Fig 1.
Topology of the Danino oscillator and HOTFM.
(a) Abstract topology of the Danino oscillator. A (LuxI) activates itself and B (AiiA), whereas B (AiiA) represses A. (b) Abstract topology of the HOTFM. The topology remains largely the same, except that B undergoes negative autoregulation via self repression. It is suggested that here that this self-repression be mediated by optogenetics, using two component light response regulation systems such as [31, 32].
Fig 2.
Dynamics of the Danino oscillator.
Concentration versus time plots for the levels of AiiA, LuxI, internal AHL and external AHL.
Fig 3.
In the biological implementation of the circuit, the LuxI is placed under the Lux promoter, similar to that in the Danino oscillator. The AiiA gene, however, is placed under a Lux-Tet promoter which can be activated by AHLs produced by LuxI and repressed by the lambda repressor. TC1 and TC2 represent the two genes of the two component light response systems, CcaS-CcaR and UirS-UirR. pLR denotes the light response promoter. Thus, in the actual circuit, there would be two copies of the TetR repressor, one under pLR1 (pCPCG2) and the other under pLR2 (pcsiR1). The output has been connected across AiiA via sfGFP under the same controls as AiiA. The hammerheads represent repression while the arrows represent activation of the particular promoter where the hammerheads and arrows end.
Fig 4.
Absolute Synchronization Error (ASE) and Rate of Synchronization.
Plots for the Absolute Synchronization Error (ASE) and Rate of Synchronization (r) for HOTFM and the Danino Oscillators. Solid curves represent ASE (Red curve for Danino and Blue curve for HOTFM). The dotted curves are obtained after fitting an exponential envelope of the form EASE = a*exp(−r*t) to the ASEs (Red curve for Danino and Blue curve for HOTFM). The values for the parameters are adapted from [23], except for γA and γH; γA = 23 and γH = 0.023. For the equations unique to HOTFM, Eqs 5 and 6, the hill function parameters for the two component system are adapted from [31] for CcaS-CcaR. TetR is used as the repressor downstream of Ccas-CcaR with a repression coefficient K = 100 and degradation constant γ = 1.5.
Fig 5.
Two parameter bifurcation for HOTFM in the repression coefficient (K)-degradation cefficient for repressor (γ) space.
Two parameter bifurcation for HOTFM against the parameters for the repressor downstream of the two component light system (CcaS-CcaR); the x-axis corresponds to the repression coefficient K and y-axis corresponds to the degradation coefficient γ. The boundary of the shaded region represents the branch of hopf solutions; the unshaded region below the hopf branch corresponds to parameter values for which there exits a stable steady state and oscillations are absent. Whereas, the shaded region above the hopf branch represents parameter values for which stable oscillations exist. a) The two parameter bifurcation plot with the shaded region representing a heatmap for the period of oscillation. At each point in this region, the color represents the period of oscillation. Points farther away from the hopf branch in the shaded region show higher period (lower frequency), while the points closer to the hopf branch exhibit lower period (higher frequency). b) The two parameter bifurcation plot with the shaded region representing a heatmap for the period of oscillation. At each point in this region, the color represents the period of oscillation. Points farther away from the hopf branch in the shaded region show higher amplitude, while the points closer to the hopf branch exhibit lower amplitude.
Fig 6.
Frequency and amplitude variation for HOTFM implemented with TetR repressor.
Variation in the frequency and amplitude of oscillation for AiiA with changing value of the degradation coefficient γ for HOTFM when TetR is used as the repressor downstream of the two component CcaS-CcaR. The repression coefficient K assumes a value of 100 for TetR [51]. At low values of the degradation coefficient (below the hopf branch) oscillations die and there is presence of a stable steady state. However, as the degradation rate increases and the hopf branch is crossed both amplitude and time period increase with increasing degradation coefficient. On the other hand, the period overshoots before settling to values equivalent to the Danino oscillator. Time period and amplitude have a largely linear relationship with increasing γ. a) Variation in time period with increase in degradation coefficient γ for AiiA. b) Variation in amplitude of oscillation with increase in degradation coefficient γ for AiiA. c) Time period vs amplitude as the degradation coefficient γ is increased. d) Numerically integrated solution for HOTFM at different values of the degradation coefficient γ.
Fig 7.
Phase plane plots for the different numerically integrated solutions shown in Fig 6d. AiiA concentration is plotted on the x-axis while LuxI concentration is plotted on the y-axis. Equivalent to the results from the time series plots, the phase plane plots show the variation in behavior with increasing degradation coefficient γ.
Fig 8.
Robustness analysis for IASE in the γI-γA parameter space.
Robustness analysis with respect to Integrated Absolute Synchronization Error (IASE). Two parameter bifurcation plots are presented for HOTFM and the Danino oscillators in the γI-γA parameter space for IASE. The blue (red) curve respresents the hopf branch for HOTFM (Danino). Region above the hopf branch corresponds to parameter values where oscillations occur, while for regions below the hopf branch oscillation dies and steady state solution arises. For a comparative analysis of robustness between HOTFM and Danino we consider 520 points in the region of oscillations common to both HOTFM and Danino. At each point the system of equations (containing a linear array of 6 cells for both HOTFM and Danino) is numerically integrated for 500 minutes. From the obtained solutions IASE is calculated. For points which have IASEDanino − IASEHOTFM > 0, HOTFM is more robust compared to Danino, while for IASEDanino − IASEHOTFM < 0 Danino is more robust. a) Heatmap representing IASEDanino − IASEHOTFM at the 520 points in the oscillatory region common between HOTFM and Danino. Positive values suggest that HOTFM is more robust for IASE compared to Danino and vice versa. b) Heatmap for part a. thresholded at the value 0. Regions with positive values in a. have dark red color here, while regions with negative values in a. are colored blue. c) Heatmap for the effect sizes for individual points calculated using bootstrap estimates for the distribution for IASEDanino − IASEHOTFM Robustness Analysis for IASE. If IASEDanino − IASEHOTFM > 0, effect size is reported for greater robustness of HOTFM, while if IASEDanino − IASEHOTFM < 0, effect size (cohen’s d) is reported for greater robustness of Danino. The color scheme is as follows—Blue (negligible effect, d < 0.2), Cyan (small effect, 0.2 ≤ d < 0.5), Yellow (medium effect, 0.5 ≤ d < 0.8) and Dark Red (large effect, d ≥ 0.8). d) Heatmap for the presence or absence of 0 in the 95% bootstrapped confidence interval. Dark red means absence of 0 while blue represents presence.
Fig 9.
Robustness analysis for Rate of Synchronization (r) in the γI-γA parameter space.
Robustness analysis with respect to Rate of Synchronization (r). Two parameter bifurcation plots are presented for HOTFM and the Danino oscillators in the γI-γA parameter space for r. The blue (red) curve respresents the hopf branch for HOTFM (Danino). Region above the hopf branch corresponds to parameter values where oscillations occur, while for regions below the hopf branch oscillation dies and steady state solution arises. For a comparative analysis of robustness between HOTFM and Danino we consider 520 points in the region of oscillations common to both HOTFM and Danino. At each point the system of equations (containing a linear array of 6 cells for both HOTFM and Danino) is numerically integrated for 500 minutes. From the obtained solutions r is calculated. For points which have rDanino − rHOTFM < 0, HOTFM is more robust compared to Danino, while for rDanino − rHOTFM > 0 Danino is more robust. a) Heatmap representing rDanino − rHOTFM at the 520 points in the oscillatory region common between HOTFM and Danino. Negative values suggest that HOTFM is more robust for r compared to Danino and vice versa. b) Heatmap for part a. thresholded at the value 0. Regions with negative values in a. have dark red color here, while regions with positive values in a. are colored blue. c) Heatmap for the effect sizes for individual points calculated using bootstrap estimates for the distribution for rDanino − rHOTFM Robustness Analysis for Rate of Synchronization (r). If rDanino − rHOTFM < 0, effect size is reported for greater robustness of HOTFM, while if rDanino − rHOTFM > 0, effect size (cohen’s d) is reported for greater robustness of Danino. The color scheme is as follows—Blue (negligible effect, d < 0.2), Cyan (small effect, 0.2 ≤ d < 0.5), Yellow (medium effect, 0.5 ≤ d < 0.8) and Dark Red (large effect, d ≥ 0.8). d) Heatmap for the presence or absence of 0 in the 95% bootstrapped confidence interval. Dark red means absence of 0 while blue represents presence.
Fig 10.
Robustness analysis for IASE in the γH-γA parameter space.
Robustness analysis with respect to Integrated Absolute Synchronization Error (IASE). Two parameter bifurcation plots are presented for HOTFM and the Danino oscillators in the γH-γA parameter space for IASE. The blue (red) curve respresents the hopf branch for HOTFM (Danino). Region above the hopf branch corresponds to parameter values where oscillations occur, while for regions below the hopf branch oscillation dies and steady state solution arises. For a comparative analysis of robustness between HOTFM and Danino we consider 1510 points in the region of oscillations common to both HOTFM and Danino. At each point the system of equations (containing a linear array of 6 cells for both HOTFM and Danino) is numerically integrated for 500 minutes. From the obtained solutions IASE is calculated. For points which have IASEDanino − IASEHOTFM > 0, HOTFM is more robust compared to Danino, while for IASEDanino − IASEHOTFM < 0 Danino is more robust. a) Heatmap representing IASEDanino − IASEHOTFM at the 1510 points in the oscillatory region common between HOTFM and Danino. Positive values suggest that HOTFM is more robust for IASE compared to Danino and vice versa. b) Heatmap for part a. thresholded at the value 0. Regions with positive values in a. have dark red color here, while regions with negative values in a. are colored blue. c) Heatmap for the effect sizes for individual points calculated using bootstrap estimates for the distribution for IASEDanino − IASEHOTFM Robustness Analysis for IASE. If IASEDanino − IASEHOTFM > 0, effect size is reported for greater robustness of HOTFM, while if IASEDanino − IASEHOTFM < 0, effect size (cohen’s d) is reported for greater robustness of Danino. The color scheme is as follows—Blue (negligible effect, d < 0.2), Cyan (small effect, 0.2 ≤ d < 0.5), Yellow (medium effect, 0.5 ≤ d < 0.8) and Dark Red (large effect, d ≥ 0.8). d) Heatmap for the presence or absence of 0 in the 95% bootstrapped confidence interval. Dark red means absence of 0 while blue represents presence.
Fig 11.
Robustness analysis for Rate of Synchronization (r) in the γH-γA parameter space.
Robustness analysis with respect to Rate of Synchronization (r). Two parameter bifurcation plots are presented for HOTFM and the Danino oscillators in the γH-γA parameter space for r. The blue (red) curve respresents the hopf branch for HOTFM (Danino). Region above the hopf branch corresponds to parameter values where oscillations occur, while for regions below the hopf branch oscillation dies and steady state solution arises. For a comparative analysis of robustness between HOTFM and Danino we consider 1510 points in the region of oscillations common to both HOTFM and Danino. At each point the system of equations (containing a linear array of 6 cells for both HOTFM and Danino) is numerically integrated for 500 minutes. From the obtained solutions r is calculated. For points which have rDanino − rHOTFM < 0, HOTFM is more robust compared to Danino, while for rDanino − rHOTFM > 0 Danino is more robust. a) Heatmap representing rDanino − rHOTFM at the 1510 points in the oscillatory region common between HOTFM and Danino. Negative values suggest that HOTFM is more robust for r compared to Danino and vice versa. b) Heatmap for part a. thresholded at the value 0. Regions with negative values in a. have dark red color here, while regions with positive values in a. are colored blue. c) Heatmap for the effect sizes for individual points calculated using bootstrap estimates for the distribution for rDanino − rHOTFM Robustness Analysis for Rate of Synchronization (r). If rDanino − rHOTFM < 0, effect size is reported for greater robustness of HOTFM, while if rDanino − rHOTFM > 0, effect size (cohen’s d) is reported for greater robustness of Danino. The color scheme is as follows—Blue (negligible effect, d < 0.2), Cyan (small effect, 0.2 ≤ d < 0.5), Yellow (medium effect, 0.5 ≤ d < 0.8) and Dark Red (large effect, d ≥ 0.8). d) Heatmap for the presence or absence of 0 in the 95% bootstrapped confidence interval. Dark red means absence of 0 while blue represents presence.
Table 1.
Aggregate robustness metrics for IASE in the γI-γA parameter space.
Table 2.
Aggregate robustness metrics for Rate of Synchronization r in the γI-γA parameter space.
Table 3.
Aggregate robustness metrics for IASE in the γH-γA parameter space.
Table 4.
Aggregate robustness metrics for Rate of Synchronization r in the γI-γA parameter space.
Fig 12.
Distribution of ratio of robust region for IASE and r compared against random assignment of robustness.
Probability density plots for comparing the ratio of number of points for which HOTFM is more robust compared to Danino for IASE and r against random assignment of robustness. The distribution for For the observed ratio of the number of points where HOTFM is more robust are obtained using boostrap sampling with replacement repeated 1000 times. For the density for the random case, bootstrap samples are drawn 1000 times assuming that both HOTFM and Danino are equally likely to be more robust than the other topology. a) Probability density plots for IASE for the observed and random case in the γI-γA parameter space. b) Probability density plots for r for the observed and random case in the γI-γA parameter space. c) Probability density plots for IASE for the observed and random case in the γH-γA parameter space. d) Probability density plots for r for the observed and random case in the γH-γA parameter space.
Fig 13.
Bootstrap distributions (dotted curves) of IASE and r for indivudual points for Danino (red) and HOTFM (blue).
Actual distributions (solid curves) generated by the numerically integarting the system 1000 times are also shown for Plots of bootstrap probability distributions for IASE and r for Danino (red) and HOTFM (blue). Green lines correspond to the observed values while the other dotted lines represent the 95% confidence intervals. a) Distributions for IASE. b) Distributions for (r).
Fig 14.
When two colonies belonging to the Danino oscillator and oscillating HOTFM are mixed, we might get beats at the interface, if the HOTFM is in a state where it has slightly higher frequency of oscillation Here we see the aggregation of AiiA and external AHL signls from the two oscillators at their interface. The upper panels of the plot show the concentration of AiiA and AHL for cell numbers 100 and 101 (on either side of the interface) over time. The lower panels show the interference pattern caused due to the interference of these two waveforms at the interface (between cell nos. 100 and 101). Thus the lower left panel shows what the concentration profile would have been, had there been an AiiA molecule present at the interface of the two cells. The lower right panel simply shows the external AHL concentration. Both of these show beat like patterns over time.