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Operons are often described using all or nothing language. A repressor binding to the operator is usually presented as "turning off" the regulated genes. Case in point, Scitable at Nature.com says:

In addition to being physically close in the genome, these genes are regulated such that they are all turned on or off together. ... The ability to turn ... genes on or off as a group therefore provides an efficient way to quickly adapt to environmental changes. ... mutations affecting the promoter can prevent all of the operon's genes from being expressed, ...

The thing that confuses me is that I've also heard genetics aren't all or nothing, that genes can only be "down regulated" or "up regulated", not truly turned on and off.

Obviously, differing amounts of moderate levels of whatever effector is modulating the repressor could lead to different levels of downstream expression. But, what about situations where nothing is inactivating the repressors? Will the regulated genes be (for all intents and purposes) be turned off? Additionally (and relatedly), is it even a realistic scenario for an operator to always (or almost always) be bound to by such a repressor?

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There are several mechanisms by which the expression of a gene can be completely turned off. Certain network architectures can ensure foolproof repression (for e.g. by using multiple repressors parallely or additional epigenetic silencing mechanisms). Bistable switches can also ensure robustness of expression in a way that small fluctuations in the transcription factor levels do not lead to expression of the gene. Sometimes the activator molecules need to co-operatively activate the expression. In such conditions the activation graph is sigmoidal with respect to activator concentration. All these mechanisms can prevent leaky expression.

Some genes do have a basal expression but it is not necessary for all cases. However, basal expression is absolutely necessary for proper functioning of the lac operon (see this post).

Having said this, there is always some non-zero probability of expression of a gene when it is actually turned off. This probability may be very low but it would not be zero. This small accidental expression may or may not have any significant downstream effects.

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