Biological regulatory systems are exposed to frequent and sometimes extreme variations in external conditions or the concentrations or activities of key components. The ability of these systems to function normally in the face of such perturbations is called robustness. If we understand a complex system to the extent that we can reproduce its behavior with a computational model, then the robustness of the system can be assessed by determining how well its normal function persists following changes in various parameters, such as rate constants and component concentrations. We have already seen, for example, how the presence of negative feedback reduces the sensitivity of the steady state to changes in the values of the system’s parameters (see Figure 8–77). Considerations of robustness also apply to dynamic behaviors. Thus, for example, when discussing negative feedback, we described how the behavior of a system tends to become more oscillatory as the number of components that constitute the feedback loop increases. If we use different values of the parameters in models derived for systems like those in Figure 8–78, we find that the system with the longer loop tends to exhibit stable oscillations within a much broader range of parameters, indicating that this system provides a more robust oscillator. We can perform similar calculations to determine the ability of different systems to achieve robust bistability arising from positive feedback. Thus, one benefit of computational models is that they allow us to probe the robustness of biological networks in a systematic and rigorous way.
Two Transcription Regulators That Bind to the Same Gene Promoter Can Exert Combinatorial Control
Thus far, we have discussed how one transcription regulator can modulate the expression level of a gene. Most genes, however, are controlled by more than one type of transcription regulator, providing combinatorial control that allows two or more inputs to influence the expression of one gene. We can use computational methods to unveil some of the important regulatory features of combinatorial control systems.
Consider a gene whose promoter contains binding sites for two regulatory proteins, A and R, which bind to their individual sites independently. There are four possible binding configurations (Figure 8–83A). Suppose that A is a transcription activator, R is a transcription repressor, and the gene is only active when A is bound and R is not bound. We learned earlier that the probability that A is bound and the probability that R is not bound can be determined by the equations in Figure 8–84A. The product of these two probabilities gives us the probability of gene activation.
This example illustrates an AND NOT logic function (A and not R) (see Figure 8–83A). Maximal activation of this gene is accomplished when [A] is high and [R] is zero. However, intermediate levels of gene activation are also possible depending on the levels of A and R and also on the binding affinities of [A] and [R] for their respective sites (that is, KA and KR). When KA » KR, even a small concentration of [A] is capable of overcoming repression by R. Conversely, if KA « KR, then much more [A] is needed to activate the gene (Figure 8–84B and C).
Many other logic functions can govern combinatorial gene regulation. For example, an AND logic gate results when two activators, A1 and A2, are both required for a gene to be transcribed (Figures 8–83B and 8–84D). In E. coli cells, the AraJ gene controls some aspects of arabinose sugar metabolism: its expression requires two transcription regulators, one activated by arabinose and the other activated by the small molecule cAMP (Figure 8–84E).
KA[A]fraction of A bound = fraction of R not bound =
P(A,R) = . = concentration of R concentration of R concentration of A concentration of A