Controlling the occurrence sequence of reaction modules through biochemical relaxation oscillators

TL;DR

This paper provides a design scheme of biochemical oscillator models in order to generate periodical species for the order regulation of these reaction modules and takes the case of arbitrary multi-module regulation into consideration.

Abstract

Embedding sequential computations in biochemical environments is challenging because the computations are carried out by chemical reactions, which are inherently disordered. In this paper we apply modular design to specific calculations through chemical reactions and provide a design scheme of biochemical oscillator models in order to generate periodical species for the order regulation of these reaction modules. We take the case of arbitrary multi-module regulation into consideration, analyze the main errors in the regulation process under \textit{mass-action kinetics} and demonstrate our design scheme under existing synthetic biochemical oscillator models.

Figures (10)

• Figure 1: Diagram for the phase plane of $\Sigma_{xy}$. The dotted green curve represents the critical manifold $\left \{ (x,y): y=\varphi(x) \right \}$ and the red lines describe the approximate position of the oscillation orbit $\Gamma_{\epsilon_1}$. $P_1, P_2, P_3$ and $P_4$ are the four initial points, and the point $E(\ell,\varphi(\ell))$ lying on the middle part of critical manifold is the unique unstable equilibrium in the first quadrant.
• Figure 2: Simulation result for the concentrations of the four signals $V_1$, $V_2$, $V_3$ and $V_4$. The four signals are symmetrical as Definition \ref{['def1']} with $T \approx 26.874$.
• Figure 3: A brief glance on the phase plane under a general $\varphi(x)$ with the unique unstable equilibrium $E$ and the approximate position of the oscillation orbit $\Gamma_{\epsilon_1}$ described by the connected red lines for a regulation task of $m$ modules.
• Figure 4: Simulation result of the Example \ref{['example:M4']}. Values of $s_1$ and $s_2$ are oscillating between $1$ and $2$, which is consistent with our expectation.
• Figure 5: Simulation result of values of $s_1$ and $s_2$ based on different choices of $T$ in Example \ref{['example:M4']}. The values of $s_1$ and $s_2$ with the same period are marked in red and blue separately, and the values with different periods are distinguished by lines of different shapes.

Theorems & Definitions (18)

• Example 3.1
• Definition 3.2: symmetrical clock signals
• Lemma 3.3
• Proof
• Lemma 3.4
• Proof
• Theorem 3.5
• Remark 3.6
• Lemma 4.1: Shi2022
• Lemma 4.2