadian clock . However, it is still not completely understood how the rhythmicity of the circadian clock is transmitted to its output genes. This may occur either directly by binding of clock proteins to their target genes or indirectly via signal transduction chains. One possibility to maintain the rhythmicity along such a signal transduction chain could be via slave oscillators that are driven by the circadian core oscillator and shape their oscillatory profile due to negative auto-regulation. Colin Pittendrigh already proposed in 1981 that “ any feedback loop in the organism is a potential slave oscillator, and if the circadian pacemaker can make input to the loop, the slave will assume a circadian period and become part of the temporal program that the pacemaker drives” . Genetic variation in such a slave oscillator can change its properties, e.g. the phase relation to the core oscillator, and thus the organisms' “ temporal program is open to evolutionary adjustment”  without the need for change in the core oscillator itself. Since driven by the core oscillator, the slave oscillator does not have to share all of the core oscillator's properties: It is not necessary that the slave oscillator exhibits independent self-sustaining oscillations, shows temperature compensation, or gains direct input from light , . On the other hand, an indispensable pre-requisite of a slave oscillator is that it must not to act in any way back onto the core oscillator.
The two RNA binding proteins Arabidopsis thaliana Glycine Rich Protein 7 and 8 (AtGRP7 and AtGRP8), also known as Cold and Circadian-Regulated 2 and 1 (CCR2 and CCR1), respectively, have been proposed to represent such a molecular slave oscillator –. These proteins share 77 percent of sequence identity and contain an approximately 80 amino acid long RNA-recognition motif at the amino-terminus and a carboxy terminus mainly consisting of glycine repeats , . The transcripts of both genes undergo circadian oscillations with evening peaks. The maximum of AtGRP8 slightly precedes that of AtGRP7 by 1–2 hours . The AtGRP7 protein oscillates with a four hour delay compared to its transcript . In plants constitutively over-expressing CCA1  or LHY , AtGRP7 mRNA oscillations are dampened under constant light conditions, approaching the trough value of their corresponding oscillations in wild type plants, and thus suggesting that the transcription of AtGRP7 is rhythmically repressed rather than activated by these partially redundant core oscillator genes. Apart from this transcriptional regulation AtGRP7 also negatively auto-regulates the steady-state abundance of its own mRNA via a post-transcriptional mechanism . When AtGRP7 protein levels are high, an alternatively spliced transcript is produced at the expense of the fully spliced mRNA . This alternative splice form is generated through the use of an alternative 5′ splice site and retains part of the intron. Due to a premature termination codon this alternatively spliced transcript cannot be translated into functional protein and is rapidly degraded via the nonsense-mediated decay (NMD) pathway , . Since AtGRP7 binds to its own transcript in vitro and in vivo, this alternative splicing likely is promoted by direct binding of AtGRP7 to its own pre-mRNA , . AtGRP8 also auto-regulates itself and both proteins cross-regulate each other by the same mechanism. Our regulatory network is therefore composed of two auto-regulatory negative feedback loops, interlocked with each other and driven by the circadian core oscillator, as depicted in Figure 1.
A) The circadian core oscillator is synchronized to the rhythm of a given external zeitgeber signal. It drives the slave oscillator composed of AtGRP7 and AtGRP8 (since the core oscillator genes LHY/CCA1 are assumed to inhibit the transcription of AtGRP7 and AtGRP8). AtGRP7 and AtGRP8 negatively auto-regulate and cross-regulate each other. B) The negative auto-regulation and cross-regulation involves an alternative splicing mechanism coupled to NMD : The AtGRP7 pre-mRNA consists of two exons (green), separated by an intron (yellow) and bounded by the and untranslated region (UTR) (gray). Its mature mRNA, with the intron completely spliced out, can produce functional protein (red). Both AtGRP7 as well as AtGRP8 protein can bind the AtGRP7 pre-mRNA and induce the production of an alternatively spliced mRNA variant, retaining the first half of the intron. This alternatively spliced mRNA cannot produce functional protein due to a premature termination codon and is degraded via NMD.
Apart from the negative auto-regulation, AtGRP7 affects the accumulation of a suite of circadian clock regulated genes in a time-of-day dependent manner, supporting the hypothesis that it acts as a slave oscillator between the core oscillator and the clock output: Rhythmic transcripts, whose steady state abundance is reduced upon AtGRP7 overexpression, peak in the evening like AtGRP7 itself, whereas rhythmic transcripts with an elevated steady state abundance peak out of phase towards the morning .
Furthermore, it has been shown that AtGRP7 has an impact on various other physiological processes: It promotes the floral transition , plays a role in the plants innate immune system , , and is known to mediate responses to stresses such as oxidative stress, high salt, mannitol, or cold , , .
Recently, various mathematical models for the circadian core oscillator in Arabidopsis thaliana have been developed –. In this paper we model the AtGRP7 and AtGRP8 feedback loops in terms of ordinary differential equations and thus propose the first mathematical model of a molecular slave oscillator in Arabidopsis thaliana. We note that a related model of a clock-controlled system has been put forward by Salazar et al. . The molecular components of this system do not incorporate any feedback mechanism and are therefore unable to reshape their own oscillatory profile. Thus, they do not adopt all of the above mentioned specifications of a slave oscillator.
Modeling the AtGRP7-AtGRP8 Interlocked Feedback Loops
In order to model the essential layers of AtGRP7 and AtGRP8 regulation we need six dynamical variables, namely the concentrations of the pre-mRNA (), mRNA (), and protein () of AtGRP7 and AtGRP8. In the absence of any measured data that distinguish between cytoplasmic and nuclear protein concentrations, we, in particular, do not take into account that AtGRP7 and AtGRP8 localize to both the nucleus and the cytoplasm , , as it was done e.g., in . The driving force of the AtGRP7 oscillations is the periodic change in protein concentration of the core oscillator components LHY/CCA1, combined into one variable . Throughout the first part of the paper we adopt the previously established mathematical model of Pokhilko et al. . In principle, one could also use any other time periodic function or generic oscillator model that properly imitates the observed protein concentration for a given experimental situation. Two examples of this type are a modified Poincaré oscillator and the refined model of Pokhilko et al.  as considered towards the end of our paper (see section Robustness Against Variations in the LHY/CCA1 Protein Oscillations).
The original model provided by Pokhilko et al.  involves dynamical variables and parameters whose quantitative values are taken over from that paper. Likewise, we utilize the same specific initial conditions for the core oscillator as in . The externally imposed light input consists of either constant light (LL) or diurnal conditions such as 12 hours of light and 12 hours of darkness (abbreviated as ) or 8 hours of light and 16 hours of darkness (), also denoted as short day conditions. These light conditions enter our core oscillator dynamics as detailed in  (especially continuous transitions instead of binary, i.e. on–off, light-dark transitions are used). Typical examples of the protein concentrations obtained in this way are depicted as dashed lines in Figure 2. In view of the fact that the AtGRP7 mRNA steady state abundance seems not to be light-induced (unpublished data) we assume no direct light effect on the slave oscillator. This assumption is also coherent with Pittendrigh's definition, proposing that the slave oscillator could receive the light input only indirectly via the core oscillator .
Solid lines denote solutions of equations (1)–(6) for the “optimal” parameter set from Table 1: A) AtGRP7 pre-mRNA (), mRNA (), and protein () concentrations. B) AtGRP8 pre-mRNA (), mRNA (), and protein () concentrations. Dashed lines denote the protein concentration of the core oscillator gene LHY/CCA1. Shown are the last two days in 12h∶12h LD conditions () and the first four days after switching to constant light conditions (). Throughout this paper, a gray-shaded background indicates darkness.
Given the input of the core oscillator to the AtGRP7 and AtGRP8 feedback loops, we model the temporal evolution of the slave oscillator's dynamical variables , , , , , and as follows(1)(2)(3)(4)(5)(6)
Consistent with other circadian clock models –, in the first term on the right-hand-side of equation (1) we use a sigmoidal Hill repressor function, describing the negative regulation of AtGRP7 transcription by LHY/CCA1. The pertinent transcription rate of AtGRP7 is then given in terms of the maximal transcription rate , the Hill coefficient , the activation coefficient , and the LHY/CCA1 protein concentration . The loss term in equation (1) describes the normal and alternative splicing of AtGRP7 pre-mRNA. It is assumed that the AtGRP7 pre-mRNA is either spliced into its mature mRNA or into its alternative splice form, without considering any further degradation pathway. The kinetics for the splicing of the AtGRP7 pre-mRNA into its alternative splice form, promoted by the binding of AtGRP7 protein to its own pre-mRNA, is assumed to depend on the splicing coefficient and the concentrations of the AtGRP7 pre-mRNA () and protein (). Equivalent kinetics are used for the alternative splicing of AtGRP7 pre-mRNA promoted by the binding of AtGRP8 protein. Note that is the coupling parameter between AtGRP8 and AtGRP7, i.e. the impact of AtGRP8 on alternative splicing of the AtGRP7 pre-mRNA. The normal splicing of AtGRP7 pre-mRNA into its mature mRNA is supposed to depend on a splicing coefficient as well as the pre-mRNA concentration and appears as the gain term in the first part of equation (2). The second part of equation (2) describes the mRNA degradation as Michaelis-Menten kinetics that account for saturation by means of the Michaelis constant and the maximal degradation rate . A similar Michalis-Menten degradation appears in equation (3), while describes the translation of mRNA into protein. Analogous considerations apply to equations (4)–(6), modeling AtGRP8. As usual, all the kinetic parameters in (1)–(6) are tacitly restricted to positive real values.
Collecting all kinetic parameters into a vector and the six dynamical variables into a vector with components , , equations (1)–(6) can be written in the form of a parameterized non-autonomous dynamical system(7)where the explicit dependence on time is a consequence of the external driving term in (1) and (4).
In analogy to , we use the value as initial conditions for all six dynamical variables in (1)–(6). Then, we numerically solve equations (1)–(6) for 14 days under 12 h∶12 h LD (entrainment) conditions followed by 13 days under constant light (free-running) conditions (see Methods for further details). In general the solutions are different for every parameter set . As it is often the case in biological modeling, none of these parameters is known from experiments –. So, the remaining challenge is now to identify the specific parameter set for which the solution reproduces as well as possible the following known (sparse and often noisy) experimental findings: 1. Both transcripts perform periodic oscillations with the same period as the core oscillator, both under LD and LL conditions . 2. The transcript oscillations exhibit evening peaks with the peak of AtGRP8 preceding that of AtGRP7 by approximately 1–2 hours , , . The corresponding AtGRP7 protein concentrations oscillate with an approximately four hour delay compared to the transcript . 3. The amplitudes of their oscillations are roughly comparable to those of the core oscillator . 4. The waveform of the mRNA and protein oscillations have been characterized by means of experimental time series , , . 5. AtGRP7 mRNA is reduced to 50% within hours after experimentally suppressing its transcription .
In order to find an optimal parameter set, we defined a cost function (described in detail in Text S1 A) which quantifies the deviation of the corresponding solution from these experimental findings 1–5 for every given parameter set . In a next step we minimized this cost function with respect to .
The detailed optimization procedure is described in Methods. Here we only summarize the main steps: To take into account the similarity of the two paralogous proteins AtGRP7 and AtGRP8 we first sampled the parameters for a reduced system, only consisting of AtGRP7, using two million Antonov-Saleev quasi-random parameter sets. The network motif was then extended to the complete interlocked feedback loop structure, including also AtGRP8. The parameters were chosen in order to generate two identical oscillatory profiles for AtGRP7 and AtGRP8. The best one hundred solutions were then further optimized in the local neighborhood of a given parameter set using a Nelder Mead downhill simplex algorithm . This modified sampling and optimization method led to better results than the full parameter space sampling and optimization, i.e. the best solutions have a lower cost function value and thus better fit the experimental data (compare Figure 3 A (discussed in the next section) and Figure S1). It might also reflect a possible evolutionary origin of that network motif since the high sequence similarity of AtGRP7 and AtGRP8 suggests that these genes are paralogues, originating from a gene duplication event , .
A) Simulated AtGRP7 and B) AtGRP8 mRNA oscillations under 12h∶12h LD conditions (blue curves) are plotted together with the corresponding “COL_LDHH” experimental data set from the DIURNAL database (green curves with markers indicating data points), which uses Columbia wild type plants investigated under 12h∶12h LD entrainment condition with a constant temperature of . The DIURNAL database collects circadian microarray time series data based on Affymetrix chips and was normalized using gcRMA . C) Simulated AtGRP7 mRNA and D) protein oscillations under LL conditions, after entrainment under 8h∶16h LD conditions, are plotted together with the corresponding RNA and protein gel blot data taken from . In , this gel blot data was published relative to the minimal level, which was defined as 1. Note that the time axis of the experimental data was adjusted by +34 hours. This takes into account a shortcoming of the core oscillator model adopted from , namely that the phase of the simulated LHY/CCA1 mRNA oscillations under LL conditions in this core oscillator model only agrees with the corresponding data in the DIURNAL database (data sets “LL12_LDHH” and “LL23_LDHH” in ), if the time axis of those experimental data is adjusted by approximately ten hours. Since the samples in the experiments ,  underlying these data sets were collected on days two and three after transferring the plants to LL conditions, we also did not take into account the first day in LL, altogether thus amounting to a total time-adjustment of +34h. Overall, the agreement between the simulated and experimental phases, periods, and waveforms is very good.
Comparison with Experimental Results and Computational Predictions
In silico waveforms and phases are consistent with the experimental data.
As a first prominent quantity we consider the phase of an oscillating concentration in units of zeitgeber time (zt), i.e. is defined as the time an oscillation needs to reach its maximal concentration after the onset of light in the external light-dark cycle. The AtGRP7 mRNA peak under 12h∶12h LD conditions at is very close to the phase estimated from the literature  and predetermined by the cost function (see Text S1 A). The AtGRP8 mRNA peak at precedes that of AtGRP7 by approximately hours as previously shown experimentally . The AtGRP7 protein concentration is maximal at about and hours after the mRNA's peak in LD and LL, respectively, which is close to the literature value  (note that in  the relative protein concentrations were measured under LL conditions after entrainment in 8h∶16h LD conditions but the time span between the mRNA and the protein concentration peaks may be also a good approximation under the 12h∶12h LD conditions used in our simulations). Since there is no published experimental data on the AtGRP7 and AtGRP8 pre-mRNA as well as AtGRP8 protein time traces, their corresponding simulations in Figure 2 can be considered as a first theoretical prediction of our present work.
A direct comparison between our simulated and the experimental time traces , , as depicted in Figure 3, shows that the proposed model mimics the experimentally observed phases, periods, and waveforms very well. The “shoulder” observed during the declining phase of the simulated AtGRP7 mRNA concentration in Figure 3 A can sometimes be seen in experiments as well; e.g. in the data set of the DIURNAL database measured under 16h∶8h LD conditions (“long day” data set in ). In our simulations, the shape of this shoulder depends on the broadness and amplitude of the driving LHY/CCA1 protein oscillations. oscillations with a lower peak concentration, e.g. for simulations under LL conditions, lead to higher trough values and a less pronounced “shoulder“ (as one can see in Figures 2 and 3 C) due to the reduced transcriptional repression by LHY/CCA1. In experimental papers, not much attention has been paid so far to this fact but it actually could hint to the two-step transcriptional and post-transcriptional regulation of AtGRP7 (see below). A further interesting feature of the system is the fact that the peak concentration of AtGRP8 mRNA is always lower than the one of AtGRP7 mRNA (see Figure 2) which was not taken into account by our cost function (see section Parameter Estimation) but is actually observed in experiments . Furthermore, the trough values of and are always non-zero. This is consistent with experimental data given in , where a non-zero trough value was detectable among all data sets.
Importance of negative auto-regulation and reciprocal cross-regulation.
Our simulations also support the assumption that the AtGRP7 and AtGRP8 negative auto-regulation and reciprocal cross-regulation could be responsible for the experimentally observed phases and sharply peaking waveforms of the AtGRP7 and AtGRP8 mRNA oscillations. The AtGRP7 and AtGRP8 pre-mRNA and mRNA concentrations reach their trough value soon after the rise of the LHY/CCA1 protein peak and quickly recover while LHY/CCA1 is declining. Subsequently, their concentrations start to fall again although the LHY/CCA1 protein concentration is still at its trough (see Figure 2). For the network topology proposed in Figure 1 A, this is only possible due to the negative auto- and cross-regulation in equations (1)–(6): Elevated protein levels promote the generation of the alternative splice forms at the expense of mature mRNA. Upon reducing the impact of the negative auto-regulation of AtGRP7 by gradually decreasing the (alternative) splicing coefficient , we observe a phase shift of the AtGRP7 pre-mRNA and mRNA oscillations to a later time of day, see Figure S2. On top of that, the peaks of the and oscillations get increasingly broader and the previously observed ”shoulder“ of the AtGRP7 mRNA as well as the second trough in the AtGRP7 pre-mRNA progressively disappear. It is thus intuitively quite plausible that both shoulder and trough have their roots in the two-step transcriptional repression by LHY/CCA1 and the post-transcriptional auto-regulation of AtGRP7. The importance of cross-regulation for the observed AtGRP8 mRNA phase will be further discussed in the paragraph AtGRP8 oscillations appear to be subordinated to AtGRP7.
In silico half-life experiments.
As a next quantity we consider the half-life of AtGRP7 and AtGRP8 mRNA and protein. As detailed in Text S1 A, we defined the simulated half-life of a given species exactly along the lines of a previous experiment: The decay of a given species is measured after its production is interrupted, e.g. in the case of AtGRP7 mRNA by transferring the plants to a medium supplemented with cordycepin which inhibits the RNA synthesis , . A graphical illustration of the simulated half-life measurements can be seen in Figures S3 A/B.
Since in equation (2) the production of AtGRP7 mRNA depends on the normal splicing of its pre-mRNA to the mRNA and since its degradation kinetics are of Michaelis-Menten type, its half-life will depend on the initial conditions of the system. It will therefore vary, depending on the day time at which the transcription is interrupted (see Figures S3 C/D). This prediction could be tested in experiments, where transcriptional blockers, such as cordycepin and actinomycin D, are supplied at different phases of the day followed by a subsequent half-life determination.
The half-life of , obtained after the interruption of RNA synthesis two hours before the maximum is expected, is in good agreement with the corresponding experiment in  that has found a half-life between three and four hours. An analogous analysis for the AtGRP8 half-life predicts a half-life of . This shorter half-life of AtGRP8 compared to that of AtGRP7 can be partially explained by the smaller amplitude and lower peak concentrations of the oscillations. The AtGRP8 half-life has not been measured experimentally.
In order to measure the protein half-life in silico we set the parameters and in equations (3) and (6) to zero, corresponding to an inhibition of protein translation. The resulting decoupled equations for can be solved analytically. The half-life for a given initial value reads as(8)and therefore depends on the initial value , in contrast to the half-lives resulting from linear degradation kinetics. The protein half-lives over a full cycle under 12h∶12h LD conditions are shown in Figure S3 E/F. They change over the course of day and their highest values and coincide with the protein concentration maxima at and , respectively.
AtGRP8 oscillations appear to be subordinated to AtGRP7.
As described above (see end of section Parameter Estimation) we used an optimization scheme that mimics the possible evolutionary origin of the AtGRP7 and AtGRP8 interlocked feedback loops, namely a gene duplication followed by further evolution. Moreover, we assumed that AtGRP7 and AtGRP8 behave similarly (see Text S1 A). The cost function only takes into account two differences between them, namely an earlier peak of AtGRP8 mRNA compared to AtGRP7 mRNA and the fact that the AtGRP8 mRNA half-life is not known. The optimization then leads to a model that proposes splicing coefficients that fulfill the inequality , see Table 1. Equations (1) and (4) thus imply that the impact of the AtGRP8 protein on the alternative splicing of its own () and of the AtGRP7 pre-mRNA () is weaker than that of the AtGRP7 protein on the alternative splicing of its own () and of the AtGRP8 pre-mRNA (). This suggests that AtGRP8 oscillations are subordinated to those of AtGRP7. Upon adopting for AtGRP8 the same parameters as for AtGRP7 (see Table 1), apart from the constants connected to alternative splicing ( and ) and transcription kinetics (, , and ), the mRNA oscillations still behave qualitatively similar (see Figure S4) and the earlier peak of AtGRP8 mRNA persists. Altogether this suggests that the higher impact of AtGRP7 on the alternative splicing could be the essential mechanism underlying the earlier AtGRP8 mRNA peak compared to that of AtGRP7 mRNA.
Our model suggests highly saturated protein degradation.
Our analysis revealed very low activation coefficients and of the corresponding protein degradation kinetics (see Table 1). As a consequence, the corresponding protein dynamics (right hand side of equations (3) and (6)) exhibits a notable dependence on the protein concentrations and themselves only if these concentrations are extremely small ( and ). The resulting straight decay in AtGRP7 protein concentration to a value close to zero and the concomitant suspension of negative auto-regulation via alternative splicing leads to the observed fast recovery of pre-mRNA concentrations and from their trough values (see Figure 2 and equations (1) and (4)). For protein concentrations much larger than the activation coefficients ( and ) the dynamics in equations (3) and (6) are solely governed by the mRNA concentrations and . In other words the degradation kinetics of AtGRP7 and AtGRP8 proteins are highly saturated.
Our model accounts for LHY-ox, ztl, and toc1 mutant data.
Introducing LHY/CCA1 as a transcriptional repressor of AtGRP7 and AtGRP8 was motivated by experiments, showing that AtGRP7 mRNA oscillations are damped to their trough value in plants constitutively over-expressing CCA1  or LHY  under LL conditions (see Introduction). We simulated the LHY or CCA1 over-expression plants by adding a constant, unregulated transcription rate to the differential equation of the LHY/CCA1 mRNA in the core oscillator model by Pokhilko et al. . The resulting time traces of the LHY overexpression mutant indeed show the experimentally observed damping of AtGRP7 mRNA to the trough value of its corresponding wild type oscillations under LL conditions, see Figure S5 A.
Plants carrying a mutation in the gene of the F-Box protein ZEITLUPE (ZTL) were shown to exhibit AtGRP7 mRNA oscillations  and CCR2::LUC (AtGRP7::LUC) expression  with a markedly prolonged period under free-running conditions. This behavior is also visible in our model (see Figure S5 B), where we simulated the ztl null mutant by setting the production of ZTL to zero: Under LL conditions, this mutant shows self-sustained LHY/CCA1 oscillations with an approximately h longer period compared to the h wild type behavior, which in turn entrain AtGRP7 to this rhythmicity.
Similarly, a hypothetical clock mutant (as described in ), neglecting the transcriptional repression of PRR9 by TOC1 accounts for the experimentally observed short period of AtGRP7 mRNA oscillations in toc1 mutant plants under LL conditions , see Figure S5 C. This is again meditated through the experimentally observed reduced period of LHY/CCA1 oscillations . Note that the simulated toc1 null mutant, realized in the model from  by setting the production of TOC1 mRNA to zero, shows stronger damping and an unrealistically strong phase shift in LL, but still retains the experimentally observed period shortening, see Figure S5 D.
The Slave Oscillator Can Be Viewed as a Driven Bistable Toggle Switch
Two genes that mutually repress each other by transcriptional inhibition are known to constitute a genetic toggle switch – a prototypical example of a biological system showing bistability . Gardner et al. reconstructed such a toggle switch in Escherichia coli and proposed a two variable model (Gardner model) in order to explain the necessary conditions for bistability . In the system studied here, both genes, AtGRP7 and AtGRP8, also cross-regulate each other. However, the reciprocal regulation of AtGRP7 and AtGRP8 occurs at the post-transcriptional level via alternative splicing followed by nonsense-mediated decay of the alternative splice forms instead of mutual inhibition of transcription. This led us to the question whether the slave oscillator could act as a toggle switch. Therefore, we decoupled the slave oscillator from the core oscillator by setting for all times , thus neglecting the transcriptional repression of AtGRP7 and AtGRP8 by LHY/CCA1. In other words AtGRP7 and AtGRP8 are now transcribed at constant rates and , respectively, see also text below equation (6). Note that both AtGRP7 and AtGRP8 show negative auto-regulation as an additional feature not described for the toggle switch as proposed in .
As a first step, we investigated whether our decoupled slave oscillator system (i.e. equations (1)–(6) with ) can exhibit bistability. While we show in Text S1 B that the simplified model with a single AtGRP7 feedback loop can only have one fixed point (either stable or unstable), the interlocked AtGRP7 and AtGRP8 feedback loop may give rise to bistability, i.e. a scenario where two stable steady states can coexist: In order to test the system's ability to show a bistable behavior, we randomly sampled parameter sets in the same range as before. A linear stability analysis applied to every fixed point of a given parameter set (see Methods) revealed a monostable longterm behavior in of all cases, bistability in , and oscillatory behavior in . Such oscillations were not possible in the two-variable model by Gardner et al. . Moreover, we found that a tiny rest of about exhibited still other phase space structures, such as the coexistence of a stable fixed point and a limit cycle attractor.
Figure 4 A illustrates the situation when only the two parameters and are varied, while all other parameters are kept at their values from Table 1. Such variations of and are of particular interest since they effectively correspond to variations of at fixed and in equations (1) and (4): The transcription of AtGRP7 and AtGRP8 is repressed whenever and the corresponding transcription rates and (see text below equation (6)) adopt values smaller than their maximal transcription rates and .
A) The –-bifurcation diagram of the slave oscillator decoupled from the core oscillator consists of four main regions: two monostable areas (blue and green), a bistable area (red), and an area where autonomous oscillations are possible (yellow). Dashed lines indicate the directions in parameter space used for the one parameter bifurcation diagrams in Figure S6. The intersection of these lines marks the optimal parameter set from Table 1. The black curve is discussed in detail in the main text. B) Modification of the splicing coefficients and , responsible for the reciprocal cross-regulation, affects the slope of the boundaries between the bistable and the monostable regions (black: original boundaries, color: modified boundaries). C) & D) Color-coded fixed point concentrations and of AtGRP7 and AtGRP8 protein in the monostable areas. Straight lines with black and white dots are explained in the main text.
For our optimal parameter set from Table 1, the system shows bistability (see intersection of the dashed lines in Figure 4 A). Similar to the Gardner model  a bistable region separates two monostable regimes in Figure 4 A. In those two monostable regions either high AtGRP7 fixed point protein concentrations dominate over AtGRP8 fixed point protein concentrations or vice versa (Figures 4 C and D). The one parameter bifurcation diagrams, following the dashed lines in Figures 4 A, show the typical hysteretic behavior of a toggle switch (Figures S6). Intuitively understandable, the and protein fixed point concentrations increase with increasing maximal transcription rates and , respectively.
In the Gardner model  the degree of cooperativity of the reciprocal transcriptional inhibition determined the slope of the bifurcation lines and therefore the size of the bistable region. In our case, the strength of the reciprocal control of alternative splicing () has an analogous effect, as one can see in Figure 4 B. An increase of the splicing coefficient nearly exclusively alters the slope of the bifurcation line bordering the monostable region where AtGRP8 protein dominates, and similarly for .
Gonze already showed in 2010 that periodically forcing the transcription of one of the two genes in the Gardner model can induce limit cycle oscillations . Specifically, high forcing amplitudes can drive the system from one monostable region to the other by crossing the bistable regime. In our system, the LHY/CCA1 protein in equations (1) and (4) was assumed to affect both AtGRP7 and AtGRP8 transcription. We therefore have to investigate this phenomenon in a two parameter bifurcation diagram. Indeed, if we pursue the trajectory of the transcriptional rates and (see text below equation (6)) of AtGRP7 and AtGRP8 (black curved line in Figure 4 A) during one cycle under 12h∶12h LD conditions, one observes that the rhythmic transcriptional repression via LHY/CCA1 drives the system from one monostable region to the other by crossing a narrow bistable branch. This is only possible due to different kinetics of AtGRP7 and AtGRP8 transcription (see Table 1). Completely identical transcription kinetics for AtGRP7 and AtGRP8 (i.e. , , , and therefore ) would lead to a straight line of unit slope in the – bifurcation diagram instead of the curved shape, not allowing the system to reach one monostable region from the other.
Neglecting the transcriptional repression of AtGRP8 by LHY/CCA1 protein.
In our model we have assumed a transcriptional repression of AtGRP8 by LHY/CCA1 for reasons of similarity with AtGRP7. Since this was never investigated experimentally so far, we asked whether it would be possible to still reproduce the experimental findings without this hypothetical repression. In the present framework, this is tantamount to keeping constant at the value . As a result, the system moves back and forth between the black dots in Figure 4 C and D without fully crossing the bistable region. Hence we can conclude that remains at an almost constant low value and at an almost constant high value. Figure 5 A confirms this expected behavior together with a similar behavior of and , while and still exhibit appreciable oscillations (which in turn could be expected from Figures S7 A and B). In other words, we obtain a strong disagreement with the known experimental facts 1–5 (see section Parameter Estimation).
A) Solutions of equations (1)–(6) for the “optimal” parameter set from Table 1 after neglecting the repression of AtGRP8 transcription by LHY/CCA1, i.e. is held constant at the value from Table 1. B) Same as in A) after additionally increasing the maximal transcription rate to 3.38. Shown are the last two days in 12h∶12h LD conditions ().
However, this problem can be readily solved by increasing the maximal transcription rate , e.g. from to , so that the system now moves back and forth between the white dots in Figure 4 C and D, and in particular fully crosses the bistable region. As a result, an oscillatory behavior of and is recovered similar to the original oscillations in Figure 2 and likewise for the other concentrations, see Figure 5 B. The main difference is the somewhat higher maximum of the oscillations, in qualitative agreement with Figure 4 D.
Impact of saturated protein degradation on the bifurcation diagrams.
For the optimal parameter set (see Table 1) the protein degradation is highly saturated (i.e. the activation coefficients and are very small, see also end of section Comparison with Experimental Results and Computational Predictions). Consequently, the fixed point concentration in equation (20) can be approximated as if . In view of the bifurcation diagram in Figure 4 C this explains the region uniformly colored red in Figure S7 B, i.e. the fixed point concentration is nearly constant and equal to in the whole area of the dominated monostable area. According to equation (21), the pre-mRNA fixed point concentration also remains almost constant (see red region in Figure S7 A). Analogous considerations apply to , , and .
This behavior could also be of theoretical interest since a highly saturated degradation kinetics allows the system to change the value of one variable (here the protein concentration or ) while keeping all other fixed point concentrations constant. This would imply yet another potential function of a saturation kinetics in addition to its recently discussed role as an efficient mechanism of inducing delay into negative feedback loops in order to favor oscillations –.
Robustness against Variations in the LHY/CCA1 Protein Oscillations
In order to examine the effect of variations in the core oscillator input on the AtGRP7-AtGRP8 slave oscillator we substituted the core oscillator protein concentrations obtained from the model of Pokhilko et al.  by obtained from a modified Poincaré oscillator, similar to the model used in . This generic oscillator, described in detail in the section Methods, is tunable in its period and amplitude . A third parameter determines the shape of the oscillations, ranging from sinusoidal () to increasingly spiky oscillations with increasing , and a fourth parameter determines the trough value. In particular, for , , , and , the resulting oscillations are very similar to under 12h∶12h LD conditions (see black lines in Figure 6 A). Likewise, the corresponding slave oscillator dynamics differ only little from those obtained by a coupling of the AtGRP7-AtGRP8 feedback loops to the more complex core oscillator model , as one can see in Figure 6 A. In other words, we can replace the complex core oscillator model, being composed of many differential equations and parameters, by any other model which faithfully imitates the actual protein oscillations of LHY/CCA1.
A) Dashed: Same traces as shown for LD conditions in Figure 2. Solid: Same but with a core oscillator input generated by a modified Poincaré oscillator with parameters , , , and , as detailed in section Methods. B) Amplitude of the oscillations when the slave oscillator is driven by a generic Poincaré oscillator of different amplitudes and waveform parameters at fixed and . The point of intersection of the dashed curves indicates the parameters and used in A).
In particular, we verified that almost identical solutions for the slave oscillator dynamics are recovered (exemplified for by Figure S8), when we replace our original model from  by the recently published refined core oscillator model from . While shape and phase of the LHY/CCA1 protein oscillations are fairly similar in both core oscillator models, the amplitude of approximately doubles for the refined model from . As expected from equations (1) and (4), adapting the activation coefficients according to and then results in almost identical results for the slave oscillator, see Figure S8.
It is known that oscillations, governed by a hysteretic switch mechanism, exhibit oscillations with a robust amplitude, mainly determined by the height of the hysteretic loop, while being easily tunable in their period , . In order to investigate the effect of changes in the LHY/CCA1 protein concentrations , and whether our driven AtGRP7-AtGRP8 slave oscillator shows robust amplitudes for varying as well, we examined the behavior of the system for different amplitudes and waveforms of the core oscillator while keeping and constant. Figure 6 B shows the color-coded values of amplitude obtained from simulations with different and . For a given shape parameter , the amplitude of oscillations nearly stays constant after reaching a certain driving amplitude even if we further increase , i.e. the values of are strong enough to overcome the bistable region and to repress the system to a trough value of the oscillations close to zero. This threshold amplitude increases for more spiky oscillations with increasing since the timespan of the transcriptional repression becomes shorter and the systems dynamics needs time to react to the corresponding “movement” in the – bifurcation diagram in Figure 4 A (similar diagrams can be obtained for the other concentration species , , , , and ). Nevertheless, oscillations of robust amplitudes can be induced for a wide range of combinations of and (see red area in Figure 6 B).
Limitations of the Model: The lhy cca1 Double Mutant
The lhy cca1 double mutant does not express LHY and CCA1, hence the protein concentration of the core oscillator must vanish. We have shown in the previous section that the resulting autonomous dynamical system ( in equations (1)–(6)) approaches a steady state in the bistable region, see Figures 4 A and 7 A, i.e. oscillatory solutions are ruled out. This theoretical result is in contradiction to the experimental finding that the AtGRP7 transcript shows diurnal oscillations with a phase shift to dawn in the lhy cca1 double mutant –.
A) Relaxation dynamics are observed for the optimal parameter set from Table 1. Dashed lines denote the corresponding fixed points. B) After changing the AtGRP7 and AtGRP8 protein degradation rates to and , respectively, the slave oscillator develops autonomous oscillations. C) Pure noise-induced oscillations of a single cell () for the parameter set from Table 1. D) Same after averaging over an ensemble of cells. See Text S1 C/D for further details (especially the noise-strength ).
As a first possible resolution of this contradiction we considered the possibility of modifying the kinetic parameters of Table 1 without changing our model (1)–(6) itself in order to generate oscillatory solutions of the autonomous dynamics (). As demonstrated by Figure 7 B and detailed in Text S1 C this is indeed possible but the obtained periods of oscillation are prohibitively small. Moreover, tiny parameter variations in an ensemble of autonomous oscillators will lead to deviating oscillation periods and hence the oscillations average out in the longterm.
Next we considered the possibility to explain the experimental facts by means of noise effects. Indeed, noise is omnipresent in biological systems due to the probabilistic nature of molecular reactions or fluctuating environmental influences ,  and noise induced oscillations have been reported in numerous other models –. Again, as shown in Figures 7 C/D and detailed in Text S1 D, we were able to generate noise-induced self-sustained oscillations on the single cell level, but not in the ensemble.
An obvious remedy in both our attempts discussed above is to introduce coupling between the individual oscillators. However, in the experimentally relevant case of many cells the details of their mutual interaction are still not fully clarified, but a global synchronization mechanism seems unlikely –. Moreover, we note that both our attempts are also unable to explain one more experimental fact, namely the entrainment of AtGRP7 mRNA oscillations to 24h-periodic light-dark cycles in the lhy cca1 double mutant , . In conclusion, the only remaining possibility to explain the observed rhythmicity of AtGRP7 mRNA in lhy cca1 double mutants seems to include to the model (1)–(6) additional influences of the core oscillator variables (as already stated, a direct influence of light seems negligible (unpublished data)), e.g. additional transcriptional activators or inhibitors.
We introduced and analyzed a mathematical model for the molecular regulatory network of the AtGRP7 and AtGRP8 slave oscillator in Arabidopsis thaliana. Based on experimental results, we assumed that the slave oscillator gains input from the circadian core oscillator via transcriptional repression by the LHY/CCA1 proteins. Furthermore, we assumed that it shapes its oscillatory profile due to a negative auto-regulation and reciprocal cross-regulation between AtGRP7 and AtGRP8 via alternative splicing followed by nonsense-mediated decay of the alternative splice form. Although alternative splicing is abundant among circadian clock genes , , this is as far as we know the first mathematical model of a circadian clock-related molecular network that includes alternative splicing as a regulatory mechanism. We determined the model's kinetic parameters by a two-step optimization process including random sampling and an evolutionary algorithm. With the resulting optimal parameter set we could successfully reproduce most of the pertinent experimental findings such as waveforms, phases, and half-lives of the time-dependent concentrations. Furthermore, the model can account for experimentally observed mutant behavior in LHY-ox, ztl, and toc1 mutant plants. The observed AtGRP7 mRNA oscillations can be sufficiently explained through the altered behavior of the LHY/CCA1 protein oscillations in these mutants.
We note again that the slave oscillator, since it is driven by the core oscillator, does not have to share all the properties of the core oscillator such as self-sustaining oscillations, temperature compensation, or direct light input , . Indeed, we find dampened dynamics rather than independent self-sustained oscillations for the optimal parameter set from Table 1 (see e.g. Figure 7 A).
The model can also be used to predict properties not considered by our optimization procedure or properties not measured so far. It suggests a shorter half-life of AtGRP8 compared to AtGRP7 mRNA and a fast and highly saturated protein degradation of both AtGRP7 and AtGRP8. The latter finding is consistent with recent experimental results showing that AtGRP7 and AtGRP8 proteins are among those with the highest degradation rates in Arabidopsis thaliana . Furthermore, the model revealed that AtGRP7 may have a stronger impact on the alternative splicing of the AtGRP7 and AtGRP8 pre-mRNAs than AtGRP8. This may be the mechanism underlying the observed earlier peak of AtGRP8 mRNA compared to AtGRP7 mRNA.
As highlighted in  it might also be interesting to investigate the persistence of the above general predictions for parameters which differ from the optimal parameter set considered so far. Figure S9 indicates that the subordination of AtGRP8 to AtGRP7 seems to be a robust feature of the optimization procedure, while the other two features (shorter AtGRP8 mRNA half-life and saturated protein degradation) seem to be less robust.
Our modeling process also provided theoretical insight into possible mechanisms underlying the experimentally observed AtGRP7 and AtGRP8 oscillations: The slave oscillator model from equations (1)–(6) is potentially able to show bistability and indeed does so for the parameter set found by our optimization scheme, suggesting that the core oscillator basically triggers periodic switching of the slave oscillator between two monostable branches by crossing a bistable regime. Our AtGRP7-AtGRP8 slave oscillator could therefore be the first in vivo manifestation of the purely theoretical proposal of a genetic toggle switch driven by an autonomous self-sustained oscillator .
What evolutionary benefit could such a mechanism have? It is known that oscillations based on a hysteretic switch can show robust amplitudes. Indeed, our present slave oscillator also shows oscillations which are robust in amplitude for a considerable variety of different driving oscillations . The formation of a driven interlocked auto-regulatory feedback loop that originated from a gene duplication event in the case of AtGRP7 and AtGRP8, can thus lead to a system showing a hysteretic behavior and resulting, if forced with an appropriate amplitude, in oscillations with a robust amplitude.
Finally, we proposed two possible changes in the current view of the regulatory network of AtGRP7 and AtGRP8:
First, we can still reproduce the experimental findings even without the common assumption of transcriptional repression of AtGRP8 by LHY/CCA1. Up to now, the latter assumption has been justified by reasons of similarity with AtGRP7 but not by direct experimental measurements , .
Second, we have discussed modifications of the model (1)–(6) in order to reproduce the experimental behavior in the lhy cca1 double mutant. In contrast to the simulation of the lhy cca1 double mutant, the AtGRP7 transcript shows oscillatory behavior with a phase shift to dawn under entrainment conditions –. We therefore tested natural possibilities how to cure this shortcoming of the model: Two of them, namely the autonomous oscillations due to noise effects and a change of the kinetic parameters from Table 1 could be readily excluded since they cannot explain the phase locking of the AtGRP7 mRNA in the lhy cca1 double mutant to 24h-periodic light-dark cycles. We therefore concluded that additional influences of the core on the slave oscillator, on top of the transcriptional repression by LHY/CCA1, have to be incorporated to consistently explain both the wild-type and the lhy cca1 double mutant behavior.
Furthermore, it has to be taken into account that AtGRP7 influences many physiological processes: It promotes the floral transition at least partly by down-regulating the floral repressor FLC . Furthermore, it plays a role in the plants innate immune system since grp7-1 plants that do not produce AtGRP7 mRNA are more susceptible to Pseudomonas syringae , . AtGRP7 is also known to mediate responses to stresses such as oxidative stress, high salt, mannitol, or cold , , . Our modeling results could be used in future work to integrate the AtGRP7 and AtGRP8 feedback loops with these other regulatory cues.
The numerical solutions of equations (1)–(6), or equivalently of equation (7), have been obtained by using the odeint function of SCIentificPYthon which uses LSODA from the Fortran library ODEPACK. In particular, we remark that LSODA is able to identify and solve initial value problems for both stiff and non-stiff problems.
Cost Function and Parameter Estimation
In this section we provide the details of the optimization procedure as referred to in the section Parameter Estimation. Similarly as in , we started our search for an optimal fit by generating Antonov-Saleev quasi-random parameter sequences (adopting the gsl_qrng_sobol routine from the GNU Scientific Library) that were subsequently tested for their fitness (for the explicit definition of , see Text S1 A). To take into account the similarity of AtGRP7 and AtGRP8 we first sampled the parameters for a reduced system consisting only of AtGRP7, see also Text S1 B. After this random sampling step, the network motif was extended to the complete system while choosing the parameters in order to generate two identical oscillatory profiles for AtGRP7 and AtGRP8 (upon comparison of equation (1)–(6) in the main text and those in Text S1 B, all parameters have to be duplicated except for the rate constant which has to be set to the half of its previous value and then has to be identified with and ).
In the next step, we took the best one hundred values and further minimized the cost function in their local neighborhood. In order to solve this dimensional minimization problem we used the gradient-free Nelder Mead Downhill Simplex method, where an initial simplex with vertices, including the starting parameter set, “crawls” amoeba-like via shape transformations (reflection, contraction and expansion) through parameter space in the direction of lower cost . We modified the original algorithm in a way that negative and therefore biologically not meaningful parameter values were penalized by setting the cost-function value of such vertices to infinity. The starting simplex was defined by the initial parameter set and the set of vertices defined by where the 's are the unit vectors in each parameter space's direction and is a constant chosen to be in our simulations. The reflection, expansion and contraction coefficients were chosen as throughout the simulations and after the algorithm claimed to be finished it was restarted four times from the best point found in the previous run.
We also tried out a Monte-Carlo Hillclimbing method instead of the simplex optimization, which however led to worse results.
Fixed Points, the Jacobian, and Bifurcation Analysis
As detailed in the main text, the lhy cca1 double mutant can be modeled by setting in equations (1) and (4) to zero for all times . In the slave oscillator model proposed here, this is equivalent to the deletion of all links to the core oscillator. Equations (1)–(6), or equivalently (7), then define an autonomous dynamical system which is easy enough to calculate the fixed points analytically.
In principle the quartic equation (9) can be solved analytically by means of the formula of Cadano & Ferrari. We used the root finding package root of SCIentificPYthon instead. In general, we thus obtained four different solutions of the quartic equation (9).
For the optimal parameter set from Table 1 we thus obtained the following four fixed points(22)(23)(24)(25)where the last one is not biologically meaningful due to its negative concentration values.
A standard linear stability analysis based on the eigenvalues of the Jacobian matrix(26)reveals that two of the remaining fixed points (22),(23), and (24) are (locally) stable (namely and ) and one is (locally) unstable (namely ).
A similar algorithm was used to generate Figure 4 and Figures S6 and S7: For each parameter set we first calculated the four fixed points as described above. In a next step, those with negative or complex components were sorted out. Finally, we performed a linear stability analysis as described above.
Tunable Modified Poincaré Oscillator
In order to better highlight the dependence of our slave oscillator on properties like the amplitude or peak broadness of , we replaced the differential equations for the molecular core oscillator model provided by Pokhilko et al.  by an easily tunable generic oscillator in the form of a modified nonuniform Poincaré oscillator as proposed in . Its radial evolution is given by(27)therefore converging for any initial condition to the stable fixed point , amounting to the amplitude of the resulting oscillations. The phase dynamics are given by(28)where determines the shape of the oscillations, ranging from a sinusoidal () to a more and more spiky oscillator () with period(29)(30)
Note that the period depends on the choice of both parameters and . In , the model parameter in (28) was originally chosen as a small non-zero positive constant in order to make sure that never becomes zero, since for the solution of equation (28) would evolve to its fixed point in phase . For our purpose, we set(31)so that, for any given , the oscillator exhibits oscillations with a fixed period . Finally, we define(32)as the input substituting the LHY/CCA1 oscillations in (1) and (4). The extra parameter in (32) denotes the trough value of the oscillations and is set to the trough-value of the oscillations.
Simulations of the “best” parameter set, obtained via the full parameter space sampling-procedure, fit the experimental time traces worse (for a comparison, see Figure 3 A). Blue: Simulated AtGRP7 mRNA oscillations. Green: “COL_LDHH” experimental data set from the DIURNAL database, as used for Figure 3 A. The time traces were normalized to their maximal expression values, defined as 1.
A gradual decrease of the (alternative) splicing coefficient , which accounts for the negative auto-regulation of AtGRP7, shifts the phases of the AtGRP7 mRNA oscillations (A) and of the pre-mRNA oscillations (B) to a later time of day. On top of that, the peaks of the oscillations get increasingly broader.
A) & B) In silico half-life experiments for AtGRP7 (A) and AtGRP8 (B) mRNA following an experimental protocol (see main text and Text S1 A). In our model, the mRNA and protein half-lives were shown to depend on the day-time at which transcription or translation were stopped, respectively. The dark-blue and red lines denote the same AtGRP7 and AtGRP8 mRNA traces as shown in Figure 2. The light-blue and orange lines denote the dynamics after the interruption of transcription. C/D/E/F) Represented are the resulting half-lives , , , and over a full diurnal cycle (blue lines) for AtGRP7 (C) and AtGRP8 (D) mRNA as well as AtGRP7 (E) and AtGRP8 (F) protein, respectively. Dashed green lines denote the same AtGRP7 mRNA (), AtGRP8 mRNA (), AtGRP7 protein (), and AtGRP8 protein () concentrations as in Figure 2. All figures were obtained under 12h∶12h LD conditions.
Dashed: Simulations for the optimal parameter set from Table 1, identical to those of Figure 2. Solid: Even if one adopts for AtGRP8 the same parameters as for AtGRP7 (see Table 1), apart from the constants connected to alternative splicing ( and ) and transcription kinetics (, , and ), the mRNA oscillations of AtGRP7 and AtGRP8 still behave qualitatively similar. In particular, the earlier peak of the AtGRP8 mRNA persists.
Simulations of the LHY overexpression (LHY-ox) mutant (A), ztl (B), and toc1 (D) null mutants as well as a hypothetical toc1 mutant (C), where the repression of PRR9 by TOC1 is neglected, as described in . Dashed lines denote the wild type (wt) and continuous lines denote the mutant simulations of AtGRP7 mRNA (green) and LHY/CCA1 protein oscillations (black).
One parameter bifurcation diagrams of the maximal transcription rates (left) and (right), corresponding to the dashed lines in Figure 4 A/C/D. The protein concentration values and for stable fixed points are plotted in red and blue, respectively. Protein concentrations for unstable fixed points are kept in black. Dashed lines indicate the parameter values from the optimal parameter set of Table 1.
Analogously to Figures 4 C/D of the main text, we plotted the color-coded fixed point concentrations of the AtGRP7 pre-mRNA (A) and mRNA (B) as well as the AtGRP8 pre-mRNA (C) and mRNA (D) in the monostable areas of the – bifurcation diagram. The intersection of the dashed lines marks the optimal parameter set from Table 1.
Solid: Reproduction of the results for and from Figure 2. Dashed: Corresponding results after replacing the original core oscillator model from  by the refined model from  and adapting the activation coefficients according to and .
Four key features of the model dynamics (1)–(6) under 12h∶12h LD conditions for the optimal parameter set from Table 1 (Ranking) and for the 19 next best parameter sets (Ranking) resulting from the above described two-step optimization process with random initialization and subsequent evolutionary optimization. A) Two representative examples of the 20 (sub-)optimal parameter sets ( and ). As detailed in the main text, the observed general property indicates that the subordination of AtGRP8 to AtGRP7 is a robust feature of our optimization procedure. The experimentally observed earlier peak of AtGRP8 mRNA compared to AtGRP7 mRNA, i.e. (see section In silico waveforms and phases are consistent with the experimental data), is a further such robust feature. B) The half-lives and (see main text and Text S1 A) indicating that the shorter life-time of AtGRP8 mRNA compared to AtGRP7 mRNA is a less robust feature of our optimization procedure. Likewise, the depicted Michaelis constants and the peak () and trough values () of oscillations in C) () and D) () indicate that the saturation of AtGRP7 and AtGRP8 protein degradation is a less robust feature.
Function (see Text S1 A) is plotted versus different peak-trough-values . The peak-trough-values and , each leading to a cost function contribution of one, are indicated by vertical dashed lines.
A) Detailed description of the cost function. B) Analysis of the one-component posttranscriptional feedback loop. C) Search for self-sustained oscillations. D) Search for noise-induced oscillations.
We would like to thank Hanspeter Herzel and his group for very helpful discussions and Christian Heintzen for critical comments on the manuscript.
Conceived and designed the experiments: CS PR DS. Performed the experiments: CS. Analyzed the data: CS. Wrote the paper: CS PR DS.
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