Figure 1.
Illustration of a first-order Taylor linearization of a hypothetical population allometric equation for resting energy expenditure (REE): when
,
and
.
The linearization well approximates expected population REE given total values of body mass in the vicinity of
. Therefore, given ‘noisy’ sample data with mean
, the linear regression model
is an estimate of the first-order Taylor series
for the true population model
.
Figure 2.
Instantiation of the model for the contribution of individual organs and tissues to the y-intercept in linear regressions of resting energy expenditure (REE) on total body mass ().
Panels A through E depict individual organ-tissue contributions to REE scaled to in accordance with the approach and numerical values for allometric scaling coefficients units expressed in
reported in [27]. Note that the y scales differ. The REE for each organ-tissue is expressed as a first-order Taylor linearization at a specific body mass
of 0.03 kg (upper equation) of the parent allometric function (lower equation). Panel F reveals that the sum of the linearized equations equals total REE at
= 0.03 kg and very nearly equals total REE in the range 0.02≤
≤0.04 kg. The aggregate y-intercept (1.66) is the sum of the individual organ-tissue y-intercepts, while the aggregate slope (123.73) is the sum of the individual slopes. Note the particularly large contribution to the y-intercept and to whole-body REE by the liver even though it represents only ∼5% of
. Applying Eqs. 6 and 7 with
= 0.03 kg in the aggregate linear equation results in the parameters
= 60.58 and
= 0.69 of a single 2-parameter allometric equation for the whole-body EE curve. These parameter values are remarkably similar to those identified by standard log-log analysis or by non-linear regression (see text). To convert the units of a scaling coefficient to
, divide by
. To convert the slope of a Taylor series to units of
, divide by 1000; the intercept remains unchanged.
Figure 3.
Influence of the organ-tissue scaling exponent on an organ-tissue’s contribution to the positive y-intercept and to REE in linear models.
Panel A depicts the contribution to the y-intercept by the i-th individual organ-tissue REE in terms of , where
is the value of total body mass about which the Taylor linearization is performed. The contribution to the y-intercept is expressed as a multiplier of
calculated as
. The individual organ-tissue’s contribution to the y-intercept is maximized given a fixed numerical value of
when
is ∼0.70 for an animal with
= 30 g, as predicted by Eq. 12. Importantly, there is a substantial range of
values that more than double
. Panel B depicts the hypothetical effect of varying the
value on both the y-intercept and slope of the hypothetical liver REE –
relationship assuming that the allometric scaling coefficient
remains fixed at 0.36
(rescaled from 22.6
, the value reported by [27] and depicted in Figure 2A). Note that the sensitivity of the slope to variation in
suggests that group differences in the
of the liver, a small organ with a big impact on whole-body REE, could contribute to the problem of differing between-group slopes of whole-body REE in phenotyping studies.