Nonlinear Model Predictive Control For Circadian Entrainment Using Small-Molecule Pharmaceuticals

Abstract

Recent in vitro studies have identified small-molecule pharmaceuticals effecting dose-dependent changes in the mammalian circadian clock, providing a novel avenue for control. Most studies employ light for clock control, however, pharmaceuticals are advantageous for clock manipulation through reduced invasiveness. In this paper, we employ a mechanistic model to predict the phase dynamics of the mammalian circadian oscillator under the effect of the pharmaceutical under investigation. These predictions are used to inform a constrained model predictive controller (MPC) to compute appropriate dosing for clock re-entrainment. Constraints in the formulation of the MPC problem arise from variation in the phase response curves (PRCs) describing drug effects, and are in many cases non-intuitive owing to the nonlinearity of oscillator phase response effects. We demonstrate through in-silico experiments that it is imperative to tune the MPC parameters based on the drug-specific PRC for optimal phase manipulation.

Keywords: Biological control; circadian oscillator; model predictive control; phase response curve; time-varying weights.

Fig. 1.

Schematic of the core circadian gene network and effect of KL001. (A) The core circadian negative feedback loop. KL001 stabilizes nuclear CRY by reducing its degradation rate, as shown in blue. (B) Parametric infinitesimal phase response curve (PRC) for KL001-mediated stabilization of nuclear CRYs.

Fig. 2.

Example of applying MPC to re-entrain the circadian oscillator following phase shifts of +4h (AC) and −4h (D–F) applied at 12:00. (A, D) Sinusoidal representation of the phase of the environment (dashed) and the circadian clock under MPC (solid). A sinusoid is used to visualize the phase progression and avoid the complications of representing an 8-state oscillator. (B, E) Control inputs (u, shaded pink) and phase response curves (PRC, blue) for MPC examples. Because the positive phase shift in B occurs where the PRC is positive, the controller acts immediately. Despite a large error, the MPC is unable to correct the negative phase shift case in E until the PRC is negative, since the control variable is manipulated unidirectionally. (C, F) Phase difference (rad²) between external and oscillator phase.

Fig. 3.

Time to re-entrain following a phase shift is dependent on predictive horizon. Controller sampling time τ_u = 2h is fixed at throughout. (A) For a 4h horizon, a 12h phase shift applied at 12:00 requires several days to re-entrain. Here, MPC uses several intermittent phase advances, as a phase advance is immediately available after the shift and the predictive horizon does not observe the larger phase delay region. (B) For a 24h horizon, the oscillator may be re-entrained in approximately 24h, as MPC identifies the optimal solution of applying a single, large dose of KL001 to cause a 12 hour phase delay. (C) A sweep across phase shift and predictive horizon demonstrates that a short predictive horizon may result in suboptimal re-entrainment of the circadian oscillator under MPC. The regions of suboptimal entrainment occur where the controller selects the incorrect direction to phase shift the clock, as in A and B.

Fig. 4.

Time to re-entrain following a phase shift is dependent on controller sampling rate and shape of the PRC. All predictive horizons are held constant at τ_uN_p = 24h. (A) For τ_u = 2h (12 samples per 24h horizon), re-entrainment to a +4h shift may occur rapidly, as the smaller positive PRC region can be targeted. (B) For τ_u = 12h (two samples per 24h horizon), it takes multiple cycles to re-entrain, as each control step includes both a positive and negative PRC region. (C) A sweep across phase shift and sampling rate (predictive horizon τ_uN_p held constant at 24h) demonstrates that for a larger τ_u, the loss of resolution can result in a longer time to entrain. This is most explicitly evident for small positive phase shifts, as the positive region of the PRC is smaller and must be therefore more precisely targeted.