Characterizing the accuracy of circadian clocks

Thomas d’Eysmond

Circadian Clocks (CCs) are regulatory gene networks, which allow organisms to coordinate their physiological rhythms in a daily cyclic environment. These cell-autonomous molecular oscillators with an intrinsic period close to one day, have been found all across the tree of life in cyanobacteria, fungi, plants, insects, mammals, etc. They play a great variety of roles from regulation of the cell cycle, tumor susceptibility and response to chemotherapy in humans [1], to time sensing, time compensated sun-compass navigation, and social behaviors such as coordination of activity, dance language communication and division of labor in the honey-bee [2]. A decade of genetic, biochemical and functional studies on these models has identified the main components and their interactions. Based on this knowledge, a model can be established for the particular clock studied. As in most biological systems, the role of noise cannot be ignored. Indeed for some of the biochemical species of the model, the number of mRNA molecules can be low. It is therefore to be expected, that the stochastic nature of chemical reactions will influence on the precision of the clock. In addition to this intrinsic source of noise, there is also extrinsic noise caused by the fluctuations in the environment, living matter, in which CCs are embedded. One can wonder how in these conditions, CCs are still able to tick accurately. The question of how nature has solved this problem, is actually at the center of the concerns of chronobiology. An appropriate measure of the precision of a clock is the quality factor, or number of periods during which the clock is a reliable time indicator. While a good watch has a quality factor of the order of 10^4, published results [3] suggest that an isolated cell exhibiting circadian rhythms represents a clock with a quality factor of about 10.
To achieve their purpose and keep track of the day-time, CCs  are entrained by periodic signals present in the environment (light, temperature, etc). An organism isolated from external signals will stay in phase with the daily cycle for extended periods of times. It was found in different studies, that CCs possess mechanisms that confer robustness to external perturbations, for instance temperature compensation of the period [4]. Coupling between individual oscillators is also a means of enhancing the precision of the clock. In mammals for instance, the neurons of the suprachiasmatic nucleus SCN, which are coupled to light perception through visual pigments in the retina, have a strong coupling among each-other and continue to keep an accurate track of day-time long after the absence of light signals [5]. Together these cells play the role of a pacemaker and have been shown to entrain the circadian oscillations across the other tissues of the mammalian organism. But even in a more primitive life form such as the Neurospora, coupling is possible since cells have several nuclei, thus producing several repression loops through the cytoplasm. This results in more accurate and robust oscillations [6]. In situations of low coupling and low entrainment, noise at the single cell level will drive a population to desynchrony. The objective of the present project is to understand the link between the structure of circadian oscillators and their function as accurate clocks.
A natural mathematical representation of a CC model including molecular noise is the Chemical Master Equation (CME), which can be simulated at the exact level using Gillespie’s algorithm. Since the latter works at a time scale of single biochemical events it is extremely computer-time consuming. A part of this project has therefore been devoted to find robust techniques to predict the quality factor without having to simulate the full CME. In Fig. 1 one can see the result of an analysis of the Gonze-Goldbeter model [6] upon reduction of transcription rate v_s for nominal parameters . The continuous line in the oscillatory regime, on the right of the Hopf Bifurcation (HB), represents the approximation yielded by the Linear Noise Approximation, an expansion of the CME for low noise. One can see that this approximation is inaccurate for high noise or close to the HB, at which the amount of noise has to be reduced extremely in order to have a good agreement.
Figure 1 :
Secondly we tried to access the quality factor also experimentally by analyzing high quality bioluminescence imaging data [7] in the frame of our collaboration with the Schibler lab in Geneva. This data consists of videos featuring three successive circadian oscillations of Bmal1-Luc NIH3T3 mouse fibroblasts at single cell resolution, which can be appreciated in the sample picture showed in Fig. 2. These bioluminescence signals were measured under different conditions of reduced transcription rates (up to about minus 30%) in wild type and Per1 knockout cells. Upon reduction of the transcription rates it was observed [7] that sustained oscillations are maintained, but that period shortens and amplitude decreases. Furthermore the period shortening was absent in Per1 knockout cells.

Figure 2 & 3 :
The type of data to be analysed differs from [3]. In the latter, the average bioluminescence signal of an initially synchronized population of fibroblasts recorded during 19 successive circadian periods, after detrending according to cell death and infradian frequencies, is fitted by an exponentially decaying periodic signal. In our data there are only three successive periods; however time courses of individual cells can be extracted. Since the cells are unentrained and uncoupled, the same decay should be observed in the autocorrelation function of an individual cell and in the average signal of an initially synchronized population under the assumption that the cells have identical biochemical properties.                                      
To take into account that in reality cells are not identical inside the population, we assume that the phase diffusion, which occurs exponentially at a rate given by the inverse of Q, is mainly induced by the intrinsic noise in the cells. Furthermore we assume that in the presence of the extrinsic noise, the cells are in different states, but are all subjected to a type of intrinsic noise, such that the quality factor inside the population has a small variance and thus the lack of statistics due to the small amount of circadian periods featured in the movies can be compensated by averaging over a sufficiently large amount of cells.
Our analysis also differs from the one that was published [7]. In the latter each cell is assumed to have a constant period and thus a part of the intrinsic noise is lost. 
In order to analyze the data we make use of the Java software SpotTracker2D and Circadian Gene Express developed for the ImageJ application by Daniel Sage \textit{et al.}. The need for more advanced software comes from the fact that cells move and their trajectories need to be followed by the region of interest.
In Fig. 3 the result of an analysis performed with SpotTracker2D is shown. About 30 cells can be extracted from each video. In the selection process cells with overlapping trajectories are avoided.
One can see that the quality factor drops upon reduction of the transcription rates. This could be indicating that untreated cells have parameters that optimize the quality factor. By using the theoretic analysis performed on the Gonze-Goldbeter model Fig. 1 one might also conclude that this behavior of quality factor reduction upon transcription factor reduction is characteristic of oscillators in a parameter regime closer to the HB.
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