115:412/508 Proteins & Enzymes spring 2002

Active Site Titration, Inhibition

Today I want to start by distinguishing reversible and irreversible inhibi­tion.  Then I’m going to cover active site titration, which is a special case of irre­versible inhibition and of pre-steady state kinetics, closely linked to the acyl-enzyme mechanism which I talked about last time; so I want to cover it now.  If time permits I shall continue with reversible inhibition, which is in the syllabus to be covered next week.  I’d like to cover the algebra of it now, then return to its role in Cleland’s system for prediction of kinetic effects.

One can of course consider two sorts of inhibition: reversible, in which some ligand binds non-covalently and reversibly to an enzyme to block or mod­ify its activity, and irreversible, some sort of covalent modification which is not reversible, at least on the time scale of an initial velocity experiment. Irrev­ersible modifications generally happen slowly, and if so you can see kinetic behavior change with time, though in the case of active site titration this change may be very fast.  Their effect is to remove active enzyme from the situation, lower E_t and hence V_max, so that the effect on a Lineweaver-Burk plot is to increase 1/V_max and increase the slope K_m/V_max; both effects will gene­rally increase with time, i.e. if the modification is slow enough so that you can do several complete Lineweaver-Burk plots in the course of one modification experiment, you will see increasing slopes and intercepts, still the same K_m.

My under­standing of active site titration dates back to work I did as a post-doc in 1967 and to a paper on enzyme titration by M.L. Bender and 10 coworkers published in the Journal of the American Chemical Society in 1966.  But it is only a special case of the gene­ral problem of detecting enzyme-sub­strate intermediates and measuring their rates of formation and breakdown, as described at length by Fersht.  One wants to do this to understand the inter­mediates - which may also be intermediates in protein folding - as well as for the specific purpose of active site titration, de­scribed by Fersht on pp. 155-8.

Measuring the rate of formation of enzyme-substrate complexes (non-co­valent, usually rapidly reversible) and intermediates (slower, usually irrever­sible in the sense of not going back to starting materials) are both problems in pre-steady state kinetics: one is trying to look at the formation of these enzyme forms which in the ordinary catalytic assay are present at steady-state concen­trations.  This poses several problems: 1) one is looking at the enzyme as a reac­tant, E + S € E·S, and consequently needs to use substrate quantities of enzyme in order to do this; 2) the reactions are usually faster than our usual enzyme assay procedures can handle, since it takes a couple of seconds to mix enzyme and substrate, put the cuvette in the spectrophotometer, and turn it on; 3) the math is somewhat more complicated, it leads to differential equations, and I never took differential equations.  So I depend on Fersht and others for the math- although I don't trust his book to be typographically correct; for instance, eqn. 4.20 on p. 141 contains an extra factor of k_f; compare eqn. 4.23 on the next page.  Fersht describes on pp. 135-138 technical ways of handling fast reactions; one can sometimes use other tricks to slow a reaction down to be able to meas­ure it in an ordinary spectrophotometer, and indeed the most straightforward way, though also not technically completely simple, is to carry out the reaction at a very low temper­ature, as low as -40° in dimethyl sulfoxide - water mixtures, to slow down sub­sequent reactions.  The amount of enzyme used depends on the sensitivity of measurement; using a spectrophotometrically measurable product such as p-nitrophenol one needs product concentration and thus enzyme concen­tration of the order of 10^-5 m, 0.25 mg/ml in the cuvette if the enzyme is chym­otrypsin with a mol. wt. of 25,000.  Fluorimetric measurements lower the level needed by a couple of orders of magnitude, and still greater sensitivity can be achieved with radioactive reagents.

Let us start from the acyl-enzyme mechanism,

EOH + RCOOR' E·RCOOR'  EO-OCR + R'OH

EO-OCR + H₂O  EOH + RCOO^- + H⁺

where R'OH in the original and frequent case is p-nitrophenol - more generally call it P₁.  Assume for the moment - in the most useful cases it is true - that k₃ is essentially = 0, the enzyme accumulates as the acyl-enzyme.  You can see that one stoichiometric equivalent of p-nitrophenol is released at the k₂ step, the acylation.  The molar concentration of enzyme active sites is thus measured as equal to the concentra­tion of p-nitrophenol released.  Of course if k₃ is not 0 and free enzyme is released, it goes back to the beginning and releas­es another equi­valent of p-nitrophenol.  The plot of product released vs. time then approaches a straight line with a positive slope, corresponding to the steady-state rate of the enzyme's action, but with a non-zero intercept on the product axis at the zero time of mixing of enzyme and substrate; see Fersht Fig. 4.10, p. 156.  This non-zero intercept is approximately equal to the molar concentration of enzyme [E]₀, and is called the burst (of p-nitrophenol release) and symbolized p.

p is equal to [E]₀ only if [S]₀>>K_m(app) and k₂>>k₃, though the latter condition favors the former because K_m(app) = , where K_s = , the dissociation constant of the E·S com­plex, and if k₂>>k₃ K_m(app) is << K_s.  In the case I worked with, the reaction of trypsin with p-nitrophenyl p'-guanidinobenzoate, I could calculate indirectly that K_m(app) was about 10^-11 m.  More generally, how­ever, p = [E]₀ ; a plot of vs. will yield as y inter­cept In practice, if you are using a titrant where [S]₀ is not large compared to K_m(app), as may be the case if both binding and solubility are poor, you do this plot once, determine the fudge factor by which p at some usable [S]₀ must be multiplied to give the true [E]₀, and ever afterward use that [S]₀ and that fudge factor.

What I have just described is called enzyme titration, the measurement of the concentration of active enzyme as the production of one molar equivalent of product P₁.  Obviously this is less sensitive than a catalytic assay; what are its advantages?  For one thing, the assay is standardized in terms of the extinction coefficient, fluorescence yield, or specific activity of some small molecule, which can be determined unequivocally, without being subject to the effects of pH, temperature, activators, etc. to which the catalytic activity is subject.  However, if the product is p-nitrophenol, one must remember that its pK_a is 7.15; the pH of the assay solution must be controlled very carefully, since it affects the effec­tive molar extinction coefficient of the product.  When I did this I used veronal buffer at pH 8.3, better than a full pH unit above the pK_a; veronal because unlike amine buffers it does not react with nitrophenyl esters non-enzymatically - but it is a 'controlled substance' that you now cannot readily buy.

One is also not subject to uncertainty as to whether 'pure' enzyme was really fully active; inactive enzyme is the most difficult thing to purify away from active enzyme.  The purity of the enzyme can be defined as

If one is carrying out a chemical modification reaction, a catalytic assay has an ambiguity: a preparation retaining 20% of the initial activity may be all a form with 20% of native activity, due to decrease in V_max or increase in K_m; or it may be 20% unmodified enzyme, 80% completely inactive enzyme; or anything in between.  Active site titration removes the ambiguity, as it measures all enzyme with any catalytic activity (100% in the first case, 20% in the second), and thus allows distinction between the two cases.  The same argument applies to mutant forms of the enzyme created by site-specific mutagenesis, and to variations in rate among isozymes.

There are also enzyme titrations which involve measurement of some stoichiometric complex of enzyme with a small molecule, for instance the com­plex of liver alcohol dehydrogenase with NAD⁺ and pyrazole.  In this case one needs to know the extinction coefficient or fluorescence yield of the complex.  If the key ligand (here pyrazole) is tightly enough bound, one can titrate excess enzyme with known amounts of ligand and determine the molar extinction coef­ficient of the complex by assuming that at low ligand concentrations all the ligand is bound and the change in absorbance represents a stoichiometric amount of E·NAD⁺·pyrazole formed.

Now let us return to the curved part of the plot, where the concentration of product P₁ is still approaching the straight line whose equation is P₁ = p + k_catt.  Take first the case where k₃ = 0 and [P₁] approaches a horizontal line.  The reac­tion is just EOH + S Æ_E-P₂ + P₁, a first-order chemical reaction.  The rate of this reaction (assuming [S]₀>>[E]₀) is just = -bt, where b is an observed rate constant - the papers on titration use this symbol.  Then [E]_t = [E]₀e^-bt, or ln[E] = ln[E]₀ - bt.  [E] in this case is the amount of enzyme remaining unmodified, the distance between the plot of [P₁] and the horizontal line with y intercept = p.  Since p = [E]₀, at least in this case, and [E] = p - [P₁], we can write ln(p-[P₁] )= lnp - bt, and b, the rate constant of acylation, is determined.  If k₃ > 0, there is turn­over of the enzyme, the plot of [P₁] approaches a straight line whose slope is the rate of turnover k_cat, which we earlier found to be .  One then plots ln[p + k_catt - [P₁]_t] vs. t to get the observed rate constant b.

Fersht describes this situation more generally as two consecutive irrever­sible reactions, using k₁ and k₂ where I use k₂ and k₃.  In my terms [B] = (EO-COR] = e^-k₂^te^-k₃^t; but I can't figure out how this comes out positive when k₂ > k₃.  Perhaps there is a typographical error, the denominator should be k₂ – k₃.  But this is the more general equation describing the concentration as the steady state decays as well as when it forms.

The observed rate constant b is related to substrate concentration in much the same way as v in simple Michaelis-Menten kinetics.  It is defined as

b= k₃ + = k₃ + = k₃+ = , where

the denominator in the second term of the first expression represents how much of [E]₀ is present as the non-covalent E·S complex as acylation starts.  This can be simplified if one stipulates that [S]₀>>K_m(app), or [S]₀>>, or [S]₀(k₂+k₃) >> K_sk₃, so that one can eliminate the last term in the equation above, leaving b ≈ .  This can be inverted like a Lineweaver-Burk plot, = + ; a plot of vs. will have intercept and slope . Given any two of these quantities one can determine the third.  If you can determine from steady state kinetics the K_m(app) = K_s, which = k₃ times the slope of this plot, you can determine k₃ in this way.  If k₃ is too small to be determined in this way, as with trypsin and p-nitrophenyl p'-guanidinobenzoate where K_m(app) is about 10^-11 m, you can isolate the acyl enzyme and measure its rate of hydrolysis, which is k₃, as the rate of recovery of activity - it took two days for full recovery in this case.  K_s is determined by dividing slope by intercept, exactly as with a Lineweaver-Burk plot.

Note how in this expression the intercept is the sum of the two rate con­stants.  This is similar to the situation when determining the rate constants k_f, k_r of a reversible reaction; the observed rate constant, analogous to b above, is the sum of the two constants, even when you are observing the first stage of approach to equilibrium.  Fersht calls this observed rate constant 1/t, the reci­procal relaxation constant, a term which comes from temperature jump studies where a sudden change in temperature changes the equilibrium of the reaction and the approach to the new equilibrium is observed.  In a simple A€B reaction the individual rate constants k_f, k_r cannot be determined unless what he calls an amplitude factor is also known, analogous to above.  But in a case where there is also substrate, such as binding of substrate to enzyme, if this is slow enough to be observed at least by stopped-flow methods, the on rate de­pends also on the substrate concentration, 1/t = k_off + k_on[S]; a plot of the observed rate constant vs. substrate concentration will have slope k_on, inter­cept k_off; see Fig. 4.7, p. 144 of Fersht.

It may be difficult to determine the rate constant b, even with stopped-flow instrumentation.  But in presence of a competitive inhibitor I, whose K_i, dissociation constant for EI€E+I, has been determined by competition with the ordinary catalytic reaction, one can slow down the reaction.  The expression for 1/b is = + .  One can keep [S] constant and vary [I], the plot of vs [I] has intercept + and slope .  Multiplying the latter by K_i gives , which can be subtracted from the intercept; if you still have an appreciable quantity you can calculate k₂+k₃, which is approxim­ately k₂.  Knowing k₂ and k₃ (from turnover) you can calculate K_s and thence K_m(app); you then know if [S]₀>>K_m(app) and the extrapolation of 1/vs. 1/[S]₀ is unnecessary.  The value of K_m(app) = 10^-11 m for p-nitrophenyl p'-guanidinobenzoate as titrant for trypsin was calculated in this way.

Another use of an inhibitor is described by Fersht on pp. 220-221.  The inhibitor proflavin increases its fluorescence when it binds to an enzyme active site, decreases it when it is displaced.  The E·proflavin complex is in equilibri­um with free E, and thus with E·S; this equilibrium is reached immediately upon addition of S.  But as E·SÆE-S, the acyl-enzyme or equivalent intermed­iate, enzyme is pulled out of the other complexes, including E·proflavin.  Thus a further decrease in proflavin fluorescence is seen, whose observed rate constant is the same observed constant b described above.  An advantage is that the same inhibitor can be used with many substrates, to characterize the K_s and k₂ in each case, without having to make new nitrophenyl esters.  Displacement of a fluores­cent ligand can be used similarly to measure the dissociation constant of other ligands from the enzyme, even metal ions which displace the fluorescent ion terbium; I will give you a sheet on that at a later class.