Mass balance 4.3 minerals carbon dioxide water dioxygen nitrogen-waste organics food structure reserve product flux of compound i chemical index for element i in compound j for all compounds j DEB model specifies organic fluxes Mineral fluxes follow from mass balance Extendable to more elements/compounds compounds

Mass-energy coupling 4.3 compounds organics food structure reserve product powers assimilation dissipation growth chemical potential of E yield of compound i on j coupler of compound i to power j for faeces: Decomposition of mineral fluxes into contributions from 3 basic energy fluxes: Organic fluxes are linear combinations of 3 energy fluxes

Energy balance Dissipating heat can be decomposed into contributions from 3 basic energy fluxes chemical potentials (energy-mass couplers) mass-energy couplers fluxes of compounds 3 basic energy fluxes (powers) chemical indices minerals organics

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 4.1  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 4.7 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 5 animal heterotrophphototroph symbiosis plant

Interactions of substrates 5.1

Photosynthesis 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 7.5 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 8.0 Life span 10 log a Volume 10 log m 3 earth whale bacterium water molecule life on earth whale bacterium ATP

Invariance property 8.1 The parameters of two individuals can differ in a very special way such that both individuals behave identically at constant food density if they start with the same values for the state variables (reserve, structure, damage) At varying food density, two individuals only behave identically if all their parameters are equal

Inter-species body size scaling 8.2 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 8.1 K 2 =K 1 z+X(z-1){J Xm } 2 ={J Xm } 1 z[p M ] 2 =[p M ] 1 {p T } 2 ={p T } 1 L b2 = L b1 {p Am } 2 ={p Am } 1 z[E G ] 2 =[E G ] 1 h a2 = h a1 L p2 = L p1 [E m ] 2 =[E m ] 1 z  2 =  1  R2 =  R1 K 2 =K 1 z{J Xm } 2 ={J Xm } 1 z[p M ] 2 =[p M ] 1 {p T } 2 ={p T } 1 L b2 = L b1 z{p Am } 2 ={p Am } 1 z[E G ] 2 =[E G ] 1 h a2 = h a1 L p2 = L p1 z[E m ] 2 =[E m ] 1 z  2 =  1  R2 =  R1 K saturation constant {J Xm } max spec feeding rate [p M ] spec maint. costs {p T } spec heating costs L b length at birth{p Am } max spec assim rate [E G ] spec growth costs h a aging acceleration L p length at puberty [E m ] max reserve capacity  partitionning fraction  R reprod. efficiency z: arbitrary zoom factor for species 2 relative to species 1: z = L m2 /L m1 invariance property (at food density X) primary scaling parameters

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 contribution from structure and reserve If reserves allocated to reproduction hardly contribute: 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 8.2.2