Fig. 4

Dynamics of the FB only and combined FF/FB systems. a A (proportional) negative feedback alone (blue curve), upon perturbation, always results in offset from the original value. A comparable combined FF/FB system also may exhibit offset; however, the I1-FFL in the FF/FB system can be tuned for perfect adaptation. The FF/FB system may also have a more rapid approach to steady state. b The addition of an I1-FFL to the FB system may result in more stable dynamics. For this choice of parameters, the FB only system is unstable, while the FF/FB system is stable. c Heatmap of the real part of principal eigenvalue (α) for the FB only system. Black curve is the contour for which α = 0, which indicates the stability limit of the FB only system. Smaller values of K₄ result in an unstable system (see part B). Here and elsewhere, values of α for other contours (red curves) are indicated directly on plot. d Same as C, but for the combined FF/FB system. e Heatmap of Δα, the difference between the real parts of the principal eigenvalues for the two systems. On the right side of the zero contour, the values of Δα are greater than zero, indicating the FF/FB system has a more rapid approach to steady state. Black and white curves indicate the stability boundaries for the FB only and FF/FB systems, respectively (see also parts C, D)