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<h1 class="top">Limiting rate in Michaelis-Menten kinetics</h1>
<p class="header">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.</p>
<p class="shout"><br>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.
<h2 class="framed">Description of the problem</h2>
<p>For an enzyme obeying Michaelis–Menten kinetics the rate <i>v</i> may be written
in terms of the substrate concentration [S] and parameters <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i> and
<acronym class="deja" title="Michaelis constant"><i>K</i><sub>m</sub></acronym>
as <i>v</i> = <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i>[S]/(
<acronym class="deja" title="Michaelis constant"><i>K</i><sub>m</sub></acronym>
+ [S] ). It is then a matter of
elementary arithmetic to show that the rate does not <q>reach</q> the limit <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i> at any
attainable value of [S], and remains noticeably far from it at quite high values of [S]. For example,
[S] = 10
<acronym class="deja" title="Michaelis constant"><i>K</i><sub>m</sub></acronym>
gives <i>v/<acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i> = 10/11, or barely over 90%. It is equally a matter
of simple arithmetic to see that the whole curve looks something like this:
<div class=centred>
<h4><img src="images/truemm.gif" width=530 height=307
alt=" Accurate sketch of the curve defined by the Michaelis–Menten equation "></h4></div>
<p>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
<acronym class="deja" title="Michaelis constant"><i>K</i><sub>m</sub></acronym>
is very close to
the <i>v</i> axis, does the value of <i>v</i> get very close to <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i> at
the right-hand end of the graph, and even then it does not <em>reach</em> it.
<div class="sidebar-medium">
<p class="in-sidebar"><strong>Other errors</strong><br>
<p class="in-sidebar"><a href="stereo0.htm">Stereochemical prefixes</a>
<p class="in-sidebar"><a href="liverhk0.htm">Liver hexokinase</a>
<p class="in-sidebar"><a href="effic0.htm">Metabolic <q>efficiency</q></a>
<p class="in-sidebar"><a href="zwitter.htm">Uncharged amino acids</a>
<p class="in-sidebar"><a href="rls0.htm">Control of glycolysis</a>
<p class="in-sidebar"><a href="glycogen0.htm">Structure of glycogen</a>
<p class="in-sidebar"><a href="choles.htm">Plant cholesterol</a>
</div>
<p>Despite this, many textbooks show the curve approaching the limit much too quickly, suggesting that
<i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i> can be directly measured. For example, the curve drawn in Fig. 6.8 on p. 140 of
<a href="howler.htm#campbell">Campbell and Farrell</a> suggests that <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i>
is reached at [S] about 5<acronym class="deja" title="Michaelis constant"><i>K</i><sub>m</sub></acronym>,
and they state (p. 148) that <q>the constant rate at saturation is the <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i><sub>max</sub> for
the enzyme, and the value of <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i><sub>max</sub>
can be roughly estimated from the graph.</q> The addition of the word <q>roughly</q>, 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, <q>It is quite difficult to estimate <i><acronym class="deja" title="Maximum velocity (or limiting rate)">V</acronym></i>
because it is an asymptote, and the value is never reached with any finite substrate concentration</q>, is a considerable improvement over what appeared in earlier
editions.
<p><a href="howler.htm#boyer">Boyer</a> 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 6<acronym class="deja" title="Michaelis constant"><i>K</i><sub>m</sub></acronym> is shown as about 97% of
<acronym class="deja" title="Maximum velocity (or limiting rate)"><i>V</i><sub>max</sub></acronym>, 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 <q>The Michaelis–Menten curve can be used to estimate
<acronym class="deja" title="Maximum velocity (or limiting rate)"><i>V</i><sub>max</sub></acronym> and
<acronym class="deja" title="Michaelis constant"><i>K</i><sub>M</sub></acronym> as shown in Figure 6.4.</q>
The sentence that follows, <q>However, an accurate value of
<acronym class="deja" title="Maximum velocity (or limiting rate)"><i>V</i><sub>max</sub></acronym> is difficult to measure because it requires very
high levels of substrate that cannot be easily achieved experimentally</q> is clearly not intended to be taken seriously, as it is immediately
followed by complete nonsense: <q><acronym class="deja" title="Maximum velocity (or limiting rate)"><i>V</i><sub>max</sub></acronym> is
usually estimated from the Michaelis–Menten graph and the value of <acronym class="deja" title="Michaelis constant"><i>K</i><sub>M</sub></acronym>
is determined as the concentration of substrate that yields an initial rate of
<sup>1</sup>/<sub>2</sub> <acronym class="deja" title="Maximum velocity (or limiting rate)"><i>V</i><sub>max</sub></acronym>.</q>
<p>The curve in Fig. 6.4 of <a href="howler.htm#mckee">McKee and McKee</a> (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 3<acronym class="deja" title="Michaelis constant"><i>K</i><sub>m</sub>,</acronym> 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.
<p>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. <a href="howler.htm#stryer">Berg, Tymoczko and Stryer</a> 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.
<h2 class="framed">Textbook checklist</h2>
<div class="table">
<table>
<tr>
<td valign="top"><a href="howler.htm#boyer">Boyer</a></td>
<td valign="top"><img src="images/redball.gif" alt="Bad" width="14" height="14" title="Major error"></td>
<td valign="top">Grossly inaccurate curve</td>
<td valign="top">p. 131</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#campbell">Campbell and Farrell</a></td>
<td valign="top"><img src="images/redball.gif" alt="Bad" width="14" height="14" title="Major error"></td>
<td valign="top">Grossly inaccurate curve</td>
<td valign="top">p. 140</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#garrett">Garrett and Grisham</a></td>
<td valign="top"><img src="images/greenbal.gif" alt="OK" width="14" height="14" title="OK"></td>
<td valign="top">Curve appears accurate</td>
<td valign="top">p. 413</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#horton">Horton <cite>et al.</cite></a></td>
<td valign="top"><img src="images/greenbal.gif" alt="OK" width="14" height="14" title="OK"></td>
<td valign="top">Curve appears accurate</td>
<td valign="top">pp. 127, 725</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#lehninger"> Nelson and Cox</a></td>
<td valign="top"><img src="images/greenbal.gif" alt="OK" width="14" height="14" title="OK"></td>
<td valign="top">Curve appears accurate</td>
<td valign="top">pp. 203, 205</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#mckee">McKee and McKee</a></td>
<td valign="top"><img src="images/redball.gif" alt="Bad" width="14" height="14" title="Major error"></td>
<td valign="top">Grossly inaccurate curve including a true maximum</td>
<td valign="top">p. 170</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#mathews">Mathews, van Holde and Ahern</a></td>
<td valign="top"><img src="images/yellowba.gif" alt="Poor" width="14" height="14" title="Partially incorrect"></td>
<td valign="top">Accurate on p. 377; grossly inaccurate on p. 385 and 387</td>
<td valign="top">p. 377, 385, 387</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#stryer">Stryer</a></td>
<td valign="top"><img src="images/yellowba.gif" alt="Poor" width="14" height="14" title="Partially incorrect"></td>
<td valign="top">Inaccurate curve on p. 197, accurate on p. 200</td>
<td valign="top">pp. 197, 200</td>
</tr>
<tr>
<td valign="top"><a href="howler.htm#voet">Voet and Voet</a></td>
<td valign="top"><img src="images/greenbal.gif" alt="OK" width="14" height="14" title="OK"></td>
<td valign="top">Accurate curve</td>
<td valign="top">p. 479</td>
</tr>
</table></div>
<hr>
<p><a href="howler.htm">Other common errors in textbooks</a>
<p><a href="howler.htm#boox">List of books considered</a>
<hr>
<div class="sidebar-tail">
<p class="in-tail">
Page created on 14 October 2005 (originally 13 April 2000)<br>
Last update: 22 December 2008<br>
Last significant update: 14 October 2005<br>
Comments and corrections to this page are welcome, and may be sent to <a href="mailto:acornish@ibsm.cnrs-mrs.fr">Athel Cornish-Bowden</a><br>
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