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Life history events in DEB theory for metabolic organisation Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam [email protected] http://www.bio.vu.nl/thb/ Oslo, 2008/03/13 Life history events in DEB theory for metabolic organisation Contents: • What is DEB theory? • Homeostasis • Standard model & calorimetry • Allocation • Embryonic development • Unexpected links • Body size scaling relationships • Parameter estimation Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam [email protected] http://www.bio.vu.nl/thb/ Oslo, 2008/03/13 Dynamic Energy Budget theory for metabolic organization • consists of a set of consistent and coherent assumptions • uses framework of general systems theory • links levels of organization scales in space and time: scale separation • quantitative; first principles only equivalent of theoretical physics • interplay between biology, mathematics, physics, chemistry, earth system sciences • fundamental to biology; many practical applications Empirical special cases of DEB year author model year author model 1780 Lavoisier multiple regression of heat against mineral fluxes 1950 Emerson cube root growth of bacterial colonies 1825 Gompertz 1889 DEB theory is axiomatic, 1951 Huggett & Widdas Survival probability for aging based on mechanisms temperature dependence of Arrhenius 1951 Weibull physiological rates not meant to glue empirical models foetal growth 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 1973 Droop reserve (cell quota) dynamics 1910 1920 Since many empirical models bilinear functional response microbial product formation 1959 Leudeking & Piret to binding be special cases of DEB theory Cooperative hyperbolic functional response Hill turn out 1959 Holling von Bertalanffy growth ofthese 1962 maintenance in yields of biomass behind models support DEB theory Pütter the data Marr & Pirt individuals 1927 Pearl 1928 Fisher & Tippitt 1932 Kleiber logistic population growth This makes DEB theory very tested against data Weibull aging water loss in bird eggs 1974 well Rahn & Ar respiration scales with body 3/ 4 1932 digestion 1975 Hungate DEB theory weight reveals when to expect deviations root growth of tumours development of salmonid embryos Mayneord 1977 Beer & Anderson from cube these empirical models Homeostasis strong homeostasis constant composition of pools (reserves/structures) generalized compounds, stoichiometric contraints on synthesis weak homeostasis constant composition of biomass during growth in constant environments determines reserve dynamics (in combination with strong homeostasis) structural homeostasis constant relative proportions during growth in constant environments isomorphy .work load allocation ectothermy homeothermy endothermy supply demand systems development of sensors, behavioural adaptations Standard DEB model food feeding defecation faeces assimilation reserve somatic maintenance growth structure 1- maturity maintenance maturation reproduction maturity offspring Definition of standard model: Isomorph with 1 reserve & 1 structure feeds on 1 type of food has 3 life stages (embryo, juvenile, adult) Extensions of standard model: • more types of food and food qualities reserve (autotrophs) structure (organs, plants) • changes in morphology • different number of life stages Three basic fluxes • assimilation: substrate reserve + products linked to surface area • dissipation: reserve products somatic maintenance: linked to surface area & structural volume maturity maintenance: linked to maturity maturation or reproduction overheads • growth: reserve structure + products Product formation = A assimilation + B dissipation + C growth Examples: heat, CO2, H2O, O2, NH3 Indirect calorimetry: heat = D O2-flux + E CO2-flux + F NH3-flux volume, m3 3.7.2 Bacillus = 0.2 Collins & Richmond 1962 time, min Fusarium = 0 Trinci 1990 time, h volume, m3 volume, m3 hyphal length, mm Mixtures of V0 & V1 morphs Escherichia = 0.28 Kubitschek 1990 time, min Streptococcus = 0.6 Mitchison 1961 time, min -rule for allocation vL2 k M L3 Ingestion rate, 105 cells/h O2 consumption, g/h Respiration Length, mm • large part 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 Ingestion fL2 Length, mm Length, mm Cum # of young Reproduction 3.5 Growth: d L rB ( L L) dt Von Bertalanffy Age, d Initial amount of reserve Initial amount of reserve E0 follows from • initial structural volume is negligibly small • initial maturity is negligibly small • maturity at birth is given • reserve density at birth equals that of mother at egg formation Accounts for • maturity maintenance costs • somatic maintenance costs • cost for structure • allocation fraction to somatic maintenance + growth Mean reproduction rate (number of offspring per time): R = (1-R) JER/E0 Reproduction buffer: buffer handling rules; clutch size Embryonic development 3.7.1 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 3 J O ,G 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 3.7.1 ml O2 d-1 ml CO2 d-1 Respiration ontogeny in birds altricial Troglodytes aëdon age, d precocial Gallus domesticus age, d Observations: just prior to hatching • respiration shows a plateau in precocial, not in altricial birds • pore size and frequency in egg shell is such that O2 flux has constant resistance Conclusion: ontogeny is constrained by diffusion limitation in precocial birds (Rahn et al 1990) DEB theory: reserve dynamics controls ontogeny (same pattern in species without shells) Minimization of water loss causes observed constant flux resistance weight, g Foetal development Mus musculus time, d 3.7.1 Foetes develop like eggs, but rate not restricted by reserve (because supply during development) Reserve of embryo “added” at birth Initiation of development can be delayed by implantation egg cell Nutritional condition of mother only affects foetus in extreme situations Data: MacDowell et al 1927 d For E0 : V vV 2 / 3 ; V (0) 0; V (t ) (vt / 3)3 dt Pupal development pupal weight, mg 17 °C time, d green-veined white butterfly, Pieris napi Data from Forsberg & Wiklund 1988 pupa = embryo in DEB theory • no uptake of resources • start of development with very small amount of structure • initiation & termination linked to maturity Metamorphosis The larval malphigian tubes are clearly visible in this emerging cicada They resemble a fractally-branching space-filling tubing system, according to Jim Brown, but judge yourself …. Java, Nov 2007 scaled reserve scaled maturity Reduction of initial reserve scaled age 3.7.1 1 0.8 0.5 scaled struct volume scaled age scaled age Daphnia Length, mm 1/yield, mmol glucose/ mg cells O2 consumption, μl/h DEB theory reveals unexpected links Streptococcus 1/spec growth rate, 1/h respiration length in individual animals & yield growth in pop of prokaryotes have a lot in common, as revealed by DEB theory Reserve plays an important role in both relationships, but you need DEB theory to see why and how Primary scaling relationships assimilation feeding digestion growth mobilization heating,osmosis turnover,activity regulation,defence allocation egg formation life cycle life cycle aging {JEAm} {b} yEX yVE v {JET} [JEM] kJ R [MHb] [MHp] ha max surface-specific assim rate Lm surface- specific searching rate yield of reserve on food yield of structure on reserve energy conductance surface-specific somatic maint. costs volume-specific somatic maint. costs maturity maintenance rate coefficient partitioning fraction reproduction efficiency volume-specific maturity at birth volume-specific maturity at puberty aging acceleration maximum length Lm = {JEAm} / [JEM] Kooijman 1986 J. Theor. Biol. 121: 269-282 Scaling of metabolic rate 8.2.2 Respiration: contributions from growth and maintenance Weight: contributions from structure and reserve 3 Structure l ; l = length; endotherms lh 0 intra-species inter-species maintenance lh l l 3 lh l l 3 growth l 2 vl3 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 Metabolic rate 2 curves fitted: 0.0226 L2 + 0.0185 L3 0.0516 L2.44 Log metabolic rate, w O2 consumption, l/h slope = 1 endotherms ectotherms slope = 2/3 unicellulars Length, cm Intra-species (Daphnia pulex) Log weight, g Inter-species 3.7 Von Bert growth rate -1, d length, mm Growth at constant food time, d Von Bertalanffy growth curve: L(t ) L ( L Lb ) exp( rB t ) rB1 3k M1 3δM L / v L fLm fVm1/ 3 / δM 3δM / v 3k M1 ultimate length, mm t L Lb L time Length L. at birth ultimate L. rB v kM δM von Bert growth rate energy conductance maint. rate coefficient shape coefficient Von Bertalanffy growth rate von Bert growth rate, a-1 8.2.2 10log 25 °C TA = 7 kK 10log rB 3 / kM 3 V 1/ 3 ultimate length, mm /v 1 10log 3 / kM 3V 1/ 3 /v ultimate length, mm 1 V1/ 3 At 25 °C : maint rate coeff kM = 400 a-1 energy conductance v = 0.3 m a-1 V 1/ 3 ↑ V 1/ 3 (a) V1/ 3 (V1/ 3 Vb1/ 3 ) exp( rB a) Vb1/ 3 rB1 ↑ 0 a Length at puberty 8.2.2 Clupoid fishes Clupea • Brevoortia ° Sprattus Sardinops Sardina Sardinella + Engraulis * Centengraulis Stolephorus Data from Blaxter & Hunter 1982 Length at first reproduction Lp ultimate length L Feeding rate 8.2.2 Filtration rate, l/h slope = 1 Mytilus edulis poikilothermic tetrapods Data: Winter 1973 Data: Farlow 1976 Length, cm Intra-species: JXm V2/3 Inter-species: JXm V 8.2.2 log scaled age at birth log scaled initial reserve Scaling relationships log scaled length at birth log zoom factor, z log zoom factor, z approximate slope at large zoom factor log zoom factor, z Two-sample case: D. magna 20°C Optimality of life history parameters? measured quantities primary pars Standard DEB model (isomorph, 1 reserve, 1 structure) reserve & maturity: hidden variables measured for 2 food levels primary parameters DEB tele course 2009 http://www.bio.vu.nl/thb/deb/ Cambridge Univ Press 2000 Free of financial costs; some 250 h effort investment Program for 2009: Feb/Mar general theory April 18-22 symposium in Brest Sept/Oct case studies & applications Target audience: PhD students We encourage participation in groups that organize local meetings weekly Software package DEBtool for Octave/ Matlab freely downloadable Slides of this presentation are downloadable from http://www.bio.vu.nl/thb/users/bas/lectures/ Audience: thank you for your attention Stig Omholt: thank you for the invitation