OBSOLETE!

This page is now *obsolete*,

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

on account of lack of access to recent editions of textbooks.

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.

For an enzyme obeying Michaelis–Menten kinetics the rate *v* may be written
in terms of the substrate concentration [S] and parameters *V* and
*K*_{m}
as *v* = *V*[S]/(
*K*_{m}
+ [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
*K*_{m}
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
*K*_{m}
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
*K*_{m}
, and she explicitly states (p. 148) that the constant rate at saturation is the

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 *V*_{max} for
the enzyme, and the value of *V*_{max} can be estimated from the graph.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
*K*_{m}
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
*K*_{m}
. 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
*K*_{m}
) compounded by a very unusual one: although the legend correctly states that

, the *K*_{m}
is the substrate concentration at which the rate is half-maximal*v* for this concentration is
drawn at a value of less than 0.4*V,* 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.

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 |

Other common errors in textbooks