that the first term in parentheses in equation 5a is not 0/0 and,therefore, the correct limit corroborates their results;
that I have performed an analysis `in a regime where the equation does not hold';
that a geometric constant is absorbed into the Lagrange multiplierλ k, and, therefore, the distinction between a cube or a sphere is irrelevant;
that I have rehashed mistaken ideas from other authors and ignored the answers given to these authors.
In relation to (1), I agree with SEW that I have mistranscribed the first term in the equation, and I apologize for this. Therefore, that term is not 0/0, only the other two are, and the limit that SEW present is correct.
As given by West et al. (West et al.,1997), the volume Vc of a capillary isπ rN2lN and, by the assumption of the case nγβ2=1, the result is the one presented above: Vb=VcnN(N+1). This prevents West et al. from obtaining a linear relationship of Vb with body mass in a general case. It has nothing to do with applying or not applying L'hospital's rule. Moreover, in contrast to what SEW state in their comments, this computation of Vbis not `the most critical claim made by Chaui-Berlinck'. This is,simply, another source of inconsistency in the West et al.(West et al., 1997) model.
In relation to (2), all I can say is that in my manuscript(Chaui-Berlinck, 2006) I present a step-by-step reasoning based on what West et al. stated in their paper (West et al., 1997). Every step in the analysis can be easily tracked by means of figure 1 in the manuscript (Chaui-Berlinck,2006). The impedance for each type of flow was analyzed by the equations put forward by West et al. themselves(West et al., 1997). For example, in my section `Impedances and resistances to flow'(Chaui-Berlinck, 2006), it is clearly demonstrated that West et al. cannot obtain the necessary ratio between radii, β=n–1/2, a conclusion that other authors, in a much deeper analysis, have drawn as well (e.g. Painter et al., 2006).
In relation to (4), mea culpa. I agree with the criticism that I did not cite their response to other authors. The following explanation to the`service volume' issue can be found in Savage et al.(Savage et al., 2004): `As pointed out in the quotations below, WBE clearly state that only the characteristics of the capillaries themselves are assumed to be invariant. Nevertheless, K & K incorrectly interpreted this size-invariance to mean that each capillary must supply a constant volume of tissue' (underlines have been added by me). However, West et al. (West et al.,1997) state: `The network must branch so that a group of cells, referred to here as a “service volume,” is supplied by each capillary. Because rklk and the total number of branchings N is large, the volume supplied by the total network can be approximated by the sum of spheres whose diameters are that of a typical kth-level vessel, namely 4/3π(lk/2)3Nk.'(underlines have been added by me). Thus, what the authors didn't realize is that when they concluded that `the volume serviced by each capillary must scale as M1/4...' (West et al., 1997), they made themselves free from their own assumptions. Simply, they have two non-coincident explanations for `service volume'.
As I show here, the problem is not with my mathematical literacy. And it is not with my geometric skills. It is also not with those who reviewed my manuscript. The problem lies in a model that was, and still is, presented to the audience as a complete and general structure for dealing with almost all topics related to biological scaling. Such a model is fully discussed in my manuscript, which presents a number of fundamental issues that have been avoided more than answered by SEW.
In conclusion, the request for the retraction of `A critical understanding of the fractal model of metabolic scaling'(Chaui-Berlinck, 2006) is, at least, delusional.