Biphasic amplitude oscillator characterized by distinct dynamics of trough and crest

Biphasic amplitude dynamics (BAD) of oscillation have been observed in many biological systems. However, the specific topology structure and regulatory mechanisms underlying these biphasic amplitude dynamics remain elusive. Here, we searched all possible two-node circuit topologies and identified the core oscillator that enables robust oscillation. This core oscillator consists of a negative feedback loop between two nodes and a self-positive feedback loop of the input node, which result in the fast and slow dynamics of the two nodes, thereby achieving relaxation oscillation. Landscape theory was employed to study the stochastic dynamics and global stability of the system, allowing us to quantitatively describe the diverse positions and sizes of the Mexican hat. With increasing input strength, the size of the Mexican hat exhibits a gradual increase followed by a subsequent decrease. The self-activation of input node and the negative feedback on input node, which dominate the fast dynamics of the input node, were observed to regulate BAD in a bell-shaped manner. Both deterministic and statistical analysis results reveal that BAD is characterized by the linear and nonlinear dependence of the oscillation trough and crest on the input strength. In addition, combining with computational and theoretical analysis, we addressed that the linear response of trough to input is predominantly governed by the negative feedback, while the nonlinear response of crest is jointly regulated by the negative feedback loop and the self-positive feedback loop within the oscillator. Overall, this study provides a natural and physical basis for comprehending the occurrence of BAD in oscillatory systems, yielding guidance for the design of BAD in synthetic biology applications.

Figure 1

Identifying the core oscillator in two-node systems with random circuit analysis. (a) Schematic of the workflow. Left panel: enumeration of topologies with two nodes. Node $X$ is used to receive the input signals and node $Y$ is the output. Middle panel: stochastic analysis of the Latin hypercube sampling procedure is used to produce random parameter sets for each topology. Right panel: The robust oscillation signal of the node $Y$ is detected numerically. $Y_{\max }$ and $Y_{\min }$ are the oscillation crest and trough of $Y$, respectively. (b) Time series of a representative deterministic model for each of the 19 possible topologies, M1–M19. (c) Probabilities of parameter sets that yield robust oscillations for the topologies of M10, M14, and M19.

Figure 2

Identification of the essence for oscillation in two-node oscillators. (a) Parameter distributions of all the identified deterministic models that can produce oscillation in topologies M10, M14, and M19. (b) Three representative time series of nodes $X$ and $Y$ selected from oscillators M10, M14, and M19 (top panels), along with the violin plot of the ratio, which is defined as FWHM/period of nodes $X$ and $Y$ (bottom panels). The top and bottom of the gray box denote the 75th and 25th percentiles, respectively. The black point within the gray box represents the mean value. The peripheral curve of the violin plot describes the probability density distribution, and also the maximum and minimum values. (c) The schematic diagram of the essence of three oscillators. Here, the dynamics of activator $X$ is significantly faster than repressor $Y$. The thick red lines indicate the fast dynamics, and the thin black line indicates the slow dynamics. The dashed-line circle indicates the self-action of node $Y$ that may be activation, inhibition, or absent.

Figure 3

A concrete example of a deterministic model of topology M10 to characterize BAD of oscillation. (a) The variation of oscillation amplitude $Y$ with input $I$ increases in a deterministic oscillation model. (b) Time series of $Y$ when $I=0.1$, 0.4, and 0.7 in the deterministic oscillation model. (c) The potential landscape of the system when $I=0.1$, 0.4, and 0.7, respectively (top panels), and the corresponding vertical views of the landscapes (bottom panels). The vector field is indicated by the black arrows. The trajectories with red arrows show that the solutions converge to a limit-cycle oscillation after initial transient. The oscillation crest and trough of $Y$ in the trajectories are represented by magenta squares and black circles, respectively. (d) Crest $Y_{\max }$ and trough $Y_{\min }$ of $Y$ change as the increase of input $I$. € Analysis of the core parameters $k_1, k_2$, and $k_3$ in regulating $H$/BAD. (f) Phase diagram of $H$ in $k_1-k_2$ parameter spaces.

Figure 4

Statistical analysis of ${10}^4$ deterministic models that satisfy the criterion of BAD for topology M10. (a) The distributions of oscillation amplitude of $Y$ under different input strength. (b) Bar chart of the oscillation crest $Y_{\max }$ and trough $Y_{\min }$ at oscillation-starting stimulus (SS), medium stimulus strength (MS), and oscillation-ending stimulus (ES) for topology M10, respectively (bottom panels). (c) Different oscillation behaviors of $Y$ in response to different input strengths. SOSS, small oscillation at small stimulus strength; LOMS, large oscillation at medium stimulus strength; SOLS, small oscillation at large stimulus strength. (d) A special case to illustrate the time courses of $Y$ responding to input with the intensity $I=0.004t$. (e) The violin plots of statistical results of oscillation crest $Y_{\max }$ and trough $Y_{\min }$ on more than ${10}^4$ parameter sets of topology M10, respectively. The mean of the trough $Y_{\min }$ increase linearly with the increasing stimulation intensity, while the mean of the crest $Y_{\max }$ increase nonlinearly.

Figure 5

Mechanism analysis of the linear response of the oscillation trough to the input strength. (a) The probability distribution of term ${(1-Y_{\min })}^{n_3}/[{(1-Y_{\min }]}^{n_3}+J_3^{n_3})$ at $I=0.2$. (b) The violin plots of $X\_Y_{\min }$ under different input strengths. (c) Schematic diagrams of dissecting the influences of each term on $X\_Y_{\min }$. (d) The violin plots for the area marked with a, b, c, and d in Fig. 5, respectively. (e) The probability distribution of term $X\_Y_{\min }^{n2}/\left(X\_Y_{min}^{n2}+J_2^{n2}\right)$, showing that most of the values are equal to 1. (f) The probability distribution of $K$. (g) The probability distribution of $k_2{k}_3/k_{\mathrm{inh}Y}$. The left-$Y$ axis indicates probability (bar plot) and the right-$Y$ axis presents the cumulative probability (blue line plot). (h) The change of $R{C}_{\mathrm{slope}\_}T$ (variation of the slope of oscillation trough) with the fold change of $k_2$. (i) Comparison of slope of the oscillation trough calculated by simulation and theoretical value 1/$k_2$ with more than ${10}^4$ parameter sets of topology M10. Pearson correlation coefficient $r=0.99$.

Figure 6

Analysis of the nonlinear response of the oscillation crest to the input strength. (a) The probability distribution of term ${(1-Y_{\max })}^{n_3}/[{(1-Y_{\max })}^{n_3}+J_3^{n_3}]$ at $I=0.2$. (b) The violin plots of $X\_Y_{\max }$ under different input strengths. (c) Schematic diagrams of dissecting the influences of each term on $X\_Y_{\max }$. (d) The violin plots for the areas marked with e, f, g, and h in Fig. 6, respectively. (e) The probability distribution of term $X\_Y_{\max }^{n2}/\left(X\_Y_{\max }^{n2}+J_2^{n2}\right)$, showing that most of the values are equal to 1. (f) and (g) The changes of $R{C}_{\mathrm{slope}\_}SS$ (variation rate of the slope of the oscillation crest at the oscillation-start stimulus) and (variation rate of the slope of the oscillation crest at the oscillation-ending stimulus) with the fold change of each parameter.