Gene regulation is often described as an "on or off' phenomenon: a gene is either fully expressed or not expressed at all. In fact, repression and activation of a gene involve ligand-binding reactions, so genes can show intermediate levels of expression when intermediate levels of regulatory molecules are present. For example, for the $E$. coli lac operon, consider the binding equilibrium of the Lac repressor, operator DNA, and inducer (see Fig. 28-8). Although this is a complex, cooperative process, it can be approximately modeled by the following reaction (R is repressor; IPTG is the inducer isopropyl- $\beta$ -D-thiogalactoside):
$$\mathbf{R}+\mathbf{I P T G} \stackrel{K_{\mathrm{d}}=10^{-4} \mathrm{M}}{\rightleftharpoons} \mathbf{R} \bullet \mathbf{P T G}$$
Free repressor, $R,$ binds to the operator and prevents transcription of the $l a c$ operon; the $R$ IPTG complex does not bind to the operator and thus transcription of the lac operon can proceed.
(a) Using Equation $5-8,$ we can calculate the relative expression level of the proteins of the $l a c$ operon as a function of [IPTG]. Use this calculation to determine over what range of [IPTG] the expression level would vary from $10 \%$ to $90 \%$
(b) Describe qualitatively the level of lac operon proteins present in an $E .$ coli cell before, during, and after induction with IPTG. You need not give the amounts at exact times - just indicate the general trends.
Gardner, Cantor, and Collins (2000) set out to make a "genetic toggle switch"-a generegulatory system with two key characteristics, $A$ and $B$, of a light switch. (A) It has only two states: it is either fully on or fully off; it is not a dimmer switch. In biochemical terms, the target gene or gene system (operon) is either fully expressed or not expressed at all; it cannot be expressed at an intermediate level.
(B) Both states are stable: although you must
use a finger to flip the light switch from one state to the other, once you have flipped it and removed your finger, the switch stays in that state. In biochemical terms, exposure to an inducer or some other signal changes the expression state of the gene or operon, and it remains in that state once the signal is removed.
(c) Explain how the $l a c$ operon lacks both characteristics $A$ and $B$. To make their "toggle switch," Gardner and coworkers constructed a plasmid from the following components:
ori An origin of replication
$a m p^{\mathrm{R}}$ A gene conferring resistance to the antibiotic ampicillin OPlac The operator-promoter region of the $E$. coli lac operon
$\mathrm{OP}_{\lambda} \quad$ The operator-promoter region of $\lambda$ phage
lacI The gene encoding the lac repressor protein, LacI. In the absence of IPTG, this protein strongly represses $\mathrm{OP}_{/ a c} ;$ in the presence of IPTG, it allows full expression from $\mathrm{OP}_{/ a c}$
rep$^{\text {ts }}$ The gene encoding a temperature-sensitive mutant $\lambda$ repressor protein, repts. At 37 $^{\circ} \mathrm{C}$ this protein strongly represses $\mathrm{OP}_{\lambda} ;$ at $42^{\circ} \mathrm{C}$ it allows full expression from $\mathrm{OP}_{\lambda}$
$G F P$ The gene for green fluorescent protein (GFP), a highly fluorescent reporter protein (see Fig. $9-16$ )
T $\quad$ Transcription terminator
The investigators arranged these components, as shown in the following figure, so that the two promoters were reciprocally repressed: OP $_{\text {lac}}$ controlled expression of rep$^{\text {ts }},$ and OP\lambda controlled expression of lacl. The state of this system was reported by the expression level of $G F P,$ which was also under the control of $\mathrm{OP}_{\text {lac }}$.
(d) The constructed system has two states: GFP-on (high level of expression) and GFPoff (low level of expression). For each state, describe which proteins are present and which promoters are being expressed.
(e) Treatment with IPTG would be expected to toggle the system from one state to the other. From which state to which? Explain your reasoning.
(f) Treatment with heat $\left(42^{\circ} \mathrm{C}\right)$ would be expected to toggle the system from one state to the other. From which state to which? Explain your reasoning.
(g) Why would this plasmid be expected to have characteristics $A$ and $B$ as described above?
To confirm that their construct did indeed exhibit these characteristics, Gardner and colleagues first showed that, once switched, the GFP expression level (high or low) was stable for long periods of time (characteristic B). Next, they measured the GFP level at different concentrations of the inducer IPTG, with the following results.
They noticed that the average GFP expression level was intermediate at concentration $\mathrm{X}$ of IPTG. However, when they measured the GFP expression level in individual cells at $[\mathrm{IPTG}]=\mathrm{X},$ they found either a high level or a low level of GFP - no cells showed an intermediate level.
(h) Explain how this finding demonstrates that the system has characteristic A. What is happening to cause the bimodal distribution of expression levels at $[\mathrm{IPTG}]=\mathrm{X} ?$