Figure 8–75 How promoter occupancy depends on the binding affinity of a transcription regulator protein. (A) The fraction of a binding site that is occupied by a transcription repressor R is determined by an equation that is similar to the one we used for a transcription activator (see Figure 8–72E), except that in the case of a repressor we are interested primarily in the unbound fraction. (B) For a transcription activator A, half of the promoters are occupied when [A] = 1/KA. Gene activity is proportional to this bound fraction. (C) For a transcription repressor R, gene activity is proportional to the unbound fraction of promoters. As indicated, this fraction is reduced to half of its maximal value when [R]=1/KR. (D) As in the case of the transcription activator A (see Figure 8–74), 0.5 0.5 we can derive equations to assess the timing of protein X production as a function of repressor concentrations.
modules are linked to produce larger circuits that we call network motifs, which can produce surprisingly complex and biologically useful responses whose properties become apparent only through mathematical analysis. A particularly common and important network motif is the negative feedback loop, which can have dramatically different functions depending on how it is structured.
We take as a first example a network motif consisting of two linked modules (Figure 8–76A). Here, an input signal initiates the transcription of gene A, which produces a transcription activator protein A. This activates gene R, which synthesizes a transcription repressor protein R. Protein R in turn binds to the promoter of gene A to inhibit its expression. This cyclical organization creates a negative feedback loop that one can intuitively understand as a mechanism to prevent proteins from accumulating to high levels. But what can we learn about negative feedback loops, and their value in biology, by using mathematics to model them?
The negative feedback loop in Figure 8–76A can be modeled using Equation 8–7 (see Figure 8–75D) for the repression of gene A and Equation 8–5 (see Figure 8–74B) for the activation of gene R. Thus, for proteins A and R, we use the set of differential equations (Equation set 8–8) shown in Figure 8–76B. The two equations in this set are coupled, which means that they must be solved together to describe d[A] ˜A.mA [A] = ˜R.mR – 1 0.5 0 fraction steady-stateprotein level
Figure 8–76 A simple negative feedback motif. (A) Gene A negatively regulates its own expression by activating gene R. The product of gene R is a transcription repressor that inhibits gene A. (B) Equation set 8–8 can be solved to determine the dynamics of system components over time. (C) A system with negative feedback (blue) reaches its steady state faster than a system with no feedback (red). The plots indicate the levels of protein A, expressed as a fraction of the steady-state level. The blue line reflects the solution of Equation set 8–8, which includes negative feedback of gene A by the repressor R. The red line represents the solution when the rate of synthesis of A was set to a constant value that is unaffected by the repressor R. (C) time the behavior of A and R over time for any value of the input. As before, we plug in values for the parameters (βR, τR, etc.) and then use a computer to determine the values of [A] and [R] as a function of time after a sudden input activates gene A.
The results reveal several important properties of negative feedback. First, rather surprisingly, negative feedback increases the speed of the response to the activating input. As shown in Figure 8–76C, the system with negative feedback reaches its new steady state faster than the system with no feedback.