Multi-Substrate Reactions

Before beginning the description of kinetics of reactions involving more than one substrate, I should make a few general points:

Talking about this involves developing a vocabulary, in which words have a specific meaning - for instance, the varied substrate means that whose concentration varies along the x axis of a Lineweaver-Burk plot, while the changing substrate is that whose concentration changes from line to line on the plot.  This vocabulary is summarized in the hand-out "A Glossary of Clelandese."

I present the material in the form, "This is the mechanism, therefore these are the kinetic consequences."  In reality, of course, we obtain the results by experimentation, and try to deduce the kinetic mechanism from the results.  However, this way of pre­senting the material fits the scientific model - a mechanism is hypothesized, experi­ments are done to test the mechanism, and on the basis of the results the hypothesis is strengthened, modified, or discarded.  Certain experiments are basic - varying one sub­strate at changing concentrations of the other - while others will be sug­gested by the mechanism hypothesized at that point.  One should develop alternative hypotheses which would explain the results obtained to date, and devise experiments which will discriminate between them.  In teach­ing we can discuss simple mechanisms and ways to distinguish between them, but in nature there may be added complications to befud­dle the matter, although one can generally eliminate these by suitable experi­ments.  There are more levels of experimental procedure, such as isotope exchange and kinetic isotope effects, which are however beyond the limits of this course.

We are here concerned with reactions in which two or more sub­strates A, B, C etc. bind to the enzyme and eventually yield two or more products P, Q, R, etc.  We call the first substrate to bind in the mechanism as written A, and the first product off P; but it should be obvious that the purpose of these kinetic analyses is to determine which actual substrate is first to bind, which actual product is first off, etc.  We will be con­cerned with what kinds of mechanism - both in the arbitrary sense of order of binding to the enzyme and in the chemical sense - are found and what sorts of experiment we do to identify them.

Most of these experiments take the form of varying the concentra­tion of one substrate at changing fixed concentrations of another ligand - substrate, inhibitor, acti­vator.  As stated earlier, these results are best treated statistically to get the best values of Km and Vm, or slope and intercept of Lineweaver-Burk plots, though the lines defined thus can then be represented as L-B plots.  It is the value of Cleland's treatment of kine­tics, which we in the main will follow, that it enables you to select experi­ments and draw conclusions without unraveling the complete equation, and to a large extent without quantitative evaluation of rate constants, which in any case depends on understanding the kinetic mechanism.

By kinetic mechanism we mean the order of binding of substrates and release of products, and whether the enzyme goes through any added forms such as acyl-enzyme in the course of the reaction.  There are two basically different types of mechanism, the sequential and ping-pong mechanisms.  In the sequential mechanism all substrates bind to the enzyme before any products are released:

                                                           A      B                          P     Q
                                                           Ø      Ø                          ≠      ≠ 

                                                E———————————————E
                                                              EA     EAB € EPQ

The chemical reaction normally takes place in the central complex, the conversion of EAB to EPQ; central complexes are those which can proceed forward or back in the reaction sequence only by releasing a ligand or by isomerizing into another central com­plex.  Steady state kinetics, the meas­urement of rate of production of products by a catalytic amount of enzyme, cannot give information about the interconversion of central complexes, unless one complex, but not the other, binds a specific inhibitor.  This isn't very likely; more usually, we study the combination of inhibitors with stable enzyme forms, the "ground state" forms which must bind a sub­strate to go ahead or back in the reaction sequence, or with transitory complexes such as EA and EQ.  I think of stable enzyme forms as the floor, transitory complexes as one or more steps up, and central complexes as the top level which can be reached.  In a three-substrate mechanism EAB would be a transitory complex.

A ping-pong mechanism is one in which at least one product leaves the enzyme before at least one substrate binds:

                                                           A               P         B               Q
                                                           Ø                ≠          Ø                ≠

                                                E———————————————E
                                                            EA€FP                  FB€EQ

For instance, E might be a transaminase, E-pyridoxal phosphate, which reacts with an amino acid A, releases a keto acid product P, and is then in stable enzyme form F, enzyme-pyridoxamine phosphate.  A second keto acid B then reacts and leaves as amino acid Q.  Such reactions are also called substituted-enzyme reactions, since the enzyme passes through some substituted form symbolized F in the course of the reac­tion - usually, but not necessarily, a covalent substitution.  The acyl-enzyme mechanism is thus a ping pong mechanism, but with water, whose concentration normally is not varied, as substrate B.

Another approach to the mechanism now known as ping-pong was initiated by Barker, Doudoroff and Hassid in 1947, when they showed that sucrose phosphorylase acting on sucrose to yield fructose and glucose-1-phosphate gave G-1-P with the same conformation at the anomeric C atom, a, as the sucrose.  Assuming, as is chemically reasonable, that the reaction proceeds by some attacking group coming in from the side of the anomeric C atom opposite to the leaving group, which would result in inver­sion of conformation at the anomeric C atom, it was obvious that the overall reaction pro­ceeds by two such displacements, of fructose by a side chain of the enzyme and of the enzyme side chain by phosphate.  Thus it involves an enzyme-substrate intermediate, and thus because fructose leaves before phosphate attacks, it is a ping-pong reaction.  Twenty years later, and now thirty-five years ago, the glucosyl-enzyme intermediate was isolated by Abeles and shown to have the predicted b conformation at the ano­meric carbon atom.

However, we cannot always use either retention of configuration or isolation of an enzyme-substrate covalent intermediate as proof of a ping-pong reaction.  Instead we use a general kinetic approach, varying the concentration of either substrate at chang­ing fixed levels of the other and plotting the determined velocities in L-B form.  Note that the same data can be used either way, with either substrate as the varied one and the other as the changing one.  A ping-pong reaction will give parallel lines, with no variation in slope as the concentration of the changing substrate changes; a sequential reaction, on the other hand, will give lines which converge somewhere to the left of the y axis.  The rate equation, for v as a function of substrate concentration, for a simple two-substrate mechan­ism has the form

v = ; dividing top and bottom by AB,

v = .

These ø terms are observed rate constants, different combinations of the individual rate constants - k1, k-1, etc. - for different mechanisms.  They are determined simply by replotting the slopes and intercepts of the primary plots vs. 1/the concentration of the changing substrate, which is a basic technique in enzyme kinetics (draw primary plots and indicate slopes and intercepts for replot).  1/Vm = the intercept of the replot of inter­cepts; øb = slope of intercepts (if primary plots have A as changing variable); øa = inter­cept of slopes (limit of Ka/Vm); øab = slope of slopes.  Cleland writes the rate equation

v = ;

then øab = , øb = , øa = ; Ka and Kb are here the Michaelis con­stants for substrates A and B at a saturating concentration of the other substrate, and Kia is in a sequential mechanism the dissociation constant of the first substrate to bind to free enzyme.  However, to gener­alize to other cases, it is called an inhibition constant for substrate A.  If you were running the reverse reaction, P + Q --> A + B, and added A, A would be a competitive inhibitor of the reaction, with its Ki = Kia.

To return to the question of sequential vs. ping-pong mechanisms:  the rate equation as first written can be inverted to give

= + + + ;

for any given value of [B] a plot of 1/v vs 1/[A] has intercept + , slope (øa + ), so that the slope changes with B.  In the ping-pong mechanism, however, the term in øab turns out to be missing - because B does not bind to a complex containing A, nor A to one containing B - and thus the slope term is not affected by B and is constant.

A ping-pong reaction involving three substrates and three products could be of two types.  It could be on, off, on, off, on, off - what is called hexa-Uni ping pong be­cause all steps are unimolecular, there are never two successive binding events or

A        P            B         Q            C         R           two successive dissociations.

Ø        ≠             Ø          ≠             Ø          ≠           Or it could involve binding of

E————————————————————E     two substrates, release of one
                              F                        G                              product, binding of one substrate,
release of two products - which Cleland would call Bi Uni Uni Bi, two on, one off, one

           A         B         P            C         Q         R           on, two off.  Here a plot with A varying,

           Ø         Ø          ≠             Ø          ≠          ≠           B chang ing would give converging

E———————————————————E      lines, while plots with A or B varying, C   F          changing would give parallel lines.  On the other hand, a fully sequential ter-reactant mechanism - all sub­strates on before any products off - would give con­verging lines with all combina­tions (although the converg­ing nature is sometimes hard to see).  There is one exception: if B is saturating, plots of A varying, C changing (or vice versa) will be parallel.  We shall see why later.

What are the rules behind all this?  (Turn to p. 3 of the handout, Pre­diction by Inspection.)  We have already used the terms variable - vary­ing along the x axis - and changing along the y axis.  We then look at the effects of the mechanism on the slope and y intercept of Line­weaver-Burk plots - these would be interchanged in a Woolf plot, [S]/v vs [S], or would be the x intercept and y intercept in an Eadie plot, v vs, v/[S].  The y intercept in a Lineweaver-Burk plot is 1/the velocity meas­ured if the concen­tra­tion of the varying substrate is infinite.  To change the intercept a change in the con­centration of the changing compound must change the reaction rate even when the concentration of the varying compound is infinite.  This is normally true of a second substrate, which binds to an en­zyme form other than that to which the varying sub­strate binds; A binds to free E, B binds to EA.  There is only one case where the level of one substrate does not affect the velocity at a saturat­ing level of the other: if E and A are at rapid equilibrium E + A € EA, and B then displaces the reaction to the right by combining with and removing EA, EA + B --> EAB -> products, saturating B will pull all the enzyme into the EAB form even at very low A concentration, so long as it exceeds the molar concentration of enzyme.  The concentration of A then ceases to affect the reaction rate.  This can happen in two ways: A might be an activating metal ion, and simply stay on the enzyme as product goes off and B comes on again, never having a chance to escape; or, A might be in rapid equilibrium on and off the enzyme, k1, k-1 >> kcat, while B binds to EA and at saturation makes all E EAB.  If k1 is not >>kcat the complete reaction will continually produce free E by product dissoci­ation, and the rate will still depend on the balance between E and EA and hence on [A].

To repeat the general rule: changing the concentration of a chang­ing compound affects the intercept of a Lineweaver-Burk plot if it binds to an enzyme form other than that to which the varying substrate binds.

The effect of a changing compound on the slope of a Lineweaver-Burk plot is a little more complicated: the changing compound and the varying compound must bind to the same enzyme form - this applies only to inhibitors and activators - or to forms which are connected by steps which are readily reversible under the assay conditions, rever­sibly connected for short.  For instance, forms E and EA are readily intercon­nec­ted by the binding and dissociation of substrate A.  Forms may be not reversibly connected in three ways: 1) by a thermodynamically irreversi­ble step - not very frequent; 2) by dissociation of a product between them - this is an irreversible step for initial velocity conditions when by definition product is not present at an appreciable concentra­tion, unless it has been specifically added; 3) by presence of a substrate at satur­ation, which is an irreversible step because if that substrate does dissociate momentarily, it comes back on before anything else can happen.  This is an important technique in three-substrate reactions: as previously mentioned, in a sequential three substrate reaction saturation with B, which makes the points of binding of A and C not reversibly connected, results in parallel lines in Lineweaver-Burk plots of A varying, C changing (or vice versa).  This is best done by running the experiments at several levels of B and then extrapolating to a saturating level of B.  Consider what this means: you run 125 assays (in duplicate at least, 250 really), 5 concentrations of A x 5 concentra­tions of B x 5 concentrations of C.  You take all the assays at one C concentration and plot 1/v vs. 1/[B] varying, [A] changing (draw).  From this primary plot you take the int­ercepts, where B is saturating; these represent the points on one line of a Lineweaver-Burk plot with A varying, C changing.  The intercepts from another primary plot at another C concentration are the points on another line of the plot with A varying, C changing.  Are these lines parallel, or converging?  If they are parallel, the substrate varied in the primary plots is indeed B.

We can now see how, given a kinetic mechanism written down, we can predict the consequences for slope and intercept of Lineweaver-Burk plots (or other forms).  The two-substrate ping pong mechanism predicts that plots of 1/v vs. 1/[A] at changing levels of B, or vice versa, will not show slope effects, because the forms to which A and B bind are not rever­sibly connected since the dissociation of a product comes between them.  One has to remember to look forward as well as back in a mechanis­tic sequence - for instance, adding product Q will affect the slope of a plot of 1/v vs. 1/[A], because they bind to the same enzyme form, free E.  In a sequential two-substrate reaction Q will also affect the slope of a plot of 1/v vs. 1/[B], since B binds to EA which is reversibly connected to free E.

So far we have discussed only non-branching mechanisms, i.e. those which fol­low only a single pathway.  Unfortunately, not all mechanisms are so simple.  Any se­quential mechanism may be random, i.e. either A or B may bind first, and either product may dissociate first.  A sequential mechanism may be random on only one side or on  A         B         P         Q         both.  A ping pong mechanism with

                                                                                          two sub-strates has no possibilities

E———        EA        ————      EQ        ———E   for randomization, unless you go the
                                                                                          whole hog and say the mechanism

                      EB                               EP                           can be either ping- pong or sequen­

                 B         A                     Q         P                       tial.  This is kin­etically possible and analyzable, but not likely in the real world, because it would require two alternative chemical mechanisms to be present sim­ultaneously.

A branched mechanism can in principle give curved Lineweaver-Burk plots, due to terms in the square of one or both substrate concentra­tions.  But you must remem­ber the first law of kinetics: kinetic experi­ments do not prove a mechanism, they only disprove other mechanisms, since the true mechanism may always be more complex than the experi­ments show.  For instance, a mechanism may in fact be random, but show no curvature of Lineweaver-Burk plots - indeed, Cleland says he has never yet seen a plot which is non-linear simply because the mechanism is ran­dom - there are at least five other ways to get a non-linear concave-downward Lineweaver-Burk plot!  In practice, random mechanisms are at either end of the scale; either they are close to sequential, most reaction following one pathway, or they are rapid equilibrium random, in which the forms E, EA, EB and EAB all interconvert rapidly compared to the rate of going ahead to prod­ucts.  This will be the case either if the interconversion of central com­plexes - which is normally the chemical interconversion in the reac­tion - is rate-limiting, or if some later step, such as an enzyme isomeriza­tion before a product dissociation, is rate-limiting.  These are the case respectively with yeast and horse liver alcohol dehyd­rogenases.  As I analyze it, you could get curvature only if steps before the central complex were rate-limiting, one alternative being faster than the other but with higher Km.  In the example, if A bound well to free enzyme, but EA had to isomerize slowly before binding B, while B bound poorly to free enzyme, but EB would bind A and yield the central complex EAB without that slow step, then you could get a concave-downward Lineweaver-Burk plot of 1/v vs 1/[B], because the reaction would shift to a more rapid pathway at high B.  But you can see that it takes a very special and unlikely set of conditions for this to happen.

As I mentioned earlier, if you think you have curvature, it is best observed in an Eadie plot, and even then you should plot the divergences from a straight line and see whether they are consistent.