Flexible dynamics of two quorum-sensing coupled repressilators

Genetic oscillators play important roles in cell life regulation. The regulatory efficiency usually depends strongly on the emergence of stable collective dynamic modes, which requires designing the interactions between genetic networks. We investigate the dynamics of two identical synthetic genetic repressilators coupled by an additional plasmid which implements quorum sensing (QS) in each network thereby supporting global coupling. In a basic genetic ring oscillator network in which three genes inhibit each other in unidirectional manner, QS stimulates the transcriptional activity of chosen genes providing for competition between inhibitory and stimulatory activities localized in those genes. The “promoter strength”, the Hill cooperativity coefficient of transcription repression, and the coupling strength, i.e., parameters controlling the basic rates of genetic reactions, were chosen for extensive bifurcation analysis. The results are presented as a map of dynamic regimes. We found that the remarkable multistability of the antiphase limit cycle and stable homogeneous and inhomogeneous steady states exists over broad ranges of control parameters. We studied the antiphase limit cycle stability and the evolution of irregular oscillatory regimes in the parameter areas where the antiphase cycle loses stability. In these regions we observed developing complex oscillations, collective chaos, and multistability between regular limit cycles and complex oscillations over uncommonly large intervals of coupling strength. QS coupling stimulates the appearance of intrachaotic periodic windows with spatially symmetric and asymmetric partial limit cycles which, in turn, change the type of chaos from a simple antiphase character into chaos composed of pieces of the trajectories having alternating polarity. The very rich dynamics discovered in the system of two identical simple ring oscillators may serve as a possible background for biological phenotypic diversification, as well as a stimulator to search for similar coupling in oscillator arrays in other areas of nature, e.g., in neurobiology, ecology, climatology, etc.

Figure 1

A repressilator genetic network with QS feedback. Lower case (a, b, c) are mRNA, and upper case (A, B, C) are expressed protein repressors. AI is the autoinducer molecule which diffuses through the cell membrane.

Figure 2

Comparison of circuit behavior (blue dots) and mathematical model (red line) for (a) Hill inhibition for $n=3.17$, $\alpha =218$, and (b) QS activation for $\kappa =21$, $\alpha =135$. Panel (b) also shows $S/(1+S)$ as a green line.

Figure 3

Effect of increasing the QS activation $\kappa$. Numerical $Q$ continuations for two QS-coupled repressilators ($n=2.8$, $\alpha =204$) showing the values of proteins $B_i$ for steady states and the amplitudes of oscillations for limit cycles. HB, Andronov-Hopf bifurcation; LP, limit point; BP, branch point (symmetry-breaking bifurcation). Stable (thick red) and unstable (thin black) steady state, stable (thick green) and unstable (thin blue) limit cycle. Two red lines between LP3 and LP2 correspond to values $B_1$ and $B_2$ for IHSS. (a) $\kappa =12$. Inset shows just the homogeneous branch. (b) $\kappa =17$. Inset shows just the inhomogeneous branch. (b) Two green lines between LP5 and HB2 are amplitudes $B_1$ and $B_2$ for stable IHLC, and two blue lines between LP5 and HB3 are for unstable IHLC.

Figure 4

Circuit measurements for two QS-coupled repressilators ($n=2.8$, $\alpha =204$, $\kappa =17.8$). (a) $Q$ continuation showing stable steady-states (red) and stable limit cycles (green). (b) Time series showing APLC for $Q=0.4$. $B$ values were calculated from measured voltages. Note that the APLC is continuously stable from $Q=0$ to 1.15.

Figure 5

Oscilloscope screenshots of transitions between stable dynamics in the circuit analog of two QS-coupled repressilators ($n=2.8,\alpha =204,\kappa =17.8$). The two traces (blue and red) are the protein $B$ voltages of the repressilators. $Q$ was varied slowly by adjusting a potentiometer. (a) IHLC to APLC for decreasing $Q$ at 0.25. (b) IHSS to HSS for increasing $Q$ at 0.6. (c) HSS to APLC for decreasing $Q$ at 0.15. (d) Low-B-HSS to high-B-HSS for increasing $Q$ at 1.2. Note that in the purely homogeneous case of panel (d), the red trace is not visible because it is covered by the blue trace.

Figure 6

Numerical APLC branch for various $\kappa$. $n=2.8,a=204$. Note that for $n=2.8$ the APLC is continuously stable for all $\kappa$.

Figure 7

Stable (thick green) and unstable (thin blue) APLC branch $n$-continuations. Black dots indicate TR (torus) bifurcations where APLC looses stability. $\kappa =15,\alpha =175$.

Figure 8

Numerical $Q$ continuations as in Figs. 3 and 6 for $n=3.0,\alpha =190$. (a) All regimes. $\kappa =15$. (b) APLC for various $\kappa$. Note the region of unstable APLC (thin blue) between the TR bifurcations for the smaller $\kappa$.

Figure 9

Circuit $Q$ continuations of all stable branches. $n=3.0,\alpha =190,\kappa =17.8$. Note the break in stability in the APLC branch from $Q=0.8$ to 1.15. Stable steady state (red), stable limit cycles (green).

Figure 10

Numerical $n$-continuation for $\alpha =175,\kappa =4,Q=0.8$ showing coexistence of stable IPLC and stable APLC. Stable LC (thick green), unstable LC (thin blue), and stable HSS (thick red).

Figure 11

Numerical $Q$ continuation for $n=4,\alpha =175,\kappa =4$ showing coexistence of stable IPLC and stable APLC. Stable LC (thick green), unstable LC (thin blue), stable SS (thick red), unstable SS (thin black).

Figure 12

Time series demonstrates transition from HSS to IPLC in circuit for decreasing $Q$ at 0.85 (LP1); $n=4,\alpha =182,\kappa =4.2$. The two traces (blue, red) for the protein B voltages of the two repressilators are nearly identical.

Figure 13

Oscilloscope screenshots of time series and FFT demonstrating (a) stable APLC with distinct frequencies in FFT at $Q=0.6$ and (b) CO with continuum FFT at $Q=0.9$. Protein B voltage traces (blue, red) in top portion, FFT of red trace in bottom portion. $n=3.0,\alpha =190,\kappa =17.8$. Stable APLC returns for $Q>1.1$ (see Fig. 9).

Figure 14

Phase plots of coexisting dynamics for two QS-coupled repressilators ($n=3.0,\alpha =200,\kappa =15,Q=0.935$). (a) 5:5 limit cycle. (b) Complex oscillations. Initial conditions are different for the two panels.

Figure 15

Distributions of return times collected from the Poincaré sections of trajectories of 5:5LC (a) and CO (b) in the presence of small noise. $n=3.0,\alpha =200,\kappa =15,Q=0.935$.

Figure 16

Q-continuations of 5:5LC, stable (solid) and unstable (dashed), for $\alpha =200$ and 270, $n=3,\kappa =15$. For $\alpha =200$ the entire 5:5LC coexists with complex oscillations (COs). For $\alpha =270$ the violet lines ending at small circles indicate regions of COs. BP, branch point (broken symmetry bifurcation); PD, period doubling bifurcation; LP, limit point.

Figure 17

Time series comparing the same LC dynamics in numerical simulations (a, b) and in circuit measurements (c, d). (a) 3:3LC numerical. $n=3.19$, $\kappa =15$, $Q=0.6$. (b) As1:2LC numerical. $n=3.2$, $\kappa =15$, $Q=1.092$. (c) 3:3LC measured. $n=3.2$, $\kappa =7.3$, $Q=1.05$. (d) As1:2LC measured. $V_{th}=16$ mV and $R_{\mathrm{hill}}=2.9k\mathrm{\Omega }$.

Figure 18

$Q$ continuations of stable (green) and unstable (dashed) LC, and stable SS (red) for $n=3.15$, $\alpha =225$, $\kappa =10$. Narrower windows of stable high-period LCs not shown. For clarity only the lower amplitude oscillation of asymmetric LCs is shown. The first period-doubled branches for As1:2 and As2:3 are shown (stable, green; unstable, orange dot-dash). Regions of “simple” antiphase complex oscillations (broken violet) exist near the torus bifurcations (TR) of the APLC branch.

Figure 19

Numerical time series of chaotic attractors for QS-coupled repressilators ($n=3.15,\alpha =225,\kappa =10$). (a) Branch switching of the repressilators resulting in symmetric chaos, $Q=0.95$. (b) As chaos as a result of no switching, $Q=0.87$.

Figure 20

Electronic circuit used to model a single gene in the QS-coupled repressilator circuit. Output $V_i$ is the concentration of expressed protein, and $V_{i-1}$ is the inhibitor concentration. Circuit parameters $V_{\mathrm{cth}}$ and $R_{\mathrm{hill}}$ are adjusted to obtain the desired Hill function inhibition.

Figure 21

Circuit for repressilator with QS feedback. The repressilator consists of the closed ring of genes A, B, and C. The quorum sensing loop is from $B$ through $S_1$ to $C$. Protein B creates autoinducer $S_1$ via the voltage-controlled current source $I\left(B\right)$, and $S_1$ activates production of C via $I\left(S\right)$. Each “gene” triangle corresponds to the single gene circuit in Fig. 20. $S_2$ is the contribution from a second repressilator (not shown), and $S_{\mathrm{ext}}$ is the autoinducer concentration in the external medium. Coupling strength $Q$ is controlled by resistor $R_Q$.

Figure 22

Numerical $Q$ continuations for $n=2.8$ demonstrating robustness of dynamics over wide range of $\alpha$. Compare with Fig. 3. (a) $\alpha =180,\kappa =18$. (b) $\alpha =250,\kappa =15$. Solid thick (thin) lines are stable (unstable) attractors.

Figure 23

${\beta }_1$ continuations of APLC branch, stable (green) and unstable (blue dashed). $n=2.8,\kappa =17,\alpha =204,{\beta }_2={\beta }_3=0.1$.

Figure 24

Evolution of sequential period maps: (a) $Q=0.6784$ (quasiperiodic regime), (b) 0.725 (simple chaos), (c) 0.7387, (d) 0.75, (e) 0.824, (f) 0.928. $n=3.15,\alpha =225,\kappa =10$.

Figure 25

Complex oscillations time series for $Q=0.928,n=3.15,\alpha =225,\kappa =10$. Pieces of As1:2LC and 3:3LC are evident.

Figure 26

Bifurcation diagram showing coexistence of As4:4LC and the stable end of a 5:5LC, in addition to the stable end of a different As4:4 branch.

Figure 27

Phase portraits of coexisting limit cycles in periodic windows at $Q=0.714$ in the presence of additional noise. (a) Asymmetrical 4:4LC. (b) Symmetric 5:5LC. $n=3.15,\alpha =225,\kappa =10$.

Figure 28

Time series of chaos composed of alternating polarities of asymmetric LC. (a) $Q=0.82$ has As2:3LC pieces, (b) $Q=0.925$, and (c) $Q=0.95$ have a variety of AsLC pieces.

Figure 29

Switching “polarities” of unstable As1:2LC measured in circuit. $n=3.2,\alpha =148,\kappa =22,Q=0.9$.

Figure 30

$Q=0.8368$ Intermittency near LP of As1:2LC.

Figure 31

$Q=0.7528$. Intermittency near LP of As2:3LC.