115:412/508 Proteins and Enzymes                                                                                     spring 2002

Enzyme Kinetics I (2002)

Although you should all have had basic enzyme kinetics in a General Biochemistry course, I shall begin at the beginning, as a review and to ground you better for the higher levels we reach. 

Enzyme kinetics is the study of the dependence of reaction rate on the concentrations of substrates, inhibitors, effectors, etc.  It is therefore restricted to enzymes among biologically active proteins, though other proteins may be substrates of enzyme-catalyzed reactions.  It may be further divided into steady-state kinetics, in which the amount of enzyme is catalytic and one is observing the rate of product appearance or substrate disappearance over a time measured in minutes, and pre-steady state kinetics, in which one is observ­ing the interac­tion of the first molecule of substrate with the enzyme, usually but not always in millisecond time periods.  I shall discuss mostly steady-state kinetics, with a few excursions into pre-steady state.

There are several purposes for the study of enzyme kinetics: 1) design of proper assay conditions for an enzyme; 2) the K_m for substrates usually approx­imates their in vivo concen­tration; 3) and most important, the kinetic mechanism of an enzyme - what substrate binds first, what product is released last, etc. - is important for figuring out the chemical mechanism of the enzyme, how it does what it does at the level of bond breaking and formation.  Kinetic studies, however, can only exclude pathways, while retaining others; they can never guaran­tee a mechanism, since another mechanism, more complicated but giving the same experi­mental results, is always possible, until you think of an experiment to distinguish them.

We assume that for an enzyme to catalyze a reaction, the substrate S must bind to the enzyme, then be transformed into a product P (we start with one-substrate reactions, work up to multi-substrate reactions):

E + S E·S E·P E + P

We can then write rate constants for the formation and breakdown of the intermediate com­plexes E·S and E·P.  (Note that I express non-covalent complexes of enzyme and ligand with a dot, E·S; covalent complexes are expressed with a dash, E-S.)  At this stage we assume that k₃>>k₂, k₃ is not at all rate-limiting and the product complex E·P does not accumulate; this is not always the case, but we start with the simplest possibility.

Michaelis and Menten based their derivation on several assumptions: first, the reaction scheme expresses the distribution of enzyme forms: we can't measure the amounts of indivi­dual forms E, E·S, E·P, but we can measure their sum, [E]_t = [E] + [E·S] + [E·P] ≈ [E] + [E·S].  This is the conservation equation, which may seem obvious but is important in any derivation of a kinetic mechanism.

Second, it is usually assumed that the substrate concentration [S] is much greater than the enzyme concentration [E]_t.  This isn't really a necessary assumption, as is shown in the paper by Chaplin in the handout, nor is it neces­sarily true in the cell; but it makes derivations much easier to assume it, to assume that the working substrate concentration [S] is the same as what you put in the cuvette [S]₀, is not changed appreciably by the amount binding to the enzyme.  Certainly it is likely to be true in an in vitro assay.

A third assumption is what we call initial velocity conditions, what you have when you have just started the reaction by bringing enzyme and substrate together.  As a reaction pro­ceeds and you use up substrate and build up product, the reaction can slow down for several reasons: you approach equilibrium, and the rate of the reverse reaction becomes appreciable; you exhaust substrate, [S] ‡ [S]₀, and the rate drops; the product inhibits the reaction in some way; or the enzyme begins denaturing.  So we assume we are before this situation, in parti­cular that [P] = 0 so that no back reaction can happen, and [S] = [S]₀.  You may remem­ber the difference between continuous and stop-time assays, that in the latter you have to demonstrate for how long the rate is actually linear with time; this is ensuring that you are still in initial velocity conditions.  Sometimes this is not so, for instance when you are using an expensive radioactive substrate and have to use an appreciable fraction of it to get a meaningful measurement - I have a paper by Lichtenthaler which expands on this for hydroxymethylglut­arylCoA reductase - and this can be corrected for, see the paper by Glick et al. in your handout - but again for initial study it should be.

A fourth assumption was not in Michaelis and Menten's original derivation, but makes the description more general.  It is called the Briggs-Haldane steady state assumption: that during the initial velocity period, after the first moment when the first S is binding to E, the rates of formation and breakdown of the enzyme-substrate complex are equal, = 0.  It may be likened to the state of Indiana: though people come into Indiana from Ohio, and pass out of it into Illinois, at substantial rates, these rates are equal, and consequently the popula­tion of Indiana remains constant.  This is of course only an approximation, fully true only for the period before [P] becomes large enough to slow the net rate of reaction - the initial velocity period.  The absolute equation is
= k₁[S]([E]_t-[E·S]) - (k_-1+k₂)[E·S] = k₁[S][E]_t - (k₁[S]+k_-1+k₂)[E·S], or [E·S] = , a differential equation which is not absolutely evaluable.  Hence the approximation.

The rate of formation of the enzyme-substrate complex is k₁[E][S].  Break­down occurs in two ways, backward to free substrate and enzyme, rate k_-1[E·S], and forward to E·P, rate k₂[E·S], which is the rate of product formation v, what we actually measure.  So we write k_i[E][S] = (k_-1+k₂)[E·S].  We now use the conservation equation to substitute the known quan­tity [E]_t for the unknown [E], which = [E]_t-[E·S]: k₁[E]_t[S] - k₁[E·S][S] = (k_-1+k₂)[E·S].  Group all the terms in [E·S] on the right: k₁[E]_t[S] = (k_-1+k₂)[E·S] + k₁[E·S] = [E·S]{k_-1+k₂+k₁[S]}.  We now have an expression for [E·S], by dividing both sides by the coefficient of [E·S]:
 [E·S] = , or dividing top and bottom by k₁, [E·S] = .

Now we go back to our definition of the overall velocity v as k₂[E·S] and substitute this expression into it: v = .  This is the famous Michae­lis-Menten equation.  is a sort of dissociation constant, the Michaelis constant, abbreviated K_m.  Note that its is not the same as the dissociation constant of the E·S complex, K_s = , although in the deriva­tion by Michaelis and Menten, without the Briggs-Haldane steady state assumption, it was the same; it approaches being the same as k_-1 becomes much larger than k₂, but the steady state assumption includes the possibility that k₂ is much greater than k_-1, that once E·S is formed it usually proceeds to products.  (This probability that E·S will go ahead, at least to the next complex, is called 'commitment' and is used in advanced kinetics.)  In this simple mechanism, if k₂ is appreciably large com­pared to k_-1 the Michaelis constant K_m is greater than the disso­ciation constant K_s.  (In our next mechanism it is likely to be less than K_s).  K_m has the dimen­sions of concentration, like a dissociation constant. 

Consider the situation if [S] is very large, much larger than K_m.  The equation then approaches v = , and [S] cancels out.  Thus as [S] becomes very large v approaches a maximum velocity, v Æ k₂[E]_t, and we use this as a definition, V_max = k₂[E]_t, with a capital V.  (Note that equilibrium-type constants such as K_m are always capitalized, rate constants are small k.)  We can then state the Michaelis-Menten equation in a simpler form, v = .  V_max, also stated V_m, is when the enzyme is fully saturated by substrate, working as hard as it can.  The rate is then zero-order in [S] - not affected by small changes in substrate concentration - but it is always first order in enzyme concentration [E]_t.

Consider the situation when the substrate concentration [S] is equal to the value of K_m.  Then v = = V_max/2.  From this we have the defini­tion of K_m as that substrate concentration at which the reaction rate will be half the maxi­mum velocity.  In reactions with more than one substrate it is assumed that all other substrates are at saturating concentration.

Finally consider the situation when [S] is small, less than K_m. In this case the equation approaches v = - the rate is first order in [S], with a propor­tion­ality constant V_m/K_m.

We can now see what a plot of v vs [S] looks like: it starts out increasing linearly with [S], as would be the case for an uncatalyzed reaction.  Gradually the plot drops away from the straight line extrapolation, and at high [S] v very grad­ually approaches a constant value V_m.  It is however very difficult to estimate V_m this way - particularly since other things may happen at high [S] and to estimate K_m as that [S] where v = V_m/2.  A plot of this shape is called in mathe­matics a rectangular hyperbola, and when a plot of v vs [S] follows this it is called hyperbolic.  Later we shall see instances where it is not such a simple hyperbola.  Always remember that this is a plot of v vs. [S]; never confuse it with plots of [P] vs. time which level off as equilibrium is approached, though they may look similar.

To obtain values for K_m and V_m we convert the Michaelis-Menten equation into one of several linear forms, which give a straight line as [S] is varied.  (In actual practice one now often obtains K_m and V_m by direct non-linear least squares fitting of the Michaelis-Menten equation to the data, which is most accu­rate, but one then generally plots out the result in one of the linear forms.  The Cleland treatment of multi-substrate reactions describes everything in Lineweaver-Burk plots, however their slope and intercept are determined.)

The commonest, as I'm sure you know, is the Lineweaver-Burk equa­tion, obtained by inverting the Michaelis-Menten equation: = + = + . The y inter­

cept of the plot of this equation (diagram), often referred to simply as the intercept, the value of 1/v when 1/[S] = 0 - which is to say, when [S] is infinite - is 1/V_m.  The slope, as the equation shows, is K_m/V_m.  The x intercept, when 1/v = 0, is -1/K_m, since if [S] = -1/K_m, the second term of the equation is ·-and = -1/V_m, cancelling out the first term 1/V_m.  Never­theless we usually calculate K_m by dividing the slope by the intercept 1/V_m, perhaps because a calculator's least squares program gives you the slope and intercept of the plot.

The Lineweaver-Burk plot is the most commonly used, partly because the independent and dependent variable are clearly separated, but it has one draw­back.  The point furthest to the right on the plot tends to have the greatest influ­ence in determining where the line is drawn, whether you use a least squares program or your eyes.  A small error in v can cause a large error in 1/v.  Yet this is also the smallest value of v, at the smallest [S], and most prone to error (since v is ≈ proportional to [S] in this range).  It can be shown that a point at [S] = K_m/3 has 81 times the influence of the position of the line of a point at [S] = 3K_m.

For this reason alternative linear forms are sometimes used, particularly if you are determining an approximate line by linear least squares.  If you multi­ply both sides of the L-B equation by [S], you get = + and plot [S]/v vs [S]; the plot now has the intercept = K_m/V_m and slope = 1/V_m, the reverse of what they are in the Lineweaver-Burk plot.  This is called a Woolf or Haines plot.  In this case the points at medium [S] have most influence on the position of the line, but only 3x that for those at extremes; this makes it the best first approx.

Thirdly, you can multiply both sides of the L-B plot by V_m: V_m/v = 1 + and then by v: V_m = v + ·v, and rearrange: v = V_m - v.  This gives a plot of v vs. v/[S], called an Eadie plot (diagram). The y intercept is V_m, the slope is -K_m.  This plot is best for showing subtle but significant deviations from linearity; or if you have very good data you use the Eadie plot to show off how good it is.

This is a description of the rate equation for a single substrate reaction; the equation will have the same general form, and K_m and V_m values will be derived, no matter how many intermediate complexes are present.  Only the com­position of K_m, what rate constants are in it, will vary; I'll show you an example in a minute, for the so-called acyl-enzyme mechanism.  Complete rate equations can be derived for two- and three-substrate mechanisms, but involve large numbers of terms - the full equation for a two-substrate, two-product fully random mechanism would use all the blackboards all the way around a class­room bigger than this and a full period just to write it - Cleland is said to do this in graduate General Biochemistry at Wisconsin - and the chance of miswrit­ing some term becomes very large.  For reversible reactions these equations include terms in product, which can be left out when [P] = 0, as is normal in initial velo­city experiments unless some finite concetration of one product is included in order to test the mechanism by observing its influence on the rate.  Cleland has rules for predicting such effects, which simplify life profoundly com­pared to using the full equations; we shall look at these rules in a later lecture.  Still larger equations are written either by using a set of geometrical rules, the King-Altman procedure, or a computer program just to write the equation!  Dr. Kulikowski in Computer Science has been interested in writing programs which would take all the data and tell you the mechanism, without looking at plots, but the data are rarely good enough to do this without further experiments.

The next mechanism to consider is the acyl enzyme mechanism, gener­ally thought of for proteases and esterase which have a covalent acyl-enzyme intermediate, but kinetically the mechanism for any case in which two products are released in different steps, with either no second substrate, a second sub­strate cannot be varied, i.e. water, or one binding to the enzyme after the first product has left (variation of the concentration of this second substrate is a fur­ther complication which we shall address later).

For a protease the mechanism is                                    H₂O
E + RCONHR' _E·RCONHR'  Æ E-COR + R'NH₂ Æ E __RCOO^- + H⁺.  More general­ly E + S₁ E·S₁ Æ E-S' + P₁ Æ E + P₂, where E-S' is usually a covalent compound of enzyme with part of the substrate, though the kinetic mechanism does not require this, a tightly bound non-covalent intermediate will give the same steady-state kinetic patterns.  The enzyme conservation equation is [E]_t = [E] + [E·S] + [E-S'], v = k₂[E·S] = k₃[E-S'] (an assumption that the enzyme does turn over, rather than accumulating as E-S'), and K_s = = k₁/k_-1.  Then [E] = , [E-S'] = [E·S], [E]_t = [E·S](+ 1 + , [E·S] = .  Since v = k₂[E·S], v = = .  There is point in converting this to a more complex expression: v = = = .  This has the form of the Michaelis-Menten equation, but the composition of the constants, in terms of rate constants, is different: V_m = k_cat[E]_t, where k_cat = , and if either k₂ or k₃ is much larger than the other, k_cat reduces to the smaller of the two!  k_cat may be generally defined as V_m/[E]_t, the overall rate constant of the enzyme-catalyzed reaction.  The apparent K_m, call it K_m(app), = K_sk₃/k₂+k₃, so that if k₂ >> k₃ K_m(app) is much smaller than K_s!  Also, note that = ; the value of this ratio is used as a measure of enzyme specificity, since both a high k_cat and a low K_m(app) indicate a good substrate, as compared to another substrate of the same enzyme; this is a general indication of specifi­city for enzymes acting on more than one substrate, not limited to this enzyme.  However, we shall see later in the course that it is not desirable for K_m or K_s to be too low, for reasons of the energetics of the overall reaction.