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.

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

_{>}→

*n*

^{–1/3}, corresponding to

*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.htmlor 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.

*x*=1 is found by taking the limit

*x*→1 using L'hospital's rule:

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

*x*as

*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:

*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,

*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.

## References

**Brown, J. H., West, G. B. and Enquist, B. J.**(

**Chaui-Berlinck, J. G.**(

**Dodds, P. S., Rothman, D. H. and Weitz, J. S.**(

**Kozlowski, J. and Konarzewski, M.**(

**Muller-Landau, H.**(

*F1000 Biol.*http://www.f1000biology.com/article/id/1061698.

**Savage, V. M., Gillooly, J. F., Woodruff, W. H., West, G. B.,Allen, A. P., Enquist, B. J. and Brown, J. H.**(

**West, G. B., Brown, J. H. and Enquist, B. J.**(