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## Limiting rate in Michaelis-Menten kinetics

OBSOLETE!

as it has proved impossible to keep it up to date after about 2005,

This page discusses the approach to the limit of the curve defined by the Michaelis–Menten equation and is one of a series that discuss common errors in current textbooks of biochemistry.

### Description of the problem

For an enzyme obeying Michaelis–Menten kinetics the rate v may be written in terms of the substrate concentration [S] and parameters V and Km as v = V[S]/( Km + [S] ). It is then a matter of elementary arithmetic to show that the rate does not reach the limit V at any attainable value of [S], and remains noticeably far from it at quite high values of [S]. For example, [S] = 10 Km gives v/V = 10/11, or barely over 90%. It is equally a matter of simple arithmetic to see that the whole curve looks something like this:

#### A 12 x 12 grid is used here to make the arithmetic trivial to do in the head, but with a calculator one can easily draw the curve with any scales, and it always looks something like what is shown. Only if a very extended scale is used for [S], so that the value for Km is very close to the v axis, does the value of v get very close to V at the right-hand end of the graph, and even then it does not reach it.

Despite this, many textbooks show the curve approaching the limit much too quickly, suggesting that V can be directly measured. For example, the curve drawn in Fig. 5.6 on p. 145 of Campbell suggests that V is reached at [S] about 5 Km , and she explicitly states (p. 148) that the constant rate at saturation is the Vmax for the enzyme, and the value of Vmax can be estimated from the graph. This is not, therefore, just a case of carelessly leaving the task of drawing the curve to a professional artist without first explaining what is needed. The curve drawn to illustrate this statement in Fig. 5.7 is reasonably accurate. On pp. 148–149, the text states that it is quite difficult to determine a single point at which the rate levels off—hardly surprising, given that no such point exists.

In Fig. 6.5 of McKee and McKee (p.125) the values of Km and V are not explicitly marked, but assuming the horizontal broken line is intended to represent the limit one can estimate that it is reached at a substrate concentration of about 5 Km . The same error is shown more explicitly in the plots illustrating inhibition types on p. 127. The book uses the term Michaelis–Menten plot for a method on plotting that was not used by Michaelis and Menten.

Fig. 8.7 of Zubay (p. 205) shows the usual error (97.5% saturation at a substrate concentration of about 13 Km ) compounded by a very unusual one: although the legend correctly states that Km is the substrate concentration at which the rate is half-maximal, the v for this concentration is drawn at a value of less than 0.4V, an error that is easily visible to the eye without measurement, and thus outside any reasonable interpretation as artistic licence.

A puzzling feature of this story is that some authors appear able to draw a reasonably accurate curve on one page but then draw it very inaccurately on another, in some cases on nearly consecutive pages.

### Textbook checklist

 Abeles, Frey and Jencks *redball Mainly accurate curve on p. 82, though carelessly drawn at very high substrate concentration; grossly inaccurate on p. 118 pp. 82, 118 Campbell *redball Grossly inaccurate curve p. 145 Garrett and Grisham *greenball Curve appears accurate p. 360 Horton et al. *yellowball Curve appears accurate in the text, but not in the Solutions section pp. 127, 725 Lehninger, Nelson and Cox *redball Very inaccurate curve pp. 212, 214 McKee and McKee *redball Grossly inaccurate curve p. 125 Mathews, van Holde and Ahern *yellowball Accurate on p. 377; grossly inaccurate on p. 385 and 387 p. 377, 385, 387 Stryer *yellowball Accurate curve on p. 192, inaccurate on p. 189 pp. 189, 192 Voet and Voet *greenball Accurate curve p. 352 Zubay *redball Inaccurate curve, the error compounded by incorrect labelling p. 205