Parameterized resetting model captures dose-dependent entrainment of the mouse circadian clock

Abstract

The phase response curve (PRC) represents the time-dependent changes in circadian rhythm phase following internal or external stimuli. However, this time dependence complicates PRC measurement and quantification owing to its variable shape with changing stimulus intensity. Our previous work demonstrated that resetting a desynchronized circadian clock (singularity response, SR) simplifies the analysis by requiring only amplitude and phase parameters. In this study, we construct a comprehensive model for phase resetting in the mouse circadian clock by converting PRCs into SR parameters. We analyze single-cell PRCs and show that the SR amplitude parameters for different stimulus concentrations follow the Hill equation. Additionally, the model predicts the combined effects of multiple stimuli and pre-treatment (background) on phase response by simple addition or subtraction of individual SR parameters. Experimental validation using SR measurements in mouse cells and tissues confirms the model's accuracy. This study demonstrates that SRs facilitate PRC quantification and reveal simple rules governing phase resetting properties under various conditions using SR parameters.

Fig. 1. PRC deformation and SR parameters.

a Phase response in a circular limit cycle model. The solid circle represents the pre-stimulus state, the dashed line represents the post-stimulus state, and the arrows represent stimulus-induced phase shifts. The input strength F = 0.5, and the input direction Φ = π/2 radians. The angle from the center point represents the phase of the circadian rhythm, and the center point indicates a singular point. b Deformation of the phase response curve depends on the input strength. The input strength is represented by the length of the arrows in (a). The input direction (Φ) is π/2 radians. Filled and open circles indicate stable and unstable states, respectively. c The maximum phase shifts induced by the PRC (blue line) and the SR amplitude (red line) are plotted against the input stimulus strength. d The phases at which the maximum responses occur in the PRC (blue line) and the SR phase (red line) are plotted against the input stimulus strength. PRC phase response curve, SR singularity response.

Fig. 2. Dose-response curves for SR parameters.

a The SR amplitude is plotted against the concentration of lithium chloride (LiCl), forskolin, and corticosterone. Each curve represents a fitted curve, and the corresponding parameters are shown in the illustration. b The relationship between SR amplitude and phase is shown for each stimulus. Each line represents a fitted line, and the corresponding approximate expressions are shown. c PRCs for forskolin at indicated concentrations (black points) and the estimated PRCs based on the SR parameters obtained from the dose-response curves (red curves). PRC phase response curve, SR singularity response.

Fig. 3. Predicting SR parameters for combined stimuli.

a Prediction model for mixed stimuli based on SR parameters for each single stimulus. b Predicted SR amplitude of the combination of different types of stimuli: dexamethasone (DEX) and phorbol 12-myristate 13-acetate (PMA), DEX and forskolin (FK), and cobalt chloride and lithium chloride. c Predicted SR phase for the combination of different types of stimuli. d Prediction model for mixed stimuli based on the equivalent concentrations. e Prediction of the SR amplitude for the combination of the same types of stimuli (hydrocortisone (HYD) and corticosterone (CORT)). Red and blue points indicate the estimated amplitude using the sum of equivalent concentrations (a) and SR parameters (d), respectively. Blank circles indicate the results of the combination of different types of stimuli (b). f Measured and predicted PRCs for mixed stimuli (black points), and predicted PRCs based on the predicted SR parameters (red curves). PRC phase response curve, SR singularity response.

Fig. 4. Background effects on SR parameters.

a Prediction model of the background effect. Dashed arrows represent SR amplitude for background stimuli (BK). Solid arrows represent the SR amplitude induced by the additional stimulus. b Attenuation of SR amplitude depends on background concentration. c The decay of background concentration during the culture. Solid and dashed lines indicate the concentration of a stimulating chemical or molecules in the medium without and with decay, respectively. d SR amplitude for forskolin and dexamethasone with background effects as a function of fold change. e, f Model predictions of background effect without (e) and with (f) background decay. SR singularity response, BK background stimuli.

Fig. 5. Evaluation of SR parameters by direct measurement of SR in cultured cells.

a Normalized bioluminescence of PER2::LUC at SR measurement using dexamethasone (DEX), forskolin (FK), and corticosterone (CORT). The concentration of each line corresponds to that in (b) and (c). b Dose-response curve of SR amplitude. Each parameter was obtained by fitting it to the Hill equation. c Change in SR phase with different concentrations. The blank circle is individual data, and the filled square is the mean value. d, e Changes in SR amplitude for mixed stimuli of the combination of DEX and FK (d), and DEX and CORT (e). f, g SR amplitudes (f) and phase (g) for the combination of DEX and FK are estimated by the sum of SR parameters for each stimulus. h, i SR amplitudes for the combination of DEX and CORT are estimated by the sum of SR parameters (h) and equivalent concentration (i). Filled circles indicate the results of single stimuli, and blank circles show the results of mixed stimuli. j, k Change in SR amplitude for DEX with DEX background in experiment (j) and model (k). l, m Change in SR amplitude for 100 nM DEX (l) and 100 μM FK (m) with different background: no background, 100 nM DEX, 100 nM CORT, or 100 μM FK (*p < 0.05, **p < 0.01, ***p < 0.001; two-tailed t-test). Exact p-values are provided in Source Data. n, o Change in SR amplitude (n) and phase (o) for the combination of 100 nM DEX and mifepristone (MF). The blank circle is individual data, and the filled square is the mean value. p, q Relationship between SR phase and SR amplitude for DEX and CORT stimulation (p) and MF stimulation (q). “BK” indicates the background stimuli, and “ADD” indicates the additional stimuli. Each individual data indicates the results of independent experiments and experiments were performed three times or more. SR singularity response.

Fig. 6. Evaluation of SR parameters by direct measurement of SR in cultured tissues.

a, b Dose-response curve of SR amplitude (a) and phase (b) for dexamethasone (DEX), corticosterone (CORT), and forskolin (FK) in the lung. c, d Dose-response curve of SR amplitude (c) and SR phase (d) in white adipose tissue (WAT), spleen, and kidney. Each parameter was obtained by fitting using the Hill equation. Blank circles and filled squares indicate individual data and the mean values, respectively. In WAT, the dose-response curve was calculated excluding the value of 100 nM. e, f Changes in SR amplitude for the combination of DEX and FK (e) and DEX and CORT (f) in the lung. g, h SR amplitudes (g) and phase (h) for the combination of DEX and FK are estimated by summing SR parameters for each stimulus. i, j SR amplitudes for the combination of DEX and CORT are estimated by summing SR parameters (i) and equivalent concentration (j). Filled circles and blank circles indicate the results of single stimuli and mixed stimuli, respectively. k, l Change in SR amplitude for DEX with DEX background (pre-treatment) experiment (k) and model (l). m Change in SR phase for 100 nM DEX with 100 nM CORT or 100 μM FK background (*p < 0.05, ***p < 0.001, ns: p ≥ 0.05; two-tailed t-test). Exact p-values are provided in Source Data. n, o Change in SR amplitude (n) and phase (o) for the combination of DEX and mifepristone (MF). Blank circles and filled squares indicate individual data and the mean values, respectively. p Relationship between SR phase and amplitude for DEX and CORT stimulation. “BK” indicates the background stimuli, and “ADD” indicates the additional stimuli. Each individual data indicates the results of independent experiments and experiments were performed three times or more. SR singularity response.