<i>Cry1</i>, <i>Per2</i> and <i>Rev-erb-α</i> oscillations are most critical for circadian rhythm generation.

<p>All possible combinations of gene-subsets were analyzed for oscillating solutions by clamping the remaining genes to their respective oscillation mean values (A: one gene clamped; B: three genes clamped). Blue bars indicate the percentage of parameter sets around the default values that result in oscillating solutions. Red bars depict the median period among these solutions. Only 3 of 10 combinations of 2 genes oscillate at all, which are shown in (B). Error bars give the upper and lower quartiles for the period.</p

Effect of parameter alterations on the period (fraction of default value on logarithmic x-scale).

<p>(A) Change of <i>Per2</i> delay. (B) Change of <i>Cry1</i> mRNA degradation rate. (C) Change of Cry1 inhibition strength on Per2. (D) Change of Bmal1 activation strength on <i>Rev-erb-α</i>. The default parameter values, corresponding to 1 on the x-axis, are: <i>Per2</i> delay <i>τ</i>₃ = 3.82, <i>Cry1</i> degradation <i>d</i>₄ = 0.2, <i>Rev-erb-α</i> activation by <i>Bmal1</i> <i>actn</i>_1,2 = 3.26 and <i>Per2</i> inhibition by <i>Cry1</i> <i>inh</i>_4,3 = 0.37. Blue symbols refer to increasing parameters, whereas orange symbols refer to the reverse parameter variation (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005266#pcbi.1005266.s002" target="_blank">S2 Appendix</a> for details).</p

The repressilator comprising <i>RevErba</i>, <i>Per2</i> and <i>Cry1</i>.

<p>The relative abundancy of processes in oscillating sub-networks is mapped to the edge width. All edges of the repressilator are highly prominent among all oscillating networks, which reflects its role as the dominant source of oscillations in the model.</p

One-variable model - <i>Per2</i> self-inhibition.

<p>(A) Scheme of the one-variable model of self-inhibition of the clock gene <i>Per2</i> with explicit delay and two E-boxes (2E). (B) Observed delays and non-linearities provided by these two E-boxes (as described by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0046835#pone.0046835.e001" target="_blank">Equation (1)</a>) lead to 24 h oscillations. (C) Bifurcation analysis reveals oscillation onset at about 5.3 h. For larger explicit delays , we plot maxima and minima of the oscillation. (D) Control of the period length for different parameters shows that the explicit delay has the strongest effect on the period. Parameter values for simulations: h-1; ; ; h. Gene expression in panels B and C is represented as normalised values divided by the mean of <i>Per2</i> expression.</p

Oscillations of sub-networks.

<p>(A) Simulation of gene expression of <i>Rev-erb-α</i> and <i>Bmal1</i> with other genes (<i>Cry1</i>, <i>Per2</i> and <i>Dbp</i>) clamped to their constant mean value. Upon doubling the strength of <i>Bmal1</i> to <i>Rev-erb-α</i> activation, oscillations are rescued with a period of 24h. (B) Simulation of gene expression of <i>Rev-erb-α</i>, <i>Per2</i> and <i>Cry1</i>, with other genes (<i>Dbp</i> and <i>Bmal1</i>) clamped to their constant mean value. The period lengthens, but oscillations are retained without parameter adjustments being necessary.</p

Regulation of phase variability.

<p>Thick lines in panels A, B, C refer to the corresponding mRNA and black lines mark the production terms; coloured lines in B and C represent “modulation factors”, see text. Long half-lives lead to later peaks of mRNAs (thick lines) compared to production terms (black lines). (A) Delayed <i>Rev-erb</i>a inhibits <i>Bmal1</i> expression. (B) <i>Per2</i> and <i>Bmal1</i> modulators determine <i>Dbp</i> production. (C) RRE modulator (green) and E-box modulator (red) govern <i>Cry1</i> production. (D) Reducing the amounts of regulators <i>Bmal1</i>, <i>Per2</i>, <i>Dbp</i>, and <i>Rev-erb</i>a mimics RNAi experiments and knock-outs. The resulting phase shifts agree with experimental data (see text).</p

Two-variable model - nuclear receptor loop.

<p>(A) Scheme of the two-variable model of the <i>Bmal1</i> - <i>Rev-erb</i>a loop showing the number of relevant CCEs of each gene (2R - two RREs; 3E - three E-boxes). (B) Simulations show that 24 h oscillations with a correct phase difference can be generated by using experimentally observed explicit delays and non-linearities arising from the number of CCEs. (C) The phase difference between the two genes is effected by model parameters with different strength. (D) The waveform is controlled by the explicit delays; greater delays lead to sharper peaks. Parameter values for simulations: h-1; h-1; ; ; ; ; ; h; h. Gene expression in panels B and C is represented as normalised values divided by the mean of the expression of the corresponding gene.</p

Experimental data and model in DD and LD regimes.

<p>(A) Normalised gene expression of <i>Bmal1</i>, <i>Per2</i>, and <i>Cry1</i> under DD and LD regimes. (B) Adding a 12 h∶12 h step function that modulates <i>Per2</i> transcription increases the <i>Per2</i> amplitude (not shown) and causes phase advance of all genes except <i>Rorg</i> and <i>Cry1</i>. (C) Starting from the endpoint in panel B, increasing the explicit delay of <i>Bmal1 </i> leads to correct <i>Rorg</i> and <i>Cry1</i> amplitudes and phases without substantially changing the dynamics of the other genes.</p

Comparison of experimental data and the six-variable model.

<p>(A) Fitted experimental gene expression data of the six core clock genes in mouse liver for the DD regime. (B) Our model (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0046835#pone.0046835.e036" target="_blank">Equations (4)</a> to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0046835#pone.0046835.e041" target="_blank">(9)</a>) reproduces the experimental data for the period length, phases, amplitudes and waveforms of these core clock genes. In both panels, gene expression is normalised by dividing by the mean expression of the respective gene.</p

Gene expression of six core clock genes in mouse liver in constant darkness (DD) and 12 h∶12 h light-dark cycles (LD).

<p>Data were normalised by three reference genes and fitted by a function with 24 h and 12 h trigonometric terms (Equation (S1) in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0046835#pone.0046835.s001" target="_blank">Supplementary Information S1</a> - Fitting of trigonometric functions to gene expression data).</p