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''E. coli'' grows most efficiently on [[Saccharides|glucose]]. If glucose is available it does not consume any other sugar and also produces no enzymes that catabolizes other sugars. But if there's no glucose available it starts to use other sugars like [[Saccharides|lactose]]. Lactose is a disaccharide that can be [[Saccharides|hydrolysed]] by $\beta$-galactosidase into the monosaccharides glucose and galactose.

[[File:Betagalactosidase.png‎|thumb|center|400px|lactose is hydrolysed into glucose and galactose by the enzyme $\beta$ galactosidase]]

In order to respond optimal to environmental conditions, ''E. coli'' developed throughout evolution an increadable control mechanism. Only if lactose and no glucose is available, ''E. coli'' turns on the machinery to use lactose. This machinery contains the famous ''lac'' operon and looks as follows

[[File:lacoperon1.png|thumb|center|800px|Picture shows the ''lac'' operon, which contains the gene for $\beta$ galactosidase (''lacZ''), permiase (''lacY''), Acetyltransferase (''lacA''). $\beta$ galactosidase brakes down lactose, while permiase makes the cell membrane permiable for lactose]]

When there's no glucose around, the genes from the ''lac'' operon are transcribed [[Gene Regulation and Protein Levels|polycistronic]] and the protein $\beta$ galactosidase is made from ''lacZ''. Additionally a permiase , which makes the cell membrane permeable for lactose is expressed by ''lacY''. The third gene ''lacA'' is an Acetyltransferase, its function is still not understood.

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How does ''E. coli'' know when to express the genes on the ''lac'' operon?
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It turns out, that the ''lac'' operon contains a so called '''operator site'''. Operater sites are DNA sequences near the promoter that are bound by DNA binding molecules, the '''transcription factors'''. Transcirption factors have the ability to eather physically block the RNA polymerase from binding, in this case it's a '''repressor''' that represses the binding and decreases the transcription rate, or the transcription factor enhances the affinity for the RNA polymerase and therefore increases the transcription rate, such a transcription factor is called '''activator'''. Transcription factors are used to control the protein levels in a cell (see article [[Gene Regulation and Protein Levels]]).
[[File:lacoperon2.png|thumb|center|800px|Picture of the ''lac'' operon including control machinery. Two transcription factors (TFs) LacI and cAMPTF control the expression of the ''lac'' operon]]
The ''lac'' operon is controled by two transcirption factors
#When no lactose is available the '''repressor''' LacI binds to the repressor binding site (rs) and the transcription rate is low. If lactose is around it binds to the lac repressor LacI causing a conformational change whereby the repressor falls off. Thus lactose is the signal that deactivates the repressor, thereby activating transcription of the ''lac'' operon. This way Lac proteins are only produced when lactose is available. 
#If glucose is present in the cell, cyclic AMP (cAMP) concentration is low. cAMP is the signal for activation of the cAMP transcription factor (cAMPTF), that is an '''activator'''. Thus in presence of glucose activation of the cAMPTF is low, whereas low glucose causes high cAMP concentration and leads to high acitivation of the cAMPTF. In this manner the transcription rate is only increased when no glucose is available.

The model of the ''lac'' operon was first worked out by '''Francois Jacob, Jacques Monod''', and their colleagues in the 1960s. They recieved the nobel prize fore their work about "the discoveries concerning genetic control of enzyme synthesis" in 1965 [http://www.nobelprize.org/nobel_prizes/medicine/laureates/1965/#]. Today the model of the ''lac'' operon is also known as the '''Jacob Monod model'''.

For their studies Jacob and Monod didn't use lactose and glucose but the related chemicals X-gal and IPTG. X-gal can be broken down by $\beta$ galactosidase but does not act as signal for LacI, whereas IPTG acts as an inducer that activates the LacI repressor, but is itself not degraded.
Under normal conditions there exists a feedback from lactose to transcription rate which results in [[Dynamical Systems|coupled differential equations]]
\begin{align}
\frac{d}{dt}LacI &= f(lactose,glucose)\\
\frac{d}{dt}lactose &= g(LacI)
\end{align}
By use of X-gal and IPTG these equations can be decoupled, thus parts can be studied individually, which simplifies the analysis. For a description of the experiments with ''E. coli'' mutants, that lead to Jacob Monod model visit [http://www.ncbi.nlm.nih.gov/books/NBK21683/ this] website, or watch the [http://www.youtube.com/watch?v=00LNy0Q_i6c MIT Lecture 13]

A [http://www.ebi.ac.uk/biomodels-main/static-pages.do?page=ModelMonth%2F2006-12 mathematical model] of the ''lac'' operon (Yildirim 2003) can be found on the BioModels Database.




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Video Lectures:
MIT: Eric Lander, Robert Weinberg - Introduction to Biology [http://www.youtube.com/watch?v=00LNy0Q_i6c Lecture 13]