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Conceived and designed the experiments: PF EDS. Performed the experiments: PF. Analyzed the data: PF ND EDS. Contributed reagents/materials/analysis tools: PF ND EDS. Wrote the paper: PF EDS.

The authors have declared that no competing interests exist.

A temperature independent period and temperature entrainment are two defining features of circadian oscillators. A default model of distributed temperature compensation satisfies these basic facts yet is not easily reconciled with other properties of circadian clocks, such as many mutants with altered but temperature compensated periods. The default model also suggests that the shape of the circadian limit cycle and the associated phase response curves (PRC) will vary since the average concentrations of clock proteins change with temperature. We propose an alternative class of models where the twin properties of a fixed period and entrainment are structural and arise from an underlying adaptive system that buffers temperature changes. These models are distinguished by a PRC whose shape is temperature independent and orbits whose extrema are temperature independent. They are readily evolved by local, hill climbing, optimization of gene networks for a common quality measure of biological clocks, phase anticipation. Interestingly a standard realization of the Goodwin model for temperature compensation displays properties of adaptive rather than distributed temperature compensation.

Circadian clocks are biological oscillators which evolved to couple the internal rhythm of animals, plants and even some bacteria to the alternation of light and day. Circadian oscillators are temperature compensated, i.e. they keep a 24-h period irrespective of the temperature of the organism. This is surprising, since many biochemical parameters, including average concentration of clock proteins, vary with temperature. From dynamical system theory, we therefore expect changes in both period and relative lengths of features in the phase response curve which are not seen. We couple mathematical modelling and computational evolution of gene networks to formulate a novel explanation for temperature compensation that accords better with experimental facts than alternatives. Our model has deep mathematical connections with the process of biochemical adaptation, by which cells respond to temporal gradients of signals rather than their absolute value.

It has long been recognized

There is presently only very sparse data on how the principal steps that additively determine the 24 hr period vary with temperature

Sensory adaptation is an apt analogy for how the clock period becomes temperature invariant yet temperature entrainable, and forms the mathematical basis for our model. Assume the temperature is the ‘stimulus’ and it only enters the model in a few specific terms consistent with adaptation to the stimulus. Then as we will show the period and PRC shape are temperature independent. The response of an adaptive system to the

The temperature response of this new class of so called

We begin with a summary of experimental facts that cast doubt on the literal application of a distributed temperature compensation to circadian clocks and then introduce a series of models of increasing complexity and realism based on the idea of adaptive compensation. The

The most prevalent model of temperature compensation is also the most parsimonious in that it makes no structural assumptions about how temperature enters the network equations, and was proposed by Ruoff and Rensing

This model though parsimonious is not intuitively satisfactory in all respects, though it does not directly contradict any experiment. As noticed by Tyson and coworkers, several mutants do not appear fully consistent with distributed compensation

Other fly mutants like

The dual properties of a temperature independent period and strong entrainment by an oscillating temperature are tantamount to asserting that the time rate of change of the phase (angular velocity) around the orbit is adaptive,

Red is temperature, blue is

A temperature step

While Eq. 2 may seem very artificial, we show next that its principal features are recovered in a widely used model for temperature compensation in the

Ruoff and coworkers have used the Goodwin model

Following

If we assume that variable

the amplitude of the orbit varies with the production rates, while the period is independent of them.

the oscillator orbit undergoes a linear transformation after a temperature step if only the production rates are temperature dependent i.e.,

the phase response curve, defined by multiplying one or more coefficients by a time dependent factor, is invariant under any constant rescaling of the production terms, since the transformation from Eqs. 4–6 to 7–9 clearly applies with the temporally modulated coefficients.

These remarks then explain the results of Ruoff and coworkers

The parameters are

The linear transformation on the orbits induced by temperature and the temperature invariant PRC we derived from Eq. 7–9 seems very specific to the Goodwin model, and we would like to demonstrate that the same properties are found in a wider class of models. As explained above, temperature compensation looks formally very similar to biochemical adaptation. Thus it is natural to ask if we can build temperature compensation upon an adaptive network for temperature. To be consistent with mutants such as

Our simulations evolve both the gene network and the parameters as we have done previously

To emphasize the connection to adaptation we initialize our simulations with a simple two gene adaptive network, shown in 3A, that we evolved previously

In contrast with the model of Zimmerman et al.

The evolution optimizes the

The second part of the fitness

For the first third of the integration period,

(A) Sketch of the initial adaptive topology and its subsequent network evolution. Parameters and equations are given in Supplementary

One of the simplest models found by numerical evolution is presented in

Schematically, compensation in this model works in a way very reminiscent to biochemical adaptation in the network used to initialize evolution: variable

(A) Sketch of the model. Parameters and equations are given in Supplementary

The properties of the oscillations defined by the network in

Clocks built from an adaptive system, share a feature of the Goodwin model that the orbits for different inputs, as well as the location of the unstable Hopf fixed point, can be superimposed by a linear rescaling,

We verified numerically that the PRC are shape invariant whether derived from a strong localized decay rate applied to any of the adapted variables in

Since the fitness is linear correlation with a sinusoidal reference phase, it is maximum when the solution is itself sinusoidal and optimally remains so when the temperature is shifted, thus explaining the linear covariance of the orbits with temperature. In general the evolved models behave as if they were near the Hopf bifurcation, yet do so over a parameter range that causes a 10× change in the orbits.

We have also verified that a two-fold variation in parameters does not appreciably degrade the period compensation shown in

Thus parameters are not tuned, and their general magnitudes are easy to find by a simple local hill climbing algorithm (a.k.a. gradient search). Two other evolved networks with similar properties are presented in Supplementary

Network of

We further wondered if computational evolution is able to select for different categories of compensated clocks, where the limit cycle and PRCs depend much more significantly on the input. We modified the fitness so it continued to favor entrainment to temperature

Properties of a network evolved under this scheme is described in

(A) Sketch of the model. Parameters and equations are given in Supplementary

This network displays autonomous oscillations for input values higher than 0.1. Remarkably, while the input is changing from

(A) Sketch of the model. Parameters and equations are given in Supplementary

We have exhibited a sequence of

Properties of these models (beyond the temperature compensation and entrainment that we imposed on the evolution) are:

temperature

clock components oscillate around means that either are temperature independent (and are coupled to the adaptive variables) OR vary and buffer the temperature change (e.g., variables 2,3 vs 1 in

orbits rescale linearly with temperature, and in addition the phases that define extrema on the orbits are invariant,

the shape of the PRC is temperature independent, when defined by an augmented decay rate on the adapted variables.

Experiments from a variety of organisms are better explained by adaptive rather than distributive temperature compensation. In fly, mutations in

In saturating light the fly PRC are temperature invariant

In cyanobacteria circadian clock temperature compensation occurs through the KaiC component alone and temperature compensation persists in mutants with periods substantially different from 24 hrs

In

The situation appears less clear to us in plants, perhaps because there are many more duplicated genes in

For all models presented here, properties 1–4, when they apply, are structural : for the Goodwin model this is due to the specific forms of the equation that allowed rescaling, in the MFL model the properties derive from the specifics of the coupling to inputs, and for

Experiments that would most readily substantiate an adaptive model for temperature would be comprehensive data on the zeitgeber time of the maxima and minima of the clock components as a function of temperature. We predict their invariance, while a generic model of distributed compensation would predict that they move with temperature but of course continue collectively add up to the invariant period length. Temperature invariance of the extrema in the clock gene orbits, would suggest some degree of shape invariance in the PRC, but the later is in principal a separate prediction. The linear rescaling of orbits at different temperatures that we found in our models could be probed by time lapse imaging two out of phase clock genes. However the effect might not occur for all choices of genes if there was some saturation. In that situation the phases of extrema will be invariant and thus provide a more robust prediction.

The primary 24 h periodic pacemaker in nature is light. It is worth stressing that adaptation for light inputs themselves has been suggested in Neurospora, a phenomenon called photoadaptation

We have no definitive proposal for how almost all the temperature dependent biochemical rates disappear from the schematic or phenomenological models we are proposing for the circadian clock. We speculate that the shape of the PRC is under strong selection to remain temperature independent along with the period, and thus forces local compensation to render most model parameters temperature invariant, but leaving behind adaptive temperature dependence to allow temperature entrainment. The experimental implications of phase orbits that linearly rescale with temperature are sufficiently dramatic that their observation would render adaptive circadian models plausible though still surprising from the biochemical vantage point.

For evolutionary simulations we follow

Protein-protein interaction (PPI) are explicitly modelled using standard mass-action laws. For instance, if proteins A and B form a dimer C, the equations are:

The fitness is computed for a population of networks, typically 40 in number. The most fit half of the population is retained, and a copy of each network is mutated and added back to the population to maintain its number. Parameter changing mutations are typically ten times as likely as topology changing events. Mutations are sampled according to their intrinsic rates and the generation time is chosen such that approximately one mutation occurs per network.

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