Gould2013 - Network balance via CRY signalling controls the Arabidopsis circadian clock over ambient temperatures.
July 2018, model of the month by Matthew Roberts
Original model: BIOMD0000000564
The circadian rhythm refers to the oscillatory behaviour seen in biological activity of organisms with a period of approximately 24 hours. Accurate entrainment of circadian rhythm to environmental cues may result in, for example in plants, faster growth and optimal attraction of pollinators. The ability of plants to maintain a circadian rhythm is not only a result of successful detection of the day/night cycle but also the result of accurate temperature-compensation, as changes in temperature can have period-shortening or period-lengthening effects. Mathematical models can be used to validate hypotheses on biological mechanisms involved in the plant circadian clock pathways. Mathematical models of regulatory genes of circadian rhythm had previously focused on the effect of light. Gould et al. (2013)  hypothesised that temperature-compensation was regulated through light detection pathways. To test this hypothesis, the authors first experimented on wild-type, single (cry1) mutant and double (cry1-cry2) mutant cryptochrome photoreceptor strains of Arabidopsis, monitoring circadian rhythm using mRNA expression profiles in varied temperature conditions and varied light conditions. They then used an existing mathematical model  and fitted it to the data. Thereby they reparametrized some of the important light-dependent parameters that affect circadian period to account for changes in temperature. mRNA expression and protein abundance were simulated for the wild-type and double cryptochrome mutant strain to verify their hypothesis.
Figure 1. Circadian clock regulatory network. Figure taken and adapted from .
Gould et al. (2013)  experimented to see how light and temperature affected the period of oscillation in gene activity. The period was determined through in vivo imaging of LUC reporter gene activity under constant light and constant temperature conditions from seedlings harvested every four hours between 72-96 hours after transfer from 12L:12D conditions. The authors monitored the oscillatory period of the three genotypes in three temperature conditions under three light conditions, as illustrated in figure 2. In the Red / Blue Light (R/BL) conditions (figure 2, left), the three genotypes had similar periods at 12℃ but differences arose in 17℃ and large discrepancies occurred at 27℃. The wild-type strain roughly maintained its circadian rhythm while the single mutant period increased close to 26 hours and the double mutant period lengthened to over 27 hours. In Red Light (RL) (figure 2, centre), none of the strains maintained a circadian rhythm and the period of oscillation increased as temperature increased. In Blue Light (BL) (figure 2, right), the wild-type and single mutant strains exhibited similar periods at 12℃ and 17℃ but these periods were longer than in R/BL. At 27℃, the single mutant strain period remained constant while the wild-type strain period resumed its circadian rhythm. Only in a minority of the seedlings of the double mutant strain, the oscillation period shortened to 24 hours under BL in 27℃. The majority of double mutant strain seedlings failed to exhibit oscillatory LUC gene activity. The authors showed that the circadian period was strongly controlled by BL via cryptochromes, particularly at higher temperatures and that temperature compensation is achieved via pathways common with light input. Furthermore, the convergence of light and temperature signals occurs downstream of CRY protein accumulation since temperature compensation of circadian period was achieved without CRY1 and CRY2 protein levels changing with circadian rhythmicity. Gould et al. (2013)  used mathematical modeling to further validated their hypothesis.
Figure 2. Circadian period determined via in vivo imaging of LUC reporter gene activity of three genotypes under varied temperature conditions. Legend: squares - wild-type, diamonds - single mutant, triangles - double mutant. Light: RL - red light, BL - blue light. Columns: left - 12℃, middle - 17℃, right - 27℃ Figure taken and adapted from .
An existing model from Pokhilko et al. (2010)  that simulated plant circadian rhythm in white light at 22℃ generated oscillatory periods similar to that observed by the authors for wild-type strain in blue light at 27℃. Hence this model was used as the foundation and sensitivity analysis was performed to reveal the light-dependent parameters that significantly influenced the circadian period. Several parameters were modified to model temperature-compensation in the wild-type and double mutant strains. Effects of temperature were quantified using the Arrhenius equation. The modified parameters were involved in LHY transcription, mRNA degradation and protein translation; CCA1 transcription and mRNA degradation; TOC1 protein degradation; modified TOC1 protein degradation; PRR7 protein degradation and GI transcription. For the double mutant strain, an additional modification was made to the parameter relating to night inhibitor (NI) protein degradation. The reparametrized model simulated mRNA levels for CCA1, LHY, PRR9, TOC1 and GI. Figure 3 shows the simulation results of LHY and TOC1. Although the model fitted the wild-type data well, it was not able to adequately fit all the double mutant data.
Additionally, a Hopf bifurcation analysis explained the observation that the double mutant cryptochrome strain failed to reliably oscillate at 27℃ (as mentioned in Observations ). A system of ordinary differential equations can be expressed as a matrix. The eigenvalues of this matrix can reveal the stability of steady states to a perturbation. A complex eigenvalue means the system will show an oscillatory behaviour and the oscillation will be sustained if the real part is positive. The parameter values used to simulate the double mutant strain had eigenvalues with real part close to zero. This explained why the double mutant seedlings failed to exhibit sustained rhythmic behaviour. A second mathematical model from Pokhilko et al. (2012)  was used in an attempt to improve the fitting of the observational data. The second model included additional species to the evening loop and changed TOC1 from an activator to a repressor of LHY and CCA1. However, this model was not able to fit all the data well either. Both models experienced trade-offs between accurately simulating LHY mRNA and protein levels or PRR9, TOC1 and GI mRNA levels. This suggested that the model required further knowledge about regulatory interactions to explain the double mutant phenotype. However, the ability of the Pokhilko et al. (2010)  model to fit the wild-type data and predict the level of key clock protein with changes in temperature, illustrated that the authors suggested a valid mechanism as to how temperature-compensation and light pathways interact.
Figure 3. Observation (solid) and simulated (dotted) mRNA profiles for the wild-type (black) and cry1-cry2 (blue) mutant genotypes in 12℃(left), 17℃(middle) and 27℃(right). Figure taken and adapted from .
Gould et al. (2013)  performed a comprehensive study of Arabidopsis circadian rhythm by combining experimental approaches with mathematical modeling. Using LUC reporter for gene activity and mRNA expression, changes to the circadian rhythm were monitored in three different genotypes of Arabidopsis (Wild-type, Cry1-mutant, cry1-cry2 double mutant) under three different light conditions in three different temperature conditions. These experiments revealed that blue light signalling via cryptochromes regulated circadian rhythm in a temperature-dependent manner. Further, to test if the light detector proteins that regulate circadian period are also involved in temperature-compensation, an existing model from Pokhilko et al. (2010)  was adopted. First, by performing sensitivity analysis, the parameters that significantly affected the circadian rhythm were revealed. The model was then reparametrized to simulate temperature-compensation in wild-type and the cry1-cry2 strains. Model simulations reproduced the experimental data from the wild-type strain of Arabidopsis in blue light and varying temperature thus potentially supporting the hypothesis that temperature-compensation of circadian rhythm is achieved via photoreceptor pathways. But the model couldn't reliably reproduce the temperature-dependent effects in cry mutants, suggesting a lack of understanding in the complete regulatory processes. Nevertheless, this study is an example of how mathematical models can be reused and how a model can be used to validate biological hypotheses.
- Gould, P. D., Ugarte, N., Domijan, M., Costa, M., Foreman, J., Macgregor, D., Rose, K., Griffiths, J., Millar, A. J., Finkenstadt, B., Penfield, S., Rand, D. A., Halliday, K. J. & Hall, A. J. (2013). Network balance via CRY signalling controls the Arabidopsis circadian clock over ambient temperatures. Mol. Syst. Biol. , 9:650. doi: 10.1038/msb.2013.7.
- Pokhilko, A., Hodge, S. K., Stratford, K., Knox, K., Edwards, K. D., Thomson, A. W., Mizuno, T. & Millar, A. J. (2010). Data assimilation constrains new connections and components in a complex, eukaryotic circadian clock model. Mol. Syst. Biol. , 6:416. doi: 10.1038/msb.2010.69.
- Pokhilko, A., Fernandex, A. P., Edwards, K. D., Southern, M. M., Halliday, K. J. & Millar, A. J. (2012). The clock gene circuit in Arabidopsis includes a repressilator with additional feedback loops. Mol. Syst. Biol. , 8:574. doi: 10.1038/msb.2012.6.