Scaling relationships based onpartition coefficients & body size have similarities & interactions

Bas KooijmanDept theoretical biology

Vrije Universiteit AmsterdamBas@bio.vu.nl

http://www.bio.vu.nl/thb/

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 andwhile

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 2004Chemosphere 57: 745-753

Elimination rate & partition coeff

log P01 log P01

log

10%

sat

urat

ion

tim

e

1 film 2 filmdiffusivities

low

high

Transition: film 1,1-compartment model

slope = 0.5slope = 0.5

Kooijman et al 2004Chemosphere 57: 745-753

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 chemicalsImplication: 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 theoryfor metabolic organisationUptake of substrates (nutrients, light, food) by organisms and their use (maintenance, growth, development, reproduction) during life cycle (dynamic)

First principles, quantitative, axiomatic set upAim: Biological equivalent of Theoretical Physics

Primary target: the individual with consequences for• sub-organismal organization• supra-organismal organizationRelationships between levels of organisation

Many popular empirical models are special cases of DEB

1- maturitymaintenance

maturityoffspring

maturationreproduction

Standard DEB scheme

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

Def “standard”:• 1 type of food• 1 type of reserve• 1 type structure• isomorphy

1- maturitymaintenance

maturityoffspring

maturationreproduction

Modes of action of toxicants

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

assimilation

maintenance costs

growth costs

reproduction costs

hazard to embryo

u

tumourtumour

maint tumour induction6

6

endocr. disruption7

7

lethal effects: hazard rateMode of action affectstranslation 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 0.038 l g-1 d-1

elimination rate 0.712 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

10lo

g N

EC

, m

M

10lo

g el

im r

ate,

d-1

10lo

g ki

ll ra

te,

mM

-1 d

-1

10log Pow 10log Pow10log Pow

Slope = -1 Slope = 1Slope = -0.5

Hazard model for survival:• one compartment kinetics• hazard rate linear in internal concentration

Alkyl benzenes in PimephalesData from Geiger et al 1990

Assumption:Each molecule has same effect

QSARs for tox parameters

10lo

g N

EC

, m

M

10lo

g el

im r

ate,

d-1

10lo

g ki

ll ra

te,

mM

-1 d

-1

10log Pow 10log Pow10log Pow

Slope = -1 Slope = 1Slope = -0.5

Benzenes, alifates, phenols in PimephalesData 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 parameters1

0log

NE

C, m

M

10log killing rate, mM-1 d-1

Slope = -1

PimephalesData from Gerritsen 1997

QSARs for LC50’s

10log Pow10log Pow

10lo

g LC

50.1

4d, M

LC50.14d of chlorinated hydrocarbons for Poecilia. Data: Könemann, 1980

Primary scaling relationships

Dependent on max size

K saturation constant

Lb length at birth

Lp length at puberty

{pAm} max spec assim rate

Independent of max size

yEX yield of reserve on food

v energy conductance

[pM] volume-spec maint. costs

{pT} surface-spec maint. costs

[EG] spec structure costs

ha aging acceleration

partitioning fraction

R reproduction efficiency

maximum length Lm = {pAm} / [pM] Kooijman 1986J. Theor. Biol. 121: 269-282

Scaling of metabolic rate

intra-species inter-species

maintenance

growth

weight

nrespiratio3

32

dl

llls

43

32

ldld

lll

EV

h

structure

reserve

32 vll

l0l

0

3lllh

Respiration: contributions from growth and maintenanceWeight: contributions from structure and reserveStructure ; = length; endotherms 3l l

3lllh

0hl

Metabolic rate

Log weight, gLo

g m

etab

olic

rat

e,

w

endotherms

ectotherms

unicellulars

slope = 1

slope = 2/3

Length, cm

O2 c

onsu

mpt

ion,

l

/h

Inter-speciesIntra-species

0.0226 L2 + 0.0185 L3

0.0516 L2.44

2 curves fitted:

(Daphnia pulex)

Von Bertalanffy growth rate

vVkr MB /3/3 3/11

At 25 °C : maint rate coeff kM = 400 a-1

energy conductance v = 0.3 m a-1

25 °CTA = 7 kK

10log ultimate length, mm 10log ultimate length, mm

10lo

g vo

n B

ert

grow

th r

ate

, a-1

)exp()()( 3/13/13/13/1 arVVVaV Bb

3/1V

a

3/1V

3/1bV

1Br

↑

↑0

SimilaritiesQSAR 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 length2 (inter-specifically)

• elimination via reproduction: max reprod mass flux length2 (inter-specifically)

• chemical composition: reserve capacity length4 (inter-specifically) in some taxa reserve are enriched in lipids

• chemical transformation, excretion is coupled to metabolic rate metabolic rate scales between length2 and length3

• juvenile period length, abundance length-3 , pop growth rate length-1

links with risk assessment strategies

InteractionsQSAR body size scaling

DEB tele course 2007

http://www.bio.vu.nl/thb/deb/

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. 2006 on DEB applications to bivalves

Software package DEBtool for Octave/ Matlab freely downloadable

Slides of this presentation are downloadable from http://www.bio.vu.nl/thb/users/bas/lectures/

Cambridge Univ Press 2000

Audience: thank you for your attention

Organizers: thank you for the invitation