Why does allosteric binding produce a sigmoidal curve?

Question

Allosteric binding is where the enzyme can be regulated through having ligands bind onto somewhere that is not the active site. This will then induce a conformational change on the active site, hence preventing binding and catalysis.

Often after a long pathway, the final product can act as the modulator and bind at the allosteric site of the first enzyme.

My question is why does the curve now change to look sigmoidal instead of hyperbolic?

I get that allosteric binding should affect the shape - but this is not what I expected. What I do not understand is why does the velocity decrease so sharply at low concentrations of substrate?

Allosteric binding should mean that at small amounts of substrate, there should be very little amounts of reaction. Thus, there are small amounts of product and hence very little allosteric binding. Thus the reaction should go at full speed, looking just like the diagram on the left. It is only at higher concentrations of substrate that the modulating effect would kick in.

Where is my reasoning wrong?

Answer 1

The curve on the right describes only a special case of allosterism in which the binding of the substrate to the allosteric site increases the activity of the enzyme. The velocity vs [substrate] plot is sigmoidal because when the enzyme binds substrate at the allosteric site, the active site of the enzyme works better. It is very common for people who do not understand the basic chemistry of allosterism to generalize this idea to all allosteric enzymes, but this is quite wrong. The sigmoidal plot is pretty much extrapolated from the binding of oxygen to hemoglobin, one of the first cases of allosterism described in the literature, but only a FEW allosteric enzymes behave this way.

This plot describes the dependence of the reaction velocity of phosphofructokinase on [ATP], one of its two substrates. This is very old data (Frieden et al, JBC, 1976) and this is clearly not a sigmoidal curve! ATP binding to the allosteric site decreases the enzyme velocity! It is unfortunate that most "instructors" do not bother to check the math nor the literature.

Answer 2

Why does allosteric binding produce a sigmoidal curve?

There is no tight link between allosteric binding and deviation from simple Michaelis-Menten kinetics.

Cooperative behavior of multi-domain proteins

What I do not understand is why does the velocity decrease so sharply at low concentrations of substrate?

The simplest model for the sigmoidal dependence of enzyme rates on substrate concentration (i.e. positive cooperativity) is a multi-subunit enzyme with multiple substrate binding sites. The enzyme has two states, both individually displaying Michaelis-Menten kinetics. However, the one with more efficient catalysis (larger $k_\mathrm{cat}$) is favored at higher substrate concentration. This is similar to hemoglobin switching to a higher affinity conformation as soon as some binding sites are occupied. This conformational change leads to a higher slope of rate vs. substrate concentration (or for hemoglobin, binding vs. ligand concentration) at the mid-point of the transition, allowing larger response of enzyme rates to substrate concentration (i.e. more sophisticated regulation). For an example, see ATCase. The enzyme has 6 active sites in different subunits, with conformational change explaining cooperativity. Complicating matters, it also has allosteric sites binding to other ligands, for additional regulation.

I get that allosteric binding should affect the shape - but this is not what I expected.

The image is misleading. For the cooperative effect in the model above, you expect binding of the substrate elsewhere (in another active site). Allosteric binding of a non-substrate ligand in an allosteric site generally does not switch the kinetics from Michaelis-Menten to non-Michaelis-Menten.

Cooperative behavior with a single active site

Glucokinase is an enzyme that shows cooperative kinetics (sigmoid rate vs substrate plot) without having multiple subunits or binding sites. This behavior is explained by slow conformational changes (in effect "memorizing" the there was a substrate recently). This means that when the number of reactions per time interval reaches a certain threshold, the enzyme will be faster because it is already in the "fast" conformational state. For a paper on possible mechanisms for this type of regulation, see doi:10.1016/j.bioorg.2011.11.001