tjalling jager dept. theoretical biology simplifying biology process-based models for toxicant...
TRANSCRIPT
Tjalling Jager
Dept. Theoretical Biology
Simplifying biologyprocess-based models for toxicant
effects and how to apply them
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
Organisms are complex …
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 …
Reproduction test
50-100 ml of well-defined test medium, 18-22°C
Reproduction test
Daphnia magna Straus, <24 h old
Reproduction test
Daphnia magna Straus, <24 h old
Reproduction test
wait for 21 days, and count total offspring …
Reproduction test
at least 5 test concentrations in geometric series …
Response vs. doseR
esp
on
se
log concentration
Contr.
Response vs. dose
NOEC
Res
po
nse
log concentration
LOEC
*
1. Statistical testing
Response vs. dose
EC50
Res
po
nse
log concentration
1. Statistical testing2. Curve fitting
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
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 …
Complexity …
Environmental chemistry …
Complexity …
air
water
sediment
naturalsoil
agricult.soil
industr.soil
emission advection diffusion degradation
Environmental media as homogeneous boxes …
Complexity …
Simplifying biology?
How much biological detail do we minimally need …
Simplifying biology?
How much biological detail do we minimally need … Too much detail …
Simplifying biology?
How much biological detail do we minimally need … Too little detail …
Simplifying biology?
How much biological detail do we minimally need … Focus on general mechanisms …
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
externalconcentration
(in time)
toxico-kineticmodel
toxico-kineticmodel
TKTD modelling
internalconcentration
in time
toxicokinetics
TKTD modelling
internalconcentration
in time
process modelfor the organism
process modelfor the organism
effects onendpoints
in time
toxicodynamics
Endpoints of interest:
survival growth reproduction …
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 ...
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
Models (process and statistical)
Models for survival
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)
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
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
Statistical model
Consider a 1-day toxicity test
p1 p2
0-1 d >1 d
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))
Statistical model
Consider a 2-day toxicity test
p1 p2 p3
0-1 d 1-2 d >2 d
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
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
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
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
Models (process and statistical)
Sub-lethal endpoints
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
Effect on reproduction
Effect on reproduction
Effect on reproduction
Effect on reproduction
Effect on reproduction
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
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
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’
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
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
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)
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
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
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 …
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
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
“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
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
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
Standard DEB animal
structurestructure
food feces
maturity maintenancesomatic maintenance
assimilation
1-
growth reproduction
maturitymaturity bufferbuffer
maturation
reservereserve
mobilisation
eggs
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
Standard DEB animal
structurestructure
food feces
maturity maintenancesomatic maintenance
assimilation
1-
growth reproduction
maturitymaturity bufferbuffer
maturation
reservereserve
mobilisation
eggs
Classical strategy data analysis
fit satisfactory?
descriptivemodel curve
experimentaldata
least squares
report EC50
yes
DEB strategy data analysis
fit satisfactory?
optimise
actualDEB model
experimentaldata
additionalexperiments
literature
educatedguesses
mechanistichypothesis
affectedparameter(s)
think
summariseconclusions
yes
DEB theory
hypothesis
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
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
Body size
Individuals are not the same– example: parameters vary between individuals
time
body
leng
th
Body size
Behaviour is stochastic– example: food encounter is a chance process
time
body
leng
th
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)
Fitting reproduction
Cumulative plot ...– observations become highly dependent ...– what error distribution is appropriate?
time
cum
ula
tive
eg
gs
Fitting reproduction
egg
s in
in
terv
al
time
Per observation interval ...– less dependence in observations
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
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 …
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?
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
Wrapping up
A short history of DEB in ecotoxicology
skip
1984
Chemicals affect the energy budget ...– effects on individuals leads to effects on populations
1993
First DEB book ...– with a chapter on ecotoxicity
ISO/OECD revision of guidelines in early 90’s
1996
DEBtox software and booklet in 1996– and 5 papers in open literature– used/adapted by a number of groups
Standard DEB animal
structurestructure
food feces
maturity maintenancesomatic maintenance
assimilation
1-
growth reproduction
maturitymaturity bufferbuffer
maturation
reservereserve
mobilisation
eggs
eggs
mobilisation
Simplified DEB animal
structurestructure
somatic maintenance
growth
maturity maintenance1-
reproduction
maturity buffer
maturation p
food fecesassimilation
reserve 1-comp. toxicokinetics
Kooijman (2010)
2010
Full DEB model for toxicants– more possible mechanisms of action– more parameters ...
2012
Revisiting the simple model ...– available data sets do not allow full DEB model– many questions do not need a ful DEB model
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)
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 ...
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)