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First published online October 19, 2007
Journal of Experimental Biology 210, 3873-3874 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.006734
Correspondence |
Comment on `A critical understanding of the fractal model of metabolic scaling'
1 Department of Systems Biology, Harvard Medical School, Boston, MA 02115,
USA
2 Department of Ecology and Evolutionary Biology, University of Arizona, Tucson,
AZ 85721, USA
3 The Santa Fe Institute, Santa Fe, NM 87501, USA
* e-mail: benquist{at}email.arizona.edu
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) |
 | (1b) |
Notice that he swaps two ß< for ß> 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 of
ß<, ß>, n and 
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 limit
ß>
n–1/3, corresponding to
(because
=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) |
1 using L'hospital's rule:
 | (3) |
 | (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 ß<=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) |
=N–
,
as originally reported in WBE. When
N 
and
constant, we have the original WBE prediction,
,
where Nc 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)
(l/2)3 is the service
volume, and the geometric constant (4/3)(1/2)3 is absorbed into
the arbitrary constant 
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.
References
Brown, J. H., West, G. B. and Enquist, B. J. (2005). Yes, West, Brown and Enquist's model of allometric scaling is both mathematically correct and biologically relevant. Funct. Ecol. 19,735 -738.[CrossRef]
Chaui-Berlinck, J. G. (2006). A critical
understanding of the fractal model of metabolic scaling. J. Exp.
Biol. 209,3045
-3054.
Dodds, P. S., Rothman, D. H. and Weitz, J. S. (2001). Re-examination of the "3/4-law" of metabolism. J. Theor. Biol. 209, 9-27.[CrossRef][Medline]
Kozlowski, J. and Konarzewski, M. (2004). Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant? Funct. Ecol. 18,283 -289.[CrossRef]
Muller-Landau, H. (2007). Evaluation for Chaui-Berlinck J. Exp. Biol. 209:3045 . F1000 Biol. http://www.f1000biology.com/article/id/1061698.[CrossRef]
Savage, V. M., Gillooly, J. F., Woodruff, W. H., West, G. B., Allen, A. P., Enquist, B. J. and Brown, J. H. (2004). The predominance of quarter-power scaling in biology. Funct. Ecol. 18,257 -282.[CrossRef]
West, G. B., Brown, J. H. and Enquist, B. J.
(1997). A general model for the origin of allometric scaling laws
in biology. Science 276,122
-126.
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 J. G. Chaui-Berlinck Response to `Comment on "A critical understanding of the fractal model of metabolic scaling'" J. Exp. Biol., November 1, 2007; 210(21): 3875 - 3876. [Full Text] [PDF]
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