1. DEB theory: a holistic view on 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/ Lewiston, 2006/07/10

2. Contents introduction surface-volume interactions homeostasis & metabolism evolution & symbiosis growth body size scaling producer-consumer dynamics Lewiston, 2006/07/10

3. Dynamic Energy Budget theory for metabolic organisation Uptake of substrates (nutrients, light, food) by organisms and the use of these substrates (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. 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) implied co-variation of parameter values: body size scaling loosely coupled metabolic modules (supply-demand spectrum) syntrophy (basis for symbioses, evolutionary perspective)

7. 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

8. 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;  = aspect ratio

9. Mixtures of changes in shape Dynamic mixtures between morphs Lichen Rhizocarpon V1- V0-morph V1- iso- V0-morph outer annulus behaves as a V1-morph, inner part as a V0-morph. Result: diameter increases  time

10. Homeostasis Homeostasis: constant body composition in varying environments Strong homeostasis  generalized compounds applies to reserve(s) and structure(s) separately Weak homeostasis: ratio reserve/structure becomes and remains constant if food or substrate is constant (while the individual is growing) applies to juvenile and adult stages, not to embryos Structural homeostasis: suborganismal structures have a constant relative size

11. 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

12. 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 submitted

13. Yield vs growth 1/spec growth rate, h 1/yield, mmol glucose/ mg cells Streptococcus bovis, Russell & Baldwin (1979) Marr-Pirt (maintenance, no reserve) DEB (maintenance & reserve) 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 Kooijman & Troost 2006 Biol Rev, to appear

14. Interactions of substrates Kooijman, 2001 Phil Trans R Soc B 356: 331-349 Synthesizing Units (SUs): generalized enzymes that follow the rules of classic enzyme kinetics but working depends on fluxes of substrates, rather than concentrations backward fluxes are small in S + E  SE  EP  E + P

15. Simultaneous nutrient limitation absence of high contents for both compounds due to damming up of reserves, low contents in structure (at zero growth) specific growth rate, d -1 vitamine B 12 10 21 mol cell -1 P-content, fmol cell -1 Pavlova lutheri, 20 ºC Data from Droop 1974 0 70 7 Kooijman 1996 Biophys Chem 73: 179-188

16. Evolution of DEB systems variable structure composition strong homeostasis for structure delay of use of internal substrates increase of maintenance costs inernalization of maintenance installation of maturation program strong homeostasis for reserve 12 345 5 67 reproduction juvenile  embryo + adult 8 Kooijman & Troost 2006 Biol Rev, to appear

17. Symbiosis product substrate

18. Symbiosis substrate

19. Steps in symbiogenesis 1 2 3 4 4 5 Internalization Free-living, clustering Free-living, homogeneous Structures merge Reserves merge Kooijman et al 2003 Biol Rev 78: 435-463

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. 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 Zonneveld & Kooijman 1993 Bull Math Biol 3:609-635

22. 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 : 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 Not age, but size: Trichopsis vittatus

23. Rules for feeding Constant number of food particles at random positions Individuals travel in straight lines to nearest food particle Speed, handling time, growth follow standard DEB rules Food particles are not “visible” if too close to other individuals distance  squared ratio of body lengths

24. time reserve density length time 1 ind 2 ind determin expectation Social interaction  Feeding

25. 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,defence[J EJ ] volume-specific maturity maint. costs allocation  partitioning fraction egg formation  R reproduction efficiency life cycle[E J b ] volume-specific maturity at birth life cycle [E J 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: 269-282

26. Scaling of metabolic rate intra-speciesinter-species maintenance growth Respiration: contributions from growth and maintenance Weight: contributions from structure and reserve Structure ; = length; endotherms Kooijman et al 2006 Sar & Qsar, to appear

27. 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 Data: Hammingsen, 1969 Intra-species 0.0226 L 2 + 0.0185 L 3 0.0516 L 2.44 2 curves fitted: Daphnia pulex; Data: Richman, 1958

28. 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

29. 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

30. Producer/consumer dynamics 0 2468 0 10 20 consumers nutrient 1.752.3 2.4 2.5 2.7 3.0 1.23 1.15 1.0 2.8 1.23 1.53 tangentfocus Hopf Bifurcation diagram isoclines homoclinic producers consumers closed for nutrient; consumer requires producer’s structure & reserve Theor Pop Biol tp appear

31. Food chains n=2 time, h glucose Escherichia coli Dictyostelium mg/ml mm 3 /ml cell vol,  m 3 X 0 (0)0.433mg. ml -1 X 1 (0)0.361X 2 (0)0.084mm 3.ml -1 e 1 (0)1e 2 (0)1- X K1 0.40X K2 0.18 g1g1 0.86g2g2 4.43- k M1 0.008k M2 0.16h -1 k E1 0.67k E2 2.05h -1 j Xm1 0.65j Xm2 0.26 Data from Dent et al 1976 h = 0.064 h -1, X r = 1mg ml -1, 25 °C Kooijman & Kooi,1996 Nonlin. World 3: 77 - 83

32. DEB tele course 2007 http://www.bio.vu.nl/thb/deb/ Free of financial costs; some 200 h effort investment Feb-April 2007; target audience: PhD students We encourage participation in groups that organize local meetings weekly French group of participants of the DEB tele course 2005: special issue of J. Sea Res. 2006 on DEB applications to bivalves Software package DEBtool for Octave/ Matlab freely downloadable Slides of this presentation are downloadable from http://www.bio.vu.nl/thb/users/bas/lectures/ Cambridge Univ Press 2000 Audience : thank you for your attention Organizers : thank you for the invitation