Cooperative binding events can produce steep S-shaped (or sigmoidal) relationships between the concentration of regulatory protein and the amount bound on the DNA (see Figure 15–16). In this case, a number called the Hill coefficient (h) describes the degree of cooperativity, and we can include this coefficient in our equations for calculating the bound fraction of promoter (Figure 8–79A). As the Hill coefficient increases, the dependence of binding on protein concentration becomes steeper (Figure 8–79B). In principle, the Hill coefficient is similar to the number of molecules that must come together to generate a reaction. In practice, however, cooperativity is rarely complete, and the Hill coefficient does not reach
Figure 8–78 Oscillations arising from delayed negative feedback.
A transcriptional circuit with three components (A, B) is less likely to oscillate than a transcriptional circuit with five components (C, D). The X (light blue), Y (dark blue), and Z (brown) here represent transcription regulatory proteins. For the simulations in (B) and (D), the system was initiated from random initial conditions for X, Y, and Z. Oscillations are produced by a delay induced as the signal propagates through the loop.
Figure 8–79 How the cooperative binding of transcription regulatory proteins affects the fraction of promoters bound. (A) Cooperativity is incorporated into our mathematical models by including a Hill coefficient (h) in the equations used previously to determine the fraction of bound promoter (see Figures 8–72E and 8–75A). When h is 1, the equations shown here become identical to the equations used previously, and there is no cooperativity. (B) The left panel depicts a cooperatively bound transcription activator and the right panel depicts a cooperatively 0.5 bound transcription repressor. Recall from Figure 8–75B that gene activity is proportional to bound activator (left panel) or unbound repressor (right panel).
1/KR Note that the plots get steeper as the Hill (B) concentration of protein A concentration of protein R coefficient increases.
We turn now to positive feedback and its very important consequences. First and foremost, positive feedback can make a system bistable, enabling it to persist in either of two (or more) alternative steady states. The idea is simple and can be conveyed by drawing an analogy with a candle, which can exist either in a burning state or in an unlit state. The burning state is maintained by positive feedback: the heat generated by burning keeps the flame alight. The unlit state is maintained by the absence of this feedback signal: so long as sufficient heat has never been applied, the candle will stay unlit.
For the biological system, as for the candle, bistability has an important corollary: it means that the system has a memory, such that its present state depends on its history. If we start with the system in an Off state and gradually rack up the concentration of the activator protein, there will come a point where autostimulation becomes self-sustaining (the candle lights), and the system moves rapidly to an On state. If we now intervene to decrease the level of activator, there will come a point where the same thing happens in reverse, and the system moves rapidly back to an Off state. But the transition points for switching on and switching off are different, and so the current state of the system depends on the route by which it has been taken in the past—a phenomenon called hysteresis.
A simple case of positive feedback can be seen in a regulatory system in which a transcription regulator activates (directly or indirectly) its own expression, as in Figure 8–80A. Positive feedback can also arise in a circuit with many intervening repressors or activators, so long as the net overall effect of the interactions is activation (Figure 8–80B and C).