Enzymes

An enzyme is a biological catalyst, speeding up reactions without affecting the overall chemical equilibrium. Enzymes themselves work in a very simple way:

• A substrate comes along and binds to an enzyme at its active site, forming an enzyme-substrate complex. In the induced fit model, this is understood by the substrate itself inducing conformational change in the enzyme as it gets near, so as to strengthen binding between the two.
• The complex then spits out the substrate, and we form a product.

Enzyme kinetics 

There's a cool formula known as the Michaelis-Menten equation which enables you to determine several things about the enzyme's kinetics. It can be built up from two reactions:

• E + S ⇋ ES
• This is an equilibrium denoting the formation of the enzyme-substrate complex. 
• ES → E + P
• This is the reaction for the formation of product. As you can see, this reaction is irreversible. 

Each of these reactions has an associated rate law, and from this, we can determine the relationship between the rate of the enzyme and the concentration of substrate. I won't include the whole derivation here, but I might write one up and link it here another time - it's quite a lengthy derivation, though not especially complex, and I'd like to cut to the chase:

Here, v is the rate of reaction, and [S] denotes the concentration of substrate. K_M, meanwhile, is the Michaelis constant, and represents the value of [S] when the rate of reaction is exactly half of the maximum rate of reaction, or v_max. It also ends up doubling as a measure of the stability of the enzyme-substrate complex - a high K_M means a weakly bound ES - for reasons I might get into another time.

The actual graph for v against [S] looks something like this (source: University of Birmingham):

 

Here, you'll notice that v will plateau towards v_max, which actually makes loads of sense. You only have so much enzyme, eventually at high enough substrate concentrations you'll end up with too much substrate for not enough enzyme. 

However, there's one fundamental issue with this graph -  v_max, in theory. is achieved at an infinite substrate concentration. That also means we can never determine v_max, and thus also K_M, from this graph. So instead, we can rearrange the Michaelis-Menten equation as such:

• v = v_max[S]/(K_M + [S])
• 1/v = (K_M + [S])/v_max[S]
• 1/v = K_M/v_max[S] + [S]/v_max[S]
• 1/v = 1/v_max + 1/[S] x K_M/v_max 

We now end up with a linear plot, known as a Lineweaver-Burk plot, where we can determine our max rate and Michaelis constant far more easily:

That's important, because K_M and v_max, depending on inhibitors, can change.

Inhibitors - competitive, non-competitive, uncompetitive

There are three main types of inhibitors - molecules that act against an enzyme to lower the rate of catalysis:

• Competitive inhibitors bind at the active site of the enzyme, competing with the substrate.
• What this means is that the K_M value will increase, but the max rate will remain constant - same y-intercept, but a smaller x-intercept. 
• And the change to K_M makes sense, since we'd expect the ES complex to be far less stable if we have a competitor lurking around.
• Non-competitive inhibitors bind at a different site of the enzyme, not at the active site. They can therefore bind to both the enzyme or the ES complex.
• Here, K_M will be constant, but the max rate will decrease. Expect a larger y-intercept as a result.
• Uncompetitive inhibitors bind near the active site, next to the substrate, once the ES complex has formed. It will therefore only bind to the ES complex.
• Both K_M and the max rate will decrease.

These are ultimately all reversible, however. Sometimes, though, we can get irreversible inhibitors that bind to an enzyme, often at an amino acid residue at or near the active site, thus completely stopping the enzyme from functioning. This sounds like a non-competitive inhibitor, indeed their Lineweaver-Burk plots are strikingly alike, so you'd have to make a secondary plot between the max rate and total enzyme concentration to distinguish between the two. There, you'd notice the max rate decreases linearly as you increase the inhibitor concentration, unlike in non-competitive inhibitors.

Enzyme regulation

Say you want to regulate an enzyme's activity, perhaps by inducing conformational chnages that will affect their function. Here's how you can do that, using enzyme modulators:

• Co-enzymes are a type of cofactor that bind with enzymes to induce catalysis. Often, co-enzymes are required for an enzyme to function at all. Unlike substrates, which are converted into products in a reaction, co-enzymes will be regenerated. A common example of these are vitamins - several co-enzymes are either derived from, or are, vitamins.
• Metal ions can act as cofactors that loosely bind to an active site. For example, the enzyme urease may use nickel ions as a cofactor in order to hydrolyse urea into carbon dioxide and ammonia.
• Interfacial catalysis: enzymes require a substrate located on a membrane to function.
• Lipid bilayers, for example, contain phospholipids, and can be acted on by phospholipases that catalyse the hydrolysis of the ester bonds in the phospholipids.
• Substrate channeling involves transferring the product from one enzyme to the active site of another.

Allosteric enzymes

Up until this point, I've referred to rate against substrate concentration graphs as obeying the Michaelis-Menten equation, but that very much depends on the type of enzyme used. If we used an allosteric enzyme, for example, we get a sigmoidal curve:

This suggests that at a certain concentration, the rate rapidly begins to increase, unlike in the previous enzymes, where the trend is hyperbolic. And the question is obviously why?

Allosteric enzymes tend to be oligomeric, meaning they can accommodate multiple active sites. In allosteric enzymes, should a substrate bind to one active site, there will be a conformational change in the enzyme which will alter the affinity of other active sites to bind with substrates. This conformational change is said to be cooperative, where the affinity of the binding sites for a substrate will vary once the substrate itself is bound. Cooperative behaviour can be seen in the binding of oxygen with haemoglobin, and in the melting of large-chain molecules like DNA and phospholipids. 

One model - the MWC model - explaining allosteric enzymes is that the different subunits in the enzyme can be either tense and inactive (T), or relaxed and active, with a greater affinity for substrates (R). These two states exist in equilibrium with each other; with no bound substrate, T is favoured, yet once a substrate binds, R is favoured. You can only have one or the other, never simultaneously (in the MWC model specifically).

Typically, allosteric inhibitors will induce a T state, whereas allosteric activators will shift equilibrium towards the R state. Both of these will bind outside the active site, inducing a conformational change in the process.