Introduction to DEB theory

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 Daphnia von Foerster ecotox application embryos 1980 body size scaling epidemiol applications morph dynamics indirect calorimetry bifurcation analysis micro’s numerical methods food chains 1990 Global bif-analysis aging DEB 1 Synthesizing Units NECs integral formulations DEBtox 1 multivar plants adaptive dynamics tumour induction ecosystem dynamics 2000 DEB 2 adaptation organ function symbioses ecosystem Self-orginazation ISO/OECD molecular organisation

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

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 Empirical special cases of DEB

Space-time scales Each process has its characteristic domain of space-time scales system earth space ecosystem population 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 individual cell time molecule

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  volume2/3 volumetric length = volume1/3 Mucor Ceratium Merismopedia V0-morph: surface area  volume0 V1-morph: surface area  volume1

Shape correction function actual surface area at volume V isomorphic surface area at volume V Shape correction function at volume V = for V0-morph V1-morph isomorph Static mixtures between V0- and V1-morphs for aspect ratio

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

Biofilms solid substrate biomass Isomorph: V1= 0 mixture between iso- & V0-morph V0-morph: V1=  biomass grows, but surface area that is involved in nutrient exchange does not

Arrhenius relationship 2.6 ln pop growth rate, h-1 r1 = 1.94 h-1 T1 = 310 K TH = 318 K TL = 293 K TA = 4370 K TAL = 20110 K TAH = 69490 K 103/T, K-1 103/TH 103/TL

Von Bertalanffy growth Length, mm 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

defecation feeding food faeces assimilation reserve somatic maintenance maturity maintenance  1- maturation reproduction growth maturity offspring structure Basic DEB scheme

Competitive tumour growth Allocation to tumour  relative maint workload defecation feeding food faeces assimilation Isomorphy: is constant Tumour tissue: low spec growth costs low spec maint costs reserve somatic maintenance maturity maintenance  1- maint maturation reproduction u 1-u growth 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 structure tumour

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  Respiration  Ingestion rate, 105 cells/h O2 consumption, g/h Length, mm Length, mm Length, mm Reproduction  Cum # of young • 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? Growth: Von Bertalanffy Age, d Age, d

Embryonic development Crocodylus johnstoni, Data from Whitehead 1987 embryo yolk O2 consumption, ml/h weight, g time, d 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

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

Simultaneous Nutrient Limitation B12 content, 10-21 mol/cell P content, fmol/cell Specific growth rate of Pavlova lutheri as function of intracellular phosphorus and vitamin B12 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

Von Bertalanffy growth rate

Biomass composition Data Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C nHW Entropy J/C-mol.K Glycerol 69.7 Reserve 74.9 Structure 52.0 Sousa et al 2004 Interface, subm Relative abundance nOW O2 nNW Weight yield, mol.mol-1 Spec prod, mol.mol-1.h-1 Spec growth rate, h-1 CO2 Spec growth rate kE 2.11 h-1 kM 0.021 h-1 yEV 1.135 yXE 1.490 rm 1.05 h-1 g = 1 nHE 1.66 nOE 0.422 nNE 0.312 nHV 1.64 nOV 0.379 nNV 0.189 Spec growth rate, h-1

Yield vs growth Streptococcus bovis, Russell & Baldwin (1979) Marr-Pirt (no reserve) DEB 1/yield, mmol glucose/ mg cells 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 V1-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

Metabolic rate slope = 1 Log metabolic rate, w O2 consumption, l/h 2 curves fitted: endotherms 0.0226 L2 + 0.0185 L3 0.0516 L2.44 ectotherms slope = 2/3 unicellulars Log weight, g Length, cm Intra-species Inter-species (Daphnia pulex)

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, Kooi, Kooijman

Introduction to DEB theory