Figure 1.
Three idealized behaviours in response to a temperature step.
Red is temperature, blue is for the clock phase. (A) Temperature sensitivity. The clock ticking rate (and consequently its period) changes with temperature. (B) Temperature insensitivity. Nothing changes with temperature. (C) Temperature compensation. The clock runs faster just after a temperature step (explaining entrainment and phase shifts) but then returns to its initial value, so that period of the clock does not change.
Figure 2.
Scaling of the limit cycle for the Goodwin model.
The parameters are ,
,
, where
is the input, a proxy for temperature. (A) limit cycle for different values of the input in XY space. (B) linear rescaling of the orbits to [0,1] collapses them all onto a single orbit. The central dots indicate the unstable (Hopf) fixed point. Straight lines from the fixed point separate the cycle in 4 periods of equal duration. (C) PRC of the Goodwin model for different input values is invariant. PRC was computed by adding a degradation term of
in Eq. 5 for
of the period. (D) Variable X as a function of phase for the limit cycles at different temperatures, scaled as in B. The maximum of X is defines phase
for the PRC. There is perfect overlap for different input values (as shown on the scale bar for all panels) as expected.
Figure 3.
Evolution of adaptive temperature compensation.
(A) Sketch of the initial adaptive topology and its subsequent network evolution. Parameters and equations are given in Supplementary Text S1. Properties of this network are further detailed in Figure 4. The input, which models the temperature; the output
which adapts; and
(Eqs 10 11) changes substantially with a temperature step and functions as a buffer are color coded. (B) Temporal behaviour of the evolved network for two different input trajectories (the colors follow (A)). The input oscillates, undergoes a random phase shift around time
, and decays exponentially to a constant value. Note only the mean of the buffer
changes substantially with the terminal input value.
Figure 4.
Scaling of the limit cycle for first evolved model.
(A) Sketch of the model. Parameters and equations are given in Supplementary Text S1 (B) Variation of the period with input level demonstrating compensation. (C) limit cycle for different values of the input in space. Limit cycle varies over almost one order of magnitude in
while the period changes by
. The input values follow the color bar in F. (D) Rescaling the limit cycles to the unit interval in each variable shows almost perfect collapse for different input values. Circles indicate the fixed point. (E) PRC for different input values, represented by different colors. The PRC was computed by adding a degradation term of
for
for
of the period. (F)
as a function of phase for the limit cycle at different temperatures. The maximum of
is defined as phase
for the PRC. There is almost perfect overlap. Panels D–F here and the following figure, demonstrate our contention that the evolved models replicate essential properties of the Goodwin model even though there is no direct parameter rescaling.
Figure 5.
Scaling of the limit cycle for the Mixed Feedback Loop adaptive model.
(A) Sketch of the model. Parameters and equations are given in Supplementary Text S1 (B) Variation of the period as a function of input values. (C) limit cycle for different values of the input in the B-transcript, A-protein plane. Limit cycle varies over almost one order of magnitude in B-transcript level, while the period changes by a few percent. Note that the fixed point for A is adaptive (independent of input). (D) Linear rescaling of the limit cycles to the unit interval in each variable showing almost perfect collapse for different input values. Circles indicate the fixed point. Color code follows bar in panel F (E) PRC for different input values, represented by different colors. The PRC was computed by adding a degradation term of for B transcripts for
of the period (see equations in Supplementary Text S1). (F) B as a function of phase for the limit cycle at different temperatures. The maximum of B is defined as phase
for the PRC. There is almost perfect overlap.
Figure 6.
Absence of scaling for an evolved model with distributed temperature compensation.
(A) Sketch of the model. Parameters and equations are given in Supplementary Text S1 (B) Variation of the period as a function of input. (C) limit cycle for different values of the input in space. Limit cycle varies over one order of magnitude in variable 1 and 2 while the period changes at most
(D) Rescaling of those limit cycle to the unit interval for each variable. The orbits for different inputs no longer scale. Circles mark the fixed point (E)The PRC for different input values do not scale. The PRC was computed by adding a degradation term of
for variable 2 for
of the period. (F) Variable
as a function of phase for the limit cycles at different temperature. Its rescaled maximum of 1 is defined as phase
for the PRC.