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In our model, we found the all of the coupling was predominantly from the cell cycle toward the circadian clock. Indeed when zeroing the coupling $F_2$ in the model the dynamics remained almost unchanged. While this could reflect a real lack of control of the cell cycle progression by the circadian clock in our cell line, it could also be due to inherent limitations of the data and the inference methods. In particular errors on the cell cycle phase inference might render difficult to estimate its velocity. Note however that these errors equally affect the estimation of a trajectory position in the $(\theta, \phi)$-plane for $F_1$ and $F_2$. In our data there is also about two times more variations in the cell cycle durations than in the circadian intervals, also reflected by our inferred phase diffusion coefficients. This high intrinsic variability might also impede the inference of the coupling from the circadian clock to the cell cycle.  \subsection{hidden \subsection{Hidden  Markov models for phase inference} In this work we developed a novel method (TODO:true?) of inferring phases from data using hidden Markov models. We found that this method is adequate to decode real-world data, as it allows to specify a noise model as well as the underlying stochastic dynamics, and to explicitly take into account features of the data like amplitude, making the hypothesis underlying the analysis more manifest. A consequence of this is that several parameters have to be calibrated. Note however that because we learn the waveform, linking the phase to the data, from a large collection of time traces, our inferred phase is robust to transformations of the data, unlike methods based on the Hilbert transform \cite{Kralemann_2008}.