'amp. Riochem. Physiol. Vol. 83A, No. 2, pp. 243-248. 1986
Tinted in Great Britain
0300-9629.86 S3.00 + 0.00
<; 1986 Pergamon Press Ltd
SCALING OF ENERGY METABOLISM IN UNICELLULAR
ORGANISMS: A RE-ANALYSIS*
John Prothero
Department of Biological Structure. University of Washington. School of Medicine. Seattle. Washington
98195. U.S.A. Telephone: (206) 543-1860
(Received 21 May 1985)
Abstract—1. The database used by Hemmingsen (1960) lo compute energ> metabolism in unicellular
organisms was reassembled and submitted to linear (log-log) analysis. As Hemmingsen noted, this data
set includes marine zygotes, which are not unicellular organisms.
2. If no temperature correction factors are applied to the data the best-tit rearession line has a slope
of 0.698 ± 0.024.
3. Application of the temperature correction factors assumed to have been used by Hemmingsen gave
a slope of 0.756 ± 0.021. identical to the value he reported. The correlation coefficient is 0.97. The mean
scatter about the regression line exceeds 100°o.
4. A revised set of temperature correction factors gave a slope of 0.730 -0.021. suggesting that the
value of almost exactly three-quarters obtained by Hemmingsen was probably fortuitous.
5. The slope of the best-fit regression line is very sensitive to the inclusion of bacteria and flagellates.
When the data points for these organisms are omitted from the calculation the slope decreases to
0.645 ± 0.045.
6. When the data points for bacteria, flagellates and marine zygotes arc omitted, the slope drops to
0.608 ±0.025. The correlation coefficient (0.97). compared to the best-fit line reported by Hemmingsen.
is unaffected; the mean deviation about the regression line drops to 40% and the points are evenly
distributed about the regression line.
7. Because of the small number of species for which measurements have been made, the existing
database relating energy metabolism to cell size is not representative of unicellular organisms generally.
8. It is concluded that the case for a three-quarters power rule expressing energy metabolism as a
function of size in unicellular organisms generally is not at all persuasive.
INTRODUCTION
Database
A peculiar feature of Hemmingsen's analysis is
that, with few exceptions, he does not provide us with
the sources from which the data were taken. Also,
with a few exceptions, he does not provide us with the
numbers which were employed in the calculations nor
the conversion factors used. In these circumstances.
one is compelled to ferret out the primary sources
from the secondary references given by Hemmingsen
(chiefly Jahn. 1941; Smith and "Kleiber. 1950: Brand.
1953; Scholander et ai. 1952). To add to the
difficulties of reconstructing the database used by
Hemmingsen, many ofthe references are incomplete,
in that data on cell size or factors used to convert
respiration measurements to energy metabolism are
missing. Hemmingsen also corrects all of the mea
surements of energy metabolism to 20 C. but he is
gest that an empirically correct value for the slope of not explicit as to how he made this correction. In
he regression line relating energy metabolism to spite of these difficulties, it is possible to reconstruct
>ody weight in mammals will be closer to two-thirds .' the essential features of the database which Hem
han three-quarters. In this paper I report the results mingsen employed. Hemmingsen provides data in
»f a careful analysis of the database utilized by tabular form for seven points on the unicellular line
lerp.mingsen in his study of energy metabolism in (see Hemmingsen. 1950). The rest of the data were
niceilular organisms.
published only in graphical form (see Hemminasen.
1950. 1960).
The approach adopted was to reconstruct the
Document No 04324. National Auxiliary Publications
Service. American Society for Information Science, c/o database point by point, insofar as is possible, by
Microfiche Publications, Division of Microfiche Systems comparison o( each datum obtained from the pri
Corporation, P.O. Box 3513, Grand Central Station, mary literature with a point given on Hemmingsen's
NY 10163, USA.
plot (Hemmingsen. 1960). If a given point could not
Fnere is general agreement that energy metabolism is
the Jundamenta! global measure of vital activity in all
organisms. It has become widely accepted, following
the work of Kleiber (1947) and Brody (1945), that
standard energy metabolism in mammals varies as
he three-quarters power of body weight. In two
capers published in 1950 and 1960 Hemmingsen
irgued that the three-quarters power rule also applies
o poikilotherms and unicellular organisms. As a
esult of these works, the three-quarters power rule
:as acquired the status of one ofthe more remarkable
mandative generalizations in comparative biology,
decent studies (Bartels, 1982; Prothero. 1984)
mve called into question the validity of the threeiuar'.ers power rule, as applied to standard energy
uetabolism in mammals. Both of these papers sug
243
4
John Prothero
be fitted to the plot, even after the application of
different trial values for various conversion factors.
then it was rejected. Only after the database was
reconstructed in this fashion were any regression
calculations undertaken.
The interested reader can obtain a copy of the
database and the computations at small cost by
writing to the Documentation Center.
Data analysis
The data were expressed in logarthmic form and
submitted to linear least squares regression analysis
(Edwards. 1976). For each regression analysis a mea
sure of the scatter about the regression line was
computed in the form of a mean percent deviation (%
dev.). Thus, the model for the data analysis and the
mean percent deviation are given by:
.•\=<*.v?
(1)
V
% d e v = 1 0 0 / . V £ | v, - v, ' | / v „ ( 2 )
where (a,, r.) represent the coordinates of the /th
datum point, y\ is the value of the ordinate predicted
from the regression equation for the value x„ and N
is the number of points. In addition to the mean
percent deviation, the correlation coefficient and the
total number of points lying above and below each
regression line are reported.
It is convenient to adopt a measure of the weight
range covered by scaling data. I will employ the
notation pWR to denote the logarithm ofthe weight
range, where pWR is defined by:
pWR =
logarithm (maximum weight/minimum weight).
It is also useful in scaling studies to have measures of
taxonomic diversity. In this study I have identified the
kingdoms from which the (unicellular) organisms are
drawn for each regression line, as well as the number
of species represented.
\NALYTICAL RESULTS
The results of the regression analyses are sum
marized in Table 1. The measures of taxonomic
diversity are given in columns two and three, whereas
the regression characteristics per se are given in
columns five lo eleven. The nature of the data set
employed in each specific regression calculation is
indicated in column twelve. For example, in the
second row the results are presented for the case in
which no temperature corrections were applied to the
data. The large percentage deviation and the non
uniform distribution of the points about the re
gression lines (see Table 1. columns ten and eleven)
should be noted. Because of the large scatter, I have
preferred to enclose the points by a minimal convex
polygon (see Fig. 1). without showing a regression
line (see below).
In the next several sections, the details of the
individual regression calculations are discussed.
Temperature corrections
Hemmingsen corrected the measurements of en
ergy metabolism to 20 C. apparently on the basis of
the plant data of Kuijper (1910). I have attfl-3
derive as nearly as possible the tempera^
rection factors (tcf) which Hemmingsen JJ
addition, I have derived a revised set of tcf j
same data. The details of these computati-'*!
been deposited with the Documentation 3S
We see from Table 1 (row 2) that the slo J
regression line fitted to the raw data (unconj
temperature) is 0.698 ±0.024 (i.e. the slou'
significantly different from two-thirds). On il°
hand, the slope of the regression line derived;
data corrected for temperature using th.;"!
assumed to have been used by Hemnufa
0.756 ±0.021 (see Table 1, row 3). Hen'
reported an identical slope of 0.756 ±0.<;
an essentially identical intercept (log a '-'
(Hemmingsen, 1960).
Using a revised set of tcf, the slope is fou-J
0.730 ±0.021. Two points should be made. ;*
is evident that the temperature correction iC3
have a significant influence on the slop*.^
regression relating energy metabolism to cei.^
depending on the tcf, the slope varies by arx,--'
Second, the fact that Hemmingsen found a 2j
0.756 (i.e. almost exactly three-quarters) ii
garded as fortuitous.
Whether it is at all reasonable to use da^
exclusively from plants to calculate tcf . J
diverse organisms, as Hemmingsen seem-, 3
done, is questionable. It is true, however, uV "J
more appropriate set of tcf probably would 23
the results given in Table 1 (row 4) very sigr^..
Extreme values
It is well known that, where the data are *A.
distributed, the slope of a regression line mz.^
sensitive to the values at the extremes. That fc3
case here may be seen by examining the c_£
omitting 14 points (19% of the total) corre_r^
to bacteria and flagellates. The slope dr^ ~\
0.730 ± 0.021 to 0.645 ± 0.045 (compare ro-_
5, Table 1). This effect cannot be attributed &_
reduction in the number of points, since on^.T
the amoebae, which are at the other end r»ft-y
range. has no effect on the slope (see Table ../_*
In this case, the number of points is reduce,
or 27%. However, the two cases are noi-I
comparable, since omission of the bacn**^
flagellates reduces the pWR by 2.6, whereas^.'
of the amoebae only reduces pWR by 1.7. C^
we are entitled to infer that the value for tht„"
the regression line obtained by Hemmi^!
strongly dependent on the values correspoi,"
bacteria and flagellates (see also below).
Sample size
When we consider sample size in a scaling ^.
are concerned with both species diversity anc^
range. We can easily satisfy ourselves*that tht^T
employed by Hemmingsen is quite represent
the weight range exhibited by unicellular oi^
The smallest known unicellular organisms «.rickettsiae, with a minimum volume of afarw
cubic micra. The largest known unicellular oi^are protozoa, with volumes of about 150^
cubic micra (see Poindexter, 1971; Stanier.
John Prothero
□ Bacteria
© Flagellates
♦ Fungi
+ Ciiiates
X Zygotes
A AmoeDae
a 10 ,2
Cell weight (g)
Fig. I. Energy metabolism as a funclion of cell size in unicellular organisms and zygotes. The data points,
corrected to 20 C with revised temperature correction factors, have been enclosed by a minimal convex
polygon (see text).
1970). Thus. pWR for unicellular organisms is about
10. The value of pWR for the data used by
Hemmingsen is 8.6 (see Table 1. column five), or 86%
of the total, on a logarithmic scale.
On the other hand, the diversity of unicellular
species in the sample available to Hemmingsen (or
current workers) is altogether unrepresentative ofthe
variety of species of unicellular organisms thought to
exist (see Table 2). In comparing the size of the
sample used by Hemmingsen with all species of
unicellular organisms. I have not included the marine
zygotes, since these are not unicellular organisms. It
is fair to note that reliable energy metabolism data
are available for only a few species of unicellular
organisms. With the exception oi' specialized cells,
such as red blood cells, there has been very little
interest in cellular energy metabolism and cell size
during the past quarter century. Thus, the potential
database for this kind of study has not increased
significantly since Hemmingsen carried out his
analysis. The results of a recent multivariate analysis
of energy metabolism in unicellular organisms are not
strictly comparable to the results obtained in the
bivariate analvsis reported here (Robinson et al..
1983).
Subsets of the data
Thirty-three ofthe data points, or 44% ofthe total,
come from a single laboratory, that of Scholander et
al. (1952). These workers fitted a regression line I
their data, for which they reported a slope of 0.5
This compares well with the slope of 0.545 obtaine
here [note that these data were obtained from a grap
(Scholander et al.. 1952) by digitization (see Mannar
et al.. 1982)]. The mean percent deviation about th
regression line is 25% (see Table 1, row 7).
It has already been observed that the bacteria an
flagellates exert a disproportionate influence on th
slope of the regression line relating energy metabc
lism to cell weight. It is also the case that zygotes (an
small embryos) cannot be regarded as unicellul;
organisms. It is, therefore, of interest to calculate
regression line with these data omitted. The results c
this analysis are given in Table 1 (row 8). The slop
of the regression line is 0.608 ± 0.025, and the mea
deviation around the line is 40% (see Fig. 2). The da;
points are not very uniformly distributed along tl
regression line. In particular, the points correspom
ing to fungi are off by themselves at the lower end ^
the weight range. However, when we omit the tu
points corresponding to fungi, the regression charai
teristics are essentially unchanged (see Table
row 9).
Aggregated data
An obvious objection to the validity of the r<
gression lines fitted to the data by Hemmingsen is tt
large mean deviation (116-146%). It is often useful i
Taole 2. Species diversity
Number ot species
Estimated number in Hemmingsen's
of species sample Comments
SchizophMa
Protozoa
Ascomvcetes
1600
5
»
Lack
mitochondria
Scaling of energy metabolism in unicellular orcanisms
♦ Fungi
A Amoebae
+ CHiates
"slope = 0 608
10
'2
10
' ■■
10
'°
IO"9
10
8
10
7
10
6
10
5
10
4
10
3
Cell weight (g)
Fig. 2. Energy metabolism as a function of cell size in fungi, amoebae and ciliates (see Table 1. row 8).
such cases to divide the data into small groups, where
each group extends over a narrow weight range. An
average is then calculated for each group and a
regression line is fitted to the averages. Often the
effect of such aggregation will be to smooth out the
fluctuations in the data. However, when this experi
ment was carried out with the data set of Hem
mingsen. using logarithmic averages, there was no
significant effect on the mean deviation of the aggre
gated points from the best-fit line (see Table 1. row
10). A possible implication is that the scatter reflects
a qualitative dissimilarity in energy metabolism in
these different forms.
Substituted data
Zygotes and small embryos are not unicellular
organisms, as Hemmingsen noted (Hemmingsen,
1960). If one wished to include data for germ cells, it
would be more consistent with the purposes of the
study to use the data for unfertilized marine eggs,
which even though haploid (in effect like bacteria) are
at least unicellular. In the majority of the cases cited
by Hemmingsen, the effect of fertilization is to in:rease the rate of oxygen consumption. It was possi
ble to obtain data on energy metabolism in un
fertilized eggs in eleven of the fourteen cases for
which data were available for marine zygotes. The
regression characteristics for this instance are given in
Table I (row 11). The slope ofthe regression line is.
-msurprisingly, essentially unchanged, since the
joints are close to the center of the regression line.
There is. however, a large increase in the scatter; to
189%. It was because of this large scatter that Hem
mingsen chose to use the data for marine zygotes
Hemmingsen. 1960).
DISCUSSION
It has been possible to reconstruct a data set which
gives essentially identical regression parameters to
those reported by Hemmingsen. Various features of
this data set have been analyzed. It has been shown
that the deviation between three-quarters and twothirds in the slope of the regression line fitted to the
data set employed by Hemmingsen is. in part, a
function of the temperature correction factors ap
plied. A more fastidious analysis of the temperature
data employed by Hemmingsen suggests that 0.73 is
a better value for the slope ofthe regression line than
the value of 0.756 obtained by Hemmingsen. But the
temperature correction factors are based on data
drawn from the garden pea, yellow lurfin anr1 •••h<»q»
Whether it makes'sense at all to apply these results
to unicellular organisms as diverse as bacteria,
flagellates, fungi, marine zygotes, ciliates and amoe
bae, as Hemmingsen seems to have done, is open to
doubt.
A point which is perhaps not sufficiently appre
ciated in scaling studies is that the correlation
coefficient, by itself, is an inadequate measure of
whether a linear regression model is appropriate for
a given set of log-log transformed data. A high
correlation coefficient is perfectly compatible with
large systematic deviations. In addition to the
correlation coefficient, it is useful to look at the mean
percent deviation about the regression line and the
distribution of points about the line.
The large deviation—greater than 100%—
associated with the regression lines fitted to the whole
data set argues that energy metabolism in these
organisms is not well described by a linear model. We
know that bacteria lack mitochondria and other
John Prothero
membrane-bound organelles possessed by eukaryotes
(Davis et al., 1980). Furthermore, at least some ofthe
flagellates (e.g. trypanosomes) are thought not to
ingest solid matter. Thus it is possible that energy
metabolism in these forms; at least, is qualitatively
dissimilar to that in other unicellular organisms.
When we omit these forms, and zygotes, which are
not unicellular organisms in any case, we find that the
mean deviation drops to the more reasonable value
of 40",,. In this case (see Table 1. row 8). the
correlation coefficient (0.97) is identical to the value
for Hemmingsen's data set (Table 1, rows 3 and 4)
and the points are evenly distributed about the
regression line. By these objective criteria the slope of
the regression line which is least open to objections
in this data set has a slope of 0.608 ± 0.025 (see Table
1, row 8). The fact that the mean deviation about the
regression line does not decrease when we aggregate
the data also argues that the complete data set is
internally inconsistent. Hemmingsen omitted the data
for fertilized insect eggs from his calculations, appar
ently on the grounds that they did not fit the uni
cellular line (Hemmingsen, I960). No biological case
is made for this omission. Inclusion of these data
would have increased the slope, and more especially
the scatter, considerably.
At the time the measurements were made which
Hemmingsen draws upon, there was no method
available to measure the oxygen concentration in the
culture medium. In some cases, at least, the oxygen
concentration may have been sufficiently reduced
locally to adversely affect oxygen consumption.
Many other difficulties with the data base are appar
ent. Measurements were made at different tem
peratures and under different conditions. Some ofthe
measurements were made on single animals, whereas
others were made on many. Some animals were
starved for long periods (e.g.. weeks) and others were
not.
Hemmingsen noted (Hemmingsen, 1950) that it is
difficult to define standard conditions for bacteria,
but it seems likely that this remark applies to uni
cellular organisms generally, since some are photosynthetic and some not. some are motile and some
not. some ingest solid materials and some do not.
These limitations of the data set do not constitute
a criticism of Hemmingsen. who brought together the
best data available. The key question is whether the
data, and his analysis of the data, supports his
conclusions.
CONCLUSIONS
Hemmingsen's suggestion that energy metabolism
in unicellular organisms varies as the three-quarters
power of cell size is both novel and bold. If the
hypothesis ultimately proves to be correct, it will be
of fundamental significance for the understanding of
cellular energetics. But the available evidence and
Hemmingsen's analysis of that evidence in support of
a three-quarters power rule are not persuasive. A
more careful application of temperature correction
factors leads to a slope of 0.73, suggesting that the
value close to the simple fraction of three-quarters
found by Hemmingsen was fortuitous. Omission of
the data on bacteria and flagellates drops the slope to
0.645; but even then the scatter around the best-fit
regression line is substantial. Omission of the data.c
marine zygotes leads to a slope of 0.608 ± 0.025 an
a mean deviation of 40%.
There are reasonable grounds for believing t
energy metabolism in bacteria may be qualitativ,
different from that in eukaryotes. If this is the cas
there is no good reason to expect that energy meta:
olism in unicellular organisms can be adequate
represented by a single linear regression line. Until v
have a better understanding of energy metabolism
unicellular organisms, it may be best to represent ti
aggregate body of information by a convex polygo
And until energy metabolism has been determine
under standard conditions, assuming that is possiblm a much wider variety of species, it is rathdoubtful that any two-parameter representation <
the data is warranted.
Acknowledgements—I am grateful to Dr K. D. Jurgens f
his constructive comments upon a draft of this paper
thank Ms Doris Ringer for preparation of the manuscrii
This work was supported in part by a USPHS er
ROl-HL-29761 to J. Prothero.
REFERENCES
Bartels H. (1982) Metabolic rate of mammals equals ti
0.75 power of their body weight? Exp. Biol. Med. 7, 1-1
Brand Th. von (1935) Der stoffwechsel der protozoe
Ergebn. Biol. 12, 161-220.
Brody S. (1945) Bioenergetics and Growth. Reinhold, Ne
York.
Davis B. D., Dulbecco R., Eisen H. N. and Ginsberg H.
(1980) Microbiology Including Immunology and Molecule
Genetics. Harper & Row, Philadelphia.
Edwards A. L. (1976) An Introduction to Linear Regressk
and Correlation. Freeman, San Francisco.
Hemmingsen A. M. (1950) The relation of standard (basa
energy metabolism to total fresh weight of living o
ganisms. Rep. Steno meml Hosp. 4, 6-58.
Hemmingsen A. M. (1960) Energy metabolism as related I
body size and respiratory surfaces, and its evolution. Re
Steno meml Hosp. 9, 1-110.
Jahn T. L. (1941) Respiratory metabolism. In Protozc
in Biological Research. (Edited by Calkins G. N. ar.
Summers F. M.), Chap. IV. pp. 352^103.
Kleiber M. (1947) Body size and metabolic rate. Physio
Rev. 27, 511-541.
Krogh A. (1916) The Respiratory Exchange of Animals ar.
Man. Longmans Green. London.
Kuijper J. (1910) Uber den Einfluss der Temperature auf d
Atmung der Hoheren. Pflanzen. Rec. Trav. Bot. Neerlai
7, 131-240.
Mannard J., Lindsay A.. Sundsten J. and Prothero J. (1981
A ploued-to-tabular data conversion program for micro
computers. Int. J. biomed. Computing, 13, 369-373.
Poindexter J. S. (1971) Microbiology: An Introduction
Protists. Macmillan, New York.
Prothero J. (1984) Scaling of standard energy metabolism i
mammals: I. Neglect of circadian rhythms. J. theor. Bio
106, 1-8.
Robinson W. R., Peters R. H. and Zimmermann J. (198?
The effects of body size and temperature on metaboli
rate of organisms. Can. J. Zool. 61, 281-288.
Scholander P. F., Claff C. L. and Sveinsson S. L. (1952
Respiratory studies of single cells: II Observations on th
oxygen consumption in single protozoans. Biol. Bull. 10:
178-184.
Smith A. H. and Kleiber M. (1950) Size and oxyge
consumption in fertilized eggs. J. cell. comp. Physiol. 35
131-140.
Stanier R. Y., Doudoroff M. and Adelberg E. A. (1970) Th
Microbial World. Prentice-Hall. Englewood Cliffs.