Stochastic modeling.

<p>(A) Results from fitting the fibroblast data to the stochastic model, as described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0033334#s4" target="_blank">Methods</a>. Fit values with the parameter <i>s</i><1 indicate that the oscillations are noise-induced (the deterministic system would be steady state), while fit values with <i>s</i>≥1 indicate that the oscillations are self-sustained. (B) Results of simulations of the stochastic model for different values of the parameter s, fixing the amplitude of <i>z</i> to be 500 molecules (by adjusting the value of Ω with respect to <i>s</i>). The mean value of the period CV over 500 simulations is shown for each value of <i>s</i>. The period CV is minimized for <i>s</i> in the range 1.1–1.2. (C) Example of a stochastic simulation with <i>s</i> = 1.03, where Ω is chosen so that the amplitude of <i>z</i> is 500 molecules. The period CV for this simulation is 0.073.</p

Correlations among fibroblast parameters, as defined in Table 1.

<p>Pearson correlation coefficients were calculated; Spearman correlation coefficients gave very similar results.</p

Comparison of cell parameters.

<p>Parameters are defined in the caption to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0033334#pone-0033334-t001" target="_blank">Table 1</a>, with correlation coefficients given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0033334#pone-0033334-t002" target="_blank">Table 2</a>. Greater amplitude tends to be associated with reduced variability, and variability in period tends to be much less than that in amplitude.</p

Period and amplitude are remarkably stable over time.

<p>Error bars indicate mean±SD across the 80 cells on each day. The mean period changes very gradually, with a slight upward slope of 0.02 hours/day (F = 15,p<0.001), an average change of 0.1% per day. The mean amplitude decreases with a slope of −0.04 photons/min/day (F = 60,p<0.001), an average change of about 1% per day.</p

Analysis of cell periods.

<p>(A) Histogram of cell periods (mean peak-to-peak times). (B) Raster plot showing two cells with clearly different periods. In the raster plot, time of day is plotted left to right and successive days down the page, such that vertically adjacent points are 24 h apart. Each row is extended to 48 h, duplicating data in the next row, so that patterns crossing midnight can be appreciated. Thick bars designate times when the luminescence for a cell was above the mean for each row. Cell #66 with period 25.5 h is plotted in red; cell #68 with period 22.5 h is plotted in blue. Due to different circadian periods, the two cells' phase relationship changes over time. (C) Standard deviation in period over the population of cells as a function of the number of cycles used for period determination. Here period for each cell is calculated as the mean of peak-to-peak times over the indicated number of cycles. This curve is expected to decrease to the true value in proportion to one over the square root of the number of cycles used. The dashed line shows the ANOVA prediction of the true value of the standard deviation in period among the fibroblasts. Note that if all cells had the same intrinsic period, and variability of observed period was only due to stochastic fluctuations, then we would expect this graph to approach zero, rather than having a positive horizontal asymptote.</p

Assessment of cell rhythmicity.

<p>(A) Percent of cells with rhythmic 3-day windows is greater for later start times. (B) Percent of rhythmic windows increases with length of window, combining over all cells with start times spaced every 12 h. (C) Strength of rhythmicity (new metric described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0033334#s2" target="_blank">Results</a>) of 3-day windows increases over time. (D) CVs of period and amplitude decrease over time, measured using 6 consecutive peak-to-peak times and peak-to-trough amplitudes starting at the indicated day.</p

Examples of fibroblast PER2::LUC recordings.

<p>The time series for each example is shown above the corresponding analytic wavelet transform (AWT) to illustrate the variability over time in period and amplitude. Period as a function of time is indicated by the black ridge curve, while amplitude is indicated by the color scale (in photons/min). A line at period 25 h is included for reference. (A) Typical cell #11, cell area 1.12×10⁴ µm², period 24.3 h with CV 0.067, amplitude 2.94 photons/min with CV 0.35. Note that red in the AWT corresponds to cycles with high amplitude, yellow those with moderate amplitude, and blue those with low amplitude. Period variability is indicated by the black ridge curve moving up and down over time. (B) Large cell #25, cell area 1.74×10⁴ µm², period 25.3 h with CV 0.030, amplitude 12.2 photons/min with CV 0.23, exhibiting steady rhythms, with both amplitude and period varying less than in (A). (C) Cell #36 with strong oscillations except for a pause on days 13–14 (reflected by the blue region of the AWT), cell area 8.36×10³ µm², period 24.8 h with CV 0.055, amplitude 2.93 photons/min with CV 0.27.</p

Summary of descriptive statistics of the 6-week-long fibroblast recordings (<i>n</i> = 80 cells from 2 cultures).

<p>After eliminating several days at either end of the data series to avoid edge effects and initial transients, 34 cycles were used for calculating cycle length and amplitude statistics of each cell. The period of each cell's time series was calculated using maximum entropy spectral analysis (MESA) and also as mean peak-to-peak time. Coefficient of variation (CV), a dimensionless measure of variability, equals the standard deviation divided by the mean. Brightness (photons/min) equals mean intensity over 34 days of recording.</p

Illustration of new rhythmicity metric.

<p>(A) Normalized power spectral density (PSD) averaged across 115 <i>Bmal1−/−</i> cell 6-day-long PER2::LUC recordings from Ko et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0033334#pone.0033334-Ko1" target="_blank">[19]</a>. The PSD exhibits characteristics of Brownian noise, with mean slope of the regression line after logarithmic transformation equal to −2.00±0.24 (mean±SD). (B) Averaged PSD for our 80 wild-type fibroblasts using 6-day windows. (C, D) Log-log plot of the PSD and its regression line to illustrate the new rhythmicity test. A star indicates the peak circadian value <i>Y</i>₀ (omitted from the data curve and regression calculation so that we can test whether this value is significantly above the regression line corresponding to background noise), the value on the regression line at the same frequency is , and the dashed line shows the test metric . The <i>Bmal1−/−</i> cell (C) is judged arrhythmic with , whereas the wild type fibroblast (D) is significantly rhythmic.</p