1. Dynamic Energy Budget theory for metabolic organization of life Bas Kooijman Dept of Theoretical Biology Vrije Universiteit, Amsterdam http://www.bio.vu.nl/thb/deb/ Leiden, 2004/02/06 adult embryo juvenile

2. Dynamic Energy Budget theory 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 Applications in ecotoxicology biotechnology Direct links with empiry

3. Space-time scales molecule cell individual population ecosystem system earth time space 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

4. Empirical special cases of DEB yearauthormodelyearauthormodel 1780Lavoisier multiple regression of heat against mineral fluxes 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 1920Pütter von Bertalanffy growth of individuals 1959Holling hyperbolic functional response 1927Pearl logistic population growth 1962Marr & Pirt maintenance in yields of biomass 1928Fisher & Tippitt Weibull aging 1973Droop reserve (cell quota) dynamics 1932Kleiber respiration scales with body weight 3/ 4 1974Rahn & Ar water loss in bird eggs 1932Mayneord cube root growth of tumours 1975Hungate digestion 1950Emerson cube root growth of bacterial colonies 1977Beer & Anderson development of salmonid embryos

5. 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 & 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

6. Surface area/volume interactions nutrient supply to ecosystems (erosion)  surface area production (nutrient concentration)  volume food availability for cows: grass weight/ surface area food availability for daphnids: algal weight/ volume feeding rate  surface area; maintenance rate  volume isomorphs: surface area  volume 2/3 V0-morphs: surface area  volume 0 V1-morphs: surface area  volume 1 many active enzyme linked to membranes (surfaces) substrate and product concentrations linked to volumes

7. 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

8. 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

9. General assumptions State variables: structural body mass & reserves they do not change in composition Food is converted into faeces Assimilates derived from food are added to reserves, which fuel all other metabolic processes Three categories of processes: Assimilation: synthesis of (embryonic) reserves Dissipation: no synthesis of biomass Growth: synthesis of structural body mass Product formation: included in these processes (overheads) Basic life stage patterns dividers (correspond with juvenile stage) reproducers embryo (no feeding initial structural body mass is negligibly small initial amount of reserves is substantial) juvenile (feeding, but no reproduction) adult (feeding & male/female reproduction)

10. Specific assumptions Reserve density hatchling = mother at egg formation foetuses: embryos unrestricted by energy reserves Stage transitions: cumulated investment in maturation > threshold embryo  juvenile initiates feeding juvenile  adult initiates reproduction & ceases maturation Somatic & maturity maintenance  structure volume (but some maintenance costs  surface area) maturity maintenance does not increase after a given cumulated investment in maturation Feeding rate  surface area; fixed food handling time Partitioning of reserves should not affect dynamics comp. body mass does not change at steady state (weak homeostasis) Fixed fraction of catabolic energy is spent on somatic maintenance + growth (  -rule) Starving individuals: priority to somatic maintenance do not change reserve dynamics; continue maturation, reproduction. or change reserve dynamics; cease maturation, reprod.; do or do not shrink in structure

11.  -rule for allocation Age, d Length, mm Cum # of young Length, mm Ingestion rate, 10 5 cells/h O 2 consumption,  g/h 80% of adult budget to reproduction in daphnids puberty at 2.5 mm No change in ingest., resp., or growth Where do resources for reprod come from? Or: What is fate of resources in juveniles? Respiration  Ingestion  Reproduction  Growth: Von Bertalanffy

12. 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

13. Synthesizing units Generalized enzymes that 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 Concentration: problematic (intracellular) environments: spatially heterogeneous state variables in dynamic systems In spatially homogeneous environments: arrival fluxes  concentrations

14. Mitochondria Transformations: 1 Oxaloacetate + Acetyl CoA + H 2 O = Citrate + HSCoA 2 Citrate = cis-Aconitrate + H 2 O 3 cis-Aconitrate + H 2 O = Isocitrate 4 Isocitrate + NAD + = α-Ketoglutarate + CO 2 + NADH + H + 5 α-Ketoglutarate + NAD + + HSCoA = Succinyl CoA + CO 2 + NADH + H + 6 Succinyl CoA + GDP 3- + P i 2- + H + = Succinate + GTP 4- + HSCoA 7 Succinate + FAD = Fumarate + FADH 2 8 Fumarate + H 2 O = Malate 9 Malate + NAD + = Oxaloacetate + NADH + H + TriCarboxylic Acid cycle Enzymes pass metabolites directly to other enzymes enzymes catalizing transformations 5 & 7: bound to inner membrane (and FAD/FADH 2 ) Net transformation: Acetyl-CoA + 3 NAD + + FAD + GDP 3- + P i 2- + 2 H 2 O = 2 CO 2 + 3 NADH + FADH 2 + GTP 4- + 2 H + + HS-CoA Dual function of intermediary metabolites building blocks  energy substrate all eukaryotes once possessed mitochondria, most still do

15. Pathway  whole cell Fixed stoichiometries for maintenance and growth Variable overall stoichiometry Can a model for pathway kinetics give this result?

16. Constraints on pathway dynamics Amount of SU i per mol of structure: Supply flux to pathway: metabolite i enzyme i rejection, production flux spec maintenance flux spec growth rate yield coefficients (fixed) reserve density M E /M V abundance of X 0 in E abundance of S i in E, V Requirements by cell for maintenance and growth:

17. Linear pathway SU i SU i+1 : Product i : Intermediary metabolite i = If all metabolites would follow the full pathway:

18. Chain of length 1 Change in Unbounded fraction Steady state Unbounded fraction Production flux

19. Closed handshaking at all nodes Change in Unbounded fractions Steady state Unbounded fractions Production fluxes

20. Open handshaking at all nodes Change in Unbounded fractions Steady state Unbounded fractions Production fluxes

21. General handshaking Change in Unbounded fractions Steady state Unbounded fractions Production fluxes Rejection fluxes

22. Numerical matching for n=4 Product flux Rejected flux Unbounded fraction  = 0.73, 0.67, 0.001, 0.27 handshaking  = 0.67, 0.91, 0.96, 0.97 binding prob k = 0.12, 0.19, 0.54, 0.19 dissociation n SE = 0.032,0.032,0.032,0.032 # in reserve n SV = 0.045,0.045,0.045,0.045 # in structure y EV = 1.2 res/struct k E = 0.4 res turnover j EM = 0.02 maint flux n 0E = 0.05 sub in res 0 0 1 1 1 2 2 2 3 3 3 4 4 Spec growth rate

23. Matching pathway  whole cell No exact match possible between production of products and intermediary metabolites by pathway and requirements by the cell But very close approximation is possible by tuning abundance parameters and/or binding and handshaking parameters Best approximation requires all four tuning parameters per node growth-dependent reserve abundance plays a key role in tuning Kooijman, S. A. L. M. and Segel, L. A. (2004) How growth affects the fate of cellular substrates. Bull. Math. Biol. (to appear)

24. Simultaneous Substrate Processing Chemical reaction: 1A + 1B 1C Poisson arrival events for molecules A and B blocked time intervals acceptation event ¤ rejection event Flux of C: production

25. Simultaneous Nutrient Limitation Specific growth rate of Pavlova lutheri as function of intracellular phosphorus and vitamin 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) P content, fmol/cell B 12 content, 10 -21 mol/cell

26. 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 Applies to all products, heat & non-limiting substrates Indirect calorimetry (Lavoisier, 1780): heat = w O J O + w C J C + w N J N No reserve: 2-dim basis for product formation

27. Symbiosis product substrate Product formation is basic to symbioses

28. Symbiosis substrate Product formation is basic to symbioses

29. Internalization Structures merge Reserves merge Free-living, clustering Free-living, homogeneous Steps in symbiogenesis

30. Symbiogenesis symbioses: fundamental organization of life based on syntrophy ranges from weak to strong interactions; basis of biodiversity symbiogenesis: evolution of eukaryotes (mitochondria, plastids) DEB model is closed under symbiogenesis: it is possible to model symbiogenesis of two initially independently living populations that follow the DEB rules by incremental changes of parameter values such that a single population emerges that again follows the DEB rules essential property for models that apply to all organisms Kooijman, Auger, Poggiale, Kooi 2003 Quantitative steps in symbiogenesis and the evolution of homeostasis Biological Reviews 78: 435 - 463

31. 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: 269-282

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

33. Scaling of 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)

34. Von Bertalanffy growth Length, mm Age, d Arrhenius Data from Greve, 1972

35. Von Bertalanffy growth rate

36. PhD Vacancies at Dept Theor Biol VUA 3 in NWO program Computational Life Sciences possibly 6 more in 2004 (EU, STW, NWO) see http://www.bio.vu.nl/thb/http://www.bio.vu.nl/thb/ selection date: 11 Feb 2004 International tele-course on DEB applications in ecotoxicology 11 march – 15 april 2004 free registration is open see http://www.bio.vu.nl/thb/http://www.bio.vu.nl/thb/