Molecular Biology of the Cell by Bruce Alberts, Alexander Johnson, Julian Lewis, David Morgan, Martin Raff, Keith Roberts, Peter Walter by by Bruce Alberts, Alexander Johnson, Julian Lewis, David Morg

520 Chapter 8: Analyzing Cells, Molecules, and Systems
set 8–9 can again provide an answer. In Figure 8–82B, we show the solution of this
equation set for two perturbations from the upper-left steady state. For a small
perturbation, the system returns to its original steady state. But the larger perturbation
causes the system to switch to the alternate steady state. Thus, this system
can be switched from one stable steady state to the other by subjecting it to an
input (or a perturbation) that is large enough to make the other steady state more
attractive. More generally, every stable steady state has a corresponding region of
attraction, which can be intuitively thought of as the range of perturbations (of [X]
or [Y] in this example) for which the dynamic trajectories converge back to that
particular steady state, rather than switch to the other one.
The concept of a region of attraction has interesting implications for the heritability
of transcriptional states and the transition rates between them. If the region
of attraction around one steady state is large, for example, then most cells in the
population will assume this particular state. Furthermore, this state is likely to be
inherited by daughter cells, since minor perturbations, like those ensuing from
an asymmetric distribution of molecules during cell division, will rarely be sufficient
to induce switching to the other steady state. We should expect that the use
of positive feedback, coupled to cooperativity, will quite often be associated with
systems requiring stable cell memory.
Robustness Is an Important Characteristic of Biological Networks
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.