File S1 - Theory on the Dynamics of Oscillatory Loops in the Transcription Factor Networks

This file contains Figure S1-Figure S5. Figure S1. Three gene repressilator model. A1-4. Phase portraits and trajectories of TF genes A, B and C of a repressilator. Simulation settings are , , and which required a critical Hill coefficient of Cna = 2. Total simulation time is 200 (number of lifetimes of the protein product of TF gene A) and integration step is . To trigger the oscillations, we have introduced the asymmetry in the initial condition for the promoter state occupancy of TF gene A as . Oscillations starts with a time delay whose value depends of the magnitude of this disproportion in the parameter values. A5. Roots of the twelfth degree characteristic polynomial associated with the Jacobian matrix of Eqs (26) for settings given in A1. B1-2. Effects of perturbation in that is raised to ( in B2) in the time interval from 0 to 100. Increase in increases the period of oscillations of the entire system from to 24.5 and reduces the amplitudes of TF genes A and C. The amplitudes of TF genes A/B/C are such that A which are raised to in the time interval from 0 to 100. Increase in increases the period of oscillation of the entire system from to 30 and reduces the amplitudes of TF genes A and B and increases the amplitude of C and the amplitudes of TF genes are such that B increases the period of oscillation of the entire system as in B3 where the amplitudes of TF genes A/B/C are such that BFigure S2. Dynamics of three independent Goodwin-Griffith oscillators cyclically coupled. A1-3. Phase portraits of TF genes A/B/C which are three independent GG oscillators cyclically coupled through -OR- type logic as given in Figure 2C2 (without dashed lines). Simulation settings are , , and which required a critical Hill coefficient of Cna = 5 (we have set this to 6 for clarity of results). Total simulation time is 500 (number of lifetimes of the protein product of TF gene A) and integration step is . In this configuration the end-products of TF gene A and C will regulate TF genes A through A-OR-C type logic whereas the promoter of TF gene B will be regulated by the end-products of TF genes A and B through A-OR-B type logic and so on. Under identical values of all the parameters the system generates synchronized oscillations with a period of . B1-2. Effects of perturbation in . Upon introduction of perturbation in from scaled time 100 to 400 (where ) there three phases of responses. In the first phase, the system tries to resist the perturbation whereas the second phase consists of repeating elements of resistance and chaos. Upon removal of perturbation the system enters into new limit-cycle orbit in the third phase. In B2 both and are perturbed as in B1. B3-4. Effects of perturbation in which are raised to in the time interval from 100 to 400. System responds to the perturbation as in B1-2. C1-2. Trajectories of TF genes A/B/C which are three independent GG oscillators cyclically coupled through -AND- type logic as given in Figure 2C3 (without dashed lines) and their responses to perturbations in Group I control parameters. Simulation settings are , , and which required Cna = 2. Total simulation time is 500 and integration step is . Identical values of all the parameters of the system generate synchronized oscillations. Introduction of perturbation in from scaled time 100 to 400 (where , C1) affects only TF gene A whereas the orbit of other TF genes B/C seems to be stable and upon removal of the perturbation the system returns back to initial limit-cycle orbit. In C2 the parameter is perturbed to as in C1. Figure S3. Dynamics of three Goodwin-Griffith oscillators which are fully interconnected. A1-3. Phase portraits of TF genes A/B/C which are three independent GG oscillators which are fully interconnected with -OR- type logic as given in Figure 2C2 (with dashed lines). Simulation settings are , , and which required a critical Hill coefficient of Cna = 6. Total simulation time is 500 (number of lifetimes of the protein product of TF gene A) and integration step is . In this configuration all the end-products of TF gene A/B/C will regulate all the three TFs through A-OR-B-OR-C type logic. Identical values of all the parameters of the system generate synchronized oscillations. Perturbation in from scaled time 100 to 300 (where , A1) seems to make the system unstable. In A2 is perturbed to as in A1 and in A3 is perturbed to as in A1. B1-3. Phase portraits of TF genes A/B/C which are three independent GG oscillators fully interconnected with -AND- type logic as given in Figure 2C3 (with dashed lines). Simulation settings are , , and which required a critical Hill coefficient of Cna = 2. Total simulation time is 500 (number of lifetimes of the protein product of TF gene A) and integration step is . In this configuration all the end-products of TF gene A/B/C will be regulated by their complex. Identical values of all the parameters of the system generate synchronized oscillations. Introduction of perturbation in from scaled time 100 to 300 (where in B1) seems to make the system unstable. In B2 is perturbed to as in B1. In B3 is perturbed to as in B1. Figure S4. Tuning dynamics of A-OR-B and A-AND-B types of coupled oscillators with respect to perturbations in μ and w. A, C. Tuning capability of GG oscillators coupled through A-OR-B type logic as given in Figure 2B2. Default Simulation settings are , and . B, D. Tuning capability of GG oscillators coupled through A-AND-B type logic as given in Figure 2B3. Default Simulation settings are , and . Plots A and B show the variation of period, critical Cnh and amplitude with respect to changes in (iterated from 5×10−7 to 10−3 with wh = 1) whereas plots C and D show the variation of these quantities with respect to changes in wh (iterated from 0.1 to 10 with ). Here period of oscillator is measured in the number of lifetimes of TF protein A (1/γa) and amplitude is measured in terms of number of Ph/Phs. Figure S5. Tuning dynamics of A-OR-B and A-AND-B types of coupled oscillators with respect to perturbations in v and ε. A, C. Tuning capability of GG oscillators coupled through A-OR-B type logic as given in Figure 2B2. Default Simulation settings are , and . B, D. Tuning capability of GG oscillators coupled through A-AND-B type logic as given in Figure 2B3. Default Simulation settings are , and . Plots A and B show the variation of period, critical Cnh and amplitude with respect to changes in (iterated from 5×10−7 to 10−4 with εh = 1) whereas plots C and D show the variation of these quantities with respect to changes in εh (iterated from 0.7 to 8 with ). Here period of oscillator is measured in the number of lifetimes of TF protein A (1/γa) and amplitude is measured in terms of number of Ph/Phs.
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