Mass aspects & scaling Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Melbourne 2012/08/06 Contents mass aspects indirect calorimetry Synthesizing Units covariation

Macrochemical reaction eq 3.5

Three basic fluxes 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, CO 2, H 2 O, O 2, NH 3 Indirect calorimetry: heat = D  O 2 -flux + E  CO 2 -flux + F  NH 3 -flux

Synthesizing units 3.7b Generalized enzymes that process generalized substrates and follow classic enzyme kinetics E + S  ES  EP  E + P with 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 1C Poisson arrival events for molecules A and B blocked time intervals acceptation event ¤ rejection event production Kooijman, 1998 Biophys Chem 73:

Interactions of substrates 3.7.3b Kooijman, 2001 Phil Trans R Soc B 356:

Competition & inhibition

Social inhibition of x  e 3.7.4b sequential parallel dilution rate substrate conc. biomass conc. No socialization Implications: stable co-existence of competing species “survival of the fittest”? absence of paradox of enrichment x substrate e reserve y species 1 z species 2

Co-metabolism Consider coupled transformations A  C and B  D Binding probability of B to free SU differs from that to SU-A complex

Photo synthesis, respiration, inhibition

Scales of life 8a Life span 10 log a Volume 10 log m 3 earth whale bacterium water molecule life on earth whale bacterium ATP molecule

Bergmann 1847

Dwarfing in Platyrrhini Perelman et al 2011 Plos Genetics 7, 3, e MYA Callitrix Cebuella Mico Leontopithecus Aotus Saimiri Cebus g g g g 3500 g g g 130 g 180 g Callimico Saguinus Cebidae

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 Kooijman 1986 Energy budgets can explain body size scaling relations J. Theor. Biol. 121:

Body weight Body weight has contributions from structure and reserve If 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 O 2 consumption,  l/h Inter-species Intra-species L L L 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 10 log egg weight, g 10 log incubation time, d l b equal ° tube noses slope = 0.25 Data from Harrison 1975 European birds tube noses

Gestation time 8.2.2l 10 log adult weight, g 10 log gestation time, d Data from Millar 1981 Mammals * Insectivora + Primates  Edentata  Lagomorpha  Rodentia  Carnivora  Proboscidea Hyracoidea  Perissodactyla  Artiodactyla slope = 0.33 Kooijman 1986 J Theor Biol 121:

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 L p  ultimate length L 

Feeding rate slope = 1 poikilothermic tetrapods Data: Farlow 1976 Mytilus edulis Data: Winter 1973 Length, cm Filtration rate, l/h

At 25 °C : maint rate coeff k M = 400 a -1 energy conductance v = 0.3 m a °C T A = 7 kK 10 log ultimate length, mm 10 log von Bert growth rate, a -1 ↑ 0 Von Bertalanffy growth rate

Primary parameters standard DEB model