Allosteric Inhibition of Adenyl Cyclase
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Let's start off by looking at a "generic enzyme" in the form of "Pacman". The enzyme is presented with a toxin "TOX". This molecule can be conceived of as either attaching to the mouth and thus keeping away other molecules. If this attachment is only temporary and is because TOX has many chemical similarities to the substrate, it is called a competitive inhibitor. Thus if you imagine that there are the same number of TOX and substrate molecules, half of the time the mouth will be chewing uselessly on TOX, and the rest of the time it will be productively chewing on substrate. Thus the overall reaction of substrate to products is only half the rate as if there were no TOX present.
On the otherhand the TOX might blind Pacman so that it can not find its substrate. Because the TOX molecule is adhering to a different place on the enzyme, we call this "second site" inhibition - or more formally "allosteric inhibition". This concept won a Nobel Prize for Jacob and Monod.
When we look at the situation of galactose inhibition of lactase, it is a feedback competitive inhibitor. It is 'feedback' because galactose is a product of the enzyme's hydrolysis of lactose. It is 'competitive' because it is not only similar but identical to the portion of the lactose molecule that fits into Pacman's mouth.
For a first mental model of how ATP is an allosteric inhibitor of adenyl cyclase, we again turn to our Pacman. If Pacman searches for and gets an ATP into its mouth, it bites off a pyrophosphate (two phosphates joined together) and spits out cyclic-AMP (cAMP). But if Pacman gets an allosteric ATP-eyepatch and cannot see, hunting for ATP gets problematic, to say the least!
However, the above model must be thrown out because chance would still dictate that an occasional ATP would stumble into the mouth and cAMP would be made. But in reality, NO cAMP is made. So let's change our mental model so that the bottom half of Pacman's head will swivel around to the back. (Reminds you of Exorcist, doesn't it?!) At low ATP concentrations, Pacman happily chews on the ATP's it finds, but at higher ATP concentrations, one sticks to the side of its head and causes the swivel to occur. Shocked, Pacman can still "see" the ATP's floating around it, but its mouth is disjointed, and it can even make a single bite! So, never ever are any cAMP's made.
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