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 A proposal for robust temperature compensation of circadian rhythms

Figure 1
Oneparameter bifurcation diagrams for the differential equations (B1) and (B2), described in Appendix. (A) Parameter values: v
_{m} = 2, k
_{m} = 0.2, k
_{p1} = 53.36, k
_{p2} = 0.06, k
_{p3} = 0.2, K
_{eq} = 1, P
_{crit} = 0.6, J
_{p} = 0.05. (B) All rate constants increased 2fold. For each value of the bifurcation parameter, v
_{p}, we plot the value of [PER] on recurrent solutions of the differential equations (steady states and limit cycle oscillations).
Solid curve, stable steady state; dashed curve, unstable steady state. Curves labeled [PER]_{max} and [PER]_{min} indicate the range of an oscillatory solution at fixed value of v
_{p}.
At the Hopf bifurcation, the steady state changes stability and small
amplitude, stable limit cycle oscillations arise. At the SNIC
bifurcation, two steady states (a stable node and an unstable saddle)
annihilate each other and are replaced by a large amplitude limit
cycle. (A Inset)
The period of oscillation at the SNIC bifurcation is infinite, but
drops quickly to a value of ≈15 h. Superimposed on the bifurcation
diagrams are the trajectories (dashed/dotted line) generated by the
resetting hypothesis (see text). Although the locations of the
bifurcation points depend strongly on parameter values, as do the
shapes of the resetting trajectories (dashed/dotted line), the period
of the two trajectories is precisely 24 h.
(Click image to enlarge)


Figure 2
Time courses of per mRNA and protein, and the resetting parameter, v
_{p}, for the mechanism described in the text. (A) Parameter values as in Fig. 1A, plus P
_{thresh} = 2, μ = 0.0288, σ = 0.5. (B) Parameter values as in Fig. 1B, plus P
_{thresh} = 2, μ = 0.0576, σ = 0.25.
(Click image to enlarge)


Figure 3
Robustness of compensated oscillator period as a function of perturbation strength (σ_{p}) for the models LG, TH and RS. (A) The CV of the period is plotted vs. σ_{p},
indicating each model's response to test A. RS is virtually unaffected
(extremely robust), whereas LG and TH fail to compensate the period to
different degrees. (B) ΔT is averaged over all possible single reaction mutants (Test B) for a given perturbation strength, σ_{p}. We see that RS is robustly temperature compensated for such mutations (very low ΔT), whereas both LG and TH fail to compensate over the given temperature range.
(Click image to enlarge)


Figure 4
Sensitivity of oscillation (% of samples that lose oscillation) as a function of perturbation strength (σ_{p}) in test A. At σ_{p}
= 0.2, LG loses oscillation ≈47% of the time, TH ≈59% of the time, and
RS ≈40% of the time. Oscillations are most robust to small
perturbations for LG, whereas RS is considerably more robust than
either LG or TH for large perturbations.
(Click image to enlarge)

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