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Dynamic Energy Budget theory for metabolic organization of life Bas Kooijman adult Dept of Theoretical Biology Vrije Universiteit, Amsterdam http://www.bio.vu.nl/thb/deb/ Oldenburg, 2004/05/05 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 Space-time scales space Each process has its characteristic domain of space-time scales system earth ecosystem population individual cell molecule 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 time Empirical special cases of DEB year author model year author model 1780 Lavoisier multiple regression of heat against mineral fluxes 1951 Huggett & Widdas foetal growth 1889 Arrhenius temperature dependence of physiological rates 1951 Weibull survival probability for aging 1891 Huxley allometric growth of body parts 1955 Best diffusion limitation of uptake 1902 Henri Michaelis--Menten kinetics 1957 Smith embryonic respiration 1905 Blackman bilinear functional response 1959 Leudeking & Piret microbial product formation 1920 Pütter von Bertalanffy growth of individuals 1959 Holling hyperbolic functional response 1927 Pearl logistic population growth 1962 Marr & Pirt maintenance in yields of biomass 1928 Fisher & Tippitt Weibull aging 1973 Droop reserve (cell quota) dynamics 1932 Kleiber respiration scales with body weight3/ 4 1974 Rahn & Ar water loss in bird eggs 1932 Mayneord cube root growth of tumours 1975 Hungate digestion 1950 Emerson cube root growth of bacterial colonies 1977 Beer & Anderson development of salmonid embryos 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 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 volume2/3 V0-morphs: surface area volume0 V1-morphs: surface area volume1 • many active enzyme linked to membranes (surfaces) substrate and product concentrations linked to volumes Biomass: reserve(s) + structure(s) substrate(s) reserve structure Reserve(s), structure(s): generalized compounds, mixtures of proteins, lipids, carbohydrates: fixed composition Compounds in reserve(s): equal turnover times, no maintenance costs structure(s): unequal turnover times, maintenance costs Reasons to delineate reserve, distinct from structure • metabolic memory • 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 respiration patterns (freshly laid eggs don’t respire) • fate of metabolites (e.g. conversion into energy vs buiding blocks) nOW nNW Spec growth rate, h-1 kE 2.11 h-1 kM 0.021 h-1 yVE 0.904 yXE 1.35 rm 1.05 h-1 g = 1 Spec prod, mol.mol-1.h-1 Data Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C -1 nHE 1.66 nOE 0.422 nNE 0.312 •μE nHW nHV 1.64 nOV 0.379 nNV 0.189 J C Weight yield, mol.mol-1 Relative abundance Biomass composition pA pM pG 0.14 1.00 -0.49 JH 1.15 0.36 -0.42 JO -0.35 -0.97 0.63 JN -0.31 0.31 0.02 O2 CO2 Spec growth rate Spec growth rate, h-1 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) 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 -rule for allocation vL2 k M L3 Ingestion rate, 105 cells/h O2 consumption, g/h Respiration Length, mm • 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 Age, d in juveniles? vL2 kM L3 (1 g / f )kM L3p fL2 Length, mm Length, mm Cum # of young Reproduction Ingestion Growth: d L rB ( L L) dt Von Bertalanffy Age, d Embryonic development weight, g embryo yolk time, d d e e g ; d l g e l dτ l dτ 3 e g J O J O , M l J O ,G 3 d 3 l dτ O2 consumption, ml/h Crocodylus johnstoni, Data from Whitehead 1987 time, d : scaled time l : scaled length e: scaled reserve density g: energy investment ratio 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 Mitochondria TriCarboxylic Acid cycle Enzymes pass metabolites directly to other enzymes enzymes catalizing transformations 5 & 7: bound to inner membrane (and FAD/FADH2) Net transformation: Acetyl-CoA + 3 NAD+ + FAD + GDP 3- + Pi2- + 2 H2O = 2 CO2 + 3 NADH + FADH2 + GTP 4- + 2 H+ + HS-CoA Dual function of intermediary metabolites building blocks energy substrate Transformations: 1 Oxaloacetate + Acetyl CoA + H2O = Citrate + HSCoA 2 Citrate = cis-Aconitrate + H2O 3 cis-Aconitrate + H2O = Isocitrate 4 Isocitrate + NAD+ = α-Ketoglutarate + CO2 + NADH + H+ 5 α-Ketoglutarate + NAD+ + HSCoA = Succinyl CoA + CO2 + NADH + H+ 6 Succinyl CoA + GDP 3- + Pi 2- + H+ = Succinate + GTP 4- + HSCoA 7 Succinate + FAD = Fumarate + FADH2 8 Fumarate + H2O = Malate 9 Malate + NAD+ = Oxaloacetate + NADH + H+ all eukaryotes once possessed mitochondria, most still do Pathways & allocation structure structure maintenance reserve maintenance reserve structure maintenance reserve Mixture of products & intermediary metabolites that is allocated to maintenance (or growth) has constant composition Kooijman & Segel, 2004 Numerical matching for n=4 Product flux 1 2 3 4 Unbounded fraction 0 4 3 2 1 Spec growth rate Rejected flux 0 1 2 3 Spec growth rate = 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 nSE = 0.032,0.032,0.032,0.032 # in reserve nSV = 0.045,0.045,0.045,0.045 # in structure yEV = 1.2 res/struct kE = 0.4 res turnover jEM = 0.02 maint flux n0E = 0.05 sub in res 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 nSi E , nSiV and/or binding and handshaking parameters ρi , αi 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) Product Formation According to Dynamic Energy Budget theory: pyruvate, mg/l Product formation rate = wA . Assimilation rate + wM . Maintenance rate + wG . Growth rate For pyruvate: wG<0 Applies to all products, heat & non-limiting substrates Indirect calorimetry (Lavoisier, 1780): heat = wO JO + wC JC + wN JN No reserve: 2-dim basis for product formation throughput rate, h-1 Glucose-limited growth of Saccharomyces Data from Schatzmann, 1975 Symbiosis substrate product Product formation is basic to symbioses Symbiosis substrate substrate Product formation is basic to symbioses Steps in symbiogenesis Free-living, homogeneous Structures merge Free-living, clustering Internalization Reserves merge 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 Central Metabolism source polymers monomers waste/source Modules of central metabolism • Pentose Phosphate (PP) cycle glucose-6-P ribulose-6-P, NADP NADPH • Glycolysis glucose-6-P pyruvate ADP + P ATP • TriCarboxcyl Acid (TCA) cycle pyruvate CO2 NADP NADPH • Respiratory chain NADPH + O2 NADP + H2O ADP + P ATP Evolution of central metabolism in prokaryotes (= bacteria) 3.8 Ga 2.7 Ga i = inverse ACS = acetyl-CoA Synthase pathwayRC = Respiratory Chain Kooijman, Hengeveld 2003 The symbiontic nature of PP = Pentose Phosphate cycle Gly = Glycolysis metabolic evolution TCA = TriCarboxylic Acid cycle Acta Biotheoretica (to appear) Prokaryotic metabolic evolution Heterotrophy: • pentose phosph cycle • glycolysis • respiration chain Phototrophy: • el. transport chain • PS I & PS II • Calvin cycle Chemolithotrophy • acetyl-CoA pathway • inverse TCA cycle • inverse glycolysis Symbiogenesis 1.5-2 Ga 1.2 Ga Sizes of blobs do not reflect number of species Survey of organisms Myxomycota Protostelida Bikont DHFR-TS gene fusion loss phagoc.Apusozoa membr. dyn unikont mainly celllose gap junctions tissues (nervous) mitochondria bicentriolar primary mainly chitin chloroplast EF1 insertion secondary Plasmodiophoromycota Chlorarachnida Cercozoa Cercomonada chloroplast Amoebozoa Archamoeba tertiary chloroplast photo symbionts Bacteria Bacteria Rhizopoda Sporozoa Percolozoa Excavates Euglenozoa Loukozoa AlveoDinozoa lates Ciliophora chloroplasts Chytridiomycota cortical alveoli Actinopoda (brown algae) Phaeophyceae Xanthophyceae Raphidophyceae Chrysophyceae Synurophyceae Eustigmatophyceae Labyrinthulomycota Dictyochophyceae Bicosoecia Pedinellophyceae Pelagophyceae Bigyromonada Bacillariophyceae Pseudofungi (diatoms) Bolidophyceae Opalinata Prymnesiophyceae Metamonada Cryptophyceae triple roots Granuloreticulata forams Xenophyophora Basidiomycota Ascomycota fungi Glomeromycota Zygomycota Microsporidia animals animals Choanozoa Composed by Bas Kooijman (plants) Cormophyta (green algae) Chlorophyceae Plantae (red algae) Rhodophyceae Glaucophyceae 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 Lm { p A } κ / [ pM ] allocation fraction to growth + maint. (intensive) [ pM ] volume-specific maintenance power (intensive) { p A } surface area-specific assimilation power (extensive) • conclusion : { p A } Lm (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 Scaling of metabolic rate Respiration: contributions from growth and maintenance Weight: contributions from structure and reserve 3 Structure l ; l = length; endotherms lh 0 comparison intra-species inter-species maintenance lh l l 3 lh l l 3 growth l l 2 l 3 0 l0 l ls l 2 l 3 dl 3 lh l 2 l 3 dV l 3 d E l 4 reserve structure respiratio n weight Scaling of metabolic rate slope = 1 0.0226 L2 + 0.0185 L3 0.0516 L2.44 Log metabolic rate, w O2 consumption, l/h 2 curves fitted: endotherms slope = 2/3 ectotherms unicellulars Length, cm Intra-species (Daphnia pulex) Log weight, g Inter-species Length, mm Von Bertalanffy growth Data from Greve, 1972 log rB Age, d Arrhenius TA 6400 K T 1 L(t ) L ( L Lb ) e rB t L length; rB von Bert growh rate Von Bertalanffy growth rate L(t ) L ( L Lb ) e rB t rB1 3 ([ EG ] f κ[ Em ]) [ pM ]1 L f κ length [ EG ] spec growth costs func resp [ Em ] spec reserve capacity fraction [ pm ] spec maint costs