= Lac repressor
= LacI
{c}
{synonym}
The \b[lac repressor], also called <LacI>, is the regulatory <protein> that reduces <transcription> of the <lac operon> by binding a <lac operator> in <Escherichia coli>. Binding of a small-molecule inducer changes the repressor's regulatory state through <allostery>. <Allolactose> connects this switch with lactose metabolism, while <IPTG> supplies an experimentally useful inducer. The <lac repressor> is distinct from the operator <DNA> and from the <RNA polymerase> whose transcriptional output is controlled.
<Walter Gilbert> and <Benno Müller-Hill>'s 1966 isolation experiment gave the proposed regulator a biochemical measurement. Their <equilibrium dialysis> assay used radioactive <IPTG>: small free inducer could cross the membrane, while the macromolecular binding component was retained. Extra total labelled inducer on the protein side reported binding. The assay deliberately used inducer recognition rather than assuming that the exact mechanism of transcriptional repression was already established.<Isolation of the lac repressor by Walter Gilbert and Benno Müller-Hill>[Printedp.1891, assay and Design of the Experiment]{ref}
An original one-site binding model explains that readout. If the free inducer concentration is $I$, the concentration of independent binding sites is $B_T$ and the <dissociation constant> is $K_I$, the bound concentration is
$$
B=B_T\frac{I}{K_I+I}.
$$
At <equilibrium dialysis> with matched solvent conditions, the protein side contains free plus bound inducer, $I+B$, while the opposite side contains free inducer $I$. Choosing $B_T=2\,\mathrm{\mu M}$ and $I=K_I=1\,\mathrm{\mu M}$ gives $B=1\,\mathrm{\mu M}$ and total concentrations $2$ versus $1\,\mathrm{\mu M}$. Equality of the free concentration does not require equality of the total concentration. The numbers are illustrative; the native <lac repressor> has a more detailed oligomeric and allosteric structure.
A separate one-operator model connects binding to expression. If a concentration $R$ of binding-competent repressor occupies the <lac operator> with probability $\theta=R/(K_O+R)$, and occupied <DNA> produces no transcripts in this model, then
$$
r=r_0(1-\theta)=r_0\frac{K_O}{K_O+R}.
$$
Reducing $R/K_O$ from $9$ to $1$ raises the model's <transcription> rate from $0.1r_0$ to $0.5r_0$. It does not follow that every measured cellular response obeys this one-site expression: operator arrangement, additional regulators and the later stages of <gene expression> affect the output. Measuring inducer binding and measuring repression therefore test related but different parts of the mechanism.
The repressor controlling the <lac operon>. Operator binding represses <transcription>, while inducer binding shifts the conformational equilibrium towards lower operator occupancy.
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