© 2016. Published by The Company of Biologists Ltd | Journal of Experimental Biology (2016) 219, 2481-2489 doi:10.1242/jeb.138305
RESEARCH ARTICLE
Metabolic allometric scaling model: combining cellular
transportation and heat dissipation constraints
Yuri K. Shestopaloff*
Living organisms need energy to be ‘alive’. Energy is produced
by the biochemical processing of nutrients, and the rate of
energy production is called the metabolic rate. Metabolism is
very important from evolutionary and ecological perspectives,
and for organismal development and functioning. It depends on
different parameters, of which organism mass is considered to
be one of the most important. Simple relationships between the
mass of organisms and their metabolic rates were empirically
discovered by M. Kleiber in 1932. Such dependence is described
by a power function, whose exponent is referred to as the
allometric scaling coefficient. With the increase of mass, the
metabolic rate usually increases more slowly; if mass increases
by two times, the metabolic rate increases less than two times.
This fact has far-reaching implications for the organization of life.
The fundamental biological and biophysical mechanisms
underlying this phenomenon are still not well understood. The
present study shows that one such primary mechanism relates to
transportation of substances, such as nutrients and waste, at a
cellular level. Variations in cell size and associated cellular
transportation costs explain the known variance of the allometric
exponent. The introduced model also includes heat dissipation
constraints. The model agrees with experimental observations
and reconciles experimental results across different taxa. It
ties metabolic scaling to organismal and environmental
characteristics, helps to define perspective directions of future
research and allows the prediction of allometric exponents based
on characteristics of organisms and the environments they
live in.
KEY WORDS: Cellular transportation networks, Living organisms,
Metabolic rate, Morphology, Nutrients, Plants
INTRODUCTION
Consumption of energy by living organisms is characterized by a
metabolic rate, which in physical terms is power (amount of energy
per unit time). In the SI system, the unit of measure of energy is the
joule (J) and the unit of power is the watt (W). The issue of how
metabolic rate scales with the change in an organism’s mass is
of special interest for biologists (Schmidt-Nielsen, 1984), as it is tied
to many important factors influencing life organization – from
organismal biochemical mechanisms to evolutionary developments
to ecological habitats. The relationship of metabolic rate B and the
R and D Lab, Segmentsoft, 70 Bowerbank Drive, Toronto M2M 2A1, Canada.
*Author for correspondence ([email protected])
Y.K.S., 0000-0003-4251-0618
Received 28 January 2016; Accepted 1 June 2016
body mass M is mathematically usually presented in the following
form:
B / M b;
ð1Þ
where the exponent b is the allometric scaling coefficient of
metabolic rate (also allometric scaling, allometric exponent). Or, if
we rewrite this expression as an equality, it is as follows:
B ¼ aM b :
ð2Þ
Here, a is assumed to be a constant.
Presently, there are several theories explaining the phenomenon
of metabolic allometric scaling. The resource transport network
(RTN) theory assigns the effect to a single foundational mechanism
related to transportation costs for moving nutrients and waste
products over the internal organismal networks, such as vascular
systems (Brown and West, 2000; West et al., 1999, 2002; Savage
et al., 2004; Banavar et al., 2010, 2014; Kearney and White, 2012).
Another approach accounts for transportation costs of materials
through exchange surfaces, such as skin, lungs, etc. (Rubner, 1883;
Okie, 2013; see also Hirst, et al., 2014; Glazier, 2005, 2014). The
similar idea of ‘surface uptake limitation’ was applied to cells
(Davison, 1955). Dynamic energy budget theory, whose analysis
and comparison with RTN can be found in Kearney and
White (2012), uses generalized surface area and volume
relationships. Heat dissipation limit (HDL) theory (Speakman
and Król, 2010) and its discussion (Glazier, 2005, 2014; Hudson,
et al., 2013) suggests and provides convincing proofs that the
ability of organisms to dissipate heat influences their metabolic
characteristics. The metabolic-level boundaries hypothesis (Glazier,
2010) generalizes observations that allometric scaling (1) is
a variable parameter and (2) its value depends on the level of
metabolic activity in a U-shaped manner, with extrema
corresponding to low (dormant, hibernating) and high (extreme
physical exertion) metabolic activities.
The study of Darveau et al. (2002) adheres to an idea of multiple
factors contributing to the value of the allometric exponent,
considering different biomolecular mechanisms participating in
ATP turnover as the major contributors, together with a ‘layering
of function at various levels of organization’, such as different
scale levels. Allometric exponents found separately for each
contributing mechanism are combined as their weighted sum.
Using energy that is tied to particular biomolecular mechanisms as a
common denominator to unite different contributing factors is a
significant step.
Because the metabolic properties of organisms depend on many
factors and environmental conditions, the explanation of allometric
scaling should instead involve a combination of interacting
mechanisms. The hierarchy, ‘weight’ and triggering into action of
particular mechanisms should be defined by an organism’s
characteristics and interaction with its environment. At a systemic
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ABSTRACT
List of symbols and abbreviations
a
A
b
B
c
d
f
F
fl
h
Hdis
HDL
K1,2,3
kt
M
r
RTN
S
v
V
β
μ
constant in the formula B=aMb (kg1–b m2 s–3)
constant in the formula for finding the amount of nutrients
for transportation (m−1)
allometric exponent
metabolic rate (W)
cell enlargement
dimensionality of cell increase
density of nutrients per unit volume (kg m−3)
nutrients required for transportation (kg)
density of nutrients in a tube (kg m−1)
heat dissipation coefficient
dissipated heat (W)
heat dissipation limitation (theory)
scaling coefficients for transportation nutrients; index
denotes dimension
transportation costs as a fraction of transported nutrients (m−1)
mass (kg)
radius for disk and sphere (m)
resource transport network
surface area (m2)
cell volume (m3)
organism volume (m3)
a constant included in the scaling coefficient
cell mass (kg)
Journal of Experimental Biology (2016) 219, 2481-2489 doi:10.1242/jeb.138305
As in other works, the present study assumes that the amount of
nutrients that is required to support transportation networks is
proportional to the distance the substances travel along these
networks. Validity of this postulate was rather an axiomatic
proposition in previous works. In fact, several previous studies
(Shestopaloff, 2012a,b, 2014a,b, 2015; Shestopaloff and Sbalzarini,
2014) confirm this postulate through calculations of nutrient
influxes for microorganisms, and for dog and human livers.
Another indirect proof relates to stability of the growth period of
large and small organisms. It was shown previously (Shestopaloff,
2012a,b, 2014a,b) that if transportation nutrient expenditures are not
proportional to the traveled distance, then the growth periods of
large and small organisms would be substantially different, which is
not the case.
Determining the amount of nutrients required for
transportation
First, we consider organisms that grow or differ in size in one
dimension. We assume that substances are transported along the
tube. The density of nutrients per unit of length is fl (‘food’),
measured in kg m−1. Then, the mass of nutrients in a small tube’s
section with length dl is fl×dl. So, the amount of nutrients dF
required to transport the nutrients’ mass fl×dl by a distance l is:
dF ¼ kt lfl dl;
level, in the author’s view, heat dissipation (HDL) theory is of
importance. Temperature is directly tied to foundational-level
physiological and biochemical mechanisms, as they are very
sensitive to temperature.
The present study supports the multifactor nature of allometric
scaling. However, there are the following differences. (1) The
overwhelming majority of previous works promote an idea that
allometric scaling is due to systemic-level mechanisms. The present
study shows that cellular-level transportation mechanisms, although
coordinated at a systemic level, present a factor of primary
importance for metabolic allometric scaling. (2) Although the
cell’s size changes significantly less than the interspecific allometric
scaling across a wide range of animal sizes, it follows from the
present study that the transportation costs, when associated with the
cell size, present an important contributing factor into ontogenetic
differences in allometric scaling.
The present study introduces a general method for calculating
transportation costs. Then, it presents arguments in favor of the
cell size factor and associated transportation costs as important
mechanisms influencing allometric scaling, and introduces
formulas for calculating allometric exponents. Briefly, the HDL
theory is considered. Finally, an allometric scaling model
incorporating cellular transportation costs and heat dissipation
constraints is introduced.
MATERIALS AND METHODS
Nutrient distribution networks models were criticized as fairly
restrictive in applicability, problematic with regard to some initial
assumptions, and that they could not explain the known diversity
of metabolic scaling relationships (White et al., 2011, 2009,
2007; Weibel and Hoppeler, 2005; Glazier, 2005; Kozlowski and
Konarzewski, 2004). Here, I present a general method for determining
the amount of nutrients required for transportation needs, which uses
substantially less restrictive assumptions. The model is applicable both
to interspecific and intraspecific allometric scaling, although the
present study considers only intraspecific scaling.
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ð3Þ
where kt is a constant coefficient that represents the cost of
transportation as a fraction of the transported nutrients per unit
length (so that the unit of measure is m–1), and distance l is the
consequence of the postulate that transportation costs are
proportional to distance. Note that the cost of transportation of
nutrients F is also valued in nutrients and measured in kg.
Then, the total amount of nutrients required for transportation to
fill the tube of length L with nutrients is:
ðL
1
F ¼ kt lfl dl ¼ kt ft L2 :
2
ð4Þ
0
In other words, F is proportional to a square of the distance nutrients
are transported to. Assuming that the cross-sectional area of a
‘feeding pipe’ is S and the length is L, we find the volume as V=SL.
Expressing the length L=V/S and substituting this value into Eqn 4,
we obtain:
1
1
F ¼ kt fl L2 ¼ kt fl V 2 =S 2 :
ð5Þ
2
2
Similarly, we can find transportation costs for two- and threedimensional variations in size. Let us consider a disk increasing
in two dimensions, with constant height H. The disk has a
current radius r and maximum radius R. Then, we can write the
following:
dF ¼ kt f 2pr r Hdr ¼ 2pkt fHr2 dr;
ð6Þ
where f is the density of nutrients per unit of volume (kg m−3)
and kt is the cost of transportation, as in Eqn 3.
A note about integration limits: in cells, much traffic goes
from the periphery (membrane) and toward the inner regions, in
particular nuclei, and then waste and other substances go back to
the periphery and leave the cells. This justifies integration in a
radial direction, from zero to R (this assumption can be significantly
relaxed, which we proved mathematically, but do not present here).
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Doing integration, we find:
ðR
2
F ¼ 2pkt fHr2 dr ¼ pkt fHR3 :
3
ð7Þ
0
Certainly, the unit of measure of F remains kg. Taking into account
that the disk volume V=2πHR 2, and consequently R 3=V 3/2/(2πH )3/2,
we can rewrite Eqn 7 as follows:
kt fV 3=2
ð8Þ
F ¼ pffiffiffiffiffiffiffiffiffiffi :
3 2pH
In other words, transportation costs in a disk increasing in two
dimensions are proportional to 3/2 power of volume. It is interesting
to note that the amoeba, when it grows, also increases its nutrient
consumption in a manner similar to that expressed in Eqn 8
(Shestopaloff, 2012a, 2015).
The obtained result is valid for other forms increasing
proportionally in two dimensions, such as squares, rectangles and
stars, whose height remains constant.
Similarly, we can find transportation costs for organisms whose
size variation is due to proportional increase in three dimensions.
Let us consider a sphere as an example:
dF ¼ kt f 4pr2 r dr ¼ 4pkt fr3 dr:
ð9Þ
Taking into account that nutrient traffic goes mostly between the
periphery and the inner regions, we can do integration from zero to
R. Doing this, we find:
ðR
F ¼ 4pkt fr3 dr ¼ pkt fR4 :
ð10Þ
0
Because the sphere volume V=(4/3)πR 3, and consequently R 3=V/
(4π/3), we can rewrite Eqn 10 as:
F ¼ ð3=4Þ4=3 p1=3 kt fV 4=3 ¼ AfV 4=3 :
ð11Þ
Here, A=(3/4)4/3π–1/3kt is a constant (measured in m−1). Note that
this result ( proportionality to V 4/3 of transportation costs in a
sphere-like organism) is valid for other geometrical shapes that
increase proportionally in three dimensions.
The formulas above were obtained under the assumption that
cells are able to increase nutrient consumption to cover a faster
increase of transportation costs than increase of volume and,
associated with it, mass. Examples of such unicellular
organisms (amoeba, fission yeast Schizosaccharomyces pombe
and Saccharomyces cerevisiae) are presented in Shestopaloff
(2012a,b, 2015).
Transportation costs in organisms with different cell sizes
but the same mass
Let us compare nutrient transportation costs for a single cell
growing proportionally in three dimensions, with a grown volume V,
and two equal cells each with grown volume V/2, so that their total
volume is also V. The first organism, according to Eqn 11, requires
F1=AfV 4/3 nutrients for transportation. Two smaller organisms
correspondingly need:
F2 ¼ Af ðV =2Þ4=3 þ Af ðV =2Þ4=3 ¼ Af ð21=3 ÞV 4=3 :
Thus, the ratio F1/F2=21/3≈1.26>1, which means that a larger cell
needs more nutrients for transportation than two smaller cells
together with the same total mass. This is due to longer
transportation networks in the larger cell. Of course, a larger cell
may also compensate for the higher transportation costs by different
means.
Let us generalize this result for multicellular organisms with
equal masses M, but different cell sizes. We assume that
organisms differ proportionally in three dimensions. Densities
are constant and equal, so that volumes are equal too. Cell
compositions are accordingly assembled from cells with volume
v1 and v2, and v1=kv2. If masses of cells are accordingly μ1 and
μ2, it also means that μ1=kμ2. Then, using Eqn 11, we can find
the ratio K3 of transportation costs for these compositions of cells
as follows:
1=3
4=3
4=3
F2 Afv2 N2
Afv2 kN1
1
K3 ¼
¼
¼
¼
F1 Afv4=3 N1 Af ðkv2 Þ4=3 N1
k
1
1=3
m2
¼
;
ð12Þ
m1
where N denotes the number of cells and the index in K
refers to dimensionality. We took into account that N2=kN1,
because the masses of the cells’ compositions are equal
[indeed, N1=M/V1=M/(kV2), N2=M/V2, from which it follows
that N2=kN1].
Similarly, we can consider compositions of cells that grow in one
and two dimensions. Using Eqns 5 and 7 and performing similar
transformations as in Eqn 12, we obtain the following values for
ratios of transportation costs: K1=k and K2=k 1/2.
What does Eqn 12 mean? If we have two organisms with the same
mass, but composed of different sizes of cells, then to obtain the
same amount of nutrients per unit of mass, the organism with larger
cells requires more nutrients for transportation. However, in many
instances, organisms, regardless of their cell size, receive about the
same amount of nutrients per unit of mass because of the specifics
of nutrient distribution networks, such as the blood circulation
system. In this case, the transportation costs have to be scaled
according to nutrient availability. The question is, by how much?
There is no simple answer to this question, because it depends on the
organism’s biochemistry and specifics of nutrient distribution
networks, among other factors. However, we can be certain in two
things: (1) such scaling affects the total amount of nutrients; and (2)
the distribution of nutrients between transportation needs and other
biochemical activities (meaning the ratio) will remain, because
transportation costs represent an intrinsic factor to cells of a
particular type and size. In other words, the decrease of nutrients
used for transportation will proportionally reduce the total amount
of transported nutrients (of course, including the nutrients used for
transportation). The metabolic rate, in turn, is proportional to the
total amount of consumed nutrients. This last assumption is a
reasonable one, because we consider the same organisms, whose
metabolic processing efficiency per unit of nutrient mass should not
vary much.
Influence of cell size on metabolic rate – organisms with
different masses
Let us consider two compositions (1 and 2) of cells. Within each
composition, cells have the same size, but cell sizes between
compositions are different. As discussed in the previous section, the
metabolic rate B is proportional to the total amount of nutrients T,
which in turn is proportional to the number of cells, which in our
case also means proportionality to mass of a given organism. First,
we assume that the amount of nutrients used for transportation is
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RESEARCH ARTICLE
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scaled according to Eqn 12, that is by a factor (μ1/μ2)1/3:
1=3 B2 T2
m1
M2
¼
¼
:
B1 T1
m2
M1
ð13Þ
The masses of organisms are M1=μ1N1 and M2=μ2N2, where N1
and N2 represent the number of cells. When N2=N1=N, Eqn 13
transforms to:
B2 ¼ B1 ðm1 =m2 Þ1=3 ðM2 =M1 Þ
¼ B1 ðm1 =m2 Þ1=3 ½m2 N =ðm1 N Þ
¼ B1 ðm2 =m1 Þ2=3 ¼ B1 ðM2 =M1 Þ2=3 :
ð14Þ
In other words, the metabolic rate of organisms whose mass
increases solely because of the increase of cell size scales as a power
of 2/3 of the masses’ ratio.
When the number of cells is different, then:
B2 ¼ B1 ðm1 =m2 Þ1=3 ðM2 =M1 Þ
¼ B1 ðm2 =m1 Þ1=3 ½m2 N2 =ðN1 m1 Þ
¼ B1 ðm2 =m1 Þ2=3 ðN2 =N1 Þ
¼ B1 ðM2 =M1 Þ2=3 ðN2 =N1 Þ1=3 :
ð15Þ
Here, we took into account that M1=μ1N1 and M2=μ2N2.
According to Eqns 14 and 15, the size and number of cells are
important parameters defining metabolic scaling ontogenetically.
Allometric scaling much depends on relative size of the cells
composing the organisms. When body size correlates well with cell
size, the allometric exponent should be close to 2/3, which Eqn 14
shows. Eqn 15 presents an intermediate situation. When correlation
between the organism’s size and size of its cells is weak (cell size
remains the same), we should expect isometric scaling (b=1).
Indeed, assuming μ1=μ2=μ, we obtain from Eqn 15 isometric
scaling:
B2 ¼ B1 ðm2 =m1 Þ2=3 ðN2 =N1 Þ ¼ B1 ðN2 =N1 Þ
¼ B1 ðmN2 =mN1 Þ ¼ B1 ðM2 =M1 Þ1 :
ð16Þ
What would happen if scaling of transportation costs is different
from (μ1/μ2)1/3, as defined by Eqn 12? The last value, representing
an intrinsic factor, as we said already, will remain the same. We also
admitted that the total amount of nutrients possibly can be scaled.
Let us denote such a scaling as β. Then, the scaling coefficient is
β(μ1/μ2)1/3. Substituting this into Eqns 13–16, we obtain the same
expressions, multiplied by a coefficient β. However, this will not
change our results with regard to the allometric exponent. For
instance, in the case of Eqn 15 we obtain:
B2 ¼ bB1 ðm1 =m2 Þ1=3 ðM2 =M1 Þ
¼ bB1 ðM2 =M1 Þ2=3 ðN2 =N1 Þ1=3 :
ð17Þ
In allometric studies, such dependencies are usually represented
in a logarithmic scale. In this case, Eqn 17 becomes as follows:
2
1
ð18Þ
ln B2 ¼ ln b þ ln B1 þ lnðM2 =M1 Þ þ lnðN2 =N1 Þ:
3
3
We do not know how much the parameter β can vary. At a first
glance, there are no reasons for it to significantly change depending
on the state of the organism (e.g. torpid or physically active), but the
issue requires further study. In the case when β≈const, Eqn 18
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complies with numerous experimental observations. Indeed, the
position of regression lines often varies vertically for different
organisms, although the allometric slopes can be the same.
According to Eqn 18, the reasons could be different values of
scaling coefficients (lnβ) for different organisms, and different
values of lnB1. When the number of cells remains the same (the
cell’s mass increases proportionally to the organism’s mass), the
fourth term (1/3)ln(N2/N1) also becomes a constant. If an
organism’s mass increases as a result of both cell enlargement
and cell number, then this fourth term provides a line with a nonzero slope, which, summing with the line defined by the third term
(2/3)lnB1(M2/M1), adds to the value of the allometric exponent 2/3,
thus providing the range from 2/3 to 1.
Note that the fraction of nutrients (from the total amount of
nutrients) used for needs other than transportation will be smaller in
organisms with larger cells than in organisms with smaller cells.
Thus, fewer nutrients may be available to support other cell
functions, for instance, DNA repair mechanisms, whose
inefficiency may cause transformation to cancerous cells. It is
well known that small cells generally are significantly more resistant
to cancerous transformations. However, once such a transformation
occurs, they represent the most malicious cancer forms, which are
very difficult to cure. Although speculative at this point, this is an
observation worth exploring.
The obtained results from the present study agree with and
explain known empirical facts. Larger organisms ontogenetically
often have larger cells. In Chown et al. (2007), seven of eight ant
species increased their size primarily through the enlargement of
cells, but not through the number of cells. When this is the case,
then, according to Eqn 14, the allometric exponent is close to 2/3.
Indeed, in Chown et al. (2007), the authors found for ants that ‘…
the intraspecific scaling exponents varied from 0.67 to 1.0.
Moreover, in the species where metabolic rate scaled as mass1.0,
cell size did not contribute significantly to models of body size
variation, only cell number was significant. Where the scaling
exponent was <1.0, cell size played an increasingly important role in
accounting for size variation.’ Also, the authors confirmed that
when the cell size remains the same, the metabolic allometry is close
to isometric (which is the case in Eqn 16). So, the present results
about the role of cellular transportation costs in metabolic scaling
comply with all experimental observations of Chown et al. (2007).
Therefore, the way organisms increase their size during ontogeny
– that is, either through the increase of cell number or cell size, or a
combination of both – explains the variability of metabolic scaling
from 2/3 to 1, with a value of 2/3 corresponding to an increase by
cell enlargement, and a value of 1 corresponding to mass increase by
number of cells.
If we repeat computations similar to Eqns 12–16 for a twodimensional increase (K2=k 1/2), then the range for the allometric
exponent will be 1/2–1, and for a one-dimensional increase, 0–1. In
the last case, these have to be some exotic, if not hypothetical,
organisms composed of tube-like cells of the same width fed from
ends.
In addition to the described results from Chown et al. (2007),
previous works (Glazier, 2005, 2014) present data from different
sources that support the present findings both for the boundary
values and for transitional growth scenarios.
First, regarding organisms increasing their size by cell
enlargement: (1) nematodes have an allometric exponent b close
to 2/3 [four species, b=0.64–0.72; one species, b=0.43; one species
(low food levels), b=0.77; mean value of b for seven nematode
species is 0.677]; (2) rotifers have b close to 2/3; (3) in the later
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stages of growth, mammals increase their size through cell
enlargement, and have a smaller allometric exponent; and (4) in
the later development phase, when increase is due mostly to cell
enlargement, fishes have smaller allometric exponents.
And second, regarding organisms increasing size by cell number:
(1) small metazoans show near-isometric metabolic scaling; (2)
squid grow throughout their entire life by increasing cell numbers,
and have isometric scaling; (3) in the early stages of growth,
mammals increase their size through cell number, and have a greater
allometric exponent; and (4) fishes in the early development phase,
when increase is due to increase in cell number, have a greater
allometric exponent.
Finally, let us consider a scenario when the cells of a larger
organism are smaller than in a small organism, that is, when μ2<μ1.
Transforming Eqn 15, we obtain the following:
B2 ¼ B1 ðm1 =m2 Þ1=3 ðM2 =M1 Þ ¼ B1 ðM2 =M1 Þ1þa ;
ð19Þ
where (μ1/μ2)1/3=(M2/M1)α and, because μ2<μ1, α>0, so that the
allometric exponent is greater than 1.
Representing allometric scaling relationships
Let us compare two forms of representation of metabolic allometric
scaling relationships – Eqn 2, and the one we used in our
derivations:
B ¼ B0 ðM2 =M0 Þb ;
ð20Þ
which is effectively the same as Eqn 2, if we assume a ¼ B0 M0b .
Here, M0 is some reference mass with associated metabolic rate B0.
The allometric exponent b can be variable, as we found above.
When mass is measured in absolute values, such as in Eqn 2, the
constant a will also be measured in variable units of kg(1–b) m2 s–3
(because the units of measure of the left part in Eqn 2 can be
represented as J s–1=kg m2 s–3). The physical meaning of a for such
units of measure is not easy to define. In this regard, the approach in
Eqn 20 is more comprehensible, because B0 is always measured in
watts.
Mathematically, a reference point (B0,M0) can correspond to any
mass. It can be shown as follows. Suppose we want to substitute
another reference point with index 1. Because B1=B0(M1/M0)b, we
can write:
B ¼ B0 ðM =M0 Þb ¼ B1 ðM0 =M1 Þb ðM =M0 Þb ¼ B1 ðM =M1 Þb ; ð21Þ
which is the same as Eqn 20.
RESULTS
Thermogenesis and heat dissipation – amount of dissipated
heat
Allometric scaling includes two opposite processes existing in
inseparable unity: one is energy supply, and the other is energy
dissipation. Constraints imposed on both sides of this energy balance
are considered as possible causes of allometric scaling. Temperature
variations strongly affect biochemical processes (Hedrick and
Hillman, 2016). Could thermogenesis, heat dissipation constraints
and related changes of body temperatures be the major mechanisms
responsible for the allometric scaling? Such views were considered
in previous works (Speakman and Król, 2010; White et al., 2006;
Hudson, et al., 2013). In the first work, the authors state: ‘We contend
that the HDL is a major constraint operating on the expenditure side
of the energy balance equation’.
The heat dissipation is done via heat conduction, convection and
radiation. Dissipated heat is usually accounted for using heat
transfer equations, applied to certain expanding geometrical forms
modeling bodies of different size. Such is the method introduced in
Speakman and Król (2010). It requires evaluation of parameters
related to an insulating layer and its thermal conductivity,
accounting for the difference between the core body temperature
and the surface temperature. In addition, the complex relationships
between the size of the body and the size of its limbs have to be
taken into account. Despite of all these complexities, the authors
created a workable model. They found the value of the scaling
exponent to be 0.63, considering it rather as underestimated value.
Heat dissipation through radiation is usually assumed to be small
compared with heat conduction and convection. For example, in
Wolf and Walsberg (2000), the authors estimated that radiation
accounts for approximately 5% of heat losses for birds with
plumage. The result is not directly transferable to animals without
plumage, and for such animals probably more studies are needed.
One possible approach to address the issue could be using the
Stefan–Boltzmann equation (or Planck’s formula) for the total
energy emitted by an absolutely black body (Shestopaloff, 1993,
2011). ‘Absolute blackness’, that is, when the reflection coefficient
is equal to zero, is a reasonable approximation, because most energy
at normal body temperatures is emitted in the infrared spectrum, for
which reflection coefficients of living organisms and natural
surfaces are small, usually less than 0.02 (Vollmer and Mollman,
2010). Living organisms both emit and absorb energy, so we should
find the difference, not the total energy.
The human body can sustain the temperature difference of few
degrees Celsius with the surrounding environment for a long time.
Let us find the power difference D of emitted and absorbed
radiation, for a 2°C difference, assuming the outer temperature of a
body is 35°C, which approximately corresponds to 308 K:
4
4
D ¼ sðTbody
Tenv
Þ ¼ 6:57108 ð3084 3064 Þ
15:02 W m2 :
ð22Þ
Here, σ is a Stefan–Boltzmann constant. Calculating the total
amount E of required energy for a time period t=24 h for an average
human being with the body surface S=1.5 m2, we find:
E ¼ D t S 15:02 1:5 24 3600
1:947 106 J:
ð23Þ
According to Human (2001), an average human requires
approximately 9×106 J a day. The obtained value is
approximately 20% of this amount. In other words, the heat
dissipation through radiation can be of the order of heat losses
through conduction or convection.
Overall, heat dissipation should be considered as an influential
factor affecting allometric scaling. If the heat dissipation constraints
affect allometric scaling, then this should affect organisms living at
colder temperatures. Indeed, such higher energy and increased milk
production at colder temperatures were confirmed for mice
(Speakman and Król, 2010).
If all organisms had the same geometry, then the allometric
scaling due to heat dissipation constraints would be proportional to
surface area, that is to volume2/3. Organisms have different shapes.
For instance, elongation increases the body surface more than
volume, which can increase the allometric exponent. Besides, HDL
may have different intensity of expression in different organisms.
2485
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RESEARCH ARTICLE
Journal of Experimental Biology (2016) 219, 2481-2489 doi:10.1242/jeb.138305
So, in the real world, the HDL effect should provide a range of
values of allometric exponents.
Distribution of transportation costs across different scale
levels
Organisms transport nutrients and their by-products at different
scale levels, such as the gastrointestinal tract, the blood supply
system, the extracellular matrix, intracellular pathways, etc.
Different needs are supported by different transportation systems.
Supporting blood supply is different from transportation at the
cellular level. Both require energy, both have certain scaling
mechanisms, and so it is important to know how much energy is
spent for transportation at each scale level. Some works suggest that
the transportation costs in nutrient distribution networks are
responsible for much of the metabolic allometric scaling (West
et al., 2002; Banavar et al., 1999), of which the blood supply system
is assumed to be a major contributor. Previous work (Banavar et al.,
2010) also considers vascular networks as major contributors to
allometric scaling. According to the authors, ‘an exponent of 3/4
emerges naturally … in two simple models of vascular networks’. In
contrast, some studies, in particular Weibel and Hoppeler (2005), do
not support this idea. However, no estimation of energy
consumption by different systems was done. Let us do this.
The heart of an adult human pumps approximately 1 to 18 liters
of blood per minute, depending on the individual and the level of
physical exertion, with an average value of 3–4 l min−1. At rest or
during moderate physical activity, the blood pressure changes from
diastolic to systolic in the range of 7 to 13 cm (mercury). So, we can
assume that the heart’s ‘pump’ produces work equivalent to lifting a
certain volume of mercury to the height of 6 cm. Taking into
account that the specific weight of mercury is approximately
13.5 g cm−3, it will be equivalent to lifting the same weight of blood
to the height of 13.5×6=81 cm=0.81 m (as the blood’s specific
weight is ∼1 g cm−3). Because the systolic blood pressure is
maintained for only a fraction of the whole cycle (let us assume, half
of the cycle), then we should accordingly reduce the height to
approximately 0.4 m. Then, the work w done by the heart’s ‘pump’
during the day (24 h) can be found as:
w ¼ M gH:
ð24Þ
Here, g is the free-fall acceleration equal to 9.8 m s−2, H is height
(m) and M is mass (kg). For a blood volume from 1 to 18 l min−1,
this gives a range from 5645 to 101,606 J.
According to a previous report (Human, 2001), the daily energy
requirements for an adult of 17–18 years old with a weight of 68 kg
is 14.3 MJ. Thus, the average energy (and accordingly food)
required for the heart is approximately 0.14% of the overall average
energy consumption, for 100% heart efficiency. In reality,
efficiency is less, so that the actual energy expenditures could be
few times higher, according to Schmidt-Nielsen (1984). Blood
pressure during physical exertion can elevate as well, by a few
centimeters of mercury, but neither of these adjustments would
change our results substantially. In any case, the value is very small
compared with the total energy production by a human.
However, for our purposes, it is important to know how these
energy costs relate to the total transportation costs, but not to the
total energy production. Shi and Theg (2013) state, ‘We estimate
that protein import across the plastid envelope membranes
consumes ∼0.6% of the total light-saturated energy output of the
organelle’. Note that this is only one of many transport pathways
working in cells. Nonetheless, its relative value is more than the
2486
energy required for the propulsion of blood by a heart. In an
interview, answering a question about the secretory and signal
recognition particle pathways, one of the authors, Theg, says: ‘But if
we do make such a back-of-the-envelope extrapolation, we can
estimate that the total energy cost to a cell for all its protein transport
activity may be as high as 15 percent of the total energy
expenditure’. Another example of high transportation energy costs
in Escherichia coli (but in ATP molecules) is presented in Driessen
(1992). Note that cells carry on other transport-related activities
besides protein transport.
Of course, more such studies are needed to obtain a
comprehensive understanding of the structure of cellular
transportation costs. (Hopefully, this article will initiate interest in
such studies.) Nonetheless, based on the presented data, we can say
that the total transport expenditures in living organisms at the
cellular level significantly (tens of times) exceed the amount of
energy required for functioning of the blood supply system.
Consequently, we are left with the only possible inference that the
overwhelming fraction of transportation costs is due to cellular
biomolecular activity. Thus, if the transport networks are the factor
that significantly contributes to the phenomenon of metabolic
scaling, then it should be the cells’ transportation costs that
contribute the most.
DISCUSSION
Cellular transportation costs versus the cellular surface
hypothesis
In Davison (1955), the author introduced an idea that the size and
number of cells are limiting the metabolic rate by restricting nutrient
and waste fluxes through the cells’ surface. He obtained the range of
metabolic scaling of 2/3–1, where a value of 2/3 corresponds to an
increase in organism size that is due to cell enlargement, and a value
of 1 corresponds to growth that is entirely due to an increase in cell
number. The idea about the relationship between the allometric
exponent and cell size and number was also explored in Kozlowski
et al. (2003). The authors argue that the allometric scaling of
metabolic rate is a by-product of evolutionary diversification of
genome size, which ‘underlines the participation of cell size and cell
number in body size optimization’. The range of the exponent’s
values and relationships obtained in the cited article are similar to
Davison’s results. Numerically, the same values were obtained in
the present study. However, here the similarities end. In essence, the
‘cell surface’ approach and that of the present study (considering
cellular transportation networks) are very different. On the positive
side, this numerical similarity means that all empirical observations
supporting the main idea in Davison (1955) and Kozlowski et al.
(2003) support the results of the present study as well.
Regarding Davison’s assumption about the restrictiveness of cell
surfaces to transfer nutrients and waste products, it was shown
experimentally in Maaloe and Kjeldgaard (1966) that
microorganisms can increase nutrient flux through the membrane
by several folds instantly, once nutrients become available. Also, it
was found (Shestopaloff, 2014b, 2015) that during normal growth in
a stable nutrient environment, the amoeba increases nutrient
consumption per unit of surface by approximately 68%, while
fission yeast increases consumption by 340%. In other words, the
cell membrane capacity of these organisms has several-folds reserve
to accommodate the sharp increase of nutrient and waste fluxes
when in need. It does not mean that the surface cannot be a factor
affecting allometric scaling. Rather, it means that the surface area is
unlikely to be the universal factor for this phenomenon we are
looking for.
Journal of Experimental Biology
RESEARCH ARTICLE
RESEARCH ARTICLE
For completeness, we should mention the surface area
hypothesis, considered in Hirst et al. (2014) and Glazier et al.
(2015). The authors related surface area to allometric exponents
using pelagic invertebrates (they change their body shape during
growth, which gives a wider range of geometrical forms for testing
the surface area hypothesis). The authors showed that the RTN
method produces invalid results for these animals, and so they
concluded that the RTN model does not explain the metabolic
scaling phenomenon. However, this statement is true only for global
transportation networks, such as blood supply systems, whose
energy consumption, as we found, is small. However, if we consider
cellular transportation costs, then our results agree with the
presented experimental data, provided cell size in pelagic
invertebrates does not change much. Change of cell size during
ontogeny should be known in order to definitively answer the
question.
Journal of Experimental Biology (2016) 219, 2481-2489 doi:10.1242/jeb.138305
present study discovered that nutrient acquisition and delivery form
the primary level in the hierarchy of different factors contributing to
allometric scaling. Other factors can modify, to a certain extent, the
allometric exponent defined by cellular transportation mechanisms,
but not override it. One of the reasons is that nutrient acquisition and
transportation, considered as a single process, form the foundation of
any metabolic activity in any living organism.
Such modifications, in particular, can relate to dynamics of
specific biomolecular energy mechanisms considered in Darveau
et al. (2002). On a practical note, Eqn 17 can accommodate such
modifications, including the ones that are due to different energetic
regimes, through the parameter B1 (and possibly β, but that depends
on the particular organism). If that is not enough, still, the model has
lots of flexibility to adapt inputs from other contributing factors. So,
here we also see that cellular transportation mechanisms and their
formal description allow integrating inputs from different
contributing factors related to energy production and utilization.
Cellular transportation and contributing energy pathways
Integrated model of allometric scaling
Here, we propose the model of allometric scaling, which includes
the following interacting mechanisms: Energy consumption by
cellular transportation networks, organismal maintenance and
biomass synthesis metabolism, and heat dissipation (described by
HDL theory). All of them present real influential physiological and
biochemical mechanisms.
There are many experimental examples that the metabolic activity
of unicellular organisms and cells in tissues strongly depends on
temperature. For example, the average growth time of fission yeast
S. pombe at 25°C is 269 min, while at 28°C it is 167 min
(Shestopaloff, 2015). In Glazier (2010), the author summarizes
results obtained in Ivleva (1980) for crustaceans as follows: ‘the
scaling exponent … decreases significantly with habitat temperature
(b=0.781±0.016, 0.725±0.086, 0.664±0.026 at temperatures of 20,
25 and 29°C)’.
Lower body temperatures correlate with low metabolic activity.
So, in this case, heat dissipation does not impose meaningful
constraints, and, if cell size is approximately the same for small and
larger animals, the allometric exponent should be close to 1. Indeed,
deeply hibernating animals have an allometric exponent close to 1
(Glazier, 2010; Hudson et al., 2013). However, when food supply is
not limited, then energy consumption is high, which apparently
triggers heat dissipation limits, reducing the allometric exponent
to 2/3.
Ectothermic animals produce heat as a by-product of muscle and
other biochemical activity, while endothermic animals have a
separate temperature regulating system, whose input is added on top.
This additive effect apparently increases the body temperature of
endothermic animals more than in ectothermic animals, so that HDL
trigger earlier for them. This, accordingly, reduces the allometric
exponent of endothermic animals compared with ectothermic
animals – an effect that many experimental observations confirm
(Glazier, 2005, 2014).
High levels of metabolism (for instance, during strenuous
physical activity) increase body temperature, which in turn
accelerates metabolic reactions and triggers more energyproducing mechanisms, such as anaerobic mechanisms. The need
to dissipate more heat is met by triggering different mechanisms,
such as sweating, and by auxiliary means, such as incoming air or
water flow during movement. The raising of body temperature
during physical activity increases the temperature contrast between
the body and the surrounding environment, which increases heat
dissipation. In addition, organisms have some tolerance to elevated
2487
Journal of Experimental Biology
Previously, we briefly discussed the approach proposed in Darveau
et al. (2002). It considers different supply and demand energy
pathways, whose allometric exponents are then combined in order to
calculate the allometric exponent for the entire organism. Such
factorization on the basis of real biochemical mechanisms is an
important step forward. However, the approach still does not give an
answer to what is the fundamental cause of allometric scaling as such.
In contrast, cellular transportation represents a general mechanism
across all living species, which defines the amount of nutrients
delivered to and consumed by organisms. Nutrient influxes for all
individual biochemical contributors, including the ones participating
in energy production and utilization, represent parts of the overall
nutrient influx. We can think of them as branched influxes distributed
from the main node, which is the total influx. The total influx, as we
found, depends on how much of the nutrients can be used for
transportation relative to other biomechanical activities. Indeed, no
delivery equals no nutrients. In this regard, cellular transportation
represents the primary mechanism, which defines the amount of
nutrients available for other biochemical activities.
What the present study found is that the specifics of cellular
transportation mechanisms, which is faster than the linear increase of
transportation costs when cells enlarge, caps the total amount of
acquired nutrients (Eqns 16 and 17) regardless of nutrient usage. It is
like when a host offers a plate of cookies to their guests, and then the
guests distribute the cookies between themselves according to their
needs and adopted social practices. However, together, the guests
cannot consume more cookies than the host offers. Fewer cookies
means a lower average number of cookies per guest. The same
situation occurs with nutrients that are brought by and under the
management of nutrient transportation mechanisms, which in this
analogy represent the host. So, the idea of combining different factors
contributing to allometric scaling through energy production very
well matches the cellular transportation mechanism affecting the
total amount of consumed nutrients and consequently organism's
metabolism. In fact, these two approaches complement each other.
The concept of combining different contributing factors also
addresses the variability of the allometric exponent that is due to
variations in the energy regime, such as the difference between the
basal and maximal metabolic rates. The model, which will be
presented in the next section, actually provides variability also
through the heat dissipation constraints, in many instances linked to
biochemical mechanisms through the temperature changes. However,
the main connection between contributing factors and cellular
transportation costs with regard to variability is the following. The
Journal of Experimental Biology (2016) 219, 2481-2489 doi:10.1242/jeb.138305
1.000
Heat dissipation coefficient h
0
Allometric exponent b
0.933
0.2
0.867
0.4
0.800
0.6
0.8
0.734
0.667
1.0
0
0.2
0.4
0.6
Cell increase c
0.8
1
Fig. 1. Dependence of an organism’s allometric exponent on how cells
contribute to the organism’s mass increase, and on the organism’s heat
dissipation regime. Parameter c defines to what extent the organism's mass
increases as a result of cell enlargement, and to what extent the mass grows as
a result of increase in cell number. Dependence on mass increase of threedimensional organisms (through cell enlargement or/and increase of cell
number), and on the organism’s heat dissipation regime.
body temperatures. Taken together, all of these adaptation
mechanisms support the ability of animals to dissipate heat at
maximum metabolic activity, which results in greater values of
allometric exponents. This inference is well supported by
experimental results from Weibel and Hoppeler (2005), in which
the exponent value of 0.872 was obtained.
Quantitatively, the model is as follows. Let us introduce a
dimensionless parameter h, the ‘heat dissipation coefficient’,
changing in the range of 0–1 and defined as h=Hdis/HdisMax. Here,
Hdis is the dissipated heat and HdisMax is a maximum possible
dissipated heat under given conditions, both in watts. The words
‘given conditions’ are of importance. At high levels of physical
exertion, additional air convection as a result of movement accounts
for 80% of all heat losses in birds (Speakman and Król, 2010), so
that even when the dissipated energy is high, its fraction from the
maximum possible dissipated heat can be relatively small because
of the ‘triggering’ of additional cooling mechanisms. The water
environment in which aquatic animals live potentially allows the
dissipation of much larger quantities of heat than such animals
produce, and accordingly a higher value of the allometric exponent.
The next parameter, c, is the cell enlargement parameter, i.e. the
fraction of body mass increase that is due to cell enlargement, which
changes from 0 to 1 (0 corresponds to mass increase that is due
solely to increase in cell number, and 1 corresponds to mass
increases exclusively through cell enlargement). It is defined as the
ratio of the relative increment of cell mass, (μ/μ0)/μ0=(μ/μ0–1), to the
relative increment of the whole organism’s mass, (M/M0–1):
c ¼ ðm=m0 1Þ=ðM =M0 1Þ:
ð25Þ
Fig. 1 presents the model in graphical form. The analytical
formula describing the lines on the graph, and which can be used for
predicting the value of the allometric exponent, is as follows:
2 1
b ¼ þ ð1 hÞð1 cÞ:
ð26Þ
3 3
Evaluation of the heat dissipation coefficient presents a certain
challenge, although there are no limitations, in principle, for doing
2488
so. The parameter c can be determined from histological studies. In
the case of complex organisms, its evaluation will involve a variety
of cells of different types, and the weighted average is one of the
approaches that can be used.
The presented graph and formula embrace known metabolic
states and experimental observations, covering a wide range of
animals, whose cell sizes change in three dimensions, from torpid
and hibernating animals to the highest levels of metabolic activity.
There is no explicit incorporation of the famous value of 3/4, but it is
included in the range. In the author’s opinion, such a value could be
the result of a compromise, when an organisms’ mass increases both
through cell enlargement (which produces the value of 2/3,
according to the present study) and increase in cell number
(which produces an isometric scaling, when the allometric exponent
is equal to 1). A combination of these two mechanisms would result
in allometric exponents close to a value of 3/4.
The dimensionality of cell increase (in how many dimensions
cells increase and by how much) can be accounted for by an
additional ‘dimensionality’ parameter d, defined as follows:
d ¼ ðux þ uy þ uz Þ= maxfux ; uy ; uz g:
ð27Þ
Here, d changes from 1 to 3, and ux, uy and uz are relative length
increases along the major body axes (‘major’ in a mathematical
sense, which means axes of canonical representation of a body, such
as major axes of an ellipsoid). This third parameter d can be easily
adopted both into Eqn 26 and Fig. 1, if needed.
Conclusions
The purpose of the present study was to propose an allometric
scaling model based on real physiological and biochemical
mechanisms, which could explain known facts and predict the
allometric exponent. Descriptive models do have value, but to
create a quantitative predictive model, the real physical mechanisms
have to be considered. In the present work, the role of cellular
transportation was studied. It was found that cellular transportation
represents a substantial part of all energy expenses. The model
balances intrinsic and systemic factors, and incorporates two
different mechanisms, thus reflecting on the multifactorial nature
of allometric scaling. In the author’s view, the model potentially has
high predictive power. Of course, much work still has to be done to
make it an efficient workable instrument. Nonetheless, in its present
form, it allows us to define directions of future studies, such as
inclusion of cell size and possibly cell geometry, evaluation of the
heat dissipation regime, and determination of the biochemistry and
energy requirements of cellular transportation networks.
Acknowledgements
The author thanks the Editor, Professor H. Hoppeler, whose help in editorial matters
went beyond professional duties; Professor C. White, for valuable comments and
support of this study; the reviewers, for helpful and important comments and
suggestions, and Dr A. Y. Shestopaloff, for editing and discussions.
Competing interests
The author declares no competing or financial interests.
Funding
This research received no specific grant from any funding agency in the public,
commercial or not-for-profit sectors.
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