1. Innovations by DEB theory to understand metabolic organisation Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Bas@bio.vu.nl http://www.bio.vu.nl/thbhttp://www.bio.vu.nl/thb/ Nantes, 2005/04/22

2. Contents introduction static & production models principles surface-volume interactions respiration patterns biomass composition product formation body size scaling Nantes, 2005/04/22

3. Dynamic Energy Budget theory for metabolic organisation Uptake of substrates (nutrients, light, food) by organisms and their use (maintenance, growth, development, reproduction) First principles, quantitative, axiomatic set up Aim: Biological equivalent of Theoretical Physics Primary target: the individual with consequences for sub-organismal organization supra-organismal organization Relationships between levels of organisation Many popular empirical models are special cases of DEB

4. molecule cell individual population ecosystem system earth time space Space-time scales When changing the space-time scale, new processes will become important other will become less important Individuals are special because of straightforward energy/mass balances Each process has its characteristic domain of space-time scales

5. Empirical special cases of DEB yearauthormodelyearauthormodel 1780Lavoisier multiple regression of heat against mineral fluxes 1950Emerson cube root growth of bacterial colonies 1825Gompertz Survival probability for aging 1951Huggett & Widdas foetal growth 1889Arrhenius temperature dependence of physiological rates 1951Weibull survival probability for aging 1891Huxley allometric growth of body parts 1955Best diffusion limitation of uptake 1902Henri Michaelis--Menten kinetics 1957Smith embryonic respiration 1905Blackman bilinear functional response 1959Leudeking & Piret microbial product formation 1910Hill Cooperative binding 1959Holling hyperbolic functional response 1920Pütter von Bertalanffy growth of individuals 1962Marr & Pirt maintenance in yields of biomass 1927Pearl logistic population growth 1973Droop reserve (cell quota) dynamics 1928Fisher & Tippitt Weibull aging 1974Rahn & Ar water loss in bird eggs 1932Kleiber respiration scales with body weight 3/ 4 1975Hungate digestion 1932Mayneord cube root growth of tumours 1977Beer & Anderson development of salmonid embryos DEB theory is axiomatic, based on mechanisms not meant to glue empirical models Since many empirical models turn out to be special cases of DEB theory the data behind these models support DEB theory This makes DEB theory very well tested against data

6. Classic energetics Anabolism: synthetic pathways Catabolism: degradation pathways Duality: compounds as source for energy & building blocks From: Mader, S. S. 1993 Biology, WCB This decomposition occurs at several places in DEBs

7. Static Energy Budgets Scope for Growth From: Brafield, A. E. and Llewellyn, M. J. 1982 Animal energetics, Blackie, Glasgow C energy from food P production (growth) F energy in faeces U energy in urine R heat Numbers: kJ in 28 d Basic difference with dynamic budgets: Production is quantified as energy fixed in new tissue, not as energy allocated to growth: SEBs exclude overheads Heat includes overheads of growth, reproduction and other processes it does not quantify maintenance costs

8. Production model foodfaeces assimilation feeding defecation maintenance offspring reproduction reserve structure growth

9. 1-  maturity maintenance maturity offspring maturation reproduction Basic DEB scheme foodfaeces assimilation reserve feeding defecation structure somatic maintenance growth 

10. Production models no accommodation for embryonic stage; require additional state variables (no food intake, still maintenance costs and growth) no metabolic memory, no growth during starvation require switches in case of food shortage (reserves allocated to reproduction used for maintenance) no natural dynamics for reserve; descriptive rules for growth vs reprod. no explanation for body size scaling of metabolic rates, changes in composition of biomass, metabolic memory require complex regulation modelling for fate of metabolites (ATP vs building blocks; consistency problem with lower levels of org.) dividing organisms (with reserve) cannot be included typically have descriptive set points for allocation, no mechanisms (weight-for-age rules quantify allocation to reproduction)

11. Some DEB pillars life cycle perspective of individual as primary target embryo, juvenile, adult (levels in metabolic organization) life as coupled chemical transformations (reserve & structure) time, energy, entropy & mass balances surface area/ volume relationships (spatial structure & transport) homeostasis (stoichiometric constraints via Synthesizing Units) syntrophy (basis for symbioses, evolutionary perspective) intensive/extensive parameters: body size scaling

12. Biomass: reserve(s) + structure(s) Reserve(s), structure(s): generalized compounds, mixtures of proteins, lipids, carbohydrates: fixed composition Compounds in reserve(s): equal turnover times, no maintenance costs structure: unequal turnover times, maintenance costs Reasons to delineate reserve, distinct from structure metabolic memory explanation of respiration patterns (freshly laid eggs don’t respire) biomass composition depends on growth rate fluxes are linear sums of assimilation, dissipation and growth basis of method of indirect calorimetry explanation of inter-species body size scaling relationships

13. Change in body shape Isomorph: surface area  volume 2/3 volumetric length = volume 1/3 V0-morph: surface area  volume 0 V1-morph: surface area  volume 1 Ceratium Mucor Merismopedia

14. Mixtures of V0 & V1 morphs volume,  m 3 hyphal length, mm time, h time, min Fusarium  = 0 Trinci 1990 Bacillus  = 0.2 Collins & Richmond 1962 Escherichia  = 0.28 Kubitschek 1990 Streptococcus  = 0.6 Mitchison 1961 growing in length only

15. Respiration ontogeny in birds age, d ml CO 2 d -1 ml O 2 d -1 altricial Troglodytes aëdon precocial Gallus domesticus Observations: just prior to hatching respiration shows a plateau in precocial, not in altricial birds pore size and frequency in egg shell is such that O 2 flux has constant resistance Conclusion : ontogeny is constrained by diffusion limitation in precocial birds (Rahn et al 1990) DEB theory : reserve dynamics controls ontogeny (same pattern in species without shells) Minimization of water loss causes observed constant flux resistance

16. Embryonic development time, d weight, g O 2 consumption, ml/h ;  : scaled time l : scaled length e: scaled reserve density g: energy investment ratio Crocodylus johnstoni Data from Whitehead 1987 yolk embryo

17. Heat increment of feeding O 2 consumption increases sharply with feeding Present situation: little understood expected increase due to chemistry of protein processing only explains 10 % of observed increase Explanation by DEB theory: elemental balance for assimilation process fluxes of CO 2, H 2 O, O 2, N-waste follow from fluxes of substrate, reserve, structure, products

18. Indirect calorimetry Empirical finding (Lavoisier, 1780): heat dissipation = weighted sum of O 2, CO 2, N-waste fluxes Explanation by DEB theory : Mass & energy balances show that dissipating heat, and all mineral and organic fluxes are weighted sums of 3 basic energy fluxes (assimilation, maintenance, growth) Empirical finding for many micro-organisms : heat dissipation  O 2 flux DEB theory offers: constraints on elemental composition of reserve relative to structure involving chemical potentials of minerals and organic (generalized) compounds gives expression for proportionality factor

19. 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 0.422 n NE 0.312 n HV 1.64 n OV 0.379 n NV 0.189 k E 2.11 h -1 k M 0.021 h -1 y EV 1.135 y XE 1.490 r m 1.05 h -1 g = 1 μ E -1 pApA pMpM pGpG JCJC 0.14 1.00-0.49 JHJH 1.15 0.36-0.42 JOJO -0.35-0.97 0.63 JNJN -0.31 0.31 0.02 Entropy J/C-mol.K Glycerol69.7 Reserve74.9 Structure 52.0 Sousa et al 2004 Interface, subm

20. 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 DEB theory: Product formation rate = w A. Assimilation rate + w M. Maintenance rate + w G. Growth rate For pyruvate: w G <0 Leudeking & Piret (1959): Product formation rate = w M. Maintenance rate + w G. Growth rate Cannot explain observed pattern

21. Yield vs growth 1/spec growth rate, 1/h 1/yield, mmol glucose/ mg cells Streptococcus bovis, Russell & Baldwin (1979) Marr-Pirt (no reserve) DEB spec growth rate yield Russell & Cook (1995): this is evidence for down-regulation of maintenance at high growth rates DEB theory: high reserve density gives high growth rates structure requires maintenance, reserves do not

22. Interactions of substrates Substrate interactions in DEB theory are based on Synthesizing Units (SUs): generalized enzymes that follow the rules of classic enzyme kinetics but working depends in fluxes of substrates, rather than concentrations “concentration” only has meaning in homogeneous environments backward fluxes are small in S + E  SE  EP  E + P Basic classification substrates: substitutable vs complementary processing: sequential vs parellel Mixture between substitutable & complementary substrates: grass  cow; sheep brains  cow; grass + sheep brains  cow Dynamics of SU on the basis of time budgetting offers framework for foraging theory example: feeding in Sparus larvae (Lika, Can J Fish & Aquat Sci, 2005): food searching sequential to mechanic food handling food processing (digestion) parellel to searching & handling gives deviations from Holling type II low high

23. parameter values tend to co-vary across species parameters are either intensive or extensive ratio’s 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 Inter-species body size scaling Kooijman 1986 Energy budgets can explain body size scaling relations J. Theor. Biol. 121: 269-282

24. Primary scaling relationships Extensive parameters K saturation constant L b length at birth L p length at puberty {J Xm } max spec feeding rate {p Am } max spec assim rate [E m ] max reserve capacity Intensive parameters [p M ] volume-spec maint. costs {p T } surface-spec maint. costs [E G ] spec structure costs h a aging acceleration  partitionning fraction  R reproduction efficiency maximum length L m =  {p Am } / [p M ]

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

26. 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 0.0226 L 2 + 0.0185 L 3 0.0516 L 2.44 2 curves fitted: (Daphnia pulex)

27. West-Brown: scaling of respiration Explanation: Minimizing of transportation costs in space-filling fractally branching tube systems results in ¾ - “law” Problems: Protostomes have open circulation system, no tube system scaling of respiration also applies to protostomes Flux in capillaries is much less than in big tubes, not equal Transport rate must match peak metabolic requirements rather than standard No differentiation between inter- and intra-specific scaling Transport costs are tiny fraction of maintenance costs minimum argument is not convincing (nor demonstrated) Scaling of respiration does not explain all other scaling “laws” nor “the growth curve” of demand systems

28. These gouramis are from the same nest, they have the same age and lived in the same tank Social interaction during feeding caused the huge size difference Age-based models for growth are bound to fail;growth depends on food intake they have the same age and lived in the same tank Social interaction during feeding caused the huge size difference Age-based models for growth are bound to fail;growth depends on food intake Growth depends on food availability; no so according to Brown 2001, Nature Trichopsis vittatus Size, not age

29. Banavar: scaling of respiration Explanation: Dilution of biomass with transport material between maintenance-requiring nodes in efficient networks results in ¾ - ”law” Problems: Transport rate must match peak metabolic requirements rather than standard No differentiation between inter- and intra-specific scaling Efficiency criterion

30. 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

31. Incubation time 10 log egg weight, g 10 log incubation time, d l b equal ° tube noses slope = 0.25 Data from Harrison 1975 European birds Incubation time Egg weight Conclusion: tube noses are special, not because of their long incubation but because of their large eggs Intra-species scaling: large eggs  short time to fledging

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

33. Some innovations by DEB theory Unifies all life on earth (bacteria, protoctists, fungi/animals, plants) Links levels of organisation Explains body size scaling relationships Deals with energetic and stoichiometric constraints Individuals that follow DEB rules can merge smoothly into a symbiosis that again follows DEB rules Method for determining entropy of living biomass Biomass composition depends on growth rate Product formation has 3 degrees of freedom Explains indirect calorimetry Explains how yield of biomass depends on growth rate Quantitative predictions have many practical applications