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Tjalling Jager
Dept. Theoretical Biology
Simplifying biologyprocess-based models for toxicant
effects and how to apply them
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Contents
Introduction Dealing with complexity Toxicokinetics-toxicodynamic modelling
Models (process and statistical) Dealing with survival Dealing with sub-lethal effects
Wrapping up (Brief history of things called “DEBtox”) Concluding remarks
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Organisms are complex …
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Stressing organisms …
… only adds to the complexity
Response to a toxic stress depends on– type of toxicant– organism (species, life stage, etc.)– endpoint (survival, reproduction, etc.)– exposure duration and intensity– environmental conditions
How is this dealt with in ecotoxicology?– standardisation …
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Reproduction test
50-100 ml of well-defined test medium, 18-22°C
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Reproduction test
Daphnia magna Straus, <24 h old
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Reproduction test
Daphnia magna Straus, <24 h old
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Reproduction test
wait for 21 days, and count total offspring …
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Reproduction test
at least 5 test concentrations in geometric series …
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Response vs. doseR
esp
on
se
log concentration
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Contr.
Response vs. dose
NOEC
Res
po
nse
log concentration
LOEC
*
1. Statistical testing
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Response vs. dose
EC50
Res
po
nse
log concentration
1. Statistical testing2. Curve fitting
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If EC50 is the answer …
… what was the question?
“What is the concentration of chemical X that leads to 50% effect on the total number of offspring of Daphnia magna (Straus) after 21-day constant exposure under standardised laboratory conditions?”
Is this an interesting question?– scientifically: no– for risk assessment ...
EC50EC50
tota
loff
spri
ng
log concentration
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Practical challenge of RA
Some 100,000 man-made chemicals For animals, >1 million species described Exposure conditions are not standardised …
– multiple stress is the norm– exposed individuals are different– complex dynamic exposure situations
We cannot test all these situations …
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Complexity …
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Environmental chemistry …
Complexity …
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air
water
sediment
naturalsoil
agricult.soil
industr.soil
emission advection diffusion degradation
Environmental media as homogeneous boxes …
Complexity …
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Simplifying biology?
How much biological detail do we minimally need …
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Simplifying biology?
How much biological detail do we minimally need … Too much detail …
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Simplifying biology?
How much biological detail do we minimally need … Too little detail …
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Simplifying biology?
How much biological detail do we minimally need … Focus on general mechanisms …
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externalconcentration
(in time)
toxico-kineticmodel
toxico-kineticmodel
TKTD modelling
internalconcentration
in time
process modelfor the organism
process modelfor the organism
effects onendpoints
in timetoxicokinetics
toxicodynamics
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externalconcentration
(in time)
toxico-kineticmodel
toxico-kineticmodel
TKTD modelling
internalconcentration
in time
toxicokinetics
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TKTD modelling
internalconcentration
in time
process modelfor the organism
process modelfor the organism
effects onendpoints
in time
toxicodynamics
Endpoints of interest:
survival growth reproduction …
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SSQ =P
(yi ¡ f (xi ;µ))2
independent variable
obse
rved
var
iabl
e
To apply TKTD models ...
we also need a model for the deviations Least-squares is immensely popular ...
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The statistical model ...
... does not receive same amount of attention as process models
Reasons:– many modellers never work with experimental data– modellers don’t like/know statistics– statisticians don’t like/know realistic models
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Models (process and statistical)
Models for survival
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Why do animals die?
Observation:– not all animals die at the same time in a treatment
Why? Stochasticity
– individuals are random selection from heterogeneous population– death itself should be treated as a stochastic process
Competing hypotheses– although both may play a role– see “GUTS” (Jager et al., 2011)
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Survival TKTD
A process model can be extremely simple!
Assume:– death is a chance process at the level of the individual– there is an internal concentration threshold for effects– above the threshold, probability to die increases linearly
(scaled) internal concentration
haza
rd r
ate
blank value
NEC
killi
ng rat
e
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What about the statistics?
Least squares?– independent random errors following a continuous (normal)
distribution?
Not a good match:– discrete number of survivors– bounded between zero and 100%– number of survivors are dependent observations
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Statistical model
Consider a 1-day toxicity test
p1 p2
0-1 d >1 d
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Lik(µjy) = Pr(Y = yjµ)
Statistical model
Consider a 1-day toxicity test– assume death probabilities are independent
p1 p2
0-1 d >1 d
binomial distribution
Y » B(n;p1(µ))
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Statistical model
Consider a 2-day toxicity test
p1 p2 p3
0-1 d 1-2 d >2 d
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Statistical model
Consider a 2-day toxicity test– assume death probabilities are independent
p1 p2 p3
0-1 d 1-2 d >2 d
multinomial distribution
Y » M(n;pi (µ))P
pi (µ) = 1
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Survival analysis
Typical data set– number of live animals after fixed exposure period– example: Daphnia exposed to nonylphenol
mg/L 0 h 24 h 48 h
0.004 20 20 20
0.032 20 20 20
0.056 20 20 20
0.100 20 20 20
0.180 20 20 16
0.320 20 13 2
0.560 20 2 0
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Example nonylphenol
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (hr)
frac
tion
surv
ivin
g
0.004 mg/L0.032 mg/L0.056 mg/L0.1 mg/L0.18 mg/L0.32 mg/L0.56 mg/L
elimination rate 0.057 (0.026-0.14) 1/hr
no-effect conc. 0.14 (0.093-0.17) mg/L
killing rate 0.66 (0.31-1.7) L/mg/d
blank hazard 0 (not fitted)1/hr
elimination rate 0.057 (0.026-0.14) 1/hr
no-effect conc. 0.14 (0.093-0.17) mg/L
killing rate 0.66 (0.31-1.7) L/mg/d
blank hazard 0 (not fitted)1/hr
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Summary survival
Process models can be extremely simple– assume that death is a chance process– starts with 3 parameters
Statistical model provides a good match– multinomial distribution
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Models (process and statistical)
Sub-lethal endpoints
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Simplifying biology
How do we deal with growth and reproduction?– these are not outcome of chance processes …– we cannot be species- or stressor-specific
Organisms obey mass and energy conservation!
internalconcentration
in time
process modelfor the organism
effects onendpoints
in time
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Effect on reproduction
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Effect on reproduction
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Effect on reproduction
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Effect on reproduction
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Effect on reproduction
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Energy Budget
To understand effect on reproduction …– we have to consider how food is turned into offspring
Challenge– find the simplest set of rules ...
– over the entire life cycle ...
– similar rules for all organisms
growth
maintenance
maturation
off spring
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Quantitative theory for metabolic organisation from ‘first principles’– time, energy and mass balance– consistent with thermodynamics
Life-cycle of the individual– links levels of organisation– molecule ecosystems
Fundamental, but many practical applications– (bio)production, (eco)toxicity, climate
change, evolution …
Kooijman (2010)
DEB theory
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eggs
mobilisation
Standard DEB animal
structurestructure
somatic maintenance
growth
maturity maintenance1-
reproduction
maturitymaturity bufferbuffer
maturation p
food feces
assimilation
reservereserve
b
3-4 states8-12 parameters
system can be scaled to remove dimension ‘energy’
3-4 states8-12 parameters
system can be scaled to remove dimension ‘energy’
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Different food densities
Jager et al. (2005)
0 2 4 6 8 10 1220
30
40
50
60
70
80
90
100
time (d)
bo
dy
len
gth
(µ
m)
0 2 4 6 8 10 1220
30
40
50
60
70
80
90
100
time (d)
bo
dy
len
gth
(µ
m)
H
M
L
0 2 4 6 8 10 120
20
40
60
80
100
120
140
160
time (d)
cum
ula
tive
nu
mb
er o
f eg
gs
0 2 4 6 8 10 120
20
40
60
80
100
120
140
160
time (d)
cum
ula
tive
nu
mb
er o
f eg
gs
H
M
L
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internal concentration
DE
B p
aram
eter
NEC
blank value
internal concentration
DE
B p
aram
eter
NEC
blank value
Toxicant effects in DEB
externalconcentration
(in time)
toxico-kinetics
toxico-kinetics internal
concentrationin time DEB
parametersin time
DEBmodel
DEBmodel
repro
growth
survival
feeding
hatching
…
over entire life cycle
Affected DEB parameter has specific consequences for life cycle
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Toxicant case study
Marine polychaete Capitella (Hansen et al, 1999)– exposed to nonylphenol in sediment– body volume and egg production followed
Jager and Selck (2011)
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Control growth
Volumetric body length in control
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
3
time (days)
vo
lum
etr
ic b
od
y l
en
gth
(m
m)
0
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Control growth
Assumption– effective food density depends on body size
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
3
time (days)
vo
lum
etr
ic b
od
y l
en
gth
(m
m)
0
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Control growth
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
3
time (days)
vo
lum
etr
ic b
od
y l
en
gth
(m
m)
0
Assumption– initial starvation …
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Control reproduction
Ignore reproduction buffer …
0 10 20 30 40 50 60 70 800
500
1000
1500
2000
2500
3000
3500
time (days)
cu
mu
lati
ve
off
sp
rin
g p
er
fem
ale
0
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NP effects
Compare the control to the first dose
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
3
time (days)
volu
me
tric
bo
dy
len
gth
(m
m)
014
0 10 20 30 40 50 60 70 800
500
1000
1500
2000
2500
3000
3500
4000
time (days)
cu
mu
lati
ve o
ffs
pri
ng
pe
r fe
ma
le 014
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“Hormesis”
Requires a mechanistic explanation …– organism must obey conservation of mass and energy
Potential assumptions– decreased investment elsewhere– toxicant relieves a secondary stress– toxicant increases the food availability/quality
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NP effects
Assumption– NP increases food density/quality
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
3
time (days)
volu
me
tric
bo
dy
len
gth
(m
m)
014
0 10 20 30 40 50 60 70 800
500
1000
1500
2000
2500
3000
3500
4000
time (days)
cu
mu
lati
ve o
ffs
pri
ng
pe
r fe
ma
le
014
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NP effects
Assumption– NP affects costs for making structure
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
3
time (days)
volu
me
tric
bo
dy
len
gth
(m
m)
1452174
1452174
0 10 20 30 40 50 60 70 800
500
1000
1500
2000
2500
3000
3500
4000
time (days)
cu
mu
lati
ve o
ffs
pri
ng
pe
r fe
ma
le
1452174
1452174
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Standard DEB animal
structurestructure
food feces
maturity maintenancesomatic maintenance
assimilation
1-
growth reproduction
maturitymaturity bufferbuffer
maturation
reservereserve
mobilisation
eggs
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NP effects
Assumption– NP also affects costs for maturation and reproduction
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
3
time (days)
volu
me
tric
bo
dy
len
gth
(m
m)
0 10 20 30 40 50 60 70 800
500
1000
1500
2000
2500
3000
3500
4000
time (days)
cu
mu
lati
ve o
ffs
pri
ng
pe
r fe
ma
le
1452174
1452174
1452174
1452174
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Standard DEB animal
structurestructure
food feces
maturity maintenancesomatic maintenance
assimilation
1-
growth reproduction
maturitymaturity bufferbuffer
maturation
reservereserve
mobilisation
eggs
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Classical strategy data analysis
fit satisfactory?
descriptivemodel curve
experimentaldata
least squares
report EC50
yes
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DEB strategy data analysis
fit satisfactory?
optimise
actualDEB model
experimentaldata
additionalexperiments
literature
educatedguesses
mechanistichypothesis
affectedparameter(s)
think
summariseconclusions
yes
DEB theory
hypothesis
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Strategy for data analysis
Are we sure we have the correct explanation?
Occam’s razor Accept the simplest explanation … for now
generatepredictions
actualDEB model
testpredictions
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yi = f (ti ;µ) +N (0;¾2)
Statistical model
Common assumptions leading to least-squares: Time is “certain” Normal errors Equal variances Independent errors
time
obse
rved
var
iabl
e
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Body size
Individuals are not the same– example: parameters vary between individuals
time
body
leng
th
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Body size
Behaviour is stochastic– example: food encounter is a chance process
time
body
leng
th
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Fitting reproduction
Model– energy flux for eggs (J/d)– egg costs (J/egg)– buffer handling ...
Observations– numbers of eggs in an
interval (eggs)– often only mean available
…
reproduction
bufferbuffer eggs
First …– ignore buffer– repro rate (eggs/d)
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Fitting reproduction
Cumulative plot ...– observations become highly dependent ...– what error distribution is appropriate?
time
cum
ula
tive
eg
gs
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Fitting reproduction
egg
s in
in
terv
al
time
Per observation interval ...– less dependence in observations
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Fitting reproduction
Is this a bad fit?– not necessarily, when there is a repro buffer ...– individuals might spawn at different times ...
time
cum
ula
tive
eg
gs
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Example Folsomia candida
0 20 40 60 800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
time (d)
cubi
c ro
ot w
et w
eig
ht (
g1/3)
0 5 10 15 20 25 30 35 40
0
50
100
150
200
250
300
350
time (d)
cum
ulat
ive
off
sprin
g
Fit on individuals:– cumulative reproduction per female …– exclude time points with zero reproduction …
Body size and reproduction not independent …
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How do we proceed?
Follow individuals over time, in detail– body size over time– timing of spawning events– investment per offspring …
Resolve questions ...– between individuals:
• how variable are parameters?
• how do parameters co-vary?
– within individuals:• role of stochastic behaviour?
• linkage between endpoints?
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In the meantime ...
Don’t throw out the baby with the bath water! Process models are valuable ...
How bad is it to assume normal independent errors? That depends on ...
– homogeneity of the test population– reproduction buffer size– purpose of the study– ...
Confidence intervals suffer most
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Wrapping up
A short history of DEB in ecotoxicology
skip
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1984
Chemicals affect the energy budget ...– effects on individuals leads to effects on populations
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1993
First DEB book ...– with a chapter on ecotoxicity
ISO/OECD revision of guidelines in early 90’s
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1996
DEBtox software and booklet in 1996– and 5 papers in open literature– used/adapted by a number of groups
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Standard DEB animal
structurestructure
food feces
maturity maintenancesomatic maintenance
assimilation
1-
growth reproduction
maturitymaturity bufferbuffer
maturation
reservereserve
mobilisation
eggs
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eggs
mobilisation
Simplified DEB animal
structurestructure
somatic maintenance
growth
maturity maintenance1-
reproduction
maturity buffer
maturation p
food fecesassimilation
reserve 1-comp. toxicokinetics
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Kooijman (2010)
2010
Full DEB model for toxicants– more possible mechanisms of action– more parameters ...
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2012
Revisiting the simple model ...– available data sets do not allow full DEB model– many questions do not need a ful DEB model
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Concluding remarks 1
Eco(toxico)logy needs idealisations of biology– TKTD models:
• survival only: unified in GUTS
• sub-lethal endpoints: DEB offers platform
– much more work is needed!
TKTD requires appropriate statistical models– least-squares is not generally appropriate
For sub-lethal data ...– deviations do not represent random error
• differences between individuals
• stochastic behaviour (feeding/spawning)
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Concluding remarks 2
Current status of TKTD The use of TKTD models in ecotoxicology is ...
– rare in scientific settings– absent in risk assessment settings
Ecotoxicology focusses on descriptions ...
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More information
on DEBtox/GUTS: http://www.debtox.info
on DEB: http://www.bio.vu.nl/thb
Courses– Summercourse TKTD modelling Denmark 2012– International DEB Tele Course 2013
Symposia– 2nd International DEB Symposium 2013 on Texel (NL)