Introduction to DEB theory Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Marseille, 2005/12/15

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

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 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/ Hungate 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

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

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 V0-morph: surface area  volume 0 V1-morph: surface area  volume 1 Ceratium Mucor Merismopedia

Shape correction function at volume V actual surface area at volume V isomorphic surface area 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 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

Biofilms Isomorph: V 1 = 0 V0-morph: V 1 =  mixture between iso- & V0-morph biomass grows, but surface area that is involved in nutrient exchange does not solid substrate biomass

Arrhenius relationship /T, K -1 ln pop growth rate, h /T H 10 3 /T L r 1 = 1.94 h -1 T 1 = 310 K T H = 318 K T L = 293 K T A = 4370 K T AL = K T AH = K

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

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

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

1-  1-  u Competitive tumour growth foodfaeces assimilation reserve feeding defecation structure somatic maintenance growth maturity maintenance maturity offspring maturation reproduction tumour  uu Allocation to tumour  relative maint workload Isomorphy: is constant Tumour tissue: low spec growth costs low spec maint costs Van Leeuwen et al., 2003 The embedded tumour: host physiology is important for the evaluation of tumour growth. British J Cancer 89, maint

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 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

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

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

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, mol/cell

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:

Scaling of metabolic rate comparisonintra-speciesinter-species maintenance growth 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 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 n NE n HV 1.64 n OV n NV k E 2.11 h -1 k M h -1 y EV y XE r m 1.05 h -1 g = 1 μ E -1 pApA pMpM pGpG JCJC JHJH JOJO JNJN Entropy J/C-mol.K Glycerol69.7 Reserve74.9 Structure 52.0 Sousa et al 2004 Interface, subm

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

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,

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:

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

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 L L L curves fitted: (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:

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