115:412/508 Proteins & Enzymes spring 2002

Mechanism of Enzyme Action

We have been approaching, by kinetics and chemical modification, the central question about enzymes: how do they work?  How do they catalyze reactions?  Some of the considerations we shall cover apply also to other functions of proteins, such as electron transfer, conformational change which transmits a hormonal message through a membrane, etc.

Jencks has noted that enzyme mechanisms may be approached in three ways: general theories (and perhaps experiments based on them); small molecule model reactions; and details of specific mechanisms.  The last is the proof of the pudding, but may not be generally applicable; indeed, we shall see that different catalytic factors play roles of varying importance in different enzyme mechanisms.  In the next several lectures I shall discuss general theories and lessons from small molecules, then go on to specific mechanisms.

First, what do we mean by catalysis?  Increase of rate of a reaction, brought about by something (the catalyst) which is not consumed in the reaction, may be reused.  A reaction may be looked at as passing from one valley, representing stable reactants, over a mountain pass to another val­ley, the products.  The pass between them is called the transition state, the state from which the molecules may with equal probability go ahead to products or back to reactants.  Energy must be put in to a reaction to raise the reactants to the top of the pass, the transition state; this energy is called the free energy of activation, ∆G*, always positive because it represents energy which must be put in to reach the unlikely transition state.  The role of the catalyst is to find a lower pass over the mountain range, a pathway with a lower activation energy.

The transition state may be described as a condition of the reactants - let's just use one, call it A* - in equilibrium with the ground state A, so that K*, the equilibrium constant of activation, = [A*]/[A].  An alternative way of looking at the role of a catalyst is that it stabilizes the transition state, increases the ratio [A*]/[A].  The rate constant of the overall reac­tion, k, by which [A] is multiplied to get the rate, is proportional to the amount of the reactant in the transition state, k = (kT/h)K*, where k is Boltzmann's constant and h is Planck's constant.  The factor kT/h is the frequency of decomposi­tion of the transition state, which is the same is the vibrational frequency n of the bond breaking.  At 25° C n = 6.212 x 1012 s-1.

The ∆G* of activation is of course related to the equilibrium constant K* in the usual way, ∆G* = -RTlnK* = -RTln([A]*/[A]) = -RT(ln[A]* - ln[A]), -(∆G*/RT) = ln[A]* - ln [A], -(∆G*/RT) + ln[A] = ln[A*].  Taking antilogs of both sides, [A]e-∆G*/RT = [A*].  This relates the conc. of [A*] to the conc. of ground-state A and the difference in free energy between the ground state and the transition state.  Since the exponent is -∆G*/RT, a negative numb­er, e-∆G*/RT is a number <1, a small fraction, and the larger ∆G* is, the smaller e-∆G*/RT, and the smaller the fraction of A in the transition state.

The ∆G* may further be separated into enthalpy and entropy, ∆G* = ∆H* - T∆S*, k = (kT/h)e-∆H*/RTe∆S*/R.  The entropy term is frequently the most important; we shall return to it.

If one can measure the rate constant k of reaction as a function of temperature, one can obtain (although not terribly accurately) values for ∆G*, ∆H* and ∆S* from the following rearrangements of the above reaction: ∆G* = -RTln(k h/kT), ∆H* = -R([d lnk/d(1/T)] + T), ∆S* = R[(Td lnk/dT) + ln(k h/kT) + T].  More commonly, rate k is plotted vs 1/T, an Arrhenius plot, and a change in the slope of the plot is taken as indicating a change in the rate-limiting step of the reaction, i.e. ∆G* has one value in one temperature and another value in another temperature range.

The catalyst may decrease the activation energy to reach, or increase the stability of, the same sort of transition state reached in the uncatalyzed reaction, or it may pro­vide an entirely different, if usually more complica­ted, pathway of reaction.  It is gener­ally assumed that some chemical path­way can be observed, for the reaction of small molecules, which is analog­ous to the enzyme-catalyzed reac­tion.  This approach is covered in the chemistry course Bio-organic Mechanisms; we are here concerned primar­ily with studying the rates of enzyme-catalyzed reactions, and obtaining from then evidence which may aid in selecting the best small-molecule reaction as model.  Once one has chosen an appropriate pathway, one is defining how the transition state is stabilized compared to a simpler, less catalyzed version of the reaction.

Some examples of the effect of an enzyme on the activation energy, albeit on ∆H* rather than ∆G*, are:
2H2O2 -> 2H2O + O2:                        uncatalyzed                        ∆H* = 18.0 kcal/mole
                        + catalase                        ∆H* =   5.5 kcal/mole
casein hydrolysis:                        uncatalyzed                        ∆H* = 20.6 kcal/mole
                        + trypsin                        ∆H* = 12.0 kcal/mole
ethyl butyrate hydrolysis:uncatalyzed                        ∆H* = 13.2 kcal/mole
                        + lipase                        ∆H* =   4.2 kcal/mole.

Let's return to the equation k = (kT/h)e-∆H*/RTe∆S*/R, and more specifically to the entropy term e∆S*/R.  Entropy may be thought of as a measure of disorder, and one law of thermodynamics states that entropy tends always toward a maxi­mum, energy must be put in to a reaction to reverse disorder.  My classic example of this is that if I put out my gar­bage can the night before it is due to be collected, and a dog or a raccoon knocks it over and scatters the garbage over the street, it takes me much more energy to collect the garbage into the can again than it did the dog or raccoon to knock it over.  Any transition state is likely to be a very ord­ered state of the molecule or molecules undergoing reaction, which means that entropy is lost compared to the ground state.  If ∆S* is negative, the term -T∆S* is positive, and ∆G* is increased, making the reaction more difficult.  Furthermore, a reaction making more molecules out of fewer - an increase of disorder - will have a favorable overall entropy term, while one making fewer molecules will have an unfavorable entropy term.  This implies that the formation of a transition state in the reaction of two or more molecules, once sep­arate but brought together in reaction, has a very substantial negative entropy term - although in many cases this is partially balanced by the displacement of water mole­cules from a rela­tively ordered state around the reactants to the disordered state of bulk H2O.

This is particularly true when two or more reactants must be brought together into a precise arrangement, having started from quite random positions in solution.  The molecules must lose translational entropy - randomness of position in the solution - rotational entropy - freedom of the whole molecule & its parts to rotate - and some vibrational entropy, freedom of the parts of a molecule to vibrate with respect to each other.  As Figure 2-5 from Fersht (handout) shows, two molecules condensing to form one lose the translational and rotational entropy of three degrees of freedom each, although the latter at least is compensated by internal rota­tion of the parts of the larger molecule.  The table in the handout shows that the translational entropy change involved in bringing a molecule from just anywhere to somewhere specific is 120 to 150 joules/degree.mole (29 to 36 cal/degree.mole, or at 25° C = 298° K 8.6 to 10.7 kcal/mole as an energy term) for a one molar solution.  Entropy increases only slightly with molecular weight - the range given is for molecules of mol. wt. 20 to 200 - but decreases with increasing concentration, since "anywhere" is a larger fraction of possible locations as coincentration increases, so the change in order is less.

So when two molecules get together to form one there is an entropy loss in forming the adduct, which is seen both in the thermodynamic free energy change of forming the final product and in the free energy of acti­vation of the reaction.  For two molecules becoming one at 25° C the total entropy loss, translational and rotational,  translates into an unfavorable free energy change of 13 to 14 kcal/ mole, though slightly offset by new internal rotational and vibrational entropy.

One way this is seen is that a catalytic group in the same molecule is immensely more effective than one in a separate molecule, because no entropy penalty, or only a small one involving the reduction of internal rotational freedom, must be paid in forming the transition state.  For in­stance, as shown at the left of the handout, the hydrolysis of p-nitrophenyl acetate is catalyzed by free acetate, with a second-order rate constant of 4x10-6s-1m-1.  Mono-p-nitrophenyl succinate hydrolyzes with a first-order rate constant of 0.8 s-1.  The ratio between these two rates is 2x105m, i.e. the other COO- in the succinate acts like acetate at a concen­tration of 2x105 m, which is clearly impossible.  In effect, in the succinate case we are see­ing close to the 'real' catalytic effectiveness of the succinate, without pay­ing the tremendous entropy penalty for bringing in a separate acetate ion into the transition state.  Succinate still has rotational entropy around the bond between the CH2 groups, so some ordering, and loss of entropy, still occurs in the transition state.  For instance, if at the left of the other side of the handout we look at the rates of hydrolysis of monoesters of glutarate (with free rotation around two C-C bonds), succinate (with rotation around one bond) and endo-norbornenyldicarboxylic acid (with no free rotation), the rates increase 230-fold for each loss of a free rotation, which corre­sponds to 3.22 kcal/mole at 25° C or 10.8 cal/degree.mole of entropy.

Bruice and Benkovic have compiled average values for the entropy of activa­tion of comparable uni- bi- and ter-molecular reactions, as shown in the right two-thirds of the back side of the handout.  They conclude that the entropy of activation divided by the kinetic order, the second column of numbers, is approximately constant, about 4.4 cal/degree.mole.  This means that a reduction in the kinetic order of the reaction - making it intramolecular rather than intermolecular - will reduce the entropy of activation by 4 to 5 cal/degree.mole, and increase the rate by about 1000-fold. Of course it took more energy to synthesize the more complex mole­cules which have intramolecular catalysis, because they are more ordered and contain less entropy.  The point is that because this order is built into these compounds, there is less difference in entropy between the reactants and the transition state, and thus less entropy of activation, which is this difference.

We can now see that a major part of the role of the enzyme in catalys­is is to separate the entropy penalty from the activation energy of the reaction per se.  One cannot escape the entropy penalty entirely, but with enzymes the entropy penalty is paid in the formation of the enzyme-substrate complex, in which energetically favorable binding interactions with negative ∆H - charge attraction, van der Waals contacts, hydrogen bonding - are formed which compensate for the unfavorable entropy change.  Not only can two or more molecules react with each other without paying the entropy penalty in forming the transition state, but groups on the enzyme can be involved in chemical cata­lysis of the reaction, in ways we'll discuss later.  These catalytic involvements in small-molecule catalysis - such as the examples with n = 3 in Bruice & Benkovic's table - must accelerate the reaction more than 1000-fold to overcome the entropy penalty and display net acceleration of the reaction.  Binding to an enzyme allows a number of catalytic groups to act, each raising the rate another 100-1000 fold, without any entropy penalty for their involve­ment, until an otherwise unlikely reaction such as the rearrangement of succinate to methylmalonate can take place at a reasonable rate.  Thus the decrease of activation entropy, thanks to binding, allows the expression of other mechanisms of catalysis.  Of course a tremendous entropy penalty had to be paid in the synthesis of the large, highly ordered enzyme molecule, but only once, the enzyme molecule can be used for many, many reactions, while in small-molecule catalysis the transition state complex must be formed anew for each reaction.  You could say that this is "the meaning of life" - the living system synthesizes catalysts which diminish the payment of entropy of activation by the reusability of the catalysts.

Another point, of which much has been made by Koshland, is that molecules do not react just anywhere on their surfaces; only when precise portions of their electron orbitals are brought into contact will fruitful reaction take place.  We may symbolize by drawing the reacting molecules as spheres with only some small area on the surface where reaction will be fruitful.  This area over the total surface area of the sphere is a factor 1/q; the smaller the area over which reaction is fruitful, the larger is q.  Of course the other reacting molecule also has a limited fruitful area, and the probability of reaction upon contact is 1/qax1/qb, or the rate enhancement obtained by binding reactants A and B to an enzyme in such arrangement - Koshland called it "orbital steer­ing" - is qaqb.  Koshland sug­gested that rate enhancements of up to 106 fold were thus obtainable, but one can calculate that this would require fruitful areas to be only one square degree on the surface, and it is rather easy to prove from infrared absorption studies that bonds actually bend five degrees or so from their rest position quite easily without breaking, which implies that if you brought reacting molecules together even five degrees off from the best alignment they would still have at least a 50% chance of forming the new bond stably, which in turn implies that the fruitful area is ten degrees across or 78 square degrees.  Consequently, this "orbital steering", or orien­tation as it is also called, is suggested to supply catalytic factors of the order of 100 to 1000, not 106-fold, and Koshland stopped being able to dine out on the idea.

Another physical effect is strain, which was originally suggested to be the bending or stretching of a substrate molecule toward the transition state conforma­tion when it bound to the enzyme, thus decreasing the enthalpy of formation of the transition state as well as the entropy.  The classic example of this was the deform­ation of the ring of N-acetylglucos­amine when it binds to lysozyme, toward the flat­tened, half-chair form which the protonated intermediate transition-state-like structure assumes in the acid catalyzed reaction (draw out).  Fersht disagrees with this de­scription, saying that the flattening out occurs only on going into the transition state, the enzyme selects the rare but naturally occurring flat­tened conformation and stabil­izes it by forming new binding interactions and relieving unfavorable interactions which occur with the 'ground' state.  The distinction is a little like that between specific acid or base catalysis, in which a proton is trans­ferred to or from the substrate before going to the transition state, and general acid or base catalysis, in which at least partial transfer occurs while going to the transition state.

The description of this effect with which Fersht would agree is that binding complementarity is best in the transition state, the enzyme stabilizes the transition state by the improvement of the interac­tions with the substrates and holds them in the transition state long enough to greatly improve the chance that it will proceed to products.  The improvement of the binding inter­actions provides negative ∆G which balances the positive (unfavorable) ∆G of distorting the sub­strate to the transition state structure.  Note that improvement of binding inter­actions quite distant from the catalytic site can be used in stabilizing the transition state; the classic example is elastase, a protease which acts very poorly, low Vmax, on simple substrates such as tosyl-alanine ethyl ester; Vmax as well as binding improves as the substrate gets larger, being best for an N-blocked tetrapeptide.

This effect is studied by the use of transition state analogs: a structure for the transition state of the reaction is guessed at, and a compound with similar structure - if not always similar atoms, the transition state analog for chymotryptic hydrolysis of a substrate like phenylethanecarboxylic acid esters is phenylethane­boronic acid, with -B(OH)2 replacing -COOH - is synthesized and shown to bind much better than the actual substrates, because less of the binding energy is used to approximate the transition state, so that the net ∆G of binding is greater.  Analogy: what one wants to hear and agree with is better received than bad news which must change your behavior.

A final physical factor is microenvironmental effects, really a facilitation of chemical effects.  Ionic groups and dipoles are more effective, interact more strongly with other ionic groups and dipoles, when they are isolated in hydro­phobic areas - like oases in a desert.  Perutz has pointed out that polar residues are not found in the interior of proteins, as determined by X-ray crystallography, unless there is a good reason in terms of function such as catalysis.  (However, Dr. Kahn now argues that enzymes adjust their flexibility - generally needed for conformational changes during catalysis - by having some buried charged groups.  Prot­eins are no more stable than they have to be for the temperature at which their organism lives.  One can make a more stable protein by eliminating such buried charges, but it will be stiffer, less cata­lytic at moderate temp­eratures, even if highly active at higher temperatures.)  Such an ionic group or dipole is likely to be a much more effective catalyst than one shielded by water molecules in aqueous solution; it is known that polar salts dissolved in non­polar solvents can catalyze reactions by several orders of magnitude, and reagents such as crown ethers (for which Charles Petersen received the Nobel Prize in 1987), which chelate a cation effectively in a nonpolar solvent, allow the free anion to reract effectively in such solutions (for instance, KMnO4 in benzene with crown-18).  The environmental effect of the enzyme would also include disrupting the solvation 'shells' of substrate molecules allowing them to interact more readily with each other as well as with groups on the enzyme.