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.
The pages in this series are in the course of revision in the light of
changes in textbooks since they were originally prepared in 2000. I am very grateful
to Dr. Edward Behrman, Department of Biochemistry, Ohio State University, for his
help, without which the revision would not be possible. As there is some interest in
checking how textbooks have evolved in five years, the original pages will be left in place
with the same names as the new pages apart from addition of a zero at the end of the
name; for example, the old version of asympt.htm is asympt0.htm.
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 × 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. 6.8 on p. 140 of
Campbell and Farrell suggests that V
is reached at [S] about 5Km,
and they state (p. 148) that the constant rate at saturation is the Vmax for
the enzyme, and the value of Vmax
can be roughly estimated from the graph.
The addition of the word roughly
, which was not in earlier editions of the book, suggests
a rather half-hearted attempt to correct the problem. The curve drawn to illustrate this statement in Fig. 6.9 is reasonably accurate, and the text at the bottom
of p. 142, It is quite difficult to estimate V
because it is an asymptote, and the value is never reached with any finite substrate concentration
, is a considerable improvement over what appeared in earlier
editions.
Boyer provides an example of the problem in all its unreconstructed state. Not only does Fig. 6.4 (p. 133) illustrate a
curve in which the rate at about 6Km is shown as about 97% of
Vmax, but the accompanying text makes it clear that this is an error
of understanding, not just of carelessly leaving the task of drawing the curve to a professional artist without first explaining
what is needed, by stating that The Michaelis–Menten curve can be used to estimate
Vmax and
KM as shown in Figure 6.4.
The sentence that follows, However, an accurate value of
Vmax is difficult to measure because it requires very
high levels of substrate that cannot be easily achieved experimentally
is clearly not intended to be taken seriously, as it is immediately
followed by complete nonsense: Vmax is
usually estimated from the Michaelis–Menten graph and the value of KM
is determined as the concentration of substrate that yields an initial rate of
1/2 Vmax.
The curve in Fig. 6.4 of McKee and McKee (p.170) is as bad as any in relation to excessively rapid approach to saturation, and it includes the additional (and much less common) error of showing a real maximum at a substrate concentration of about 3Km, with a (very slight) decrease in rate at higher concentrations. The curve on p. 172 does not show the maximum, but in other respects is at least as bad as the one on p. 170.
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. Berg, Tymoczko and Stryer exemplify this tendency, and also illustrate another, more worrying feature, that even in the 5th edition of a hugely successful text that appeared in its 1st edition more than a quarter-century ago the authors apparently have little interest in ensuring that the basic information they supply to students is correct.
| Boyer |  |
Grossly inaccurate curve | p. 131 |
| Campbell and Farrell | |
Grossly inaccurate curve | p. 140 |
| Garrett and Grisham |  |
Curve appears accurate | p. 413 |
| Horton et al. | |
Curve appears accurate | pp. 127, 725 |
| Nelson and Cox | |
Curve appears accurate | pp. 203, 205 |
| McKee and McKee | |
Grossly inaccurate curve including a true maximum | p. 170 |
| Mathews, van Holde and Ahern |  |
Accurate on p. 377; grossly inaccurate on p. 385 and 387 | p. 377, 385, 387 |
| Stryer | |
Inaccurate curve on p. 197, accurate on p. 200 | pp. 197, 200 |
| Voet and Voet | |
Accurate curve | p. 479 |
Other common errors in textbooks