1 Introduction. Maria Luisa Guerriero. Jane Hillston

Transcription

1 Time-Series Analysis of Circadian Clocks: Quantifying the Effect of Stochasticity on the Distribution of Phase, Period and Amplitude of Circadian Oscillations Maria Luisa Guerriero Centre for Systems Biology at Edinburgh University of Edinburgh, UK Jane Hillston School of Informatics and Centre for Systems Biology at Edinburgh University of Edinburgh, UK Circadian clocks are gene regulatory networks present in most organisms that help them to adapt to the 24-hour day/night cycle. The effect of stochasticity on the behaviour of circadian clocks is well known and several works have shown that stochastic models generally give a better description of experimental data compared to deterministic models. Most of the existing work show this by visual comparison of time-series data obtained from simulations with time-series data obtained from laboratory experiments, often in terms of qualitative features such as oscillations dampen or oscillations are persistent, instead of providing a precise quantitative measure of the stochastic effects. Here we report some of our investigations on the use of time-series analysis techniques for computing quantitative measures such as the distribution of phase of oscillations. 1 Introduction Circadian clocks are gene regulatory networks which, by means of interlocking transcriptional feedback loops involving a small number of genes, enable organisms to adapt to the day/night cycles. The behaviour of circadian clocks has two essential features: (i) in the absence of external stimuli (e.g. constant light) the amounts of the involved proteins oscillate rhythmically with a period of approximately 24 hours (circadian); (ii) in the presence of external stimuli (e.g. light on/off) the oscillations adjust their rhythm to the external stimuli (entrainment). Due to the low copy numbers in which most of the involved molecules are generally present, stochasticity plays a major role in the behaviour of circadian clocks, in particular in terms of robustness of the free running (i.e. not entrained) clock. Several mathematical and computational models have been proposed in the past to describe the behaviour of circadian clocks of different organisms (e.g. the plant Arabidopsis thaliana [4] the fungus Neurospora crassa [5, 1], and the alga Ostreococcus tauri [6, 2]). Several issues need to be addressed when investigating the effect of stochasticity comparing experimental and modelling data. First, the current limitations in available laboratory technology imply that single cell experimental data are rarely available and, instead, experimental data are generally ensemble data obtained across a large population of cells. Second, experimental observations are generally carried over short periods of time (generally a few days) because of the cost and technical difficulties in maintaining the organisms in constant experimental conditions. Third, there are also issues coming from data analysis: for instance it is not straightforward to measure phase, period and amplitude of oscillations in noisy data (such as both experimental data and stochastic simulation results). The Centre for Systems Biology at Edinburgh is a Centre for Integrative Systems Biology (CISB) funded by BBSRC and EPSRC, reference BB/D019621/1. Submitted to: PASTA 2010 c M. L. Guerriero & J. Hillston This work is licensed under the Creative Commons Attribution License.

2 The aim of this research is to investigate the use of time-series analysis techniques coming from signal processing for computing summary measures of the quality of the circadian clocks related to the regularity of the oscillations, such as the distribution of phase and the stability of period and amplitude. 2 A model of the circadian clock of Ostreococcus tauri In [2] we presented a Bio-PEPA model of the circadian clock of the green alga Ostreococcus tauri based on the deterministic model introduced in [6]. The focus there was to investigate the differences between the continuous-deterministic and the discrete-stochastic approaches. Several analysis techniques, settings and light conditions were considered there. The clock network involves two genes, TOC1 and LHY: both their mrnas and proteins are modelled explicitly (each protein is present in two different forms) and are involved in a negative feedback loop with 5 light inputs. A schematic representation of the model is shown below, and Fig. 1 reports the deterministic behaviour of the system in the considered conditions, as shown in [2]. Level LHY mrna TOC1 mrna Total LHY Total TOC1 Level LHY mrna TOC1 mrna Total LHY Total TOC Time (hours) Time (hours) Figure 1: Deterministic solution of the model: LL (left), LD (right). 3 Time-series analysis of the model Here we use the same model presented in [2], and we focus on time-series data obtained from stochastic simulation and deterministic solution, and consider only two light conditions: constant light (LL) and 24-hour cycles of alternating 12 hours of light and 12 hours of dark (LD). All the results shown in the following refer to the total amount of LHY protein (Total LHY = LHY c + LHY n). Other species have a similar behaviour. First, let us define a few basic terms which will be used in the rest of the paper.

3 Peak: highest value obtained by the variable in one oscillation. Trough: lowest value obtained by the variable in one oscillation. Phase: time of the day at which peak occurs. Period: peak-to-peak distance between two oscillations. Amplitude: difference between peak and trough values. Though all these concepts are basic, it is worth pointing out a couple of issues often make their measurement tricky. First, experimental data are generally very noisy, they only refer to a few days of observation, and the same experiment is rarely repeated more than a couple of times due to the high costs. Moreover, noise in data can be of various types: variations due to differences in the observed cells/organisms, random variations due to system stochasticity, and experimental errors. Even when considering data obtained from simulations, where the only type of noise considered is the one due to stochasticity and despite the fact that simulations could be easily generated for extremely long observation periods and repeated multiple times, the stochastic variations could be so high that these measurements can still be problematic. In the following, we show a possible way to identify peaks and troughs in noisy data and to use them to compute the distribution of phase, period and amplitude in time-series data. 3.1 Peaks/troughs identification and the computation of clock measures There is a major difference in the deterministic behaviour of the free running and the entrained clock: quickly dampened oscillations in the first, permanent oscillations in the latter (see Fig. 1 and [2]). From these deterministic time-series results it is possible to easily measure peak/trough values and the time at which they occur, and from those compute phase, period and amplitude of oscillations. Fig. 2, for instance, shows the behaviour of the LD system, which is a perfectly regular clock with period = 24h, phase 2 and amplitude 4. Figure 2: Measure of phase/period/amplitude on the deterministic solution of the model. Fig. 3 shows the same results computed on stochastic simulation time-series data. Especially when dealing with the noisy data coming from a single run, the first issue is to define what peaks and troughs are and how to distinguish them from random fluctuations. The method we have used here to find the

4 peaks shown in the figure is the following (analogously for troughs), assuming we have equally spaced time-series data points x 0...x 9600 for the time interval [0,960] hours. 1. Bin the time points into 1-hour-long slots and find the maximum in each hour [max 0...max 960 ]. This step smooths the data getting rid of the small fluctuations. 2. Find the local maxima in [max 0...max 960 ]: local maxima are data points x i [max 0...max 960 ] which are higher than a given number of their neighbours (the simplest case is x i 1 < x i > x i+1 ); we have also set a minimum threshold that local maxima must pass in order to be considered peaks. Figure 3: Distribution of phase/period/amplitude on a single time-series in LD: single SSA run (left) and average over 50 SSA runs (right). Note the different x-axes scales in the two panels. As shown in Fig. 3, phase, period and amplitude on stochastic data are obtained as distributions over the whole time-series. The distributions computed from time-series data which are averages of multiple runs are narrower and getting closer to the deterministic value. Fig. 4, instead, shows the time-dependent distribution of phase, period and amplitude computed over 50 independent SSA runs at both an early time and a late time for the free-running clock (LL). Contrary to what is observed in the deterministic behaviour (Fig. 1), which shows dampening of oscillations, the oscillations shown in the single SSA run are persistent, and indeed the distributions of phase, period and amplitude at early time t=2 and at later time t=99 do not exhibit substantial differences. 3.2 Autocorrelation functions and phase distribution The autocorrelation function of a time-series measures the similarity of the time-series with a shifted version of itself. For perfectly periodic signals, it is a function which oscillates between +1 and -1, with the same period as the data, and with the highest value at time t=0. For noisy signals, the autocorrelation dampens exponentially, and the speed of dampening is a measure often used in signal processing to quantify the noise and phase diffusion in oscillations. For a signal which is only random noise the autocorrelation immediately reaches values very close to 0. The half-life of the autocorrelation is defined as the time it takes for the autocorrelation function to dampen to values smaller than half its maximum (i.e. 0.5). An application of autocorrelation to biochemical oscillators can be found in [3]. Fig. 5 shows the autocorrelation function of both the deterministic and the stochastic time-series for the entrained clock LD. As expected from the considerations above, the perfectly periodic ODE results lead to a perfectly periodic autocorrelation function; the SSA function is similarly regular, but the noise

5 Figure 4: One single SSA run (top) in LL and the time-dependent distribution of phase/period/amplitude on multiple time-series for time t = 2 (middle) and t = 99 (bottom). Figure 5: Time-series and autocorrelation function in LD: ODE (left), average of 50 SSA runs (right). causes the autocorrelation to oscillate between values around +0.5 and -0.5 instead of +1 and -1. The half-life of the autocorrelation function is 72 hours. Fig. 6 shows the autocorrelation function for the free running clock LL. In this case, both for ODE and SSA the autocorrelation function dampens, though we can see that in the single run it does not go below the 95% confidence bounds (the green lines, which mark values of the autocorrelation functions which are so low that they are indistinguishable from random noise), thus confirming what we have observed earlier about the persistence of a periodic (though noisy) behaviour. The half-life of the autocorrelation is 25h, which means that after only one cycle the phase distribution is already evident. 4 Conclusions and future work In this work we have shown some of the preliminary results we have obtained by applying statistical and data analysis techniques to the analysis of simulation data for the Ostreococcus tauri circadian clock model. We are still working on fully understanding how to properly employ autocorrelation functions and related techniques to gain further insight into the simulation results. We are planning to apply this

6 Figure 6: Time-series and its autocorrelation function in LL: ODE (top), a single SSA run (bottom left), and the average of 50 SSA runs (bottom right). approach to data coming from lab experiments as well, and to use this in other circadian clock models. References [1] O.E. Akman, J.C.W. Locke, S. Tang, I. Carré, A.J. Millar, and D.A. Rand. Isoform switching facilitates period control in the Neurospora crassa circadian clock. Mol. Sys. Biol., 4:64, [2] Ozgur E. Akman, Maria Luisa Guerriero, Laurence Loewe, and Carl Troein. Complementary approaches to understanding the plant circadian clock. CoRR, abs/ , [3] Pierre Gaspard. The Correlation Time of Mesoscopic Chemical Clocks. J. Chem Phys., 117: , [4] J.C.W Locke, L. Kozma-Bognar, P.D. Gould, B. Fehér, E. Kevei, F. Nagy, M.S. Turner, A. Hall, and A.J. Millar. Experimental validation of a predicted feedback loop in the multi-oscillator clock of Arabidopsis thaliana. Mol. Sys. Biol., 2:59, [5] J.J. Loros and J.C. Dunlap. Genetic and molecular analysis of circadian rhythms in Neurospora. Annu. Rev. Physiol., 63: , [6] J.S. O Neill, G. van Ooijen, C. Troein, L. Dixon, F.-Y. Bouget, F. Corellou, and A. Millar. Circadian rhythms persisting in the absence of transcription in a eukaryote. Submitted, 2009.