Fig 1.
Molecular architecture of the mammalian circadian clock.
Two transcription-translation feedback loops drive the oscillations of the mammalian circadian clock at the cellular level. We show the architecture for the core clock species of a single cell, where the grey oval represents the nucleus of the cell. The negative loop (warm colors, solid arrows) contains the mRNAs Period and Cryptochrome and their associated proteins. The positive loop (cool colors, dashed arrows) contains Rev-erb, Ror, and Bmal mRNAs and the associated proteins. Hollow circles represent mRNAs and filled circles represent proteins of the different clock species. Arrowheads represent increased production, while flat arrows represent inhibited production. The dashed lines in the positive feedback loop represent interactions that were not included in the Hirota model [1] but that were added to our extended dual loop model to provide additional control targets acting on the core circadian clock architecture.
Table 1.
Action of known small molecules on the molecular clock.
Several small molecules are known to impact the clock by acting on core clock model species. These actions may be represented by changing a corresponding parameter of the model.
Table 2.
Definition of the symbols for the species of our model.
Table 3.
The desired values of the model features with the weights used in the cost function compared with the values of these features produced by the model fit with the lowest cost parameter set.
Table 4.
The parameter set with lowest cost. Because we did not fit the model explicitly to a specific period or to a specific concentration of one of the model species and instead only used relative time differences and relative abundances, the units for both time and species concentration are arbitrary. The amplitude and period can both be arbitrarily scaled to specific experimental results. Units for the rates, denoted with a “v” are arbitrary concentration units per arbitrary time units.
Fig 2.
Output oscillations of the model.
The output oscillations in expression concentrations of the model parameterized with the lowest cost parameter set show that the model produces a limit cycle. Each model species is represented by a different color line. The x-axis is in arbitrary time units since time can be rescaled to any desired period. From the expression curves, one can visualize the different model features, including peak-trough ratios, relative abundances, and phase differences, used to fit the parameters. Despite similar underlying physiological processes (transcription, translation, and degradation) corresponding to similar equations, each species has a different resulting waveform, presenting a potentially important model outcome for comparison to other models as experimental techniques improve to have better time-resolution for these expression curves.
Table 5.
Performance on model validation tests.
Comparison of the performance of different models on different validation tests. To analyze period (T) changes to increased expression levels of some species, we examine the period sensitivity, expressed as partial derivatives, to the corresponding parameter. Positive sensitivities correspond to an increased period.
Fig 3.
Period sensitivities to model parameters in each loop.
For every parameter in the model, we plotted the period sensitivity calculated locally at the optimally fit parameter value versus the fit parameter value. The period sensitivity to the parameters in the negative feedback loop (blue) were orders of magnitude greater than those of the positive feedback loop (orange). Since these sensitivities were infinitesimal, the period sensitivity was largely independent of the magnitude of the fit parameter and depended much more strongly on which feedback loop the parameter governs.
Fig 4.
Correlation of amplitude sensitivities of different species to model parameters.
For each model species, we computed the amplitude sensitivity for each of the model parameters. The positive correlations between the amplitude sensitivities for each species showed that the amplitude of different species are most sensitive to similar parameters. As expected, these correlations were strongest between the mRNA and protein of the same species. Strong correlations were also seen within the negative feedback loop. These correlations suggested that the targets which are most effective in controlling amplitude for a single species will similarly be strong targets in affecting the amplitude of the other clock species.
Fig 5.
Phase and amplitude response curves to small molecule inputs.
The phase response curves (left) and amplitude response curves of Bmal1 and BMAL1 (right) to perturbations in parameters corresponding to the action of known small molecules (Table 1).
Fig 6.
Use of phase only linear approximation to predict phase changes.
The phase changes that were observed in the full model compared to the estimated phase changes using the linear approximation from the ipPRC (Eq 20) for a single input (left) or pairs of parameters (right) corresponding to known inputs were strongly correlated, validating the use of this approximation in the MPC algorithm and suggesting that nonlinear interaction terms are not significant for these pairs of inputs.
Fig 7.
Comparison the MPC algorithm using different control inputs.
The output of the MPC simulation using the linear approximation from the ipPRC (Eq 20) for different combinations of parameters. A. KL001 (single input on the negative loop), B. Longdaysin (single input on the negative loop), C. KL001 and FBXW7-α (multi-input on the negative and positive loop), D. Longdaysin and FBXW7-α (multi-input on the negative and positive loop), E. KL001 and Longdaysin (multi-input on the negative loop). Top panels in each subfigure show the phase trajectory of the model (green) in comparison to the reference (dotted gray). Other panels show the PRC of the control input (dashed lines) and control delivered (solid lines) for KL001 (red), Longdaysin (cyan), and FBXW7-α (blue).
Table 6.
Settling time of simulation for different inputs.
Comparison of the performance of different input combinations with the MPC algorithm in our simulated scenario on the settling time in the simulation for the 5 h advance and 11 h delay.