Point anJ Counterpoint: partitioning ratios in derivation of rate equations point Misuse of Partitioning Ratios in the Derivation of Rate Equations: The pH Dependence of Chymotrypsin Catalysis Dexter 0. Northrop University of Wisconsin Madison. WI 53706
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paper by Breslow entitled "Partitioning Ratios and a Biochemical Kinetic Dilemma, The pH Dependence of Chymotrypsin Catalysis" (1)offers a short-hand approach to extracting rate equations from abbreviated kinetic mechanisms based on the partitioning ratios of Cleland (2).Unfortunately, the approach leads to rate equations that differ from those obtained by rigorous methods and applies a narrower definition of the steady-state assumption. Students and teachers should be alerted to these problems and urged not to study this paper as a general guide to deriving rate equations, but rather as an exercise regarding the assumptions employed in chemical versus enzyme kinetics. The discussion bepins bv assipnine the acvlation sequence of chymotryp& cathysis as rate deter&ning with no defmition as to what is meant bv the ohrase. The oossibility of identifying a rate determi&ng step in chemikl reactions (3) or in enzvme-catalvzed reactions (4) has been challenged, and the dimculties in attempting to formulate a practical definition of the conceot have been examined in detail (51, with no consensus to date (6).Secondly, no distinction is made between the terms Yrate" and "rate constant", which leads to several misconceptions. For example, the mechanism i n Figure 1 is identified as the ratedetermining portion of the chymotrypsin-catalyzed hydrolysis of peptide substrates, whose rate is therefore considered eoual to k.+: but it can be readilv shown aleebraically that'the ra&nstant for an enzyGatic turnover is often smaller than any of the individual rate constants (or net rate constants, see below) of the operative kinetic mechanism including a so-called rate determining step (5). Similarly, the vertical axis of Figure 2 is labeled Pelative k,, for acvlation" which is incomoatible with convention: k i t is unkersally accepted as th; apparent rate constant for a complete enzymatic turnover and cannot be used to describe a portion thereof. On the other hand, the rate for a complete turnover applies equally to all portions, because by definition, steady-state catalysis requires that all steps must proceed a t exactly the same rate. More seri&s is an error in the rate equation used to describe the rate determining acylation, reproduced'in eq 1:
For its deiivation, Breslow envokes a steady-state approximation and states: "Rate of formation of the acyl enzyme product = rate of formation.of the intermediate Itimes the fraction of the intermediate that goes to prod-
counterpoint The Correct use of Partitioning Ratios in the Derivation of Rate Equations Ronald Breslow Columbia Univerrity New York. NY 10027
The adjacent paper by Northrop (1) makes a number of incorrect statements about my previous paper. The bottom line is that my. . Daoer . (2)is correct. It has been understood-and confirmed-by myriad experts in the field over theyears. Most simificantlv, it was understood and itscorr&ness confirmed by the leiding expert in the field of the partitioning approach to steady-state kinetics. Northrop and I agreed to submit our dispute to this expert, who stated (correspondance on file with the editor of this Journal) that with my assumption that the tetrahedral intermediate is a t steady-state-an assumption stated and justified in my paper-my conclusions are indeed correct. I urge readers to consult my original paper. It makes two points, both important to chemical education. 1) The partitioningtreatment of steady-statekinetics is more intuitive, and is mmpletely mathematically equivalent to the older algebraic approach. It is not a short-cut in the sense of being incomplete, but it is easier to use. All experts in enzyme kinetics (except one) with whom I have
discussed this matter were aware of this, but many other teachers are not. For applicationsofthisapproach to RNA cleavaee. . ~ . " . see ref. (3). 2) Application ofsteady-statekinetics to the acylation step in chymntrypsin catalysis makes it clear why a bell-shaped curve should not be seen wthm the observable pH range. ~
The wint is that the intermediate that I label as "I-" in -my paper consists of a tetrahedral intermediate along with a protonated imidazole. It is in equilibrium with the ES
The expected aependence of the rate constant for catalyzed acy alion of chyrnotryps n by a Dound amide substrate vs. the operat ng pH. At a h~gneno~ghpH the oeprotonated lnterrneolate wt I cecome the dominant form--no onger a steaoy-stateintermealate--an0 me rate constant will fall. However, the pH~atwhich this occurs is above the observed range, because of the instabilitvofa tetrahedral adduct relative to an amide. Volume 70 Number 12 December 1993
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noi nt . continued uct." This statement confuses a rate with a rate constant by leaving out a reference to reactant concentration. The equation;however, contains a concentration term in the form of [ES complexlllml but fails to define how the term relates to the total concentration of reactant, [EJ,or how the relationship of [ESYIEJ might change as a function of pH. Both the statement and equation ignore the contribution of the rate of reformation of the ES complex, which is reouired for the steadv-state a s s m t i o n . The correct use of partitioning ratios i s described b; Cleland (2)includes this contribution and differentiates between rates and rate constants, for according to Cleland, "the rate is the reciprocal of the sums of the reciprocals of net rate constants times the total enzyme concentration." Applied to the mechanism in Breslow's Figure 1, Cleland's method yields:
where the net rate constants are defined as,
complex, and its relative concentration is determined not only by the pK of the imidazole but alio by the unfavorable eq&libriGto form the tetrahedral intermediate. The expected dependence of the rate constant for acylation is shown in the figure. The second half of the bell curve caunot be seen in normal pH ranges because it is shifted to high pH by the unfavorable equilibrium to convert the amide substrate to a tetrahedral intermediate. We explained all this verbally in our original paper; the figure just makes the original explanation clearer. Two aspects of the Northrop paper merit comment. Nowhere in our paper did we confuse rates with rate constants, and it is significant that Northrop still cannot cite a real example. Our paper was concerned only with the acylation step of chymotrypsin hydrolysis, so we did not, of course, treat the entire multistep sequence that he discusses. The additional lesson in all this--for students and others-is that one should never accept a critic's restatement of what was said in a previous paper without checking back with the original. Literature Cited
k,' = k,
1. Nalthmp.D. J.Chem.Edue. 1899.70,OOW. 2.Bredow R. J Chem. Educ IS+,67, 228. 3.Breslow,R. Pmc. N-L h a d SCL.USA 199%90. 1208.
Hence,
which differs significatly from eq 1 in that eq 1 contains onlv the first of the two net rate constants found in eo 2. he first net rate constant describes the &sappearan& of an ES com~lexinto a turnover. or 4ESlldt: .the second net rate constkt describes the r e h a t i o n of another ES complex, dependent upon a turnover, or +d[ESlldt. Because steady-state turnovers require that d[ESYdt = 0,both net rate constants are needed to describe dPldt. Stated another way, the steady-state assumption requires that d[I-lldt = 0.In the convention of chemical kinetics, this is assured by assuminging that the intermediate is a minor species, or [I-]
