To illustrate how positive feedback can generate stable states, let us focus on a simple positive feedback loop containing two repressors, X and Y, each of which inhibits expression of the other (Figure 8–81A). As we saw with Equation set 8–8 (Figure 8–76B) earlier, we can create differential equations describing the rate of change of [X] and [Y] (Equation set 8–9, Figure 8–81B). We can further modify these equations to include cooperativity by adding Hill coefficients. As we did earlier, we can then create equations for calculating the concentrations of [X] and [Y] when the system reaches a steady state (that is, when (d[X]/dt ) = 0 and (d[Y]/dt ) = 0; Equations 8–10 and 8–11, Figure 8–81C).
Equations 8–10 and 8–11 can be used to carry out an intriguing mathematical procedure called a nullcline analysis. These equations define the relationships between the concentration of X at steady state, [Xst], and the concentration of Y at steady state, [Yst], which must be simultaneously satisfied. We can plug in different values for [Yst] in Equation 8–10, and calculate the corresponding [Xst] for each of these values. We can then graph [Xst] as a function of [Yst]. Next, we repeat the process by varying [Xst] in Equation 8–11 to graph the resulting [Yst]. The intersections of these two graphs determine the theoretically possible steady states of the system. For systems in which the Hill coefficients hX and hY are much larger than 1, the lines in the two graphs intersect at three locations (Figure 8–81D). In other systems that have the same arrangement of regulators but different parameters, there might only be one intersection, indicating the presence of only a single
Figure 8–80 Positive feedback of a gene onto itself through serially connected interactions. A sequence of activators and repressors of any length can be connected to produce a positive feedback loop, as long as the overall sign is positive. Because the negative of a negative is positive, not only circuit (A) and (B) but also circuit (C) create positive feedback.
= ˜X.mX – dt 1 + (KY[Y])hY ˜X = ˜Y.mY – GENE Y dt 1 + (KX[X])hX ˜Y (B) concentration of X = ˜X.mX.˜X = ˜Y.mY.˜Y
Figure 8–81 A graphical nullcline analysis. (A) X inhibits Y and Y inhibits X, resulting in a positive feedback loop. (B) Equation set 8–9 can be used to determine the rate of change in the concentrations of proteins X and Y. (C) Equations 8–10 and 8–11 provide the concentrations of proteins X and Y, respectively, when these concentrations reach concentration of X a steady state. (D, E) Blue curves (called nullclines) are plots of [Xst]calculated from Equation 8–10 over a range of concentrations of [Yst]. Red curves indicate values of [Yst] calculated from Equation 8–11 over a range of concentrations of [Xst]. At an intersection of the two lines, both [X] and [Y] are at steady state. For plot (D), the binding of both proteins to their target gene promoters was cooperative (hX and hY much larger than 1), resulting in the presence of multiple intersections of the nullclines––suggesting that the system can assume multiple discrete steady states. In plot (E), the binding of protein X to the promoter of gene Y was not cooperative (hX close to 1), resulting in only one nullcline intersection and thus just one likely steady state.
steady state. For example, when there is a low cooperativity of protein X binding to the promoter of gene Y (that is, a small Hill coefficient, hX, in Equation 8–11), the plot of [Y] is less curved (Figure 8–81E), and it is less likely that there will be multiple intersections of the two curves.