Sensitivity analysis of circadian entrainment in the space of phase response curves

Abstract

Sensitivity analysis is a classical and fundamental tool to evaluate the role of a given parameter in a given system characteristic. Because the phase response curve is a fundamental input--output characteristic of oscillators, we developed a sensitivity analysis for oscillator models in the space of phase response curves. The proposed tool can be applied to high-dimensional oscillator models without facing the curse of dimensionality obstacle associated with numerical exploration of the parameter space. Application of this tool to a state-of-the-art model of circadian rhythms suggests that it can be useful and instrumental to biological investigations.

Figures (8)

• Figure 1: (A) Circadian oscillators are viewed as open dynamical systems with input $u$ and output $y$. (B) The unforced system exhibits autonomous rhythms that occur with a period close to $24$ hours. (C) The periodically forced system adapts the organism rhythms through entrainment ($1\mathord{:}1$ phase-locking) with the 24-hours stimulus associated with earth's rotation.
• Figure 2: The asymptotic phase map $\Theta:\mathcal{B}(\gamma)\rightarrow\mathbb{S}^1$ associates with each point $x_q$ in the basin $\mathcal{B}(\gamma)$ a scalar phase $\Theta(x_q)=\theta$ on the unit circle $\mathbb{S}^1$ such that $\lim_{t\rightarrow+\infty} \left\|\phi(t,x_q,0) - \phi(t,x_p,0)\right\|_2 = 0$ with $x_p = x^\gamma(\theta/\omega)$.
• Figure 3: Entrainment is studied by applying weakly connected oscillator theory to the feedforward interconnection between an artificial oscillator generating the input and the actual oscillator.
• Figure 4: The Leloup-Goldbeter model accounts for several regulatory processes identified in circadian rhythms of mammals. Reproduction of a figure from Leloup:2003cp.
• Figure 5: Normalized robustness measures $\overline{R}_\omega$ (angular frequency) and $\overline{R}_{q}$ (iPRC) reveal the distinct sensitivity of three distinct genetic circuits (Cry, Per, and Bmal1). Each point is associated to a particular parameter. The three lines are regression over the parameters of the three gene loops. The dashed bisector indicates the positions at which both measures of robustness are identical. Only parameters associated with the Cry-loop exhibit low angular frequency and high iPRC sensitivities. The color code corresponds to different subsets of parameters associated to different loops (see the text for details).

Theorems & Definitions (5)

• Definition 2.1
• Definition 2.2
• Remark 1
• Remark 2
• Remark 3