Edited by: Joshua L. Heazlewood, The University of Melbourne, Australia
Reviewed by: Roeland M. H. Merks, Centrum Wiskunde & Informatica, Netherlands; Manfred Goedel, Ludwig Maximilians University Munich, Germany
*Correspondence: Didier Gonze
This article was submitted to Plant Systems and Synthetic Biology, a section of the journal Frontiers in Plant Science
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The circadian clock is an endogenous timekeeper that allows organisms to anticipate and adapt to the daily variations of their environment. The plant clock is an intricate network of interlocked feedback loops, in which transcription factors regulate each other to generate oscillations with expression peaks at specific times of the day. Over the last decade, mathematical modeling approaches have been used to understand the inner workings of the clock in the model plant
The circadian clock is an endogenous timekeeper that allows organisms to anticipate the day/night cycle, thus improving their fitness (Green et al.,
The first conceptual models of the oscillator in the model plant
Other approaches that are not based on ODEs were also developed, such as the Boolean model described in Akman et al. (
A more recent trend is the study and modeling of
In this work, we present an ODE-based model for
Given the large number of known clock components and the high degree of redundancy between some of them, we chose to merge very similar genes into single variables, which allowed us to include many well-characterized genes without needing several sets of almost identical equations. This approach is not novel, and has successfully been used with several clock genes, in particular
The dawn-phased genes
In our model, the expression of
The Evening Complex (EC) is made up of three proteins,
In light/dark cycles,
In the model, we have assumed that ELF4 or LUX must be the limiting factor in the formation of the EC, and have not included ELF3 explicitly. One equation represents the mRNA of
The
The PRRs are transcriptional repressors, acting to inhibit
The circadian phenotypes of the various
At the transcriptional level,
At the post-translational level, all of the PRRs are degraded faster in the dark than in the light but the proteins mediating that effect are different: PRR5 and TOC1 are tagged for degradation by ZEITLUPE (ZTL) (Más et al.,
In the model, the transcription of
The degradation of PRR5 and TOC1 is known to be mediated by two other proteins, GI and ZTL. The degradation rate is not constant throughout the night. Instead, it starts out slow at dusk, when a large fraction of ZTL proteins are sequestrated by GI, and accelerates as the night progresses, when more ZTL is released from the complex (Fujiwara et al.,
It should be noted that several previous models described an activation cascade where PRR9 induces
The transcription of the morning genes
The model consists of nine ODEs (See Supplementary Materials). Eight equations describe the temporal evolution of the mRNA and protein levels of the main clock genes, grouped into four groups, each of which represents a pair of clock genes: CL (
The simple model captures the main features of the wild-type clock and the main clock mutants, in both entrained and free-running conditions. In contrast to earlier models of similar complexity, this model also responds well to unusual light cues, including extreme photoperiods and non-24 h light/dark cycles.
We first present the time profiles, obtained after optimization, for the wild-type clock under various LD cycles. As shown in Figure
The simulated expression of clock genes varies according to the entraining photoperiod, as observed experimentally. Figure
Figure
In order to evaluate the ability of the model to reproduce the observed defects (defined as any change in gene expression levels, phase, or free-running period length) associated with a loss-of-function of the main clock genes, we looked at those features in single and double mutants. For three of the four gene pairs in the model, there are two single mutants with the same or very similar defects, and a double mutant with qualitatively similar but more pronounced version of the same phenotype.
The fourth pair,
Simulated single mutants were produced by dividing the relevant mRNA synthesis rate by 2. For simulated double mutants, the synthesis rate was divided by 10. The results of the double mutant simulations are shown in Figure
Experimentally, in light/dark cycles,
Figure
Experimentally, the
Figure
Experimentally, the
As seen in Figure
Experimentally, the effects of the
In simulations, lowering the synthesis rate of the EC shortens the free-running period. This is coherent with the reported short period of reduced-function
In our model, light affects every gene, whether at the transcriptional, translational, or post-translational levels. Those multiple light-sensing points are all backed up by experimental data, as described in our model building section, and allow for complex responses to a wide range of light conditions.
Both at the experimental and simulation levels, changing the length of the photoperiod leads to changes in the phase and amplitude of expression of many clock genes. The relationship between the phase of maximum (or minimum) expression of a gene and the photoperiod can be characterized and even quantified using the concept of dawn- and dusk-sensitivity (Edwards et al.,
The response of the model variables in photoperiods ranging from 3 to 21 h in increments of 3 h is shown in Figure
Experimentally,
Edwards et al. (
The simulated
Finally, although neither
In addition to the wild type, Figure
As discussed above, our model can be reliably entrained to a 24 h cycle regardless of the length of the light period. Figure
This feature of our model is coherent with the existing literature, as it includes a complex network structure and multiple light inputs, both of which have been shown to be important factors in the ability of a system to respond to light cues (Troein et al.,
An oscillator has a range of entrainment, that is, a range of periods it can be entrained to. Frequency demultiplication is a particular phenomenon that happens when the clock is subjected to a short cycle that is outside of its range of entrainment, but has a frequency that is close to an integer multiple of the free-running frequency. In such a case, the clock can skip cycles and entrain not to the short period but to a multiple of it. The wild-type Arabidopsis clock has been shown to oscillate with a 24 h period for at least several cycles following a switch from standard 24 h (12L:12D) conditions to a 12 h (6L:6D) (Kolmos et al.,
The model is fully capable of reproducing this behavior: as shown in Figures
Another hallmark of a functional circadian clock is the ability to be entrained by skeleton photoperiods, in which the organism is maintained in darkness except for pulses of light at dawn and dusk. Experimentally, the clock in
One of the motivations for developing clock models, aside from understanding the core circadian oscillator itself, is to investigate the many clock-controlled metabolic and physiological aspects. To ensure the ability of our minimal model to capture the general features of such processes, we have developed a small output module to model hypocotyl growth.
Hypocotyl growth is gated by light and the circadian clock through a coincidence mechanism. The clock controls the expression of the partially redundant genes
This output module keeps the same minimalist approach as the rest of the clock, consisting of three equations and nine parameters. The three modeled variables are
Figure
Our minimal model is able to reproduce many of the key features of the
The number of equations and parameters in our model (9 equations and 34 parameters) is similar to that of the very first one-loop model proposed by Locke et al. (
The ability of our minimal model to respond to various photoperiods is similar to that of the most recent clock models, published by Pokhilko et al. (
While the current results are more qualitative than quantitative, the model could be expanded to either split the gene pairs into individual variables or include more detailed descriptions of various effects, especially of post-translational regulations affecting protein stability and activity. This would allow a more quantitative description of the clock response to environmental stimuli, at the cost of reintroducing complexity into the system.
Nonetheless, our simple model is capable of driving an equally simple output module to reproduce the experimentally measured hypocotyl length of the wild type plant and predict that of several mutant lines. This demonstrates the potential use of our model in studying both the core circadian clock and clock-dependent processes. The model could also be coupled to more output modules, including clock-controlled slave oscillators (as done by Schmal et al.,
All simulations were done in MATLAB and XPPAUT (Ermentrout,
During the optimization process, the clock was simulated in 8L:16D conditions for a total of 384 h, then released into free-running conditions for 300 h. The purpose of the first 360 h of simulation is to allow the system to reach its limit cycle; the scoring algorithm used only the last light/dark cycle to compute the score in entrained conditions. Similarly, the first 100 h in free-running conditions were discarded, as they contain transient effects, and the score was computed using the last 200 h of simulations. The wild type and the four single mutants (obtained by halving the relevant mRNA synthesis rates) were simulated in 8L:16D and continuous light. Additionally, the wild type was simulated in continuous darkness.
Because the model is very simple and meant to be more qualitative than quantitative, the optimization function was similar to the one defined in Locke et al. (
Total RNA was extracted and purified using Aurum total RNA mini kit (Bio-Rad). After NanoDrop RNA quantification (Thermo Scientific), reverse transcription was done with the GoScript Reverse cDNA Synthesis kit (Promega). Quantitative PCR (qPCR) reactions were performed in a final volume of 10 μl using VeriQuest SYBR Green qPCR Master Mix (Affymetrix) with the PikoReal Real-Time PCR System (Thermo Scientific): preincubation at 95°C for 7 min, 40 cycles at 95°C for 15 s, 60°C for 60 s followed by melting curves. Primers are listed in Supplementary Table
JD, JL, and DG conceived the model. JD performed the numerical simulations. CH and QX grew and harvested the plants. QX performed the qPCR. All authors analyzed the results. JD wrote the manuscript. All authors read and approved the manuscript.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
We thank Geneviève Dupont for fruitful discussions. This work was supported by the program Actions de Recherche Concertée (ARC2012-2017) launched by the Division of Scientific Research, Ministry of Science and Education, French community of Belgium. CH is an F.R.S.-FNRS research associate.
The Supplementary Material for this article can be found online at: