Mass aspects & scaling

Bas KooijmanDept theoretical biology

Vrije Universiteit AmsterdamBas@bio.vu.nl

http://www.bio.vu.nl/thb

Melbourne 2012/08/06

Contents

• mass aspects• indirect calorimetry• Synthesizing Units• covariation

Macrochemical reaction eq 3.5

Three basic fluxes 4.3.1

• assimilation: substrate reserve + products

linked to surface area

• dissipation: reserve products

somatic maintenance: linked to surface area & structural volume

maturity maintenance: linked to maturity

maturation or reproduction overheads

• growth: reserve structure + products

Product formation = A assimilation + B dissipation + C growth

Examples: heat, CO2, H2O, O2, NH3

Indirect calorimetry: heat = D O2-flux + E CO2-flux + F NH3-flux

Synthesizing units 3.7b

Generalized enzymes that process generalized substrates and follow classic enzyme kinetics E + S ES EP E + Pwith two modifications:• back flux is negligibly small E + S ES EP E + P• specification of transformation is on the basis of arrival fluxes of substrates rather than concentrations In spatially homogeneous environments: arrival fluxes concentrations

Transformation A → B

Michealis-Menten (Henri 1902)Holling type II (Holling 1957)

Classification of behavioural modes: free & bound

Simultaneous Substrate Processing 3.7c

Chemical reaction: 1A + 1B 1CPoisson arrival events for molecules A and B

blocked time intervals

• acceptation event¤ rejection event

production

production

Kooijman, 1998Biophys Chem73: 179-188

Interactions of substrates 3.7.3b

Kooijman, 2001Phil Trans R Soc B356: 331-349

Competition & inhibition

Social inhibition of x e 3.7.4b

sequential parallel

dilution rate

subs

trat

e co

nc.

biom

ass

conc

.

No

soci

aliz

atio

n

Implications: stable co-existence of competing species “survival of the fittest”? absence of paradox of enrichment

x substratee reservey species 1z species 2

Co-metabolism 3.7.5

Consider coupled transformations A C and B DBinding probability of B to free SU differs from that to SU-A complex

Photo synthesis, respiration, inhibition

Scales of life 8a

Life span

10log aVolume

10log m3earth

whale

bacterium

water molecule

life on earth

whale

bacteriumATP molecule

30

20

10

0

-10

-20

-30

Bergmann 1847

Dwarfing in Platyrrhini 8.1.2

Perelman et al 2011 Plos Genetics 7, 3, e1001342

24

.82

0.2

MYA

CallitrixCallitrix

CebuellaCebuella

MicoMicoLeontopithecusLeontopithecus

AotusAotus

SaimiriSaimiri

CebusCebus

780-1250 g

400-450 g480-700 g

400-535 g

3500 g

700-1000 g

200-400 g

130 g

180 g

CallimicoCallimico

SaguinusSaguinus

Ceb

idae

Inter-species body size scaling

• parameter values tend to co-vary across species• parameters are either intensive or extensive• ratios of extensive parameters are intensive• maximum body length is allocation fraction to growth + maint. (intensive) volume-specific maintenance power (intensive) surface area-specific assimilation power (extensive)• conclusion :• write physiological property as function of parameters (including maximum body weight)• evaluate this property as function of max body weight

]/[}{ MAm ppL

}{ Ap

][ Mp

mA Lp }{

Kooijman 1986 Energy budgets can explain body size scaling relationsJ. Theor. Biol. 121: 269-282

Body weight

Body weight has contributions from structure and reserveIf reserve allocated to reproduction hardly contributes:

Scaling of metabolic rate

Metabolic rate

Log weight, g

Log metabolic rate,

w

endotherms

ectotherms

unicellulars

slope = 1

slope = 2/3

Length, cm

O2 consum

ption,

l/h

Inter-speciesIntra-species

0.0226 L2 + 0.0185 L3

0.0516 L2.44

2 curves fitted:

(Daphnia pulex)

Incubation time: intra-species

Eudyptes first lays a small egg, then a large one, which hatches earlier if fertile

It can rise one chick only

If all parameters are the same, maturity at birth is reached earlier with big initial reserve

Incubation time: inter-species

10log egg weight, g 10log egg weight, g

10lo

g in

cuba

tion

time,

d

10lo

g in

cuba

tion

time,

d

lb equal° tube noses

slope = 0.25

Data from Harrison 1975

European birdstube noses

Gestation time 8.2.2l

10log adult weight, g

10lo

g ge

stat

ion

time,

d

Data from Millar 1981

Mammals* Insectivora+ Primates Edentata Lagomorpha Rodentia Carnivora Proboscidea Hyracoidea Perissodactyla Artiodactyla

slope = 0.33

mL

396.0

weightbirth

weightadult timegestationactualtimegestation

3/1

Kooijman 1986J Theor Biol 121: 269-282

Length at puberty

L, cm

L p,

cm

Clupea• Brevoortia° Sprattus Sardinops Sardina

Sardinella+ Engraulis* Centengraulis Stolephorus

Data from Blaxter & Hunter 1982

Clupoid fishes

Length at first reproduction Lp ultimate length L

Feeding rateslope = 1

poikilothermic tetrapodsData: Farlow 1976

Mytilus edulisData: Winter 1973

Length, cm

Filt

ratio

n ra

te, l

/h

At 25 °C : maint rate coeff kM = 400 a-1

energy conductance v = 0.3 m a-1

25 °CTA = 7 kK

10log ultimate length, mm 10log ultimate length, mm

10lo

g vo

n B

ert

grow

th r

ate

, a-1

a↑0

Von Bertalanffy growth rate

Primary parametersstandard DEB model