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The use of models in DEB research
Bas KooijmanDept theoretical biology
Vrije Universiteit AmsterdamBas@bio.vu.nl
http://www.bio.vu.nl/thb/
Nijmegen, 2005/02/23
Contents• DEB theory introduction
• Scales in space & time
• Empirical cycle
• Dimensions
• Plasticity in parameters
• Stochastic vs deteriministic
• Dynamical systems
Nijmegen, 2005/02/23
Dynamic Energy Budget theoryfor metabolic organisation
Uptake of substrates (nutrients, light, food) by organisms and their use (maintenance, growth, development, reproduction)
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
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 Survival probability for aging 1951 Huggett & Widdas foetal growth
1889 Arrhenius temperature dependence of physiological rates
1951 Weibull 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 bilinear functional response 1959 Leudeking & Piret microbial product formation
1910 Hill Cooperative binding 1959 Holling hyperbolic functional response
1920 Pütter von Bertalanffy growth of individuals
1962 Marr & Pirt maintenance in yields of biomass
1927 Pearl logistic population growth 1973 Droop reserve (cell quota) dynamics
1928 Fisher & Tippitt
Weibull aging 1974 Rahn & Ar water loss in bird eggs
1932 Kleiber respiration scales with body weight3/ 4
1975 Hungate digestion
1932 Mayneord cube root growth of tumours 1977 Beer & Anderson development of salmonid embryos
DEB theory is axiomatic, based on mechanisms not meant to glue empirical models
Since many empirical models turn out to be special cases of DEB theory the data behind these models support DEB theory
This makes DEB theory very well tested against data
Some DEB pillars• life cycle perspective of individual as primary target embryo, juvenile, adult (levels in metabolic organization)
• life as coupled chemical transformations (reserve & structure)
• time, energy & mass balances
• surface area/ volume relationships (spatial structure & transport)
• homeostasis (stoichiometric constraints via Synthesizing Units)
• syntrophy (basis for symbioses, evolutionary perspective)
• intensive/extensive parameters: body size scaling
1- maturitymaintenance
maturityoffspring
maturationreproduction
Basic DEB scheme
food faecesassimilation
reserve
feeding defecation
structurestructure
somaticmaintenance
growth
molecule
cell
individual
population
ecosystem
system earth
time
spac
e
Space-time scales
When changing the space-time scale, new processes will become important other will become less importantIndividuals are special because of straightforward energy/mass balances
Each process has its characteristic domain of space-time scales
Modelling 1• model: scientific statement in mathematical language “all models are wrong, some are useful”
• aims: structuring thought; the single most useful property of models: “a model is not more than you put into it” how do factors interact? (machanisms/consequences) design of experiments, interpretation of results inter-, extra-polation (prediction) decision/management (risk analysis)
Modelling 2• language errors: mathematical, dimensions, conservation laws
• properties: generic (with respect to application) realistic (precision) simple (math. analysis, aid in thinking) plasticity in parameters (support, testability)
• ideals: assumptions for mechanisms (coherence, consistency) distinction action variables/meausered quantities core/auxiliary theory
Dimension rules• quantities left and right of = must have equal dimensions
• + and – only defined for quantities with same dimension
• ratio’s of variables with similar dimensions are only dimensionless if addition of these variables has a meaning within the model context
• never apply transcendental functions to quantities with a dimension log, exp, sin, … What about pH, and pH1 – pH2?
• don’t replace parameters by their values in model representations y(x) = a x + b, with a = 0.2 M-1, b = 5 y(x) = 0.2 x + 5 What dimensions have y and x? Distinguish dimensions and units!
Models with dimension problems• Allometric model: y = a W b
y: some quantity a: proportionality constant W: body weight b: allometric parameter in (2/3, 1) Usual form ln y = ln a + b ln W Alternative form: y = y0 (W/W0 )b, with y0 = a W0
b
Alternative model: y = a L2 + b L3, where L W1/3
• Freundlich’s model: C = k c1/n
C: density of compound in soil k: proportionality constant c: concentration in liquid n: parameter in (1.4, 5) Alternative form: C = C0 (c/c0 )1/n, with C0 = kc0
1/n
Alternative model: C = 2C0 c(c0+c)-1 (Langmuir’s model)
Problem: No natural reference values W0 , c0
Values of y0 , C0 depend on the arbitrary choice
Allometric functions
Length, mmO2 c
onsu
mpt
ion,
μl/
h
Two curves fitted:
a L2 + b L3
with a = 0.0336 μl h-1 mm-2
b = 0.01845 μl h-1 mm-3
a Lb
with a = 0.0156 μl h-1 mm-2.437
b = 2.437
Model without dimension problem
Arrhenius model: ln k = a – T0 /Tk: some rate T: absolute temperaturea: parameter T0: Arrhenius temperature
Alternative form: k = k0 exp{1 – T0 /T}, with k0 = exp{a – 1}
Difference with allometric model: no reference value required to solve dimension problem
Arrhenius relationship
103/T, K-1
ln p
op g
row
th r
ate,
h-1
103/TH 103/TL
r1 = 1.94 h-1
T1 = 310 KTH = 318 KTL = 293 K
TA = 4370 KTAL = 20110 KTAH = 69490 K
}exp{}exp{1
}exp{
)( 11
TT
TT
TT
TT
TT
TT
r
TrAH
H
AH
L
ALAL
AA
Biodegradation of compoundsn-th order model Monod model
nkXXdt
d
1)1(10 )1()(
nn ktnXtX
ktXtXn
0
0
)( kXt /0
}exp{)( 0
1
ktXtXn
n
akXaXt
nn
1
1)(
111
00
XK
XkX
dt
d
ktXtXKXtX }/)(ln{)(0 00
ktXtXXK
0
0
)( }/exp{0 KktXt
}/exp{)( 0
0
KktXtXXK
aKkakXaXt ln)1()( 1100
; ;
X : conc. of compound, X0 : X at time 0 t : time k : degradation rate n : order K : saturation constant
Biodegradation of compoundsn-th order model Monod model
scaled time scaled time
scal
ed c
onc.
scal
ed c
onc.
Plasticity in parameters
If plasticity of shapes of y(x|a) is large as function of a:
• little problems in estimating value of a from {xi,yi}i
(small confidence intervals)
• little support from data for underlying assumptions
(if data were different: other parameter value results, but still a good fit, so no rejection of assumption)
Stochastic vs deterministic models
Only stochastic models can be tested against experimental data
Standard way to extend deterministic model to stochastic one: regression model: y(x| a,b,..) = f(x|a,b,..) + e, with e N(0,2)Originates from physics, where e stands for measurement error
Problem: deviations from model are frequently not measurement errorsAlternatives:• deterministic systems with stochastic inputs• differences in parameter values between individualsProblem: parameter estimation methods become very complex
StatisticsDeals with• estimation of parameter values, and confidence in these values• tests of hypothesis about parameter values differs a parameter value from a known value? differ parameter values between two samples?
Deals NOT with• does model 1 fit better than model 2 if model 1 is not a special case of model 2
Statistical methods assume that the model is given(Non-parametric methods only use some properties of the given model, rather than its full specification)
Dynamic systemsDefined by simultaneous behaviour of input, state variable, outputSupply systems: input + state variables outputDemand systems input state variables + outputReal systems: mixtures between supply & demand systemsConstraints: mass, energy balance equationsState variables: span a state space behaviour: usually set of ode’s with parametersTrajectory: map of behaviour state vars in state spaceParameters: constant, functions of time, functions of modifying variables compound parameters: functions of parameters
Embryonic development 3.7.1
time, d time, d
wei
ght,
g
O2 c
onsu
mpt
ion,
ml/
h
l
ege
dτ
d
ge
legl
dτ
d
3
3,
3, l
dτ
dJlJJ GOMOO
; : scaled timel : scaled lengthe: scaled reserve densityg: energy investment ratio
Crocodylus johnstoni,Data from Whitehead 1987
yolk
embryo
C,N,P-limitation
Nannochloropsis gaditana (Eugstimatophyta) in sea waterData from Carmen Garrido PerezReductions by factor 1/3 starting from 24.7 mM NO3, 1.99 mM PO4
CO2 HCO3- CO2 ingestion only
No maintenance, full excretion
N,P reductions N reductions
P reductions
79.5 h-1
0.73 h-1
C,N,P-limitation
half-saturation parameters KC = 1.810 mM for uptake of CO2
KN = 3.186 mM for uptake of NO3
KP = 0.905 mM for uptake of PO4
max. specific uptake rate parameters jCm = 0.046 mM/OD.h, spec uptake of CO2
jNm = 0.080 mM/OD.h, spec uptake of NO3
jPm = 0.025 mM/OD.h, spec uptake of PO4
reserve turnover rate kE = 0.034 h-1
yield coefficients yCV = 0.218 mM/OD, from C-res. to structure yNV = 2.261 mM/OD, from N-res. to structure yPV = 0.159 mM/OD, from P-res. to structure
carbon species exchange rate (fixed) kBC = 0.729 h-1 from HCO3
- to CO2
kCB = 79.5 h-1 from CO2 to HCO3-
initial conditions (fixed) HCO3
- (0) = 1.89534 mM, initial HCO3- concentration
CO2(0) = 0.02038 mM, initial CO2 concentration
mC(0) = jCm/ kE mM/OD, initial C-reserve density mN(0) = jNm/ kE mM/OD, initial N-reserve density mP(0) = jPm/ kE mM/OD, initial P-reserve density
OD(0) = 0.210 initial biomass (free)
Nannochloropsis gaditana in sea water
Vacancies at VUA-TB
• PhD 4 yr: 2005/02 – 2009/02 EU-project Modelkey Effects of toxicants on canonical communities
• Postdoc 2 yr: 2006/02 – 2008/02 EU-project Modelkey Effects of toxicant in food chains
• PhD 4 yr: 205/06/01 – 2009/06/01 EU-project Nomiracle Toxicity of mixtures of compounds
Further reading
Basic methods of theoretical biology
freely downloadable document on methods http://www.bio.vu.nl/thb/course/tb/
Data-base with examples, exercises under construction
Dynamic Energy Budget theory http://www.bio.vu.nl/thb/deb/