Co-existing feedback loops generate tissue-specific circadian rhythms

The analysis of tissue-specific data-based models of the gene regulatory network of the mammalian circadian clock reveals organ-specific synergies of feedback loops.


S3 Scoring parameter sets for global optimization
Extracting features from experimental data We fitted harmonic models to experimental data [1] that is publicly available for different tissues with a circadian resolution of 2 h over two days. Using such models, measurements are approximated to yield more reliable estimates of amplitudes and phases. The fitted models have the form given in Equation S3-1. Thus, the functions contain harmonics and have curve shapes with wide troughs and narrow peaks. Such a shape is well suited for the measured time series.
The fit parameters a, b and c are obtained by nonlinear regression. Example fits of harmonic models to liver data are shown in Figure S3 The scoring function The complete scoring function incorporating period, phases and fold changes is given in Equation S3-2. Differences between simulated ( ⋅ sim ) and experimentally measured values ( ⋅ exp ) are weighted by tolerances (tol ⋅ ).
As phase sim and phase exp relative phase differences to Bmal1 are used. Thus, there are four phase differences.
The fold changes are calculated as log 2 max min . For experimental values we use maxima and minima (peaks and troughs) derived from harmonic fits to the data. In this way, measurement errors of individual points at peaks or troughs are reduced since all points contribute to the fits.
There are 10 terms in total (1 period + 4 phases + 5 amplitudes). If differences between data and fit are equal to the tolerances we get a score of 10. Thus, we consider a score of 10 as a reasonable cutoff. Figure S3-2 shows an example fit with a score close to 10.

Tolerances
We use a tolerance of tol period = 0.1 h for the period, reflecting typical experimental deviations in mice WT data. For relative phases we compare measurements from different experiments [1][2][3] and derive a tolerance of tol phase = 1 h. Figure S3-3 shows that relative phases measured in the three experiments are indeed comparable.
Fold changes are less consistent between the different experiments. To define a reasonable tolerance we therefore split the Zhang et al. data set into day 1 and day 2 and compare the deviation between these days. Since fold changes vary strongly between genes, we define five tolerances (one for each gene) based on the median differences between the days: tol f oldch(Bmal1) = 0.

Cuts through the fitness landscape
Application of the scoring function to all parameter combinations yields a 35 dimensional landscape with troughs and peaks. Using global optimization with VFO (see Supplement S4) and Particle Swarm Optimization we search troughs with minimal score. In Figure S3-4 we present a representative selection of cuts through the fitness landscape along one parameter axis. For each depicted parameter, cuts are shown for models fitted to SCN and liver. The parameter values are taken from successful optimization runs and the found minimum is marked in red.
This depiction presents useful information to judge the quality of fits and the ability to identify unique parameter values. Most cuts have a nearparabolic shape and clear trough as in the first 2 columns of Figure S3-4. Only some cuts have minima located in less steep troughs, as for example in row 1, column 3. In fits with less good scores, as for example in row 1, sometimes also a boundary is reached after which rhythms vanish (dotted lines).
In just a few cases there is no clear optimum (e.g. last plot in row 1). Interestingly, Cry1 ⊣ Per2, a regulation that is part of the repressilator, has a clear trough for the fit to liver data which contains a repressilator in its rhythm generating set of loops. However, it shows no discernible optimum for the SCN fit that does not involve this regulation as an essential part of its oscillation generating mechanism. In each plot one parameter of an optimal model fit is varied from 0.2 to 5 times the optimal value (red line). Transitions to regions with no oscillation are marked by dotted lines.