Scaling relationships based on partition coefficients & body size have similarities & interactions Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Lyon, 2006/05/10

Contents toxicokinetic models one-compartment, film toxic effects DEB theory QSARs body size scaling similarities interactions Lyon, 2006/05/10

1-compartment model For a given external concentration as function of time:

1,1-compartment model compound can cross interface between media with different rates vice versa interface medium i medium j

1,1 compartment model Suppose and while Conclusion: relationship between par values follows from model structure

n,n-compartment models compound can cross interface between media with different rates vice versa sub-layers with equal rates for all sub-layers

film models Steady flux approximation Kooijman et al 2004 Chemosphere 57:

Elimination rate & partition coeff log P 01 log 10% saturation time 1 film 2 film diffusivities low high Transition: film  1,1-compartment model slope = 0.5 Kooijman et al 2004 Chemosphere 57:

Concentration ranges of chemicals too little def: variations in concentration come with variations in effects enough def: variations in concentration within this range hardly affect physiological behaviour of individuals too much def: variations in concentration come with variations in effects e.g. water concentration can be too much even for fish no basic difference between toxic and non-toxic chemicals “too little” and “enough” can have zero range for some chemicals Implication: lower & upper NEC for each compound

Effects on organisms Chemicals, parasites, noise, temperature affect organisms via changes of parameters values of their dynamic energy budget these values are functions of internal concentrations Primary target: individuals some effects at sub-organism level can be compensated (NEC) Effects on populations are derived from that on individuals individuals interact via competition, trophic relationships Parameters of the energy budget model individual-specific and (partly) under genetic control

Models for toxic effects Three model components: kinetics external concentration  internal concentration example: one-compartment kinetics change in target parameter(s) internal concentration  value of target parameter(s) example: linear relationship physiology value of parameter  endpoint (survival, reproduction) example: DEB model

Dynamic Energy Budget theory for metabolic organisation Uptake of substrates (nutrients, light, food) by organisms and their use (maintenance, growth, development, reproduction) during life cycle (dynamic) 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

1-  maturity maintenance maturity offspring maturation reproduction Standard DEB scheme foodfaeces assimilation reserve feeding defecation structure somatic maintenance growth  Def “standard”: 1 type of food 1 type of reserve 1 type structure isomorphy

1-  maturity maintenance maturity offspring maturation reproduction Modes of action of toxicants foodfaeces assimilation reserve feeding defecation structure somatic maintenance growth    assimilation   maintenance costs   growth costs   reproduction costs   hazard to embryo uu tumour maint tumour induction 6 6 endocr. disruption 7 7 lethal effects: hazard rate Mode of action affects translation to pop level 8

Simplest basis: Change  internal conc that exceeds internal NEC or with Change in target parameter Rationale effective molecules operate independently approximation for small effects

Effect on survival Effects of Dieldrin on survival of Poecilia killing rate l  g -1 d -1 elimination rate d -1 NEC 4.49  g l -1 Hazard model for survival : one compartment kinetics hazard rate linear in internal concentration

QSARs for tox parameters 10 log NEC, mM 10 log elim rate, d log kill rate, mM -1 d log P ow Slope = -1 Slope = 1Slope = -0.5 Hazard model for survival : one compartment kinetics hazard rate linear in internal concentration Alkyl benzenes in Pimephales Data from Geiger et al 1990 Assumption: Each molecule has same effect

QSARs for tox parameters 10 log NEC, mM 10 log elim rate, d log kill rate, mM -1 d log P ow Slope = -1 Slope = 1Slope = -0.5 Benzenes, alifates, phenols in Pimephales Data from Mackay et al 1992, Hawker & Connell 1985 Assumption: Each molecule has same effect Hazard model for survival : one compartment kinetics hazard rate linear in internal concentration

Covariation of tox parameters 10 log NEC, mM 10 log killing rate, mM -1 d -1 Slope = -1 Pimephales Data from Gerritsen 1997

QSARs for LC50’s 10 log P ow 10 log LC50.14d,  M LC50.14d of chlorinated hydrocarbons for Poecilia. Data: Könemann, 1980

Primary scaling relationships Dependent on max size K saturation constant L b length at birth L p length at puberty {p Am } max spec assim rate Independent of max size y EX yield of reserve on food v energy conductance [p M ] volume-spec maint. costs {p T } surface-spec maint. costs [E G ] spec structure costs h a aging acceleration  partitioning fraction  R reproduction efficiency maximum length L m =  {p Am } / [p M ] Kooijman 1986 J. Theor. Biol. 121:

Scaling of metabolic rate intra-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)

Von Bertalanffy growth rate At 25 °C : maint rate coeff k M = 400 a -1 energy conductance v = 0.3 m a °C T A = 7 kK 10 log ultimate length, mm 10 log von Bert growth rate, a -1 ↑ ↑ 0

Similarities QSAR  body size scaling 1-compartment model: partition coefficient (= state) is ratio between uptake and elimination rate DEB-model: maximum length (= state) is ratio between assimilation and maintenance rate Parameters are constant for a system, but vary between systems in a way that follows from the model structure

uptake, elimination fluxes, food uptake  surface area (intra-specifically) elimination rate  length -1 (exposure time should depend on size) food uptake  structural volume (inter-specifically) dilution by growth affects toxicokinetics max growth  length 2 (inter-specifically) elimination via reproduction: max reprod mass flux  length 2 (inter-specifically) chemical composition: reserve capacity  length 4 (inter-specifically) in some taxa reserve are enriched in lipids chemical transformation, excretion is coupled to metabolic rate metabolic rate scales between length 2 and length 3 juvenile period  length, abundance  length -3, pop growth rate  length -1 links with risk assessment strategies Interactions QSAR  body size scaling

DEB tele course Free of financial costs; some 250 h effort investment Feb-April 2007; target audience: PhD students We encourage participation in groups that organize local meetings weekly French group of participants of the DEB tele course 2005: special issue of J. Sea Res on DEB applications to bivalves Software package DEBtool for Octave/ Matlab freely downloadable Slides of this presentation are downloadable from Cambridge Univ Press 2000 Audience : thank you for your attention Organizers : thank you for the invitation