Lecture 3 Implications of theory

Mass & energy balance The standard DEB model specifies fluxes of 4 organic compounds food, faeces, stucture (growth), reserve (including reproduction) The fluxes of 4 mineral compouds (CO 2, H 2 O, O 2, NH 3 ) follow from conservation of chemical elements C, H, O, N and strong homeostasis The standard DEB model assumes that only food is limiting Dissipating heat follows from conservation of energy and strong homeostasis (constant chemical potentials)

Method of indirect calorimetry Empirical origin (multiple regression): Lavoisier 1780 Heat production = w C CO 2 -production + w O O 2 -consumption + w N N-waste production DEB-explanation: Mass and heat fluxes = w A assimilation + w D dissipation + w G growth Applies to CO 2, O 2, N-waste, heat, food, faeces, … For V1-morphs: dissipation  maintenance

Mass fluxes  flux notice small dent due to transition maturation  reproduction At abundant food: growth ceases at l = 1 allocation to reproduction use of reserve not balanced by feeding in embryo

Methanotrophy Yield coefficients Y and chemical indices n depend on (variable) specific growth rate r ACAC Assim (catabolic) A Assim (anabolic)010 MMaintenance010 GCGC Growth (catabolic)010 GAGA Growth (anabolic)001 CCarbon HHydrogen40203 OOxygen02120 NNitrogen00001 symbolprocessX: methane C: carbon dioxide H: waterO: dioxygenN: ammoniaE: reserve V: structure For reserve density m E = M E /M V (ratio of amounts of reserve and structure), the macroscopic transformation can be decomposed into 5 microscopic ones with fixed coefficients rate Yield coefficients T Chemical indices

Methanotrophy spec growth rate, h -1 X/O N/O C/O flux ratio, mol.mol -1 spec flux, mol.mol -1.h -1 C E N X O X: methane C: carbon dioxide O: dioxygen N: ammonia E: reserve j EAm = 1.2 mol.mol -1.h -1 y EX = 0.8 y VE = 0.8 k M = 0.01 h -1 k E = 2 h -1 n HE = 1.8 n OE = 0.3 n NE = 0.3 n HV = 1.8 n OV = 0.3 n NV = 0.3 chemical indices Kooijman, Andersen & Kooi Ecology, to appear

Biomass composition Data Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C n HW n OW n NW O2O2 CO 2 Spec growth rate, h -1 Spec growth rate Spec growth rate, h -1 Relative abundance Spec prod, mol.mol -1.h -1 Weight yield, mol.mol -1 n HE 1.66 n OE n NE n HV 1.64 n OV n NV k E 2.11 h -1 k M h -1 y EV y XE r m 1.05 h -1 g = 1 μ E -1 pApA pMpM pGpG JCJC JHJH JOJO JNJN Entropy J/C-mol.K Glycerol69.7 Reserve74.9 Structure 52.0 Sousa et al 2004 Interface, subm

Product Formation throughput rate, h -1 glycerol, ethanol, g/l pyruvate, mg/l glycerol ethanol pyruvate Glucose-limited growth of Saccharomyces Data from Schatzmann, 1975 According to Dynamic Energy Budget theory: Product formation rate = w A. Assimilation rate + w M. Maintenance rate + w G. Growth rate For pyruvate: w G <0

1 Reserve – 1 Structure

2 Reserves – 1 Structure

Reserve Capacity & Growth low turnover rate: large reserve capacity high turnover rate: small reserve capacity

Multivariate extensions animal heterotrophphototroph symbiosis plant

Interactions of substrates

Photosynthesis 2 H 2 O + 4 h  O H e - CO H e -  CH 2 O + H 2 O CO 2 + H 2 O + light  CH 2 O + O 2

Simultaneous nutrient limitation Specific growth rate of Pavlova lutheri as function of intracellular phosphorus and vitamine B 12 at 20 ºC Data from Droop 1974 Note the absence of high contents for both compounds due to damming up of reserves, and low contents in structure (at zero growth)

Reserve interactions Spec growth rate, d -1 P-content, fmol.cell -1 P-conc, μM B 12 -conc, pM B 12 -cont., mol.cell -1 PVitamin B 12 kEkE d -1 y XV mol.cell -1 j EAm mol.cell -1. d -1 κEκE kMkM d -1 K pM, μM Data from Droop 1974 on Pavlova lutheri P(μM)B 12 (pM)

Steps in food Growth of Daphnia magna at 2 constant food levels time, d 0 d7 d14 d21 d length, mm Only curves at 0 d are fitted Notice slow response gut content in down steps Steps up Steps down

Growth on reserve Optical Density at 540 nm Conc. potassium, mM Potassium limited growth of E. coli at 30 °C Data Mulder 1988; DEB model fitted OD increases by factor 4 during nutrient starvation internal reserve fuels 9 hours of growth time, h

Growth on reserve Growth in starved Mytilus edulis at 21.8 °C Data Strömgren & Cary 1984; DEB model fitted internal reserve fuels 5 days of growth time, d growth rate, mm.d -1

Protein synthesis spec growth rate, h -1 scaled spec growth rate RNA/dry weight, μg.μg -1 scaled elongation rate Data from Koch 1970 Data from Bremer & Dennis 1987 RNA = w RV M V + w RE M E dry weight = w dV M V + w dE M E

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

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 : (so are all extensive parameters) 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:

Primary scaling relationships assimilation {J EAm } max surface-specific assim rate  L m feeding {b} surface- specific searching rate digestion y EX yield of reserve on food growth y VE yield of structure on reserve mobilization venergy conductance heating,osmosis {J ET } surface-specific somatic maint. costs turnover,activity [J EM ] volume-specific somatic maint. costs regulation,defencek J maturity maintenance rate coefficient allocation  partitioning fraction egg formation  R reproduction efficiency life cycle[M H b ] volume-specific maturity at birth life cycle [M H p ] volume-specific maturity at puberty aging h a aging acceleration maximum length L m =  {J EAm } / [J EM ] Kooijman 1986 J. Theor. Biol. 121:

Follows from: 1.maturity at birth equals a given value 2.reserve density at birth equals that of mother State variables: Parameters: Problem: Given parameter values, find Initial reserve of an egg Theory in Kooy2008

Effects of nutrition scaled res density at birth scaled length at birth scaled initial reserve scaled age at birth

Reduction of initial reserve scaled age scaled maturity scaled struct volume scaled reserve

Scaling relationships log zoom factor, z log scaled initial reserve log scaled age at birth log scaled length at birth approximate slope at large zoom factor

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 

Body weight Body weight has contributions from structure and reserve If reserve allocated to reproduction hardly contributes: intra-spec body weight inter-spec body weight intra-spec structural volume Inter-spec structural volume reserve energy compound length-parameter specific density for structure molecular weight for reserve chemical potential of reserve maximum reserve energy density

Feeding rate slope = 1 poikilothermic tetrapods Data: Farlow 1976 Inter-species: J Xm  V Intra-species: J Xm  V 2/3 Mytilus edulis Data: Winter 1973 Length, cm Filtration rate, l/h

Scaling of metabolic rate intra-speciesinter-species maintenance growth Respiration: contributions from growth and maintenance Weight: contributions from structure and reserve Structure ; = length; endotherms

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)

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