A symmetric dual feedback system provides a robust and entrainable oscillator

Abstract

Many organisms have evolved molecular clocks to anticipate daily changes in their environment. The molecular mechanisms by which the circadian clock network produces sustained cycles have extensively been studied and transcriptional-translational feedback loops are common structures to many organisms. Although a simple or single feedback loop is sufficient for sustained oscillations, circadian clocks implement multiple, complicated feedback loops. In general, different types of feedback loops are suggested to affect the robustness and entrainment of circadian rhythms. To reveal the mechanism by which such a complex feedback system evolves, we quantify the robustness and light entrainment of four competing models: the single, semi-dual, dual, and redundant feedback models. To extract the global properties of those models, all plausible kinetic parameter sets that generate circadian oscillations are searched to characterize their oscillatory features. To efficiently perform such analyses, we used the two-phase search (TPS) method as a fast and non-biased search method and quasi-multiparameter sensitivity (QMPS) as a fast and exact measure of robustness to uncertainty of all kinetic parameters.So far the redundant feedback model has been regarded as the most robust oscillator, but our extensive analysis corrects or overcomes this hypothesis. The dual feedback model, which is employed in biology, provides the most robust oscillator to multiple parameter perturbations within a cell and most readily entrains to a wide range of light-dark cycles. The kinetic symmetry between the dual loops and their coupling via a protein complex are found critically responsible for robust and entrainable oscillations. We first demonstrate how the dual feedback architecture with kinetic symmetry evolves out of many competing feedback systems.

Figure 1. Biochemical network maps of the circadian clock models with different types of loop coupling logics.

A: The single feedback model, B: the semi-dual feedback model, C: the dual feedback model, D: the redundant feedback model. The notation of CADLIVE – is used for simplifying the diagram. The dashed circle represents nucleus.

Figure 2. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the semi-dual feedback model.

A: QMPS for period, B: QMPS for amplitude. The level of protein Y was changed in the semi-dual feedback model: Y<10 nM (cross), 10 nM≤Y≤200 nM (circle), Y>200 nM (square). The single feedback model (plus) is the control model.

Figure 3. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the dual feedback models.

A: QMPS for period, B: QMPS for amplitude. The kinetic symmetry (ρ) was changed in the dual feedback model: ρ≥99 (cross), ρ = 1 (circle), ρ = 0.1 (square), ρ = 0.01 (diamond), ρ = 0 (triangle). The single feedback model (plus) is the control model. A decrease in ρ increases the kinetic symmetry.

Figure 4. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the redundant feedback models.

A: QMPS for period, B: QMPS for amplitude. The kinetic symmetry (ρ) was changed in the redundant feedback model: ρ≥99 (cross), ρ = 1 (circle), ρ = 0.1 (square), ρ = 0.01 (diamond), ρ = 0 (triangle). The single feedback model (plus) is the control model. A decrease in ρ increases the kinetic symmetry.

Figure 5. Frequency distributions of the γ values for the parameter sets that yield circadian oscillation.

The frequency distributions of the quantitative balance between X and Y loops (γ) were simulated, while changing the kinetic symmetry (ρ): ρ≥99 (cross), ρ = 1 (circle), ρ = 0.1 (square), ρ = 0.01 (diamond). A decrease in ρ increases the kinetic symmetry. γ is the quantitative balance of the X and Y feedback loops, which is defined by: , where [X(nuc)]_mean indicates the mean concentration for X in nucleus and [Y(nuc)]_mean that for Y in nucleus. When γ is close to zero, the effect of the X loop on the oscillator is weak, while the Y loop is dominant. When γ is close to 0.5, the effects of both the X and Y loops are comparable. When γ is close to one, the X loop is dominant. At ρ = 0 (perfect kinetic symmetry), γ is always equal to 0.5. The distribution for ρ = 0 is not shown.

Figure 6. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the competing models.

A: QMPS for period, B: QMPS for amplitude. The single feedback model (plus), the semi-dual feedback model with Y>200 nM (cross), the dual feedback model with ρ = 0 (circle), the redundant feedback model with ρ = 0 (square). ρ = 0 indicates perfect symmetry between two feedback loops.

Figure 7. Entrainment probability maps for different types of feedback models.

A: the single feedback model, B: the semi-dual feedback model with Y>200 nM, C: the dual feedback model with ρ = 0, D: the redundant feedback model with ρ = 0. ρ = 0 indicates perfect symmetry between the two feedback loops. The color indicates the probability of entrainment given by Eq. (9), the ratio of the parameter sets that entrain to light-dark cycles to all the parameter sets. Red color indicates a high probability, where the cycle readily entrains to the light-dark cycle; blue a low one.