Enzymes IV

Last time we analyzed a simple enzymatic reaction and we derived its kinetic parameters in the steady state. We saw that enzymes which catalyze the following reaction:

This reaction follows what is known as Michaelis-Menten kinetics, and we saw that we can relate their initial velocities (V_o) with parameters we can measure:

K_M is the Michaelis-Menten constant, and is an indication of the affinity of the enzyme for its substrate. We also defined k_cat, or the turnover of the enzyme, as the number of times that an enzyme undergoes the reaction per unit time:

Both K_M and kcat are dependent on the enzyme-substrate pair, as well as the temperature, pH, and other 'environmental' parameters of the enzymatic reactiion.

However, we cannot compare two enzymes that catalyze two different reaction using either of them. An enzyme can have high affinity for its substrate (small K_M), but a really bad kcat, so the catalytic efficiency of an enzyme is affected by both K_M and k_cat. So the catalytic efficiency is defined as:

If we consider the case in which [S] << K_M, E will be almost all free, so [E] @  [E]_T. In these conditions we will have that the initial velocity of the reaction is:

We see that the reaction turns into a second order reaction in [E] and [S], and k_cat / K_M becomes a pseudo-second order rate constant. It is, as we said before, a measurement of how efficient the enzyme is. There is an upper limit to the value of . It cannot be greater than k₁, the forward rate constant of ES-complex formation, because the conversion of S into P and its release depends on how many times the enzyme binds to the substrate. The value of k₁ is also limited by what is known as the diffusion-limit: That is, how much the molecules can move around in solution. Many enzymes have catalytic efficiencies around the diffusion-limit, and their rates are said to be diffusion-controlled (controlled by how fast the substrate ca get into the enzyme active site).

Multiple substrates

OK, so now we saw a reasonable number of definitions and situations for an enzyme involving one substrate, S. There are many enzymes that have more than one substratre, and therefore there are many steps in the enzymatic reaction. Also, it is very unlikely that both substrates (A and B) will bind the enzyme at the same time, and that the products will leave the enzyme at the same time.

Therefore, with multiple substrates we will have to analyze the order of addition and release of substrates and products. There are two major cases, sequential and ping-pong. We will analyze enzymes that work with two substrates, or bisubstrate mechanisms.

Sequential addition of substrates

Supose you have the following reaction, which involves two substrates, A and B, and two products, P and Q:

In this reaction, the substrate A goes into the enzyme active site to form an EA-complex, and it is followed by substrate B, to form the EAB complex. This complex is known as a ternary complex, and the enzymatic reaction is a sequential-ordered mechanism: There is an order that the substrates follow to go into the enzyme. We use a shorhand notation to represent reactions of this type, known as the Cleeland notation: The surface of the enzyme is represented by a line. The binding and release of substrates and products is indicated by arrows pointing towards the surface of the enzyme (binding) and away (release). Below the 'enzyme surface' line we write the different complexes that form during the reaction:

We can also have enzymes in which the order of addition of substrates does not matter, and either A or B can bind the enzyme at any point. This is known as a sequential-random mechanism:

In Cleeland notation, this reaction is represented by the following diagram:

In both cases of sequential reactions, all the substrates combine with the enzyme before any chemistry takes place. Therefore, there is full reversibility of the reaction. None of the connection between the different steps of the reaction are irreversible. Note that either or both the binding of substrates and release of products can follow an ordered or random pattern.

Ping-Pong mechanisms

What if the enzyme does chemistry with one of the substrates before the second substrate binds? This is very common, and many proteases (trypsin, for example) follow this reaction pattern: The enzyme binds with the substrate and becomes chemically modified by the substrate, generating product P. That is, we don't have E anymore, but another functional enzyme form that we call 'F', which will not bind substrate A, but only B. After B bind to F, more chemistry occurs, B takes whatever was left behind on the enzyme by A, and goes to product P. The enzyme goes back to its normal E form:

This is called a ping-pong mechanism. Using Cleeland notation, we can represent it as follows:

In enzymes following a ping-pong mechanism there is no formation of a ternary complex, because we never have more than one substrate at a time inside the enzyme active site. Furthermore, since we have a modification in the form of the enzyme (from E to F), we will have connections between the different steps of the reaction that are irreversible: Addition of Q cannot make the enzyme go back to E, and release of B from the FB complex neither.

These two (three) are the simplest cases of multisubstrate enzymes. We can have enzymes that react with 3, 4, 5, etc., substrates, and therefore we can have combinations of sequential (ordered and random) and ping-pong events in the reaction.

Kintetics in multisubstrate enzymes

Multisubstrate enzymes like the ones described above also follow Michaelis-Menten kinetics, but in this case we will have more parameters. There will be more enzyme-substrate complexes, each of them will have their own V_max and K_M, and the matematical analysis of the initial velocity equations gets really complicated. We have more steps to analyze, because our assumptions that the last step is the rate limiting step may not be entirely true. We can have several steps that all affect the rate of the reaction. Fortunatelly, you will only be punished with stuff like this if you take an enymology course at the graduate level.

However, we can analyze the double-reciprocal (LIneweaver-Burke) patterns and find out to which group of enzymatic reactions our enzyme reaction belongs. What we do in these cases is simple: We have two substrates, so the solution is to mainatain the concentration of one of them, [B], fixed while changing the concentration of [A]. Since [B] won't vary appreciably if [B] >> [E], and remembering that we are always looking at the initial rate (steady state), we obtain different plots for each [B] that we use.

In the case of sequential mechanisms, the double-reciprocal plots look like this:

This is known as an intersecting pattern. In this pattern, the slope of the curve changes. The slope was K_M / V_max in a simple reaction. Intuitively, this makes sense, because by changing the concentration of B, we are modifying the chance of having the ternary complex formed. The more [B], the more chances to get the reaction forward (increase in rate), so the apparent V_max increases.

If our enzymatic reaction goes through a ping-pong mechanism, we will get a parallel pattern double-reciprocal plot:

This also makes sense. In this case we do not have formation of a ternary complex, so the events related to A are independent of those related to B. The apparent V_max and K_M are affected equally by [B], because the first step (addition of A and formation of F) and the second step (addition of B to F) are independent. Therefore, the more B we have, the higher the apparent V_max and apparent K_M. However, they are both affected equally by [B], because both apparent constant are proportional to [B]. Therefore, while the X and Y intercepts change, the ratio K_M / V_max is maintained.

Next time we'll hopefully start with enzyme inhibition...

Prepared by Guillermo Moyna, 1999.