Circadian Clocks
Collaborators: Andrew Millar, Isobelle Carre, Domingo Salazar, Boris Shulgin, Ozgur Akman, Paul Brown.
Funding: EPSRC & BBSRC
Circadian rhythms control 24-hour rhythms of metabolism, physiology and behaviour in organisms ranging from humans to cyanobacteria. The circadian clock controls the expression of between ~6% (in the model plant, Arabidopsis thaliana) and ~100% (in a cyanobacterium) of the genome, as indicated by transcriptome analysis from several labs. The photoperiodic mechanism that responds to day length also depends on the circadian clock; photoperiodism leads to the seasonal rhythms of breeding and hibernation in animals, and of flowering and bud dormancy in plants. Consequently, circadian rhythms affect both daily and annual rhythms, with impacts in agriculture, industry and medicine, including surgery, chemotherapy and mental health.
For circadian clocks, molecular genetics revealed a key problem at the core of the known clock mechanisms: the form of all the clock gene circuits includes multiple, intertwined regulatory loops. Molecular genetics has not determined why this is so, and multiple theoretical studies have shown that the fundamental properties of circadian rhythms could be generated by a single loop. Understanding this shared feature was a necessary foundation for any analysis of species-specific circuits. Another question that has barely been addressed by biologists is how the genetic network adapts multiple properties of circadian rhythms (output phases, entrainment, temperature compensation, etc.) to selective pressures that might be conflicting. Our results directly attack these problems.
The first step in our approach is to mathematically characterise the key evolutionary goals. To do this we developed infinitesimal response curves (IRCs, see Box 1). Clock models include many parameters (biochemical rate constants, affinity constants, etc.). In any but the simplest network model, it is challenging to predict which parameters affect a specific regulatory property; each parameter or property is typically analysed in turn. We use IRCs to describe which combinations of parameters can be tuned in order to produce a specific combination of circadian properties. IRCs also allow us to determine which properties of a circadian clock can be tuned independently and which of them are strongly related. The aim is to characterize the evolutionary goals so that they are given by transparent and comparable mathematical conditions. We can then compare how the various clock models meet these conditions.
This approach allows a synthesis of various key clock properties such as robust entrainment (see Box 2), temperature compensation of period, stability of timing in the face of parameter fluctuations, tuning of clock output pathways and the effects of mutations
The second step is to analyse the flexibility that is necessary to achieve the multiple evolutionary goals revealed by the above. To make sense of this we have to define a measure of the flexibility and show how to calculate it (see Box 2). Our measure is called the flexibility dimension, d. We show that (i) there is a selective advantage in increasing d to a value where the full range of adaptive properties can be tuned and that (ii) each individual regulatory loop is rather inflexible and clocks require considerable loop complexity to get the amount of flexibility that is necessary to adapt to multiple pressures.
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IRCs tell us how changes in a given parameter ki at a given phase f of the oscillation affect a given output Qj such as period, the phases of the maxima and minima of mRNA and proteins, the amplitude of these maxima and minima, and the levels of mRNA and protein at prescribed phases. |
| When Qj is period an IRC gives the PRC for a small perturbation in the parameter ki. If light or another environmental input of interest acts by changing the value of ki we can express the usual PRC for this input using the IRCs. Light increases TIM protein degradation in Drosophila, for example, so the PRC of the Drosophila clock in response to a light pulse is the negative of the IRC for the parameter marked “TIM p2 degradation” in the figure above. Moreover, we can calculate the PRCs for complex inputs that affect multiple parameters. They give very good approximations of PRCs even for very long pulses (see figure below). |  |
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Circadian clocks must be robustly entrained by environmental cycles in light or temperature. We express this condition in terms of IRCs using the fact that they give the PRCs for the environmental signal. A informative approximate way to analyse this involves the mapping F which tells us the phase f?=F(f) of the oscillator at the end of a day if at the beginning it was f. This can be calculated from the PRC and hence the appropriate IRC as shown in the figure above. What we can deduce from this is that the basic requirement is that the relevant IRC have large amplitude and allow daily correction of the clock period to the external day length. We can then investigate the conditions necessary for robust entrainment, for example, in response to sustained or stochastic changes in the environment, to changes in nutrition, and to molecular fluctuations in the cell. |
| Flexibility dimension and relevant singular spectrum of various published models. n and s are respectively the number of dynamical variables and parameters. The four values given for d are respectively the values of the flexibility dimension d when e2 = 0.05, 0.01, 0.005 and 0.001 so that the first d principal components capture approximately 95%, 99%, 99.5% and 99.9% of the variance. |  |
Other clocks projects:
- Structure of the Arabidopsis clock. Photoperiodism and the timimg of flowering
- Global temperature compensation
- How to tune clocks models
- Statisticd from time-series
- …
Address:
Mathematics Institute
University of Warwick
Coventry CV4 7AL
United Kingdom