Post on 29-Sep-2020
Size matters
„You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom,
it gets a slight shock and walks away, provided the ground is fairly soft.
A rat is killed, a man is broken, a horse splashes. (J.B.S. Haldane, 1928: On being the right size)
Allometric principles and metabolic allometry
Marcus Clauss Clinic for Zoo Animals, Exotic Pets and Wildlife, Vetsuisse Faculty, University of
Zurich, Switzerland Wildlife Digestive Physiology Course Vienna 2013
42 D. Adams: Hitchhiker‘s Guide to the Galaxy
0.75 Max Kleiber (1932)
0
2
4
6
8
10
12
14
16
500 1000 1500 2000 2500
Mass (kg)
Co
nsu
mp
tio
n (
l/1
00
km
)
Data from ADAC (2007)
y = 0.0059x - 0.2554
R2 = 0.9616
0
2
4
6
8
10
12
14
16
500 1000 1500 2000 2500
Mass (kg)
Co
nsu
mp
tio
n (
l/1
00
km
)
Data from ADAC (2007)
y = 0.0063x0.99
R2 = 0.9622
0
2
4
6
8
10
12
14
16
500 1000 1500 2000 2500
Mass (kg)
Co
nsu
mp
tio
n (
l/1
00
km
)
Data from ADAC (2007)
Kleiber (1932)
0
5000
10000
15000
20000
25000
30000
35000
40000
0 200 400 600 800
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Kleiber (1932)
0
5000
10000
15000
20000
25000
30000
35000
40000
0 200 400 600 800
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Kleiber (1932)
0
5000
10000
15000
20000
25000
30000
35000
40000
0 200 400 600 800
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Kleiber (1932)
0
5000
10000
15000
20000
25000
30000
35000
40000
0 200 400 600 800
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Kleiber (1932)
0
5000
10000
15000
20000
25000
30000
35000
40000
0 200 400 600 800
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Kleiber (1932)
0
5000
10000
15000
20000
25000
30000
35000
40000
0 200 400 600 800
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Kleiber (1932)
y = 308x0.74
R2 = 0.999
0
5000
10000
15000
20000
25000
30000
35000
40000
0 200 400 600 800
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Kleiber (1932)
y = 308x0.74
R2 = 0.999
1
10
100
1000
10000
100000
0.1 1 10 100 1000
Body mass (kg)
Basal m
eta
bo
lic r
ate
(kJ/
d)
Allometry
Target parameter = a * body massb
Z = a * BMb
Allometry
Target parameter = a * body massb
Z = a * BMb
BM
Z
b = 1
Allometry
Target parameter = a * body massb
Z = a * BMb
BM
Z
b > 1
Allometry
Target parameter = a * body massb
Z = a * BMb
BM
Z
b < 1
Allometry
Target parameter = a * body massb
log (Z) = log(a) + b log(BM)
BM (log)
Z (
log
)
b < 1
Liver (kg) = 0.033 BW0.87
Brain (kg) = 0.011 BW0.76
Blood (kg) = 0.069 BW1.02
Muscle (kg) = 0.450 BW1.00
Skeleton (kg) = 0.061 BW1.09
Integument (kg) = 0.134 BW0.92
Gut contents (kg) = 0.093 BW1.08
(Parra 1978, Calder 1983)
Organ allometry
Basal metabolic rate
Energy production in resting awake at thermoneutrality „post-absorptive (not digesting)
Rubner s Surface Law (1883)
The basal metabolism of an animal is determined by its energy losses to the environment.
These losses are determined by the contact surface to the environment - i.e. the body surface.
With increasing body mass the ratio of surface to volume decreases.
Rubner s Surface Law (1883)
The basal metabolism of an animal is determined by its energy losses to the environment.
These losses are determined by the contact surface to the environment - i.e. the body surface.
With increasing body mass the ratio of surface to volume decreases.
Rubner s Surface Law (1883)
The basal metabolism of an animal is determined by its energy losses to the environment.
These losses are determined by the contact surface to the environment - i.e. the body surface.
With increasing body mass the ratio of surface to volume decreases.
Rubner s Surface Law (1883)
The basal metabolism of an animal is determined by its energy losses to the environment.
These losses are determined by the contact surface to the environment - i.e. the body surface.
With increasing body mass the ratio of surface to volume decreases.
6:1 24:8=3:1
Rubner s Surface Law (1883)
The volume of a geometric body increases with mass3/3 = mass1.
The surface of a geometric body increases with mass2/3 = mass0.67.
The length of a geometric body increases with mass1/3 = mass0.33.
Basal metabolic rate should scale to mass0.67.
Rubner s Surface Law (1883)
If every cell of an elephant produced the same heat as a mouse cell, the elephant would be well done within a day.
If every cell of a mouse produced the same heat as an elephant cell, the mouse would need a 20 cm-thick fur to maintain a constant body temperature.
0.67
Kleiber (1932)
0.74
Kleiber (1932)
0.74
Brody (1945): Mouse-to-Elephant-Curve
0.73
Brody (1945): Mouse-to-Elephant-Curve
0.73
Calculation of BMR (kJ/d) in vertebrates
Calculation of BMR (kJ/d) in vertebrates
mammals: 293 BM0.75
Body mass (kg)
Basal
meta
bo
lic r
ate
(kJ/
d)
Calculation of BMR (kJ/d) in vertebrates
mammals: 293 BM0.75 birds: 335 BM0.67
Body mass (kg)
Basal
meta
bo
lic r
ate
(kJ/
d)
Calculation of BMR (kJ/d) in vertebrates
mammals: 293 BM0.75 birds: 335 BM0.67
reptiles (30°C): 28 BM0.77
Body mass (kg)
Basal
meta
bo
lic r
ate
(kJ/
d)
Calculation of BMR (kJ/d) in vertebrates
mammals: 293 BM0.75 birds: 335 BM0.67
reptiles (30°C): 28 BM0.77
Body mass (kg)
Basal
meta
bo
lic r
ate
(kJ/
d)
0.67 0.75 1.00 0.50 0.25 0.00
0.67 or 0.75 ?
0.67 0.75 1.00 0.50 0.25 0.00
0.67 or 0.75 ?
Kleiber: 0.74
Brody: 0.73
Why 0.75 ?
Why 0.75 ?
Why 0.75 ?
„The widespread use and acceptance of Kleiber‘s exponent can probably be attributed to a remarkably tight regression fit (r2=0.999; n=13). To put this r2 in perspective, we randomly selected 250000 groups of 13 species from a list of 391 species compiled by Heusner (including Kleiber s data). Each group had a mass range of 3-4 orders of magnitude. Of the 250000 regressions, only four had an r2 greater than 0.9980 and none an r2 greater than 0.9990. The strength of Kleiber s exponent therefore seems to stem from an exceedingly fortuitous selection of data.
(White & Seymour 2003)
0.67 or 0.75 ?
0.67 or 0.75 ?
0.67 or 0.75 ?
Savage, West et al. (2004): 626 Species!
y = 239.05x0.7098
R2 = 0.9497
1
10
100
1000
10000
100000
1000000
0.001 0.01 0.1 1 10 100 1000 10000
Body mass (kg)
Basal m
eta
bolic r
ate
(kJ/
d)
Data from Savage et al. (2004)
y = 291.19x0.7616
R2 = 0.9955
1
10
100
1000
10000
100000
1000000
0.001 0.01 0.1 1 10 100 1000 10000
Body mass (kg)
Basal m
eta
bolic r
ate
(kJ/
d)
Savage, West et al. (2004): Body mass- binning !
Data from Savage et al. (2004)
White & Seymour (2003): 626 Species!
y = 239.05x0.7098
R2 = 0.9497
1
10
100
1000
10000
100000
1000000
0.001 0.01 0.1 1 10 100 1000 10000
Body mass (kg)
Basal m
eta
bolic r
ate
(kJ/
d)
Data from Savage et al. (2004)
White & Seymour (2003): exclude large animals!
y = 239.05x0.7098
R2 = 0.9497
y = 215.88x0.6713
R2 = 0.9196
1
10
100
1000
10000
100000
1000000
0.001 0.01 0.1 1 10 100 1000 10000
Body mass (kg)
Basal m
eta
bolic r
ate
(kJ/
d)
Data from Savage et al. (2004)
Basal metabolic rate
Energy production in resting awake at thermoneutrality „post-absorptive (not digesting)
White & Seymour (2003): exclude large animals!
Large animals are mainly herbivores Large animals contain a high proportion of body
mass as gut contents with an active but flora Large animals are rarely post-absorptive (even if a
cow fasts for a day, it still has digesta in its rumen)
White & Seymour (2003): exclude large animals!
Large animals are mainly herbivores Large animals contain a high proportion of body
mass as gut contents with an active but flora Large animals are rarely post-absorptive (even if a
cow fasts for a day, it still has digesta in its rumen)
tElephant = tMouse (KMElephant0.25 / KMMouse
0.25)
tMouse = 3 hours tElephant = 53 hours
White & Seymour (2003): exclude large animals!
Large animals are mainly herbivores Large animals contain a high proportion of body
mass as gut contents with an active but flora Large animals are rarely post-absorptive (even if a
cow fasts for a day, it still has digesta in its rumen)
tElephant = tMouse (KMElephant0.25 / KMMouse
0.25)
tMouse = 3 hours tElephant = 53 hours
Voluntary food intake at the zoo
Evans & Miller (1968)
Energy expenditure in real life (field metabolic rate)
Nagy et al. (1999)
Endogenous protein losses
Brody (1945)
Mineral maintenance requirements
Rucker & Storm (2002)
y = 0.32x0.75
R2 = 0.98
1
10
100
1000
10000
100000
10 100 1000 10000 100000 1000000 10000000
Body mass (g)
En
do
gen
ou
s f
aecal C
a lo
sses (
mg
/d)
Data from Kamphues et al. (1986), Shoe et al. (1992), Meyer & Coenen (2002), Clauss et al. (2003, 2005)
Endogenous faecal calcium losses
Endogenous vitamin C synthesis
Rucker & Steinberg (2002)
Interim result
A large number of parameters that have a connection to metabolism scale allometrically to body mass
- with an exponent of about 0.67-0.75 (but also above or below this)
y = 239.05x0.7098
R2 = 0.9497
1
10
100
1000
10000
100000
1000000
0.001 0.01 0.1 1 10 100 1000 10000
Body mass (kg)
Basal m
eta
bolic r
ate
(kJ/
d)
Good correlation but enormous variance!
Data from Savage et al. (2004)
Good correlation but enormous variance!
y = 239.05x0.7098
R2 = 0.9497
1
10
100
0.001 0.01 0.1
Body mass (kg)
Basal m
eta
bolic r
ate
(kJ/
d)
Data from Savage et al. (2004)
Good correlation but enormous variance!
0
200
400
600
800
1000
1200
0.001 0.01 0.1 1 10 100 1000 10000
Body mass (kg)
rel. B
MR (
kJ/
kg
0.7
5/d
)
293 kJ/kg0.75/d
Data from Savage et al. (2004)
What determines the relative BMR?
Adaptation to climate zone: increase with latitude
McNab (2002)
What determines the relative BMR?
Adaptation to climate zone: increase with latitude Adaptation to habitat:
increase in marine mammals
McNab (2002)
What determines the relative BMR?
Adaptation to climate zone: increase with latitude Adaptation to habitat:
increase in marine mammals decrease in subterranean mammals
McNab (2002)
What determines the relative BMR?
Adaptation to climate zone: increase with latitude Adaptation to habitat:
increase in marine mammals decrease in subterranean mammals
Taxonomy: marsupials always lower than eutherians
McNab (2005)
What determines the relative BMR?
Adaptation to climate zone: increase with latitude Adaptation to habitat:
increase in marine mammals decrease in subterranean mammals
Taxonomy: marsupials always lower than eutherians Adaptation to diet: higher the more digestible the
food? (higher in carnivores)
McNab (2002)
Interim result
The relative BMR of different animal groups varies with various taxonomic, geographic, anatomical and physiological conditions.
In order to recognize outliers, a knowledge of the fundamental allometric relationship is necessary.
!?
The Acid Test and Allometry
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant?
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant? Dose cat: 0.15 mg/kg Dose elephant: 0.10 mg/kg
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant? Dose cat: 0.15 mg/kg 3 kg => 0.45 mg total dose Dose elephant: 0.10 mg/kg 2970 kg => 297 mg total dose
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant? Dose cat: 0.15 mg/kg 3 kg => 0.45 mg total dose Dose elephant: 0.10 mg/kg 2970 kg => 297 mg total dose
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant? Dose cat: 0.15 mg/kg 3 kg => 0.45 mg total dose Dose elephant: 0.10 mg/kg 2970 kg => 297 mg total dose
0
50
100
150
200
250
300
350
0 500 1000 1500 2000 2500 3000
Body mass (kg)
Do
se L
SD
(m
g)
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant? Dose cat: 0.15 mg/kg 3 kg => 0.45 mg total dose Dose elephant: 0.10 mg/kg 2970 kg => 297 mg total dose same logic: man 70kg => 7 mg total dose
0
50
100
150
200
250
300
350
0 500 1000 1500 2000 2500 3000
Body mass (kg)
Do
se L
SD
(m
g)
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant? Dose cat: 0.45 mg/3 kg => 0.2 mg/kg0.75 Dose man: 7mg/70kg => 0.3 mg/kg0.75 Dose elephant: ca. 0.25 mg/kg0.75 2970 kg => 101 mg total dose
0
50
100
150
200
250
300
350
0 500 1000 1500 2000 2500 3000
Body mass (kg)
Do
se L
SD
(m
g)
!?
The Acid Test and Allometry
Lysergic acid diethylamide: its effects on a male Asiatic elephant
West et al. (1962) Science 138: 1100-1103 What dose of LSD is adequate for an elephant? Dose cat: 0.45 mg/3 kg => 0.2 mg/kg0.75 Dose man: 7mg/70kg => 0.3 mg/kg0.75 Dose elephant: ca. 0.25 mg/kg0.75 2970 kg => 101 mg total dose
0
50
100
150
200
250
300
350
0 500 1000 1500 2000 2500 3000
Body mass (kg)
Do
se L
SD
(m
g)
Liver (kg) = 0.033 BW0.87
Brain (kg) = 0.011 BW0.76
Blood (kg) = 0.069 BW1.02
Muscle (kg) = 0.450 BW1.00
Skeleton (kg) = 0.061 BW1.09
Integument (kg) = 0.134 BW0.92
Gut contents (kg) = 0.093 BW1.08
(Parra 1978, Calder 1983)
Organ allometry
0
20
40
60
80
100
120
140
160
10 100 1000 10000
Pro
port
ion
of b
ody
mas
s (%
)
Body mass (kg)
Muscles Fat Gut contents Skin (incl. fur) Skeleton Bood Internal organs
Allometry of body composition
from Hummel and Clauss (2010)
0
20
40
60
80
100
120
140
160
10 100 1000 10000
Pro
port
ion
of b
ody
mas
s (%
)
Body mass (kg)
Muscles Fat Gut contents Skin (incl. fur) Skeleton Blood Internal organs
Allometry of body composition
from Hummel and Clauss (2010)
0
20
40
60
80
100
120
140
160
10 100 1000 10000
Pro
port
ion
of b
ody
mas
s (%
)
Body mass (kg)
Muscles Fat Gut contents Skin (incl. fur) Skeleton Blood Internal organs
Allometry of body composition
from Hummel and Clauss (2010)
0
20
40
60
80
100
120
140
160
10 100 1000 10000
Pro
port
ion
of b
ody
mas
s (%
)
Body mass (kg)
Muscles Fat Gut contents Skin (incl. fur) Skeleton Blood Internal organs
Allometry of body composition
from Hummel and Clauss (2010)
Interim results
The reliability of allometric predictions depends on whether the species in question is within the body size range from which the equation was derived, or beyond it.
Summary
(Empirical) allometric functions with body mass are abundant in biology.
The explanation of these functions is mostly under debate (but the debate is interesting!).
The knowledge of these functions allows the identification of outliers and thus facilitates insight into functional and ecological correlations.
For the calculation of dosages for species for which no data exists one should use an allometric approach.
The reliability of allometric predictions depends on whether the species in question is within the body size range from which the equation was derived, or beyond it.
Vielen Dank für Ihre Aufmerksamkeit !
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