Figure 3

Numerical approximation of non-autonomous trajectories. (A) Toggle switch network. Red arrows representing auto-activation indicate time-dependence of threshold parameters a _x and a _y (see equation 1). (B) Values of auto-activation thresholds a _x and a _y are altered simultaneously and linearly over time. The graph shows the step-wise approximation of a continuous change, in this case, an increase in parameter values. Step size is taken as small as computational efficiency allows. (C) During every time step, parameters can be considered constant, and the phase portrait and (quasi-)potential landscape are calculated for the current set of parameter values. Trajectories are then integrated over the duration of the time step using the previous end point as the current initial condition. The result is mapped onto the potential surface. The four panels in (C) show examples of potential landscapes (upper panels) calculated based on sets of parameter values at time points indicated by dashed arrows from (B). Important events altering the geometry of the trajectory are indicated. Lower panels show the corresponding instantaneous phase portraits with the integrated progression of the trajectory across time steps. See Model and methods for details.