Figure 1.
Proposed network structure and mechanism of AtGRP7 and AtGRP8 auto-regulation and cross-regulation.
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 [73]: 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.
Figure 2.
Systems dynamics for the “optimal” parameter set under 12h∶12h LD and LL conditions.
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.
Figure 3.
The model properly fits experimental data.
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 [25]. 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 [22]. In [22], 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 [11], 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 [25]), if the time axis of those experimental data is adjusted by approximately ten hours. Since the samples in the experiments [74], [75] 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.
Table 1.
Optimal parameter set.
Figure 4.
The slave oscillator may represent a driven bistable toggle switch.
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.
Figure 5.
Stable oscillations can be observed even without transcriptional repression of AtGRP8.
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 (
).
Figure 6.
Systems dynamics driven by a modified Poincaré oscillator.
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).
Figure 7.
Damped, autonomous, and noise-induced oscillations after decoupling the slave from the core oscillator.
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
).