$\mathcal{I}\overset{\scriptscriptstyle\Delta}{=}f^{-1}(y)-z_{r}$, and the
second step is due to a change of variable. We emphasize that $\mathcal{I}$ is
indexed by $y$ and $z_{r}$, but to make notation light we did not make such
dependency explicit. When $f^{-1}(y)$ is a discrete set, the integration is
simply replaced by a summation.

With slight abuse of notations, let $(z,w,n,y,z_{r})$ be an instance of
$(Z,W,N,Y_{\sigma},Z_{r})$. From (83), we have
$z_{r}=(1-v_{r})z+\sqrt{v_{r}(1-v_{r})}n$ and $y=f(z+\sigma_{w}w)$. Then,

$\begin{split}\frac{1}{v_{r}}\left(z-\mathbb{E}[Z|Y_{\sigma}=y,Z_{r}=z_{r}]\right)^{2}&=\frac{1}{v_{r}}\left(z-z_{r}-\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}\right)^{2}\\\
&=\frac{1}{v_{r}}\left(v_{r}z-\sqrt{v_{r}(1-v_{r})}n-\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}\right)^{2}\\\
&=\left(\sqrt{v_{r}}z-\sqrt{1-v_{r}}n-\frac{\sqrt{v_{r}}}{{v_{r}+\sigma^{2}_{w}}}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}\right)^{2}\\\
&=\left(r-\frac{\sqrt{v_{r}}}{{v_{r}+\sigma^{2}_{w}}}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}\right)^{2},\end{split}$
(93)

where the last step is due to the definition of the r.v. $R$ in (83). Recall
that
$\mathcal{E}_{1}=\\{Z+\sigma_{\sigma}W\in\mathbb{R}\backslash\mathcal{Q}_{f}\\}$,
where $\mathcal{Q}_{f}=\\{z:f^{-1}(f(z))\text{ contains an interval}\\}$.
Conditioned on $\mathcal{E}_{1}$, $f^{-1}(y)$ is a discrete set, and so is
$\mathcal{I}\overset{\scriptscriptstyle\Delta}{=}f^{-1}(y)-z_{r}$. Hence,
conditioned on $\mathcal{E}_{1}$, the integration in the above formula is
replaced by summation over the elements in $\mathcal{I}$. Since
$y=f(z+\sigma_{w}w)$, we have $z+\sigma_{w}w\in f^{-1}(y)$. Further,
$z=z_{r}+\sqrt{v_{r}}r$, and thus

$z+\sigma_{w}w-z_{r}=\sqrt{v_{r}}r+\sigma_{w}w\in
f^{-1}(y)-z_{r}=\mathcal{I}.$

Let $\mathcal{E}_{2}$ be the event that there does not exist $x\in f^{-1}(y)$
and $x\neq z+\sigma_{w}w$ such that $|z+\sigma_{w}w|=|x-z_{r}|$. Then, on the
event $\mathcal{E}_{1}\cap\mathcal{E}_{2}$,

$\lim_{v_{r}+\sigma^{2}_{w}\to
0}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}-(\sqrt{v_{r}}r+\sigma_{w}w)=0.$

This is due to the fact that ${\mathcal{I}}$ is a discrete set and the term
with minimum exponent dominates. Hence,

$\begin{split}\lim_{v_{r}+\sigma^{2}_{w}\to
0}\frac{\sqrt{v_{r}}}{{v_{r}+\sigma^{2}_{w}}}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}-\frac{\sqrt{v_{r}}(\sqrt{v_{r}}r+\sigma_{w}w)}{v_{r}+\sigma^{2}_{w}}=0,\end{split}$

Hence, conditioned $\mathcal{E}_{1}\cap\mathcal{E}_{2}$, we have (see (93))

$\begin{split}\frac{1}{v_{r}}\left(z-\mathbb{E}[Z|Y_{\sigma}=y,Z_{r}=z_{r}]\right)^{2}&=\left(r-\frac{\sqrt{v_{r}}}{{v_{r}+\sigma^{2}_{w}}}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}\right)^{2}+o(v_{r}+\sigma_{w}^{2})\\\
&=\left(r-\frac{\sqrt{v_{r}}(\sqrt{v_{r}}r+\sigma_{w}w)}{v_{r}+\sigma^{2}_{w}}\right)^{2}+o(v_{r}+\sigma_{w}^{2})\\\
&=\left(\frac{\sigma_{w}^{2}r-\sqrt{v_{r}}\sigma_{w}w}{v_{r}+\sigma^{2}_{w}}\right)^{2}+o(v_{r}+\sigma_{w}^{2})\end{split}$

Since $\mathbb{P}(\mathcal{E}_{2}^{c})=0$, overall we have

$\begin{split}\mathbb{P}\left(T_{1}=0\right)&=\mathbb{P}\left\\{\lim_{v_{r}+\sigma^{2}_{w}\to
0}\mathbb{I}(\mathcal{E}_{1})\cdot\bigg{[}\frac{1}{v_{r}}\left(Z-\mathbb{E}[Z|Y_{\sigma},Z_{r}]\right)^{2}-\Big{(}\frac{\sigma^{2}_{w}R}{v_{r}+\sigma^{2}_{w}}-\frac{\sqrt{v}\sigma_{w}W}{v_{r}+\sigma^{2}_{w}}\Big{)}^{2}\bigg{]}=0\right\\}\\\
&=1.\end{split}$

Hence, $\mathbb{E}[T_{1}]=0$.

Analysis of $T_{2}$: Let $(z,n,w,r,y,z_{r})$ be an instance of
$(Z,N,W,R,Y_{\sigma},Z_{r})$. From (93), we have

$\displaystyle\frac{1}{v_{r}}\left(z-\mathbb{E}[Z|y,z_{r}]\right)^{2}$
$\displaystyle=\left(\sqrt{v_{r}}z-\sqrt{1-v_{r}}n-\frac{\sqrt{v_{r}}}{{v_{r}+\sigma^{2}_{w}}}\frac{\int_{\mathcal{I}}u\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}{\int_{\mathcal{I}}\mathcal{N}(u;0,v_{r}+\sigma^{2}_{w})\mathrm{d}u}\right)^{2}$
(94a)
$\displaystyle=\Big{(}\sqrt{v_{r}}z-\sqrt{1-v_{r}}n-\frac{\sqrt{v_{r}}}{{v_{r}+\sigma^{2}_{w}}}\frac{\int_{\hat{\mathcal{I}}}u\mathcal{N}(u;0,1)\mathrm{d}u}{\int_{\hat{\mathcal{I}}}\mathcal{N}(u;0,1)\mathrm{d}u}\Big{)}^{2}$
(94b)

where
$\hat{\mathcal{I}}\overset{\scriptscriptstyle\Delta}{=}\frac{\mathcal{I}}{\sqrt{v_{r}+\sigma^{2}_{w}}}=\frac{f^{-1}(y)-z_{r}}{\sqrt{v_{r}+\sigma^{2}_{w}}}$.
Let $\mathcal{E}_{3}$ be the event that $z_{r}$ is not on the boundary of
$f^{-1}(y)$. Consider the third term in (94b) under
$\mathcal{E}_{1}^{c}\cap\mathcal{E}_{3}$. From the definition of
$\mathcal{E}_{1}^{c}$, $\hat{\mathcal{I}}$ only consists of intervals. If $0$
is an interior point of $\hat{\mathcal{I}}$, we have

$\left|\int_{\hat{\mathcal{I}}}\mathcal{N}(u;0,1)\mathrm{d}u-1\right|\leq\int_{\hat{\mathcal{I}}^{c}}\mathcal{N}(u;0,1)\mathrm{d}u=O\left(e^{-c/(v_{r}+\sigma^{2}_{w}))}\right).$

where $\hat{\mathcal{I}}^{c}=\mathbb{R}\backslash\hat{\mathcal{I}}$ and $c>0$
is some constant. Similarly, for the numerator,

$\left|\int_{\hat{\mathcal{I}}}u\mathcal{N}(u;0,1)\mathrm{d}u\right|\leq\int_{\hat{\mathcal{I}}^{c}}|u|\mathcal{N}(u;0,1)\mathrm{d}u=O\left(e^{-c/(v_{r}+\sigma^{2}_{w}))}\right).$

Hence, when $0$ is an interior point of $\hat{\mathcal{I}}$, we have

$\lim_{v_{r}+\sigma^{2}_{w}\to
0}\frac{\sqrt{v_{r}}}{{v_{r}+\sigma^{2}_{w}}}\frac{\int_{\hat{\mathcal{I}}}u\mathcal{N}(u;0,1)\mathrm{d}u}{\int_{\hat{\mathcal{I}}}\mathcal{N}(u;0,1)\mathrm{d}u}=0.$

Next, we decompose
$S\overset{\scriptscriptstyle\Delta}{=}\left(z-\mathbb{E}[Z|y,z_{r}]\right)^{2}/v_{r}$
as

$S=S\cdot\mathbb{I}\left(0\in(f^{-1}(y)-z_{r})\right)+S\cdot\mathbb{I}\left(0\notin(f^{-1}(y)-z_{r})\right).$

We note that as $v_{r}+\sigma^{2}_{w}\to 0$, we have $z_{r}\to z$. Further,
$z\in f^{-1}(y)$. Therefore,

$\lim_{v_{r}+\sigma^{2}_{w}\to
0}\mathbb{I}\left(0\in(f^{-1}(y)-z_{r})\right)=1.$

We have shown in (90) that $S<\infty$. Hence,

$\begin{split}\lim_{v_{r}+\sigma^{2}_{w}\to
0}\frac{1}{v_{r}}\left(z-\mathbb{E}[Z|y,z_{r}]\right)^{2}&=\lim_{v_{r}+\sigma^{2}_{w}\to
0}S\cdot\mathbb{I}\left(0\in(f^{-1}(y)-z_{r})\right)+\lim_{v_{r}+\sigma^{2}_{w}\to
0}S\cdot\mathbb{I}\left(0\notin(f^{-1}(y)-z_{r})\right)\\\
&=\lim_{v_{r}+\sigma^{2}_{w}\to
0}S\cdot\mathbb{I}\left(0\in(f^{-1}(y)-z_{r})\right)\\\
&=\left(\sqrt{v_{r}}z-\sqrt{1-v_{r}}n\right)^{2}.\end{split}$

Since $\mathbb{P}(\mathcal{E}_{3}^{c})=0$, we have

$\begin{split}\mathbb{P}\left(T_{2}=0\right)&=\mathbb{P}\left(\lim_{v_{r}+\sigma^{2}_{w}\to
0}\frac{1}{v_{r}}\left(z-\mathbb{E}[Z|y,z_{r}]\right)^{2}\mathbb{I}(\mathcal{E}_{2})=\frac{1}{v_{r}}\left(v_{r}z-\sqrt{v_{r}(1-v_{r})}n\right)^{2}\mathbb{I}(\mathcal{E}_{2})\right)\\\
&=1.\end{split}$

∎

##  Appendix H Some Auxiliary Results

In this section, after introducing the conditional MMSE dimension
$\mathscr{D}(Z|Y)$, we present a few properties of $\mathsf{mmse}_{z}(\cdot)$
and the SE maps $\phi(\cdot)$, $\Phi(\cdot)$.

###  H.1 MMSE Dimension and information dimension

The MMSE dimension $\mathscr{D}(Z)$ defined below characterizes the high SNR
behavior of the MMSE $\mathsf{mmse}(Z,\mathsf{snr})$. Similarly,
$\mathscr{D}(Z|U)$ characterizes the high SNR behavior of
$\mathsf{mmse}(Z,\mathsf{snr}|U)$.

######  Definition 6 (MMSE dimension [64]).

The following limits, if exist, is called the MMSE dimension (resp.
conditional MMSE dimension):

$\begin{split}\mathscr{D}(Z)&=\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathsf{mmse}(Z,\mathsf{snr}),\\\
\mathscr{D}(Z|U)&=\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathsf{mmse}(Z,\mathsf{snr}|U).\end{split}$
(95)

The following lemma establishes the connection between the conditional MMSE
dimension $\mathscr{D}(Z|Y)$ and the information dimension $d(Y)$ (see Section
1.3).

######  Lemma 16.

Suppose Assumption (A.3) holds. Let $Z\sim\mathcal{N}(0,1)$ and $Y=f(Z)$. We
have

$d(Y)=1-\mathscr{D}(Z|Y).$

###### Proof.

The conditional MMSE dimension can be calculated as follows:

$\begin{split}\mathscr{D}(Z|Y)&=\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathsf{mmse}(Z,\mathsf{snr}|Y)\\\
&=\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathbb{E}\left[\left(Z-\mathbb{E}[Z|\sqrt{\mathsf{snr}}Z+N,Y=y]\right)^{2}\right]\\\
&\overset{a}{=}\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathbb{E}\left[\left(Z-\mathbb{E}[Z_{y}|\sqrt{\mathsf{snr}}Z_{y}+N]\right)^{2}\right]\\\
&\overset{\scriptscriptstyle\Delta}{=}\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathbb{E}_{Y}\left[\mathsf{mmse}(Z_{y},\mathsf{snr})\right]\\\
\end{split}$ (96)

where $Z_{y}\sim P_{Z|Y=y}=P_{Z|Z\in f^{-1}(y)}$ denotes a random variable
indexed by $y$, and $N$ is independent of $Z$. Note that
$\mathsf{mmse}(Z_{y},\mathsf{snr})\leq\mathsf{snr}^{-1}$ [30]555This is true
even when the moments of $Z_{u}$ do not exist. To see this, consider
$\tilde{Y}=\sqrt{\mathsf{snr}}Z_{u}+N$ and the linear estimator
$\tilde{Y}/\sqrt{\mathsf{snr}}$. The MSE of this linear estimator is
$\mathsf{snr}^{-1}$ and hence
$\mathsf{mmse}(Z_{u},\mathsf{snr})\leq\mathsf{snr}^{-1}$. and so
$\mathsf{snr}\cdot\mathsf{mmse}(Z_{u},\mathsf{snr})\leq 1$. Hence, by
Lebesgue’s dominated convergence theorem we have

$\begin{split}\mathscr{D}(Z|Y)&=\mathbb{E}_{Y}\left[\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathsf{mmse}(Z_{y},\mathsf{snr})\right]\\\
&=\mathbb{E}_{Y}\left[\mathscr{D}(Z_{y})\right],\end{split}$ (97)

provided that
$\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathsf{mmse}(Z_{y},\mathsf{snr})$
exists almost surely. From [64, Theorem 10 and Theorem 11],

$\mathscr{D}(Z_{y})=\begin{cases}0&\text{if }P_{Z|Y=y}\text{ is discrete}\\\
1&\text{if }P_{Z|Y=y}\text{ is absolutely continuous w.r.t. Lebesgue
measure}\end{cases}$

This implies that

$\mathscr{D}(Z_{y})=\begin{cases}0&\text{if
}y\in\mathbb{R}\backslash\mathcal{Q}_{f}\\\ 1&\text{if
}y\in\mathcal{Q}_{f}.\end{cases}$

Hence,

$\mathscr{D}(Z|Y)=\mathbb{P}\\{f(Z)\in\mathcal{Q}_{f}\\}=1-d(Y).$

∎

###  H.2 A property of $\mathsf{mmse}_{z}(v_{r})$

Note that $\eta_{z}(z_{r},y,v)$ in GLM-EP (see (6f)) is an MMSE estimator:

$\eta_{z}(z_{r},y,v)=\mathbb{E}[Z|Y=y,Z_{r}=z_{r}]=\frac{\int_{f^{-1}(y)}u\cdot\mathcal{N}(u;z_{r},v)du}{\int_{f^{-1}(y)}\mathcal{N}(u;z_{r},v)du},$
(98)

where $(Z,Z_{r})\sim\mathcal{N}(\mathbf{0},\bm{\Sigma})$ where

$\bm{\Sigma}\overset{\scriptscriptstyle\Delta}{=}\begin{bmatrix}1&1-v_{r}\\\
1-v_{r}&1-v_{r}\end{bmatrix},$ (99)

and $Y=f(Z)$. Recall that $\mathsf{mmse}_{z}(v_{r})$ is defined as

$\mathsf{mmse}_{z}(v_{r})=\mathbb{E}\Big{(}Z-\mathbb{E}[Z|Z_{r},Y]\Big{)}^{2},$
(100)

Lemma 17 below is a consequence of the covariance structure of $(Z,Z_{r})$
defined in (99).

######  Lemma 17.

Let $\mathsf{mmse}_{z}(v_{r})$ be the MMSE defined in (100). Let
$Z\sim\mathcal{N}(0,1)$, $Y=f(Z)$ and $v_{r}\in(0,1]$. We have

$\mathsf{mmse}_{z}(v_{r})=\mathsf{mmse}(Z,v_{r}^{-1}-1|Y),$

where the right hand side is a conditional MMSE defined in (2).

###  H.3 Properties of the SE maps

In this appendix, we discuss a few properties of the maps $\phi$ and $\Phi$ in
(7):

$\displaystyle\phi(v_{r})$
$\displaystyle=\left({\frac{1}{\mathsf{mmse}_{z}\left(v_{r}\right)}-\frac{1}{v_{r}}}\right)^{-1}.$
(101a) $\displaystyle\Phi(v_{l})$
$\displaystyle=\left({\frac{1}{\frac{1}{\delta}\cdot\mathbb{E}\left[\frac{v_{l}\lambda}{v_{l}+\lambda}\right]}-\frac{1}{v_{l}}}\right)^{-1},$
(101b)

where the expectation in $\Phi$ is over $\lambda$, which is distributed
according to the asymptotic eigenvalue distribution of
$\bm{A}^{\mathsf{T}}\bm{A}$, and

$\mathsf{mmse}_{z}(v_{r})\overset{\scriptscriptstyle\Delta}{=}\mathsf{mmse}\left(Z,v_{r}^{-1}-1|f(Z)\right).$

The following lemmas collect some useful properties of the MMSE [29], and the
maps $\phi$, $\Phi$.

######  Lemma 18 (Properties of $\mathsf{mmse}(Z,\mathsf{snr}|U)$).

The following hold:

  1. (i)

Assume $Z\sim\mathcal{N}(0,1)$. Then,
$\mathsf{mmse}(Z,\mathsf{snr}|U)\leq\frac{1}{1+\mathsf{snr}}$,
$\forall\mathsf{snr}>0$. Further, the inequality is strict if $U$ is not
independent of $Z$.

  2. (ii)

$\frac{\mathrm{d}}{\mathrm{d}\mathsf{snr}}\mathsf{mmse}(Z,\mathsf{snr}|U)=-\mathbb{E}\left(\mathrm{var}^{2}[Z|\sqrt{\mathsf{snr}}Z+N,U]\right)$,
where
$\mathrm{var}[Z|\sqrt{\mathsf{snr}}Z+N,U]\overset{\scriptscriptstyle\Delta}{=}\mathbb{E}[Z^{2}|\sqrt{\mathsf{snr}}Z+N,U]-\mathbb{E}^{2}[Z|\sqrt{\mathsf{snr}}Z+N,U]$,
and $N\sim\mathcal{N}(0,1)$ is independent of $(Z,U)$.

######  Lemma 19 (Properties of $\phi$ and $\Phi$).

The functions $\phi$ and $\Phi$ defined in (101) have the following
properties:

  1. (i)

$\phi(v_{r})$ is continuous and non-decreasing in $v_{r}\in(0,1)$. If $f(z)$
is not an invertible function, $\phi(v_{r})$ is strictly increasing. Suppose
that $f(Z)$ is not independent of $Z\sim\mathcal{N}(0,1)$. Then,
$0\leq\phi(v_{r})<\infty$ and $\phi(0)=0$ if $d(f(Z))\neq 0$, and
$\phi(1)<\infty$ if $\mathbb{E}[Z|f(Z)]\neq 0$;

  2. (ii)

$\Phi(v_{l})$ is continuous and strictly increasing in $v_{l}\in(0,\infty)$.
Further, $\Phi(0)=0$ and $\Phi(\infty)=1$.

###### Proof.

Proof of (i): The continuity of $\phi(v_{r})$ is due to the continuity of the
function $\mathsf{mmse}_{z}(v_{r})=\mathsf{mmse}(Z,\mathsf{snr}|Y)$, where
$\mathsf{snr}=v_{r}^{-1}-1$ [29].

We next prove that $\phi$ is strictly increasing. Differentiation yields (see
(101))

$\displaystyle\phi^{\prime}(v_{r})$
$\displaystyle=\frac{v^{2}_{r}\cdot\mathsf{mmse}_{z}^{\prime}(v_{r})-\mathsf{mmse}_{z}^{2}(v_{r})}{\left(v_{r}-\mathsf{mmse}_{z}(v_{r})\right)^{2}}.$
(102a)

Hence, we only need to prove

$\displaystyle\mathsf{mmse}_{z}^{\prime}(v_{r})$
$\displaystyle>\frac{1}{v_{r}^{2}}\cdot\mathsf{mmse}_{z}^{2}(v_{r}),\quad\forall
v_{r}\in(0,1].$ (103a)

Recall the definition

$\mathsf{mmse}_{z}(v_{r})=\mathsf{mmse}(Z,\mathsf{snr}|Y),\quad\mathsf{snr}\overset{\scriptscriptstyle\Delta}{=}v_{r}^{-1}-1,$

and the derivative formula of the conditional MMSE in Lemma 18, we have

$\mathsf{mmse}_{z}^{\prime}(v_{r})=\frac{1}{v_{r}^{2}}\cdot\mathbb{E}\left(\mathrm{var}^{2}[Z|\sqrt{\mathsf{snr}}Z+N,Y]\right),\quad\forall
v_{r}\in(0,1].$ (104)

Further,

$\mathsf{mmse}_{z}(v_{r})=\mathsf{mmse}(Z,\mathsf{snr}|Y)=\mathbb{E}\left(\mathrm{var}[Z|\sqrt{\mathsf{snr}}Z+N,Y]\right)$
(105)

Combining (104) and (105), and applying Jensen’s inequality proves
$\mathsf{mmse}_{z}^{\prime}(v_{r})\geq\frac{1}{v_{r}^{2}}\cdot\mathsf{mmse}_{z}^{2}(v_{r})$,
and equality holds only when $\mathrm{var}[Z|\sqrt{\mathsf{snr}}Z+N,Y]$ is
constant with respect to realizations of $\sqrt{\mathsf{snr}}Z+N$ and $Y$.
This is only possible when $Z_{y}\sim P_{Z|Y=y}$ is Gaussian with
$\mathrm{var}[Z_{y}]$ invariant to $y$ (including the degenerate case where
$\mathrm{var}[Z_{y}]=0$). Again, this is only possible when $f(z)$ is an
invertible function for which $Z_{y}$ is a constant and
$\mathrm{var}[Z_{y}]=0$). To summarize, when $f(z)$ is not an invertible
function, (103a) holds and so $\phi$ is a strictly increasing function.

Finally, we verify $\phi(0)$ and $\phi(1)$. First, for any $v_{r}\in(0,1)$, we
have

$\begin{split}\mathsf{mmse}_{z}(v_{r})&=\mathsf{mmse}(Z,\mathsf{snr}|Y)\quad(\mathsf{snr}=v_{r}^{-1}-1)\\\
&\overset{(a)}{\leq}\mathsf{mmse}(Z,\mathsf{snr})\\\
&=\frac{1}{1+\mathsf{snr}}\\\ &=v_{r}\end{split}$ (106)

where step (a) is from the fact that conditioning reduces MMSE [29,
Proposition 11]. Further, the inequality is strict for $v_{r}\neq 1$
($\mathsf{snr}>0$) whenever $f(Z)$ is not independent of $Z$. It follows that

$\phi(v_{r})=\left(\frac{1}{\mathsf{mmse}_{z}(v_{r})}-\frac{1}{v_{r}}\right)^{-1}\in[0,\infty),\quad\forall
v_{r}\in(0,1).$

Further, $\phi(v_{r})$ is continuously increasing in $(0,1)$ and so the limit
$\lim_{v_{r}\to 0_{+}}\phi(v_{r})$ exists (which is defined to be $\phi(0)$).
Hence, $\phi(0)\geq 0$.

Lemma 16 shows $d(Y)=1-\mathscr{D}(Z|Y)$. Hence, if $d(Y)\neq 0$, we would
have

$\mathscr{D}(Z|Y)\overset{\scriptscriptstyle\Delta}{=}\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathsf{mmse}(Z,\mathsf{snr}|Y)<1.$

Then,

$\begin{split}\phi(0)&\overset{\scriptscriptstyle\Delta}{=}\lim_{v_{r}\to
0}\phi(v_{r})\\\ &=\lim_{v_{r}\to
0}\frac{\mathsf{mmse}_{z}(v_{r})}{1-\frac{\mathsf{mmse}_{z}(v_{r})}{v_{r}}}\\\
&\overset{(a)}{=}\lim_{\mathsf{snr}\to\infty}\frac{\mathsf{mmse}(Z,\mathsf{snr}|Y)}{1-(\mathsf{snr}+1)\mathsf{mmse}(Z,\mathsf{snr}|Y)}\quad(\mathsf{snr}=v_{r}^{-1}-1)\\\
&=0\end{split}$

where step (a) follows from the definition of $\mathsf{mmse}_{z}$ below (7),
and the fact that
$\lim_{\mathsf{snr}\to\infty}\mathsf{mmse}(Z,\mathsf{snr}|Y)=0$ and
$\lim_{\mathsf{snr}\to\infty}\mathsf{snr}\cdot\mathsf{mmse}(Z,\mathsf{snr}|Y)=\mathscr{D}(Z|Y)<1$.

Finally,

$\begin{split}\phi(1)&=\left(\frac{1}{\mathsf{mmse}_{z}(1)}-1\right)^{-1}\\\
&=\left(\frac{1}{\mathsf{mmse}(Z,\mathsf{snr}=0|Y)}-1\right)^{-1}\\\
&=\left(\frac{1}{\mathbb{E}\left(\mathrm{var}[Z|Y]\right)}-1\right)^{-1}\\\
&=\left(\frac{1}{\mathbb{E}\left(\mathbb{E}[Z^{2}|Y]-\mathbb{E}^{2}[Z|Y]\right)}-1\right)^{-1}\\\
&=\left(\frac{1}{1-\mathbb{E}\left(\mathbb{E}^{2}[Z|Y]\right)}-1\right)^{-1}\quad(\mathbb{E}[Z^{2}]=1),\end{split}$
(107)

where $\mathbb{E}[Z^{2}]=1$ since $Z\sim\mathcal{N}(0,1)$. Hence, $\phi(1)\geq
0$ and $\phi(1)<\infty$ if $\mathbb{E}[Z|Y]\neq 0$.

Proof of (ii):  Similar to the proof of part (i), to prove $\Phi(v_{l})$ is
increasing, we only need to verify

$\frac{1}{\delta}\mathbb{E}\left[\left(\frac{v_{l}\lambda}{v_{l}+\lambda}\right)^{2}\right]>\left(\frac{1}{\delta}\mathbb{E}\left[\frac{v_{l}\lambda}{v_{l}+\lambda}\right]\right)^{2},\quad\forall
v_{r}\in(0,1].$ (108)

When $\delta>1$, Jensen’s inequality yields the result:

$\frac{1}{\delta}\mathbb{E}\left[\left(\frac{v_{l}\lambda}{v_{l}+\lambda}\right)^{2}\right]>\frac{1}{\delta}\left(\mathbb{E}\left[\frac{v_{l}\lambda}{v_{l}+\lambda}\right]\right)^{2}>\left(\frac{1}{\delta}\mathbb{E}\left[\frac{v_{l}\lambda}{v_{l}+\lambda}\right]\right)^{2},\quad\forall
v_{r}\in(0,1].$

For $\delta\leq 1$, note that $P_{\lambda}=(1-\delta)P_{0}+\delta
P_{\tilde{\lambda}}$ where $P_{\tilde{\lambda}}$ denotes the asymptotic
eigenvalue distribution of $\bm{AA}^{\mathsf{T}}$ (we have
$\mathbb{E}[\tilde{\lambda}^{2}]=1$). Hence, (108) can be reformulated as

$\mathbb{E}\left[\left(\frac{v_{l}\tilde{\lambda}}{v_{l}+\tilde{\lambda}}\right)^{2}\right]>\left(\mathbb{E}\left[\frac{v_{l}\tilde{\lambda}}{v_{l}+\tilde{\lambda}}\right]\right)^{2},\quad\forall
v_{r}\in(0,1],$

and holds due to Jensen’s inequality. ∎

######  Lemma 20.

If $f(z)=f(-z),\forall z$, then $\mathsf{mmse}_{z}(1)=1$. Further,
$(V_{r},V_{l})=(1,\infty)$ is a fixed point of the state evolution equations
in (7).

###### Proof.

Recall that $\mathsf{mmse}_{z}(v_{r})=\mathsf{mmse}(Z,v_{r}^{-1}-1|Y)$. Hence,
$\mathsf{mmse}_{z}(1)=\mathsf{mmse}(Z,\mathsf{snr}=0|Y)$ and

$\begin{split}\mathsf{mmse}(Z,\mathsf{snr}=0|Y)&=\mathbb{E}\Big{(}\mathbb{E}[|Z|^{2}|Y]-\mathbb{E}^{2}[Z|Y]\Big{)}\\\
&=\mathbb{E}\left(\mathbb{E}[|Z|^{2}|Y]\right)\\\
&=\mathbb{E}[|Z|^{2}]=1.\end{split}$

A simple calculation shows that $(V_{r},V_{l})=(1,0)$ is a fixed point of (7).
∎

The following lemma summarizes a few useful properties of
$\phi(v_{r},\sigma^{2}_{w})$ (which is the noisy counterpart of
$\phi(v_{r})$).

######  Lemma 21.

Define

$\mathsf{mmse}_{z}(v_{r},\sigma^{2}_{w})\overset{\scriptscriptstyle\Delta}{=}\mathbb{E}\left[\left(\mathbb{E}[Z|Y_{\sigma},Z_{r}]-Z\right)^{2}\right],$
(109)

where $Z_{r}=(1-v_{r})Z+\sqrt{v_{r}(1-v_{r})}N$,
$Y_{\sigma}=f(Z+\sigma_{w}W)$, $Z,W,N$ are mutually independent standar
Gaussian RVs. Define

$\phi(v_{r},\sigma^{2}_{w})=\left(\frac{1}{\mathsf{mmse}_{z}(v_{r},\sigma^{2}_{w})}-\frac{1}{v_{r}}\right)^{-1}.$
(110)

For any $\sigma_{w}>0$ and $v_{r}\in(0,1)$, $\phi(v_{r},\sigma^{2}_{w})$
satisfies the following:

  * (i)

$\phi(v_{r},\sigma^{2}_{w})$ is continuous and increasing in $v_{r}\in[0,1)$.
Further, $\phi(v_{r},\sigma^{2}_{w})\geq 0$;

  * (ii)

$\sigma^{2}_{w}\leq\phi(v_{r},\sigma^{2}_{w})<\infty$, $\forall
v_{r}\in[0,1)$.

###### Proof.

Part (i): Same as Lemma 19-(i).

Part (ii): We will show that $\mathsf{mmse}(v_{r},\sigma^{2}_{w})$ can be
rewritten as

$\mathsf{mmse}_{z}(v_{r},\sigma^{2}_{w})=\left(\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\right)^{2}\cdot\mathbb{E}\Big{(}U-\mathbb{E}[U|Z_{r},Y_{\sigma}]\Big{)}^{2}+\frac{v_{r}\sigma^{2}_{w}}{v_{r}+\sigma^{2}_{w}},$
(111)

where $U=Z+\sigma_{w}W$, $Y_{\sigma}=f(U)$, and
$(U,Z_{r})\sim\mathcal{N}(\mathbf{0},\bm{\Sigma})$ where

$\bm{\Sigma}=\begin{bmatrix}1+\sigma^{2}_{w}&1-v_{r}\\\
1-v_{r}&1-v_{r}\end{bmatrix}.$

From (111) we have

$\mathsf{mmse}_{z}(v_{r},\sigma^{2}_{w})\geq\frac{v_{r}\sigma^{2}_{w}}{v_{r}+\sigma^{2}_{w}},$

which together with (110) yields
$\phi(v_{r},\sigma^{2}_{w})\geq\sigma_{w}^{2}$. We next prove the boundedness
of $\phi(v_{r},\sigma^{2}_{w})$. Substituting (111) into (110) and after
straightforward calculations, we have

$\phi(v_{r},\sigma^{2}_{w})=\frac{v_{r}\cdot\mathbb{E}\Big{(}U-\mathbb{E}[U|Z_{r},Y_{\sigma}]\Big{)}^{2}}{v_{r}+\sigma^{2}_{w}-\mathbb{E}\Big{(}U-\mathbb{E}[U|Z_{r},Y_{\sigma}]\Big{)}^{2}}.$

Since conditioning reduces MMSE [30, Proposition 11], we have

$\mathbb{E}\Big{(}U-\mathbb{E}[U|Z_{r},Y_{\sigma}]\Big{)}^{2}\leq\mathbb{E}\Big{(}U-\mathbb{E}[U|Z_{r}]\Big{)}^{2}=v_{r}+\sigma^{2}_{w},$

where the inequality is strict whenever $Y_{\sigma}$ is not independent of
$U$. All together, we have $\phi(v_{r},\sigma^{2}_{w})<\infty$.

It only remains to prove (111). Let us write $Z={Z}_{r}+\tilde{Z}$, where
$\tilde{Z}\sim\mathcal{N}(0,v_{r})$ is independent of $Z_{r}$. We have

$\begin{split}\tilde{U}\overset{\scriptscriptstyle\Delta}{=}U-Z_{r}=\tilde{Z}+\sigma_{w}W.\end{split}$

Define

$\tilde{Z}_{\tilde{u}}^{\perp}\overset{\scriptscriptstyle\Delta}{=}\tilde{Z}-\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\tilde{U}.$

By construction, $\tilde{Z}_{\tilde{u}}^{\perp}\Perp\tilde{U}$. We also have
$\tilde{Z}_{\tilde{u}}^{\perp}\Perp Z_{r}$666Throughout this paper, $A\Perp B$
denotes the random variables $A$ and $B$ are independent, since
$\tilde{Z}_{\tilde{u}}^{\perp}$ a linear combination of $\tilde{Z}$ and $W$
and the latter two RVs are independent of $Z_{r}$. Also,
$\tilde{Z}_{\tilde{u}}^{\perp}\Perp Y_{\sigma}$ according to Lemma 22. Hence,

$\begin{split}\mathbb{E}[\tilde{Z}|Z_{r},Y_{\sigma}]&=\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\cdot\mathbb{E}[\tilde{U}|Z_{r},Y_{\sigma}]+\mathbb{E}[\tilde{Z}_{\tilde{u}}^{\perp}|Z_{r},Y_{\sigma}]\\\
&=\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\cdot\mathbb{E}[\tilde{U}|Z_{r},Y_{\sigma}],\end{split}$

where the last step is due to the independence of
$\tilde{Z}_{\tilde{u}}^{\perp}$ and $(Z_{r},Y_{\sigma})$ and the fact that
$\tilde{Z}_{\tilde{u}}^{\perp}$ is zero-mean Gaussian. Hence, we have

$\begin{split}\mathbb{E}\Big{(}Z-\mathbb{E}[Z|Z_{r},Y_{\sigma}]\Big{)}^{2}&=\mathbb{E}\Big{(}\tilde{Z}-\mathbb{E}[\tilde{Z}|Z_{r},Y_{\sigma}]\Big{)}^{2}\\\
&=\mathbb{E}\Big{(}\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\tilde{U}+\tilde{Z}_{\tilde{u}}^{\perp}-\mathbb{E}[\tilde{Z}|Z_{r},Y_{\sigma}]\Big{)}^{2}\\\
&=\mathbb{E}\Big{(}\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\tilde{U}+\tilde{Z}_{\tilde{u}}^{\perp}-\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\cdot\mathbb{E}[\tilde{U}|Z_{r},Y_{\sigma}]\Big{)}^{2}\\\
&\overset{(a)}{=}\left(\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\right)^{2}\cdot\mathbb{E}\Big{(}\tilde{U}-\mathbb{E}[\tilde{U}|Z_{r},Y_{\sigma}]\Big{)}^{2}+\frac{v_{r}\sigma^{2}_{w}}{v_{r}+\sigma^{2}_{w}}\\\
&=\left(\frac{v_{r}}{v_{r}+\sigma^{2}_{w}}\right)^{2}\cdot\mathbb{E}\Big{(}U-\mathbb{E}[U|Z_{r},Y_{\sigma}]\Big{)}^{2}+\frac{v_{r}\sigma^{2}_{w}}{v_{r}+\sigma^{2}_{w}}\end{split}$

where step (a) is due to the fact that
$\tilde{Z}_{\tilde{u}}^{\perp}\Perp(\tilde{U},Z_{r},Y_{\sigma})$ and
$\mathbb{E}[(\tilde{Z}_{\tilde{u}}^{\perp})^{2}]=\frac{v_{r}\sigma^{2}_{w}}{v_{r}+\sigma^{2}_{w}}$.
∎

######  Lemma 22.

Consider two independent Gaussian RVs: $Z\sim\mathcal{N}(0,\tau)$ and
$W\sim\mathcal{N}(0,1)$. Suppose $U=Z+\sigma_{w}W$ and $Y_{\sigma}\sim
P_{Y_{\sigma}}$, where $P_{Y_{\sigma}}\propto P_{U}\cdot P_{Y_{\sigma}|U}$ and
$P_{Y_{\sigma}|U}$ is an arbitrary distribution. Define
${Z}_{u}^{\perp}\overset{\scriptscriptstyle\Delta}{=}Z-\frac{\tau}{\tau+\sigma^{2}_{w}}U$.
Then, we have $Z_{u}^{\perp}\Perp(U,Y_{\sigma})$.

###### Proof.

It is straightforward to show $Z_{u}^{\perp}\Perp U$. Since $Y_{\sigma}$ is
generated from $U$, we also have $Z_{u}^{\perp}\Perp Y_{\sigma}$. ∎


# A Gentle Introduction to Scaling Laws in Biological Systems

Fabiano L. Ribeiro1* and William R. L. S. Pereira2$\diamond$

###### Abstract

This paper investigates the role of size in biological organisms. More
specifically, how the energy demand, expressed by the metabolic rate, changes
according to the mass of an organism. Empirical evidence suggests a power-law
relation between mass and metabolic rate, namely allometric law. For vascular
organisms, the exponent $\beta$ of this power-law is smaller than one, which
implies scaling economy; that is, the greater the organism is, the lesser
energy per cell it demands. However, the numerical value of this exponent is a
theme of an extensive debate and a central issue in comparative physiology. It
is presented in this work some empirical data and a detailed discussion about
the most successful theories to explain these issues. A historical perspective
is also shown, beginning with the first empirical insights in the sec. 19
about scaling properties in biology, passing through the two more important
theories that explain the scaling properties quantitatively. Firstly, the
Rubner model, that consider organism surface area and heat dissipation to
derive $\beta=2/3$. Secondly, the West-Brown-Enquist theory, that explains
such scaling properties as a consequence of the hierarchical and fractal
nutrient distribution network, deriving $\beta=3/4$.

1 Departmento de Física (DFI), Universidade Federal de Lavras (UFLA), Lavras
MG, Brazil;

2 Independent Researcher.

*<EMAIL_ADDRESS>
$\diamond$<EMAIL_ADDRESS>
##  I Introduction

When we hold a small hamster in our hands, we can feel its fast heartbeat,
with approximately 450 bpm. In turn, if we listen to the heartbeat of an
elephant, we will realize that the heart rate of this large mammal is
extremely slow, approximately 30 bpm. In fact, this higher heart rate of
hamsters reveals a higher metabolic rate of the rodent compared to that of the
elephant. But why is it greater? What does this mean and why does it happen?

The metabolic rate is the mean value of energy, per unit of time, used by an
organism to perform its vital functions. This energy is obtained from food,
water, air, light, etc. Figure (1) shows the log–log scale graph of the
empirical value of the metabolic rate $B$ (in watts) as a function of mass $M$
(in grams) of organisms of different taxonomic groups. The straight lines in
this figure show that all groups can be described individually by a power law
of the type

$B=B_{0}M^{\beta}.$ (1)

This equation was called the allometric equation by Huxley in 1932 HUXLEY1932
, where $B_{0}$ is the allometric constant and $\beta$ is the allometric
exponent. The straight lines in Figure (1) show a compatibility of the data
with the allometric law. This law covers 22 orders of magnitude (see fig.
(2)), from unicellular beings ($10^{−14}$ grams) to the largest mammals
($10^{8}$ grams), and it differs only in the values of the parameters for
certain taxonomic groups. The data present fig. (1) also show three different
regimes expressed by the scaling exponent: superlinear ($\beta>1$), linear
($\beta=1$) and sublinear ($\beta<1$).

Figure 1:  Metabolic rate as a function of body mass in different taxa, from
unicellular beings ($10^{-14}$ grams) to the largest mammals ($10^{8}$ grams)
(see also fig. (2)). A valid power law of the type $B=B_{0}M^{\beta}$
(straight line captures the trend of points) is observed for all taxonomic
groups. The parameter $B_{0}$ varies from group to group, and $\beta$ is
approximately constant and sublinear ($\beta<1$) in beings with a mass of
approximately $10^{-5}$g or higher. Protists have linear behaviour
($\beta\approx 1$), and bacteria have a superlinear behaviour ($\beta>1$). The
data were extracted directly from the references Allman1999 (homeotherms),
Hemmingsen1960 (homeotherms and poikilotherms), and shiftspnas2010
(prokaryotes and protists).  Figure 2:  Biological systems covers around 22
orders of magnitude, from unicellular beings to the largest mammals.

These facts show us that it does not matter how complex a species is, such as
in its physiology, its anatomy, or the different types of environments in
which it lives. In the end, all animals obey the same rule relating metabolic
rate with mass. In the present study, we are interested in investigating the
scaling laws behind this allometric equation. More specifically, we will
investigate the possible $\beta$ values in different types of animals, in
addition to offering explanations for these values.

The allometric equation was first perceived in 1839 by Saurus and Rameaux
Robiquet1839 . These researchers noticed this relationship when they realized
that the metabolic rate per weight decreases with increasing animal size.
Following Fourier’s lawRoselli2011 ; Ballesteros2018b , they proposed that the
metabolic rate should depend on heat dissipation by the organism. Thus, the
numerical value of the allometric exponent would be a natural response to the
release of heat by the organism and would make the relationship between the
surface area and volume of the organism valid. This idea led to a theoretical
exponent $\beta=2/3$. Further details regarding this deduction will be
presented in section (III).

At the end of the 19-th century, some experiments were performed to verify the
empirical value of the allometric exponent. For example, Rubner Rubner1883 ;
bertalanffy1957  studied dogs and in 1883 found that their energy production
per square metre of body surface is constant with the size of the animal,
which was evidence in favour of $\beta=2/3$. More careful experiments were
performed at the beginning of the 20th century. Among these studies, we
highlight the works of KroghKrogh1916  and that of Kleiber in 1932 (the best
known study) Kleiber1932 . From the data set analysed, an experimental value
of $\beta_{exp}\approx 3/4$ was observed, which differed from the theoretical
result that was accepted until then. It is just the controversy about the
scaling exponent different empirical values and theories that this work
focuses on.

The work is organized as follows. In the section (II), some empirical evidence
from the literature is presented, showing that the empirical value of the
scaling exponent is wildly inaccurate. This section also presents some groups
of theories proposed to explain such scaling properties. The section (III)
presents a model that explains the scaling properties as a consequence of heat
dissipation and organism surface area, namely Rubner model, deriving the
exponent $\beta=2/3$. In the section (IV), the theory developed by West-Brown-
Equisque based on the nutrient distribution network is presented. This theory
leads to an exponent $\beta=3/4$. The conclusion is presented in section (V).

##  II Different values of $\beta$

Currently, we know that organisms larger than $10^{-5}$g have sublinear
regimes (see Figures (1) and (3)), with some taxonomic groups best described
by $\beta=2/3$, while the vast majority are better described by $\beta=3/4$.
Figure (3), which shows the distribution of $\beta$ values for different
taxonomic groups with sufficiently large masses, shows the sublinearity of
this exponent among the analysed groups, but it also leaves doubts about the
exact value of this exponent.

The theoretical and experimental values of $\beta$ are a central issue in
comparative physiologyBallesteros2018b . Only to cite some of the many
examples in the literature, Dodds et al. dodds2001  show that rats smaller
than 10 kg are better described by an exponent of $2/3$, while for the other
sizes, the exponent of $3/4$ is better. Exponent $2/3$ well describes fish and
some types of invertebrates, such as crustaceans and mussels bertalanffy1957 ;
exponent $3/4$ fits better endothermic animals111Those that keep a constant
internal temperature. (birds and mammals) and reptiles. In contrast, organisms
with sizes between $10^{−10}$g and $10^{−5}$g, such as protists, exhibit a
linear behaviour of the allometric law shiftspnas2010 . Some types of insects
also obey this linear regime bertalanffy1957 . Finally, we have organisms
smaller than $10^{−5}g$, namely, bacteria, which exhibit superlinear behaviour
(see Figure (1)).

Figure 3:  Histograms referring to the number of species during ontogeny
(development) and the allometric exponent value for different taxonomic
groups. The histograms show the sublinearity of $\beta$ within these taxonomic
groups but also show uncertainty about its exact value. The data were taken
directly from Moses2008 ; Glazier2005b .

But what mechanisms lead to this scaling law, and why can different $\beta$
values occur? We can think of two types of explanatory approaches. The
cellular hypothesis suggests that the allometric law is the result of distinct
cellular properties in animals of different sizes. The scaling hypothesis, on
the other hand, suggests that this law would be the result of regulatory
factors of the organism as a whole.

Some empirical observations favour the scaling hypothesis. For example, in the
experiment described in Brown2007 , cells from 10 different mammals were
cultured in vitro, i.e., outside their original organisms. This study found
that all cells, regardless of the animal of origin, had the same metabolic
rate. However, note that the allometric equation (1) tells us that the
metabolic rate per cell $B/N$, where $N$ is the number of cells of the
organism, decreases with its size (because $M\sim N$, and then $B/N\sim
B/M\sim M^{\beta-1}$, with $\beta<1$). That is, when in vivo, cells of larger
organisms spend less energy than cells of smaller organisms, but outside their
original organisms, they all expend the same amount of energy. In this sense,
the cellular hypothesis loses strength in favour of the scaling hypothesis.
Other empirical examples in this direction can be found in West2002a ;
Ahluwalia2017  and in Figure (4). However, there are studies that support the
cellular hypothesis and argue that cells in vitro lose their allometry because
they are not performing their normal activities Glazier2014a .

Numerous studies have tried to explain the allometric law. Douglas S. Glazier
Glazier2014a ; Glazier2005b ; Glazier2018  classified four groups of theories
to explain it, which are based on:

  * •

Body surface area: The ratio between the total body area and the mass of an
organism would be the major determinant of its metabolic rate because heat
dissipation depends strongly on the body surface area. The mathematical
derivations of this theory lead to $\beta=2/3$; and in fact, some groups of
experimental data are compatible with this result. More details will be
presented in section (III);

  * •

Nutrient transport: Allometric law would be a consequence of the type of the
transport network that carries oxygen and nutrients to each cell of the body.
Natural selection has made the circulatory system of organisms as efficient as
possible, with the result that blood vessels decrease its diameter in a
hierarchical, fractal way to the lowest level (capillaries). This idea was
proposed by Geoffrey West, James Brown, and Brian Enquist (WBE)West1997 , who
predicted the theoretical value of $3/4$ for the allometric exponent. The
premises and results of this theory will be seen in detail in section (IV);

  * •

System composition: The theories of this group consider experimental findings
that reveal different allometric exponents for isolated organs. This has been
observed in the brain, heart, liver, kidneys, spleen, etc., and the behaviour
of the exponent of each organ is quite variedKestner1936 ; Darveau2002a ;
Glazier2014a ;

  * •

Resource demand: Allometric law would be a consequence of the energy demand of
the cells. This demand would decrease with the size of the organism. This idea
is based on the observation that the energy consumption per  in vivo cell
decreases with body mass ($\propto M^{-1/4}$), while the energy consumption in
in vitro cells does not depend on the mass of the original organism. This
absence of allometry in vitro would be because the cell is not performing its
proper routine activities (it would be in a quiescent state Glazier2014a ), so
it uses the minimum amount of energy necessary for its survival.

Figure 4:  Metabolic rate per cell ($B/M$) as a function of the organism mass
($M$), analyzed in two contexts: in vitro and in vivo. In in vivo cells, the
metabolic rate per cell decays with the mass of the organism obeying a power
law with exponent $-1/4$. In contrast, in vitro cells do not show any scale
relation, i.e. $(B/M)\sim M^{0}$ (horizontal line). Data was extracted
directly from West2002a .

Finally, what we can say about the allometric exponent is that the value $2/3$
is compatible with the theories that are based on heat dissipation, while the
exponent of $3/4$ is compatible with theories based on nutrient supply
networks. In this article, we will explore some theories to minimally explore
the possibility of explaining the scaling exponent quantitatively.

There are other empirical power laws, and consequently other exponents, that
relate certain biological variables, say $y$, with the mass of the organism.
These laws have the form $y\sim M^{\alpha}$, where $\alpha$ is a scaling
exponent, which in certain cases is numerically identical to the $\beta$
exponent of the metabolic rate, but which in other cases assumes values that
are multiples of $1/4$. That is the case when the variables in question are
related to the respiratory or circulatory systems. For example, heart rate is
related to body mass through an exponent of $\alpha=-1/4$, the time of blood
circulation through an exponent of $1/4$, etc. The fact that these exponents
are multiples of $1/4$ suggests that metabolism is the master determinant of
the other biological variables. In this sense, a change in the metabolic
exponent would lead to a systemic change in the other exponents in a kind of
cascade effect, as suggested in Darveau2002a . A non-exhaustive list of
variables related to the circulatory system and the respiratory system are
presented in the tables (1) and (2). They show the empirical and theoretical
values of the exponent $\alpha$.

$y$ (Circulatory system) |  $\alpha$ (predicted by WBE theory) |  $\alpha$ (empirical)
---|---|---
aortic radius ($r_{0}$) | $3/8=0.375$ | 0.36
aortic pressure | $0$ | $0.032$
blood velocity in the aorta ($u_{0}$) | $0$ | $0.07$
total blood volume ($V_{b}$) | $1$ | $1.00$
circulation time | $1/4=0.25$ | $0.25$
circulation distance ($\sum_{k=0}^{K}l_{k}$) | $1/4=0.25$ | not available
cardiac injection fraction | $1$ | $1.03$
heart rate | $-1/4=-0.25$ | $-0.25$
cardiac output | $3/4=0.75$ | $0.74$
number of capillaries ($N_{c}$) | $3/4=0.75$ | not available
capillary density | $-1/12=-0.083$ | $-0.095$
oxygen affinity in the blood | $-1/12=-0.083$ | $-0.089$
service volume radius | $1/12=0.083$ | not available
Krogh cylinder radius | $1/8=0.125$ | not available
peripheral resistance | $-3/4=-0.75$ | $-0.76$
Womersley number | $1/4=0.25$ | $0.25$
metabolic rate | $3/4=0.75$ | $0.74$
Table 1:  Quantities related to the circulatory system and their respective values (theoretical and empirical) of the $\alpha$ scaling exponent. The theoretical values were obtained from WBE theory, which will be presented in section (IV). Note that in some cases, this exponent is a multiple of $1/4$. The data in this table were extracted directly from West1997 ; Peters1983 ; K.Schmidt-Nielsen1984 . $y$ ( respiratory system) |  $\alpha$ (predicted by WBE theory) |  $\alpha$ (empirical)
---|---|---
lung volume | $1$ | $1.05$
respiratory rate | $-1/4=-0.25$ | $-0.26$
volume flow to lung | $3/4=0.75$ | $0.80$
interpleural pressure | $0$ | $0.004$
tracheal diameter | $3/8=0.375$ | $0.39$
air velocity in the trachea | $0$ | $0.02$
tidal volume | $1$ | $1.041$
dissipated energy | $3/4=0.75$ | $0.78$
number of alveoli | $3/4=0.75$ | not available
alveolar radius | $1/12=0.083$ | $0.13$
surface area of alveoli | $1/6=0.1666...$ | not available
surface area of the lung | $11/12=0.92$ | $0.95$
oxygen diffusing capacity | $1$ | $0.99$
total airway resistance | $-3/4=-0.75$ | $-0.70$
rate of oxygen consumption | $3/4=0.75$ | $0.76$
Table 2:  Quantities related to the respiratory system and its respective
values (theoretical and empirical) of the $\alpha$ scaling exponent. As in the
circulatory system, in some cases this exponent is a multiple of $1/4$. The
theoretical values were obtained from WBE theory, and the data in this table
were extracted directly from West1997 ; Peters1983 ; K.Schmidt-Nielsen1984 .

##  III Heat dissipation model

To stay alive, every organism must convert energy from nutrients into another
form of energy, which will be used in vital functions. According to the laws
of thermodynamics, processes such as this that convert energy from one form to
another must necessarily release heat. For an organism to stay alive, it must
eliminate/release this heat at the same rate at which it processes metabolic
energy. In this sense, the heat released by an animal can be understood as a
substrate of the energy transformation. In fact, one of the ways to quantify
the metabolic rate of an endothermic organism is to measure its heat release
rate Lighton2008 .

To begin to understand the consequences of this process of heat production
through energy transformation, we will consider two animals of very different
sizes, the mouse and the elephant. While a mouse has a mass of the order of
$2$g, the elephant has a mass of the order of $2,000,000$g (two tons); that
is, two species with a difference of six orders of magnitude in mass. In a
first approximation, we could consider that the elephant expends $10^{6}$
times much energy than the mouse. However, this reasoning has the consequence
that the amount of heat generated by the elephant would also be $10^{6}$ times
bigger than that of the mouse. Therefore, the elephant needs to eliminate all
excess heat so that it does not become too hot (which would lead to its
death). In fact, the elephant has a large contact surface with the external
world (much larger than the mouse), which allows it to dissipate much of the
heat produced. Then, the question we need to answer is: Would the size of this
surface be sufficient to dissipate the heat generated in the production of
metabolic energy?

Figure 5:  Rough approximation of the elephant as a sphere. This approximation
will allow us to estimate the volume ($\frac{4}{3}\pi r^{3}$) and the area of
the surface ($4\pi r^{2}$) of this animal.

To try to answer this question, we will consider a very simple model for the
contact surface of these animals. Let us consider (roughly) that both the
elephant and the mouse are almost spherical, as in the diagram of Figure (5).
With this approximation, it is much easier to estimate the volume ($V$) and
the area of contact with the external environment ($A$) without loss of
generality. These quantities are related to a single linear metric, the radius
$r$, through $V\approx\frac{4}{3}\pi r^{3}\sim r^{3}$ and $A\approx 4\pi
r^{2}\sim r^{2}$. These values are important for determining other properties
of the animal. For example, from the volume we can estimate the mass. After
all, the mass $M$ of the animal must be proportional to its volume, which
implies $M\sim r^{3}$. Suppose, for convenience, that the mouse has radius
$r=1$, given in any unit. Thus, for the elephant volume to be $10^{6}$ times
greater than that of the mouse, the elephant radius must be approximately
$r=100$. Of course, these values are speculative, but we are only interested
in the order of magnitude of these numbers; the lack of detail and precision
should not compromise the qualitative analysis we are interested in. With this
information, we can create Table (3) with the values related to mouses and
elephant.

The mass of the animal must be proportional to the volume, so mass of the
elephant, as well as its volume, must be $10^{6}$ times greater than the mass
of the mouse. However, the spherical surface area of the elephant is “only”
$10^{4}$ times the spherical surface area of the mouse (and not $10^{6}$
times, as is the case of volume). That is, the surface area increases with $r$
much more slowly than the volume does, which implies that larger animals have
relatively smaller surfaces. Quantitatively, we can verify, from the
relationships $A\sim r^{2}$, $V\sim r^{3}$ and $M\sim V$, that

$A\sim M^{\frac{2}{3}},$ (2)

which means that the surface area is related to the mass of the animal in a
sublinear way. In absolute terms, this means that larger animals have greater
contact surface area, but these animals have smaller contact area per unit of
mass compared to smaller animals.

This result leads us to conclude that the heat dissipation model does not
hold. The elephant generates $10^{6}$ times as much heat as the mouse but
radiates this heat on a contact surface of only $10^{4}$ times that of the
mouse. Thus, if the heat dissipation hypothesis were correct, the elephant
would be fully carbonized because it has a much smaller contact surface than
that required to dissipate all of the heat it produces mitchell .

Note that the reasoning in this section is not fully valid because we are
considering that the metabolic rate of animals is directly proportional to
body mass, which is not true. We will see this in more detail below by using
the model proposed by Rubner.

| mouses | elephants
---|---|---
$r$ | $1$ | $100$
$A$ | $\sim 1$ | $\sim 10.000$
$V$ | $\sim 1$ | $\sim 1.000.000$
$M$ | $\sim 1$ | $\sim 1.000.000$
Table 3:  Values relating to mouses and elephants in a any unit.

### The Rubner model

At the end of the 19th century, Max Rubner postulated that living organisms
evolved, by natural selection, to a state in which body mass should follow a
surface scaling law and thus be able to radiate excess heat. To understand
Rubner’s idea, consider the schematic graph of Figure (6), which follows the
principle of energy conservation and the second law of thermodynamics. Within
the metabolic process, energy from food ($E$), or any other source, is
transformed into: useful energy ($B$), that is, the energy that will be used
for the vital needs of the organism; and  heat ($Q$), which must be somehow
dissipated by the organism. Based on the conservation of energy principle,
$E=B+Q$.

Figure 6:  Scheme of energy transformation in organisms.

Rubner hypothetically considered that these three quantities ($E$, $B$ and
$Q$) scale with the mass in a similar way and obey the relationships

$E\sim M^{\beta},$ (3)

$B\sim M^{\beta},$ (4)

e

$Q\sim M^{\beta},$ (5)

in which $\beta$ is the allometric exponent. In the model considered in the
previous section, we hypothesized that metabolic energy was proportional to
the mass of the organism, i.e. $B\sim M$. That is, using the idea expressed in
that section, we were considering $\beta=1$.

To avoid the problem of carbonizing larger animals, the heat must be properly
dissipated. Let us consider that the heat $Q$ is composed of two parts: the
dissipated heat $Q_{diss}$ and the heat retained by the organism $Q_{ret}$,
and the conservation relationship is valid: $Q=Q_{diss}+Q_{ret}$. In addition,
the dissipated heat must be directly proportional to the contact surface of
the organism, i.e. $Q_{diss}\sim A$. We will then look at the ratio
$Q_{diss}/Q$, which will serve as a parameter to measure the efficiency of the
organism in dissipating heat. This ratio can give rise to extreme particular
cases:

$\frac{Q_{diss}}{Q}=\left\\{\begin{array}[]{ll}1&\Rightarrow\textrm{all heat
produced is dissipated;}\\\ 0&\Rightarrow\textrm{all heat produced is retained
(overheating).}\end{array}\right.$

As $A\sim M^{\frac{2}{3}}$ and, by hypothesis, $Q\sim M^{\beta}$, then

$\frac{Q_{diss}}{Q}=M^{\frac{2}{3}-\beta}.$ (6)

In the model proposed in the previous section, in which $\beta=1$, we have
$\frac{Q_{diss}}{Q}\sim M^{-\frac{1}{3}}$, which means that the dissipated
heat tends to zero for large $M$ (see graph (7)). This would cause overheating
in larger animals, as already discussed. However, as Rubner proposed, if
$\beta=\frac{2}{3}$ then $\frac{Q_{diss}}{Q}\sim M^{0}=1$; that is, this ratio
no longer depends on mass, the individual should not suffer from overheating
if it has a very large mass, as it can proportionally dissipate the same
amount of heat as the smaller animals.

This Rubner theory, known as the surface hypothesis, seems to be quite
coherent and was accepted for 50 years. The only problem with this theory is
that the allometric exponent predicted by it ($\beta=2/3$) is not consistent
with much of the experimental data, as verified by Max Kleiber in 1930.
$\beta_{exp}\approx 0.74\approx 3/4$ obtained from the set of data analysed by
Kleiber differs from the prediction given by the surface hypothesis. In
conclusion, although Rubner’s theory has coherent considerations, it is not
sufficient to describe the complexity of this scaling phenomenon.

We therefore need another model, another theory, that best describes the
empirical evidence. A theory that has the merit of quantitatively explaining
the exponent $3/4$ will be introduced in the next section. This theory was
developed by Geoffrey West, James Brown, and Brian Enquist, and originates
from the idea of fractal distribution networks of nutrients to the cells that
constitute the organism.

Figure 7:  Graph of the ratio $Q_{diss}/Q$ as a function of mass. For
$\beta=1$, the ratio and consequently the dissipated heat tend to zero for
large $M$, which in practice would mean overheating in larger animals. In
turn, if $\beta=2/3$, then $Q_{diss}/Q$ is independent of the mass of the
organism (horizontal line, representing $Q_{diss}/Q\sim M^{0}$ ). In this
case, the animal should not suffer from overheating if it has a very large
mass, as it can proportionally dissipate the same amount of heat as the
smaller animals.

##  IV Fractal Distribution Network Model

At the end of the 1990s, the theoretical physicist Geoffrey West and the
biologists James Brown and Brian Enquist proposed a model based on the idea of
an efficient distribution of nutrients inside an organism to explain the
allometric equation (1). This model, which we call  WBE theory, derives a
scaling law between metabolic rate and organism mass with exponent $\beta=3/4$
West1997 ; West1999 ; West2004 222 A critical and alternative review of this
model can be found in Savage2008 .. This theory has the merit of explaining
the scaling phenomenon in a quantitative way, starting from simple and
reasonable hypotheses. Before presenting this theory in detail, we will
briefly introduce the process of blood and oxygen circulation in an organism.

### Circulatory System

The circulatory system carries blood that contains all the material (glucose,
nutrients, oxygen, etc.) that each cell needs to perform its vital functions.
Blood distribution begins in the heart, a pulsating pump that ensures blood
flow throughout the body. From out of the heart comes a large-diameter vessel,
the aorta, which branches out following a hierarchy of decreasing diameter
(aorta → artery → arteriolar → capillary), seeking to reach all parts of the
body. The oxygen contained in erythrocytes (red blood cells) is transferred by
diffusion from the arterial capillary, the smallest circulatory unit, to each
of the cells. Soon after, the cell returns carbon dioxide to the same red
blood cell, which starts its return journey to the heart through the venous
vessels. The venous capillary gradually increases in diameter (venous
capillary → venula → vein) and returns to the heart.

### Respiratory System

From the heart, the blood saturated with carbon dioxide is directed to the
lungs. In the lungs, venous blood receives oxygen captured from air
inspiration. The erythrocytes again receive oxygen and release carbon dioxide
into the lungs. During the process of air inspiration, this same oxygen-filled
air travels inside tubes whose diameters gradually decrease (trachea → bronchi
→ bronchioles → alveoli). The alveolus functions as a chamber, and the entire
surface of each alveolus is surrounded by capillaries. Hence, the oxygen
molecules diffuse from the alveolus to the arterial capillary, and the carbon
dioxide exits the venous capillary and diffuses to the alveolus. The
concentration of carbon dioxide increases, and then expiration occurs.

### Hypotheses of the theory

Now that some fundamental characteristics of the circulatory and respiratory
systems have been presented, we will describe WBE theory. The theory is based
on three basic considerations (or hypotheses):

  1. 1.

Fractal distribution network: The nutrient distribution network, i.e., the
circulatory system, has a fractal branching pattern. The circulatory system
fills the entire volume of the body, carrying nutrients to each of its cells;

  2. 2.

Terminal units (e.g., cells and capillaries) do not vary with the size of the
organism: This hypothesis considers that the quantities related to the last
branch of the distribution network – the capillaries – do not vary in relation
to the body mass of the individual. These invariant quantities are, for
example, the size and mass of a cell and the length, area, and volume of
capillaries. Thus, these terminal units function as fundamental building
blocks in the construction of any type of biological organism. Some
experimental evidence supporting this hypothesis can be found in Savage2007 ;
Chan2010 ;

  3. 3.

Natural selection and energy minimization: Natural selection should favour a
distribution network that minimizes energy waste (Hamilton principle). An
inefficient network for nutrient transport should be eliminated by natural
selection.

Before analysing the consequences of these three hypotheses, let us also make
some considerations or assumptions:

  * •

Assumption 1: Total blood volume in an organism is proportional to the mass of
the organism;

  * •

Assumption 2: The metabolic rate is proportional to the blood flow through the
aorta of the organism.

Assumption 1 comes from empirical evidence (see Table (1)). This assumption
also arises from the following relationship: Given that the blood volume is
proportional to the volume of the organism and that the latter scales linearly
with mass, then the blood volume must be proportional to the mass of the
organism.

Assumption 2 is based on the idea that blood transports energy (in the form of
nutrients) to cells. As all nutrients and oxygen required for metabolic
processes are carried by the blood and always pass through the aorta, this
assumption occurs naturally.

###  IV.1  Modelling the distribution network

The nutrient distribution network (circulatory system) presents a fractal form
of branching, approximately as described in Figure (8) (upper). In a very
rudimentary way, we will suppose that these branches can be represented by the
stylized model described in Figure (8) (lower), where $k$ is an index that
represents the level of branching. Note that $k=0$ is the aorta level, while
$k=K$ is the capillary level, which implies that the network is formed by
$K+1$ branching levels. Each of the blood vessels at a given level branches
into $n$ smaller vessels. For example, in Figure (8) (lower), we have $n=2$.
That is, the number of vessels of a level has, in this particular case, two
times the number of vessels of the previous level. For convenience, $n$ will
be considered the same at all network levels. Level $k$ has $N_{k}$ vessels,
and this number can be determined from the previous level by the recurrence
formula

$N_{k+1}=nN_{k}\,.$ (7)

This implies, if $N_{0}=1$, that

$N_{k}=n^{k}$ (8)

holds.

Thus, the number of capillaries $N_{c}$ of this network, i.e., the number of
vessels in the $K$-th level, will be

$N_{c}\equiv N_{K}=n^{K}.$ (9)

Figure 8:  Upper: Fractal form of the branching of the nutrient distribution
network (circulatory system). Lower: Very simple model of the fractal form of
the circulatory system branching.

We will also model the shape of a typical blood vessel of this network. For
this, suppose that the vessels have a cylindrical shape, as shown in Figure
(9). This figure represents a blood vessel of the $k$-th level, where $l_{k}$
is the length of the vessel, $r_{k}$ is its radius, and $u_{k}$ is the mean
velocity of the blood inside that vessel. For example, we can calculate the
total distance of blood circulation ($l$) in the organism by the sum of
$l=\sum_{k=0}^{K}l_{k}$.

Figure 9:  Cylindrical representation of a blood vessel of the $k$-th level.

Just as the number of vessels can be written by a recurrence equation, we
assume that the length and radius of these vessels also can. In this sense,
one has

$l_{k+1}=\gamma l_{k},$ (10)

and

$r_{k+1}=\eta r_{k},$ (11)

where $\gamma$ and $\eta$ are the parameters that relate the subsequent
levels. Note that $\gamma$ and $\eta$ are less than 1 because the vessels of a
given level are always smaller than the vessels of the previous level
($l_{k+1}<l_{k}$ and $r_{k+1}<r_{k}$). However, the number of vessels at a
given level will always be greater than that at the previous level
($N_{k+1}>N_{k}\Rightarrow n>1$).

###  IV.2 Blood flow

In the appendix (A), we showed that the total volume of blood in the organism,
which we call $V_{b}$, can be written in terms of the parameters introduced
above by

$V_{b}\propto(\gamma\eta^{2})^{-K}.$ (12)

Consider also that $Q_{k}$ is the blood volume inside of the vessel of level
$k$, and then $\dot{Q}_{k}=\Delta Q_{k}/\Delta t$ represents the blood flow
rate flowing within this tube in a time interval $\Delta t$. This flow can
also be written as

$\dot{Q}_{k}=\frac{\textrm{blood volume in }k}{\Delta t}=\frac{(\pi
r_{k}^{2})\cdot(u_{k}\Delta t)}{\Delta t},$ (13)

where $\pi r_{k}^{2}$ is the cross-sectional area of the vessel and
$u_{k}\Delta t$ is its length ($l_{k}$).

Eq. (13) leads to

$\dot{Q}_{k}=\pi r_{k}^{2}u_{k}.$ (14)

As the fluid volume is maintained ($Q_{0}=N_{k}Q_{k}$, for any $k$), the Eq.
$\dot{Q}_{0}=N_{k}\dot{Q}_{k}$ must hold, which can be written in terms of the
quantities of the capillary level as

$\dot{Q}_{0}=N_{c}\dot{Q}_{c}=N_{c}\pi r_{c}^{2}u_{c}.$ (15)

Here, $Q_{c},\dot{Q}_{c},N_{c},l_{c},r_{c}$ and $u_{c}$ are relative to the
capillaries and therefore scaling invariant (by hypothesis 2). It is worth
adding that experimental observations in mammals suggest that $\dot{Q}_{c}$ is
the same for all species Savage2008 ; K.Schmidt-Nielsen1984 . We conclude from
Eq. (15) that

$\dot{Q}_{0}\propto N_{c};$ (16)

that is, hypothesis 2 implies that the blood flow in the distribution network
is linearly proportional to the number of capillaries in the organism.
Furthermore, by assumption 2, the above result leads us to conclude that the
metabolic rate and the number of capillaries scale linearly with each other,
that is

$B\propto N_{c}.$ (17)

From this result, we will derive the numerical value of the allometric
exponent in the next section.

###  IV.3  Deriving $\beta=3/4$

Now we are ready to make a prediction about the allometric exponent $\beta$
from the hypotheses and considerations that make up WBE theory. We have seen
by Eq. (17) that the metabolic rate is linearly related to the number of
capillaries, so given equation (9), we have $B\sim n^{K}$. In addition, by
assumption 1, we have $V_{b}\sim M$, which implies $n^{K}\sim M^{\beta}\sim
V_{b}^{\beta}$. Therefore, by result (12), we have

$n^{K}\sim(\gamma\eta^{2})^{-\beta K}.$ (18)

By taking the logarithm of the two sides of this relationship, we obtain

$\beta=-\frac{\ln n}{\ln(\gamma\eta^{2})}.$ (19)

This result tells us that it is enough to know the values of the constants
$n$, $\gamma$ and $\eta$ to calculate $\beta$. That is, the exponent of the
allometric law depends only on the constants of the fractal networks that form
the circulatory system. This result in itself is a great achievement because
it interprets the scaling law in a way completely different from the ideas
that permeated the explanations of this phenomenon for more than a century and
that were strictly based on the phenomenon of heat dissipation.

The theory goes even further because we can determine the numerical value of
$\beta$ if we consider hypothesis 3 about natural selection, which favours
distribution networks that maximizes efficiency and minimizes energy
expenditure. In fact, the network that minimizes the loss of nutrients during
transport should be the one with the lowest impedance and the one that
minimizes the reflection effects of propagation waves. The distribution
network that meets these requirements is the one that preserves the transverse
area from one vessel level to another, as shown in Figure (10). If there is
preservation of the area, then $A_{k}=nA_{k+1}$ is valid and so

$\pi r_{k}^{2}=n\pi r_{k+1}^{2}.$ (20)

Consequently, by inserting $\eta=r_{k+1}/r_{k}$ (see Eq. (11)) into the above
relationship, we obtain

$\eta^{2}=\frac{1}{n}.$ (21)

Figure 10:  Scheme of the vessels and the section that passes from one level
to another. In the image above, we have an ideal situation in which the cross-
sectional area is preserved in the passage from one level to another, i.e.,
$A_{k}=nA_{k+1}$. This prevents the blood flow being obstructed. In the image
below, in which there is no preservation of the transverse area, the flow is
partially interrupted and causes a break in blood propagation. This
inefficient type of distribution network should be eliminated by natural
selection.

This result serves to determine how the area, and consequently the radius, of
the vessels should vary between the different levels of the vascular network.
We will now see how the length of the vessels should vary between these
levels. For this, we must understand that the distribution network must be
configured to feed/serve each cell of the organism. In this sense, each of the
capillaries must feed a set of cells that fill a volume $V_{c}$, which we will
call the service volume. This volume should be such that if the organism has
$N_{c}$ capillaries, then its total volume should be $V=N_{c}\cdot V_{c}$. We
can repeat this argument for the veins at the level before the capillaries, so
that $V=N_{K-1}\cdot V_{K-1}$, where $V_{K-1}$ is the service volume of each
vein of this level, that is, the volume of capillaries that this vein should
serve. This argument can be repeated iteratively for the entire distribution
network so that the relationship

$V=N_{k}V_{k}=N_{k+1}V_{k+1}$ (22)

must be valid for any $k$. By writing the volume $V_{k}$ in terms of the
length of the vessel at this level $k$ (and by assuming its cylindrical shape
as in Figure (9)), the above recurrence equation becomes

$N_{k}\cdot\frac{4}{3}\pi\left(\frac{l_{k}}{2}\right)^{3}=N_{k+1}\cdot\frac{4}{3}\pi\left(\frac{l_{k+1}}{2}\right)^{3}.$
(23)

Inputting $n=N_{k+1}/N_{k}$ and $\gamma=l_{k+1}/l_{k}$ (Eqs. (7) and (10))
into the above equation yields

$\gamma=n^{-\frac{1}{3}}.$ (24)

Finally, by introducing the results (21) and (24) into equation (19), we
conclude that

$\beta=\frac{3}{4}.$ (25)

This result makes the WBE theory one of the most successful theories for
explaining scaling laws in biology. In fact, the result of this theory brought
a flavour of exact sciences to biology, in the sense of describing and
explaining the phenomena as a deductive consequence of some premises. In this
way, this theory inaugurates an era of systematic understanding of biology,
attempting to find general rules that are valid for a large number of
phenomena Cohen2004 ; ribeiro_tumor2017 ; Ribeiro2017 . Indeed, this is what
has happened in Physics for at least 300 years, and the results of this theory
show, albeit in a very modest way, that the life phenomenon can also be
understood more deeply.

###  IV.4 Number of capillaries

We can use this theory to determine how the number of capillaries scales with
the organism size. To do this, first consider that the metabolic rate is
directly related to blood flow (given assumption 2), so that

$B\propto\dot{Q}_{0}.$ (26)

By considering Eq. (16) and the allometric law ($B\propto M^{3/4}$), we have

$N_{c}\propto M^{\frac{3}{4}}.$ (27)

That is, according to the hypothesis and assumptions of this theory, it
follows that the number of capillaries must also obey an allometric law with
an exponent of $3/4$. However, this prediction has not been verified
experimentally. This result leads us to conclude that if this proposed theory
is valid, then the intuitive idea that the number of capillaries is linearly
proportional to the number of cells (and consequently to the mass) of the
organism is wrong.

In addition, the theory predicts an economy of scale because the bigger the
animal is, more cells are fed by one single capillary. That is, if $N/N_{c}$
is the average number of cells fed by one capillary, and $N\sim M$, then

$\frac{N}{N_{c}}\propto\frac{M}{M^{\frac{3}{4}}}=M^{\frac{1}{4}},$ (28)

which shows that $N/N_{c}$ (cell fed by one capillary) is an increasing
function of the organism size. This is an example of efficiency increasing
with size, in a similar way that happens with infrastructure in cities. In the
case of urban phenomena, the bigger the city is, the lesser per-capita
infrastructure it demands bettencourt2007growth ; ribeirocity2017 ;
joao_plosone2018 .

##  V Conclusion

This work intended to give a self-contained insight into the laws governing
the relation between some biological properties and organisms’ size,
particularly the metabolic rate and mass. We have presented a historical
perspective, passing through some data that suggest a power-law behaviour
between metabolic rate (and others metrics) and mass, namely allometric law.
We have seen strongly empirical support to this power-law relation, with
super-linear behaviour for prokaryotes, linear for protists, and sublinear for
vascular organisms. However, the exact numeric value for the scaling exponent
is quite uncertain, according to data.

Some theories to explain the allometric law’s sublinear behaviour
quantitatively were presented in details in this work. For instance, the
Rubner model, which is based on heat dissipation, comes up with a scaling
exponent $\beta=2/3$. It was the most accepted theory to explain the metabolic
rate and mass relation for more than 50 years. We then presented the WBE
theory, based on three primary premisses: i) fractal distribution network; ii)
terminal units do not vary with organism size; and iii) natural selection.
These premises lead to a scaling exponent $\beta=3/4$.

The ideas posted here illustrate science’s journey to understand, through a
mathematical theory, one aspect of life. Of course, we are still a long way
from reaching a level of mathematical description as one has today in physics,
for example. However, these theories and ideas gathered here show a giant leap
achieved in the last few decades towards a general theory that would explain
the phenomenon of life. However, of course, there is still a doubt whether
this general theory would, in fact, be achievable, given the great complexity
of biology. The next few years will bring us some information about this.

###### Acknowledgements.

This work was only made possible by the invaluable help of my student Victor
Cabral and Erika Aparecida Costa Lomeu with the figures presented. I also want
to thank the financial support from the Brazilian agencies CAPES (process
number: 88881.119533/2016-01) and CNPq (process number: 405921/2016-0).

##  Appendix A Total Blood Volume in the Organism

This appendix shows how to determine the total blood volume in the organism,
say $V_{b}$, in terms of WBE theory’s parameters. In fact, it will be shown
that

$V_{b}\propto(\gamma\eta^{2})^{-K}.$ (29)

In order to show this relation, consider $V_{k}$ as the blood volume within a
single blood vessel of the level $k$, which implies
$V_{b}=\sum_{k=0}^{K}N_{k}V_{k}$. As $V_{k}=\pi r_{k}^{2}l_{k}$ and
$N_{k}=n^{k}$ (for $N_{0}=1$), one has

$V_{b}=\pi\sum_{k=0}^{K}n^{k}r_{k}^{2}l_{k}.$ (30)

The ratio $\eta$ between radius of subsequent levels can be written as
$\eta=r_{c}/r_{K-1}$, which implies $r_{K-1}=r_{c}/\eta$,
$r_{K-2}=r_{c}/\eta^{2}$, and so on. One can then write the recurrence
relationship

$r_{k}=\eta^{-(K-k)}r_{c}.$ (31)

Similarly for the length of the vessels at the level $k$, one has the relation

$l_{k}=\gamma^{-(K-k)}l_{c}.$ (32)

This way of writing $r_{k}$ and $l_{k}$ is interesting because they are
written in terms of scale-invariant parameters ($r_{c}$ and $l_{c}$,
respectively). Returning to Eq. (30) one has

$V_{b}=V_{c}(\eta^{-2K}\gamma^{-K})\sum_{k=0}^{K}n^{k}\eta^{2k}\gamma^{k},$
(33)

where $V_{c}\equiv\pi r_{c}^{2}l_{c}$ is the volume of a capillary and,
therefore, scale-invariant. One can then write

$V_{b}\propto(\gamma\eta^{2})^{-K}\sum_{k=0}^{K}(n\eta^{2}\gamma)^{k}.$ (34)

Here, “$=$” was replaced by “$\propto$”, giving up the scale-invariant
parameters (constants).

The sum in equation above is indeed a geometric progression, with initial
value $a_{0}=1$, and common ratio $q=n\eta^{2}\gamma$. Knowing that the sum of
a finite geometric progression is $a_{0}(1-q^{K}+1)/(1-q)$, then

$V_{b}\propto(\gamma\eta^{2})^{-K}\left[\frac{1-(n\eta^{2}\gamma)^{K+1}}{1-n\eta^{2}\gamma}\right].$
(35)

Since $n$, $\eta$ and $\gamma$ are constant (by definition) and scale-
independent, the denominator term in the above equation can be omitted; that
is, we can write simply

$V_{b}\propto(\gamma\eta^{2})^{-K}\left[1-(n\eta^{2}\gamma)^{K+1}\right].$
(36)

Identifying $N_{c}=n^{K}$, one has $n^{K+1}=n^{K}n=N_{c}n$, which leads to

$V_{b}\propto(\gamma\eta^{2})^{-K}-N_{c}n\eta^{2}\gamma.$ (37)

Note that the second term to the right of the above equation has only constant
or scale-invariant quantities, and therefore one can write

$V_{b}\propto(\gamma\eta^{2})^{-K},$ (38)

demonstrating what was proposed at the beginning of this section.

One can also calculate the volume of blood in terms of the volume of blood in
the aorta. For this, the property

$r_{k}=\eta^{k}r_{0},$ (39)

$l_{k}=\gamma^{k}l_{0},$ (40)

and

$V_{0}=\pi r_{0}^{2},$ (41)

are used, which yield

$V_{b}=\sum_{k=0}^{K}N_{k}(\pi
r_{k}^{2})l_{k}=V_{0}\sum_{k=0}^{K}(n\eta^{2}\gamma)^{k}.$ (42)

Solving this geometric progression in a similar way to the previous one, one
arrives at

$V_{b}=V_{0}\left[\frac{1-(n\eta^{2}\gamma)^{K+1}}{1-n\eta^{2}\gamma}\right].$
(43)

It is the total blood volume in the organism, written in terms of blood volume
in the aorta.

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