Chaui-Berlinck recently published a paper in which he claims that the
original West, Brown and Enquist (WBE) model for metabolic scaling
(West et al., 1997) is
fundamentally flawed (Chaui-Berlinck,
2006). In particular, Chaui-Berlinck asserted that `*the
minimization procedure* [of the original WBE model] *is mathematically
incorrect and ill-posed*' and that the model `*lacks self-consistency
and correct statement*'. These are strong accusations and should,
therefore, be closely scrutinized. Unfortunately, Chaui-Berlinck's conclusions
are incorrect because of rudimentary mathematical mistakes, and, even worse,
these false conclusions are now being perpetuated in the literature. For
example, Muller-Landau (Muller-Landau,
2007), in a review for Faculty of 1000, recently drew attention to
Chaui-Berlinck's paper by stating that `*This article carefully dissects
West, Brown and Enquist's
(**1997**)
derivation of allometric scaling of metabolism. It illuminates important
logical inconsistencies and mathematical problems with the argument*'.

We note that none of the original authors nor the extended scaling community associated with the WBE model were asked to review Chaui-Berlinck's manuscript. As we show below, the entire basis of Chaui-Berlinck's paper stems from fundamental mathematical mistakes. In short, the conclusions of Chaui-Berlinck (and, subsequently, Muller-Landau) are completely incorrect. We conclude that Chaui-Berlinck's paper (Chaui-Berlinck, 2006) should be retracted.

The most egregious errors of Chaui-Berlinck are seen in his equation 5a. Specifically, Chaui-Berlinck makes two mistakes. He first mis-transcribes the original equation from WBE (West et al., 1997) and then makes a fundamental error in his calculus.

In his equation 5a, Chaui-Berlinck insists that he is carefully analyzing the mathematics of the WBE model. He obtains the quotient 0/0 in several equations and then concludes that, because of his analysis, the results of WBE are meaningless. However, Chaui-Berlinck's results of 0/0 only demonstrate both a misreading of the WBE paper and a basic mathematical error. The first mistake stems from Chaui-Berlinck incorrectly writing equation 9 from WBE as: (1a) but the correct expression, as given in WBE, is: (1b)

Notice that he swaps two β_{<} for β_{>} in
the first term inside the parentheses. For the second mistake, he then goes on
to evaluate equation 9 from WBE *in a regime where the equation does not
hold*. Thus, Eqn 1b (above)
contains expressions for geometric sums that hold only for values ofβ
_{<}, β_{>}, *n* and γ such that
and
. These expressions are
not stated explicitly in WBE because they are apparent from basic rules for
sums. The correct result can be obtained directly from equation 9 in the
original WBE paper (or Eqn 1b
here) by taking the limitβ
_{>}→*n*^{–1/3}, corresponding to
(becauseγ
=*n*^{–1/3}). Although it is true that the
numerator and denominator of the second two terms inside the parentheses both
go to zero in this limit, this does *not* equal the limit of the
fraction. From introductory calculus, the limit of the fraction as a whole can
be obtained using L'hospital's rule (for example, see
http://mathworld.wolfram.com/LHospitalsRule.html
or even a standard calculus class website such as
http://www.math.tamu.edu/~fulling/coalweb/lhop.htm).
Using L'hospital's rule simply amounts to taking the derivative of the
numerator and denominator separately and *only then* taking the limit
of the numerator and denominator in the resultant fraction.

As was done in the original WBE model, a finite geometric sum can be
expressed as:
(2)
The correct result for *x*=1 is found by taking the limit
*x*→1 using L'hospital's rule:
(3)
or going back to the original sum and recognizing that:
(4)

Unfortunately, Chaui-Berlinck's criticism did not incorporate these rules.

Now realizing that we can think of *x* as
, the sum from 0 to
*N*–1 as over the *N* levels of the branching network, and
using β_{<}=*n*^{–1/2} along with our
previous expressions for the other scaling ratios, equation 9 from WBE (i.e.
Eqn 1b here) gives the correct
result:
(5)
Here,
*N̄*=*N*–*k̄*,
as originally reported in WBE. When
*N**N̄*
and
*k̄*1,
with *V*_{c} and *N̄*
constant, we have the original WBE prediction,
,
where *N*_{c} is the number of capillaries in the organism and
is directly related to metabolic rate. Consequently, the most critical claim
made by Chaui-Berlinck is patently false.

Chaui-Berlinck makes several additional errors. In the equation at the top
of the second column on p. 3050, Chaui-Berlinck's treatment of the geometric
constants in the Lagrange multiplier calculation is not correct. Specifically,
in the original WBE model, (4/3)π(*l*/2)^{3} is the service
volume, and the geometric constant (4/3)π(1/2)^{3} is absorbed into
the arbitrary constant λ_{k}, highlighting the fact that the
distinction between a sphere and a cube does not matter for these arguments.
Lastly, Chaui-Berlinck rehashes the mistaken ideas of Dodds et al.
(Dodds et al., 2001) and
Kozlowski and Konarzewski (Kozlowski and
Konarzewski, 2004). Interestingly, Chaui-Berlinck perpetuates
these flawed arguments once more but does not cite the responses, which does
not present a balanced, fair or accurate view of the field
(Brown et al., 2005;
Savage et al., 2004). In
summary, Chaui-Berlinck's paper is riddled with mathematical mistakes that
reflect a misreading of the original WBE paper.

- © The Company of Biologists Limited 2007