Global entrainment of transcriptional systems to periodic inputs

Abstract

This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dynamical systems. Through the use of contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all their solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific cases of models of transcriptional systems as well as constructs of interest in synthetic biology. A self-contained exposition of all needed results is given in the paper.

Figure 1. Entrainment of (9) to .

Time (minutes) on the -axis. The Figure shows the behavior of (9) for (blue), (green), (red). Notice that an increase of , causes an increase of the contraction rate, hence trajectories converge faster to the system unique periodic attractor. The other system parameters are set to: ,

Figure 2. A schematic diagram of the transcriptional system modeled in (12).

As explained in , the transcriptional component takes as input the concentration of protein and gives as output the concentration of protein . The downstream transcriptional module takes as input the concentration of protein .

Figure 3. Entrainment of the transcriptional module (12).

Time in minutes on the -axis. The state of the system (green), , is entrained to both and to a repeating sequence. System parameters are set to: ,  = 1, .

Figure 4. Entrainment of the transcriptional module (26).

Time in minutes on the -axis. The system state (green), , is entrained to the periodic input (blue): . The zoom on min highlights that trajectories starting from different initial conditions converge towards the attracting limit cycle. System parameters are set to: , , , , .

Figure 5. Multiple driven transcriptional modules.

A schematic diagram of the transcriptional modules given in (12).

Figure 6. Entrainment of two-driven transcriptional modules.

Time in minutes on the -axis. Outputs (top) and (bottom) of two transcriptional modules driven by the external periodic input . The parameters are set to: , , , for module and , , for module .

Figure 7. Transcriptional cascade discussed in the text.

Each box contains the transcriptional module described by (12).

Figure 8. The Repressilator circuit.

A schematic representation of the three-genes Repressilator circuit.

Figure 9. Modular addition to the Repressilator circuit.

This module is used for forcing the original circuit with some external signal (represented by an extra-cellular molecule in the bottom panel).

Figure 10. Simulation of the Repressilator model described by (53), (54), (55).

Time (minutes) on the -axis. Behavior of when the input is applied. Notice that when no forcing is present converges to a non oscillatory regime behavior. System parameters are tuned in order to satisfy (72). Specifically: , , , , , , .

Figure 11. Increasing the amplitude of oscillations for the model described by (53), (54), (55).

Time (minutes) on the -axis. Behavior of when: (i) the input is applied; (ii) no forcing is present. System parameters are the same as that used in Figure 10, except .

Figure 12. Simulation of the Repressilator forced by some extra-cellular molecule.

Time (minutes) on the -axis. Behavior of when the input is applied. Notice that when no forcing is present, the steady state behavior is non-oscillatory. System parameters are: , , , , , .

Figure 13. Synchronization of Repressilators.

Behavior of a population of Repressilator modeled as in (80). Time (minutes) on -axs. Notice that all the circuits synchronize with a steady-state evolution having the same period as . System parameters are chosen as in Figure 11, with .

Figure 14. Simulation of counter-example.

The following randomly-chosen input and initial conditions are used: , , , , . Green: inputs are (left panel) and (randomly picked, right panel). Blue: . Note chaotic-like behavior in response to periodic input, but steady state in response to constant input.