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 Oslo 2012/02/09 -10

Contents • preliminary concepts required to link predictions to data • standard DEB model for a 1 -food, 1 -reserve, 1 -structure isomorph • implications & extensions • covariation of parameter values

Energy Budgets Basic processes Life history events • Feeding • Digestion • Storing • Growth • Maturation • Maintenance • Reproduction • Product formation • Aging • zero: start of development • birth: start of feeding start of acceleration • metamorphosis: end of acceleration • puberty: end of maturation start of reproduction All have ecological implications All interact during the life cycle Life stages embryo juvenile adult

Empirical patterns: stylised facts Feeding During starvation, organisms are able to reproduce, grow and survive for some time At abundant food, the feeding rate is at some maximum, independent of food density Growth Respiration Animal eggs and plant seeds initially hardly use O 2 The use of O 2 increases with decreasing mass in embryos and increases with mass in juveniles and adults The use of O 2 scales approximately with body weight raised to a power close to 0. 75 Animals show a transient increase in metabolic rate after ingesting food (heat increment of feeding) Many species continue to grow after reproduction has started Growth of isomorphic organisms at abundant food is well described by the von Bertalanffy The chemical composition of organisms depends on For different constant food levels the inverse von Bertalanffy growth rate increases linearly with the nutritional status (starved vs well-fed) ultimate length The chemical composition of organisms growing The von Bertalanffy growth rate of different species at constant food density becomes constant decreases almost linearly with the maximum body length Fetuses increase in weight approximately Dissipating heat is a weighted sum of 3 mass flows: proportional to cubed time CO 2, O 2 and N-waste Stoichiometry Energy Reproduction increases with size intra-specifically, but decreases with size inter-specifically

Supply-demand spectrum 1. 2. 5

Static Energy Budgets (SEBs) Differences with DEBs • overheads interpretation of respiration interpretation of urination • metabolic memory • life cycle perspective change in states gross ingested faeces apparent assimilated maintenance urine gross metabolised spec dynamic action net metabolised production work somatic activity maintenance growth products thermo reproduction regulation

Not : age, but size: 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 Trichopsis vittatus

Empirical special cases of DEB 11. 1 year author model 1780 Lavoisier multiple regression of heat against mineral fluxes 1950 Emerson cube root growth of bacterial colonies 1825 Gompertz 1891 Survival probability for aging DEB theory is axiomatic, 1951 Huggett & Widdas temperature dependence of Arrhenius 1951 Weibull based on mechanisms physiological rates allometric growth of body parts Huxleynot meant 1955 Best to glue empirical models 1902 Henri 1905 Blackman 1889 1910 1920 Michaelis--Menten kinetics 1957 Smith foetal growth survival probability for aging diffusion limitation of uptake embryonic respiration bilinear functional response 1959 Leudeking & Piret microbial product formation Since many empirical models Cooperative binding hyperbolic functional response Hill 1959 Holling turn out to be special cases of DEB theory von Bertalanffy growth of maintenance in yields of biomass Pütter 1962 Marr & Pirt individuals the data behind these models support DEB theory 1927 Pearl logistic population growth 1973 Droop reserve (cell quota) dynamics 1928 Fisher & Tippitt Weibull aging 1974 Rahn & Ar water loss in bird eggs 1932 Kleiber respiration scales with body weight 3/ 4 1975 Hungate digestion 1932 Mayneord cube root growth of tumours 1977 Beer & Anderson development of salmonid embryos This makes DEB theory very well tested against data

Biomass: reserve(s) + structure(s) Reserve(s), structure(s): generalized compounds, mixtures of proteins, lipids, carbohydrates: fixed composition Reasons to delineate reserve, distinct from structure • metabolic memory • biomass composition depends on growth rate • explanation of respiration patterns (freshly laid eggs don’t respire) method of indirect calorimetry fluxes are linear sums of assimilation, dissipation and growth fate of metabolites (e. g. conversion into energy vs buiding blocks) inter-species body size scaling relationships

Reserve vs structure 2. 3 Reserve does not mean: “set apart for later use” compounds in reserve can have active functions Life span of compounds in • reserve: limited due to turnover of reserve all reserve compounds have the same mean life span • structure: controlled by somatic maintenance structure compounds can differ in mean life span Important difference between reserve and structure: no maintenance costs for reserve Empirical evidence: freshly laid eggs consist of reserve and do not respire

Homeostasis strong constant composition of pools (reserves/structures) generalized compounds, stoichiometric constraints on synthesis weak constant composition of biomass during growth in constant environments determines reserve dynamics (in combination with strong homeostasis) structural constant relative proportions during growth in constant environments isomorphy. work load allocation thermal ectothermy homeothermy endothermy acquisition supply demand systems; development of sensors, behavioural adaptations

Body size • length: depends on shape and choice (shape coefficient) volumetric length: cubic root of volume; does not depend on shape contribution of reserve in lengths is usually small use of lengths unavoidable because of role of surfaces and volumes • weight: wet, dry, ash-free dry contribution of reserve in weights can be substantial easy to measure, but difficult to interpret • C-moles (number of C-atoms as multiple of number of Avogadro) 1 mol glucose = 6 Cmol glucose useful for mass balances, but destructive measurement Problem: with reserve and structure, body size becomes bivariate We have only indirect access to these quantities

Flux vs Concentration • concept “concentration” implies spatial homogeneity (at least locally) biomass of constant composition for intracellular compounds • concept “flux” allows spatial heterogeneity • classic enzyme kinetics relate production flux to substrate concentration • Synthesizing Unit kinetics relate production flux to substrate flux • in homogeneous systems: flux conc. (diffusion, convection) • concept “density” resembles “concentration” but no homogeneous mixing at the molecular level density = ratio between two amounts

Macrochemical reaction eq 3. 5

Synthesizing units Are 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 The concept concentration is problematic in spatially heterogeneous environments, such as inside cells In spatially homogeneous environments, arrival fluxes are proportional to concentrations

Surface area/volume interactions • 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 environment food availability for daphnids: amount of algae per volume environment • 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 for V 0 -morph V 1 -morph isomorph = actual surface area at volume V isomorphic surface area at volume V V 1 -morphs are special because • surfaces do not play an explicit role • their population dynamics reduce to an unstructured dynamics; reserve densities of all individuals converge to the same value in homogeneous environments Static mixtures between V 0 - and V 1 -morphs for aspect ratio

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

Mixtures of changes in shape 2 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

1 Evolution of DEB systems strong homeostasis for structure 2 delay of use of internal substrates 3 increase of maintenance costs 4 inernalization of maintenance 5 7 Kooijman & Troost 2007 Biol Rev, 82, 1 -30 reproduction juvenile embryo + adult animals 8 strong homeostasis for reserve installation of maturation program prokaryotes variable structure composition 6 plants 9 specialization of structure

Symbiogenesis 2. 7 Ga phagocytosis 2. 1 Ga 1. 27 Ga

ln rate Arrhenius relationship reproduction young/d ingestion 106 cells/h Daphnia magna growth, d-1 aging, d-1 104 T-1, K-1

ln pop growth rate, h-1 Arrhenius relationship 103/T, K-1 103/TH 103/TL 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

Assumptions of auxiliary theory • A well-chosen physical length (volumetric) structural length for isomorphs • Volume, wet/dry weight have contributions from structure, reserve, reproduction buffer • Constant specific mass & volume of structure, reserve, reproduction buffer • Constant chemical composition of juvenile growing at constant food

Compound parameters

Concept overview • empirical facts • supply-demand spectrum • reserve & structure • 5 types of homeostasis • body size: weight, Cmol, . . • body composition • flux vs concentration • macrochemical reactions • Synthesizing Units • surface area/volume • iso-, V 0 -, V 1 -morphs • shape correction function • evolutionary aspects • effects of temperature • auxiliary theory • compound parameters