Introduction to DEB theory Bas Kooijman Dept theoretical
Introduction to DEB theory Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Bas@bio. vu. nl http: //www. bio. vu. nl/thb/ Marseille, 2005/12/15
DEB – ontogeny - IBM 1980 Daphnia ecotox embryos application body size scaling epidemiol applications morph dynamics indirect calorimetry micro’s food chains 1990 NECs DEBtox 1 Synthesizing Units multivar plants 2000 tumour induction organ adaptation function ISO/OECD bifurcation analysis numerical methods Global bif-analysis aging DEB 1 DEB 2 von Foerster molecular organisation ecosystem dynamics integral formulations adaptive dynamics symbioses ecosystem Self-orginazation
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
Empirical special cases of DEB year author model 1780 Lavoisier multiple regression of heat against mineral fluxes 1950 Emerson cube root growth of bacterial colonies 1825 Gompertz Survival probability for aging 1951 Huggett & Widdas foetal growth 1889 Arrhenius 1902 temperature dependence of DEB theory is rates axiomatic, 1951 Weibull physiological allometric of body parts Huxleybased 1955 Best on growth mechanisms Michaelis--Menten kinetics empirical Henri not meant 1957 Smith to glue models 1905 Blackman 1910 Hill 1891 1920 1927 bilinear functional response 1959 Leudeking & Piret survival probability for aging diffusion limitation of uptake embryonic respiration microbial product formation Cooperative binding hyperbolic functional response 1959 Holling Since many empirical models von Bertalanffy growth of maintenance in yields of biomass Pütter 1962 Marr & Pirt individuals turn out to be special cases of DEB theory logistic population growth reserve (cell quota) dynamics Pearl Droop the data behind these 1973 models support DEB theory 1928 Fisher & Tippitt 1932 Kleiber 1932 Mayneord Weibull aging 1974 Rahn & Ar water loss in bird eggs This makes DEB theory very tested against data respiration scales with body digestion 1975 well Hungate weight 3/ 4 cube root growth of tumours 1977 Beer & Anderson development of salmonid embryos
Space-time scales Each process has its characteristic domain of space-time scales space 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
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 2. 2 • biosphere: thin skin wrapping the earth light from outside, nutrient exchange from inside is across surfaces production (nutrient concentration) volume of environment • food availability for cows: amount of grass per surface area environ food availability for daphnids: amount of algae per volume environ • feeding rate surface area; maintenance rate volume (Wallace, 1865) • many enzymes are only active if linked to membranes (surfaces) substrate and product concentrations linked to volumes change in their concentrations gives local info about cell size; ratio of volume and surface area gives a length
Change in body shape Isomorph: surface area volume 2/3 volumetric length = volume 1/3 Mucor Ceratium V 0 -morph: surface area volume 0 Merismopedia V 1 -morph: surface area volume 1
Shape correction function at volume V = actual surface area at volume V isomorphic surface area at volume V for V 0 -morph V 1 -morph isomorph Static mixtures between V 0 - and V 1 -morphs for aspect ratio
Mixtures of changes in shape Dynamic mixtures between morphs V 1 - V 0 -morph outer annulus behaves as a V 1 -morph, inner part as a V 0 -morph. Result: diameter increases time Lichen Rhizocarpon V 1 - iso- V 0 -morph
Biofilms solid substrate biomass Isomorph: V 1 = 0 mixture between iso- & V 0 -morph: V 1 = biomass grows, but surface area that is involved in nutrient exchange does not
ln pop growth rate, h-1 Arrhenius relationship 103/T, K-1 103/TH 103/TL 2. 6 r 1 = 1. 94 h-1 T 1 = TH = TL = 310 K 318 K 293 K TA = 4370 K TAL = 20110 K TAH = 69490 K
Length, mm Von Bertalanffy growth Data from Greve, 1972 Arrhenius Age, d
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
Basic DEB scheme food feeding defecation faeces assimilation somatic maintenance growth structure reserve 1 - maturity maintenance maturation reproduction maturity offspring
Competitive tumour growth Allocation to tumour relative maint workload food defecation feeding faeces assimilation somatic maintenance growth structure reserve maint 1 - u 1 - u tumour maturity maintenance Isomorphy: is constant Tumour tissue: low spec growth costs low spec maint costs maturation reproduction maturity offspring Van Leeuwen et al. , 2003 The embedded tumour: host physiology is important for the evaluation of tumour growth. British J Cancer 89, 2254 -2268
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
-rule for allocation Ingestion rate, 105 cells/h O 2 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? Length, mm Cum # of young Reproduction Ingestion Growth: Von Bertalanffy Age, d
Embryonic development weight, g embryo yolk time, d ; O 2 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 EP E + P with two modifications: • back flux is negligibly small E + S 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
Simultaneous Substrate Processing Flux of C: production Chemical reaction: 1 A + 1 B 1 C Poisson arrival events for molecules A and B blocked time intervals • acceptation event ¤ rejection event
Simultaneous Nutrient Limitation B 12 10 - 21 co nte mo l/c e nt, ll l/cell t, fmo P conten 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)
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
Scaling of metabolic rate Respiration: contributions from growth and maintenance Weight: contributions from structure and reserve Structure ; = length; endotherms comparison maintenance growth intra-species inter-species
Von Bertalanffy growth rate
Biomass composition n. OW n. NW Spec growth rate, h-1 k. E 2. 11 k. M 0. 021 y. EV 1. 135 y. XE 1. 490 rm 1. 05 h-1 g = 1 h-1 Sousa et al 2004 Interface, subm Weight yield, mol-1 Reserve 74. 9 Structure 52. 0 Spec prod, mol-1. h-1 Relative abundance Data Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C • μE-1 n. HW Entropy J/C-mol. K Glycerol 69. 7 JC p. A p. M p. G 0. 14 1. 00 -0. 49 JH 1. 15 0. 36 -0. 42 JO -0. 35 -0. 97 0. 63 JN -0. 31 0. 02 O 2 CO 2 Spec growth rate n. HE 1. 66 n. OE 0. 422 n. NE 0. 312 n. HV 1. 64 n. OV 0. 379 n. NV 0. 189 Spec growth rate, h-1
Yield vs growth 1/yield, mmol glucose/ mg cells Streptococcus bovis, Russell & Baldwin (1979) Marr-Pirt (no reserve) DEB spec growth rate yield 1/spec growth rate, 1/h Russell & Cook (1995): this is evidence for down-regulation of maintenance at low growth rates DEB theory: high reserve density gives high growth rates structure requires maintenance, reserves not
Synthesizing Unit dynamics SU: Generalized enzyme that operates on fluxes of metabolites Typical form for changes in bounded fractions Typical flux of metabolites for Mixing of types: Example of mixture between sequential & complementary substrates:
Interactions of substrates Kooijman, 2001 Phil Trans R Soc B 356: 331 -349
Co-metabolism Co-metabolic degradation of 3 -chloroaniline by Rhodococcus with glucose as primary substrate Data from Schukat et al, 1983 Brandt et al, 2003 Water Research 37, 4843 -4854
Size-structured Unstructured Population Dynamics Isomorphs: individual-based or pde formulation V 1 -morphs: unstructured (ode) formulation Effect of individuality becomes small if ratio between largest and smallest body size reduces This suggest a perturbation method to approximate a pde with an ode formulation Need for simplification of ecosystem dynamics
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
Scaling of metabolic rate Respiration: contributions from growth and maintenance Weight: contributions from structure and reserve Structure ; = length; endotherms comparison maintenance growth intra-species inter-species
Metabolic rate slope = 1 0. 0226 L 2 + 0. 0185 L 3 0. 0516 L 2. 44 Log metabolic rate, w O 2 consumption, l/h 2 curves fitted: endotherms ectotherms slope = 2/3 unicellulars Length, cm Intra-species (Daphnia pulex) Log weight, g Inter-species
1 -species mixotroph community Mixotrophs are producers, which live off light and nutrients as well as decomposers, which live off organic compounds which they produce by aging Simplest community with full material cycling Kooijman, Dijkstra, Kooi 2002 J. Theor. Biol. 214: 233 -254
Canonical community Short time scale: Mass recycling in a community closed for mass open for energy Long time scale: Nutrients leaks and influxes Memory is controlled by life span (links to body size) Spatial coherence is controlled by transport (links to body size) Kooijman, Nisbet 2000 How light and nutrients affect life in a closed bottle. In: Jørgensen, S. E (ed) Thermodynamics and ecological modelling. CRC, 19 -60
Self organisation of ecosystems • homogeneous environment, closed for mass • start from mono-species community of mixotrophs • parameters constant for each individual • allow incremental deviations across generations link extensive parameters (body size segregation) • study speciation using adaptive dynamics • allow cannibalism/carnivory • study trophic food web/piramid: coupling of structure & function • study co-evolution of life, geochemical dynamics , climate • adaptive dynamics applied to multi-character DEB models Troost et al 2004 Math Biosci, to appear; Troost et al 2004 Am Nat, submitted Collaboration: Metz, Troost, Kooijman
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