Parameter estimating procedures for the Michaelis-Menten model: reply to Tseng and Hsu.

In a recent account of methods of obtaining best-fit parameters of the parameters V and Km of the Michaelis-Menten equation, Tseng & Hsu (1990) appear to have misunderstood the method used by Lineweaver & Burk (1934), and to have overlooked much of the work done since 1934. The Michaelis-Menten equation represents the observed rate of v of an enzyme-catalysed reaction in terms of the two parameters V and Kin, and the substrate concentration s, Vs v =-+ e (1) Km+ s As written here [as well as by Tseng & Hsu (1990)] this equation assumes that the rate is subject to an additive error e. In particular , their finding that for the error structure they assumed better results are obtained by direct fitting than by an unweighted fit to the double-reciprocal form of eqn (1) is neither novel nor surprising. Unfortunately, their paper also introduces some new ideas and repeats some old and seriously misleading misconceptions. The methods that Tseng & Hsu (1990) call LB and ECB purport to be those of Lineweaver & Burk (1934) and of Eisenthal & Cornish-Bowden (1974) respectively, but in reality they bear little relation to them. In the case of Lineweaver & Burk (1934) the misconception is almost universal, apparently extending even to the original authors (see Burk, 1984). What Lineweaver & Burk (1934) actually did (see Lineweaver et al., 1934), was to fit the data to the double-reciprocal form of eqn (1) with each v value given a weight equal to v4: this gives a close approximation to a direct fit of eqn (1) with equal weight to each rate, and for the primitive computing equipment available in 1934, it is as good an approximation as one could expect. More recently, Cleland (1967) suggested using weights of v* as a first approximation but refining the fit iteratively with weights of ~34, where B is the value of v calculated with the values of V and Km for the previous approximation, whereas I suggested refining with weights of v213 a (Cornish-Bowden, 1976). Actually, these two suggestions differ from the ideal solution in opposite directions by about the same amount: as I have discussed elsewhere (Cornish-Bowden, 1982), refining with weights of v~33 leads iteratively to exactly the same result as direct minimization of e 2 by non-linear regression.