(*Overleaf*.) Illustration of the roles of ultrasensitivity for complex network dynamics. (*a*–*d*) Ultrasensitivity is required for bistability. (*a*) Gene *X* and *Y* form a double-positive feedback loop, where *X* activates *Y* in an ultrasensitive manner, and *Y* activates *X* in a Michaelis–Menten manner. The system is described by equations (5.1) and (5.2), and the parameters are *k*_{1} = 3, *k*_{2} = 1, *k*_{3} = 1, *k*_{4} = 1, *K*_{x} = 2, *K*_{y} = 0.5 and *n* = 1, 3 or 5. (*b*–*d*) Stability analysis using nullclines with different *n-*values. The intersection points between *X* (red) and *Y* (blue) nullclines indicate the steady states of the feedback system (solid dot, stable steady state; empty dot, unstable steady state). The system is bistable when there are three intersection points: two stable steady states and one unstable steady state in between (*c*) and (*d*). The *Y* nullclines in (*c*) and (*d*) show increasing degree of ultrasensitivity, making bistability arise easily. Reducing ultrasensitivity makes the *X* and *Y* nullclines difficult to intersect three times, leading to monostability, as illustrated in (*b*). (*e*–*h*) Ultrasensitivity helps negative feedback loops to achieve robust cellular adaptation and homeostasis. (*e*) A generic negative feedback circuit underlying cellular adaptation and homeostasis against stress. *S* represents the total stress level containing background/internal stress (*S*_{bkg}) and external stress (*S*_{ext}), thus *S* = *S*_{bkg} + *S*_{ext}. The system is described by equations (5.5)–(5.7), and the default parameters are *k*_{1} = 1, *k*_{2} = 1, *k*_{3} = 0.1, *k*_{4} = 0.1, *k*_{5} = 1.01, *k*_{6} = 0.01, *S*_{bkg} = 1 and *n* = 2. (*f*,*g*) Adaptive response of controlled variable *Y* and underlying induction of anti-stress gene *G* under persistent external stress at various levels (*S*_{ext} = 1, 2 and 3). Dashed lines are baseline levels of *Y* and *G* in the absence of *S*_{ext}. (*h*) Adapted steady-state levels of *Y* with respect to various levels of *S*_{ext}. In the open-loop case (*R*_{loop} = 0), the response is linear (grey line). As *R*_{loop} increases by setting Hill coefficient *n* = 1, 2 and 3, the respective response (red, green and blue curves) becomes increasingly subsensitive, indicating improved adaptation and more robust homeostasis. To maintain the same basal level of *G*, *k*_{5} = 0.02, 0.11, 1.01 and 10.01 for *n* = 0, 1, 2 and 3, respectively. (*i*–*l*) Ultrasensitivity is required for a negative feedback loop to generate sustained oscillation. (*i*) Genes *X* (red) and *Y* (blue) form a negative feedback loop, where *X* activates *Y* in an ultrasensitive manner, and *Y* inhibits *X* linearly with a time delay. The system is described by equations (5.10) and (5.11), and the parameters are *k*_{1} = 1, *k*_{2} = 1, *k*_{3} = 1, *k*_{4} = 1, *K* = 3, *τ* = 5 and *n* = 1, 2 or 3. *τ* denotes the time delay from *Y* to *X*. Initial *X* = 3 and *Y* = 0.5. (*j*–*l*) As the Hill coefficient *n* increases from 1 to 3, the feedback system tends to oscillate better. Small *n-*values only give rise to damped oscillation, whereas large *n-*values lead to sustained oscillation.

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