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The use of models in DEB research

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The use of models in DEB research

Bas Kooijman

Dept theoretical biology

Vrije Universiteit Amsterdam

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 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

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

Empirical special cases of DEBSome 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

feeding

food

faeces

assimilation

reserve

somatic

maintenance

maturity

maintenance

1-

maturation

reproduction

growth

maturity

offspring

structure

Basic DEB schemeSpace-time scales

Each process has its characteristic domain of space-time scales

system earth

space

ecosystem

population

When changing the space-time scale,

new processes will become important

other will become less important

Individuals are special because of

straightforward energy/mass balances

individual

cell

time

molecule

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 W0b
- 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 = kc01/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

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

O2 consumption, μl/h

Length, mm

Model without dimension problem

Arrhenius model: ln k = a – T0 /T

k: some rate T: absolute temperature

a: parameter T0: Arrhenius temperature

Alternative form:

k = k0exp{1 – T0 /T}, with k0 = exp{a – 1}

Difference with allometric model:

no reference value required to solve dimension problem

Arrhenius relationship

ln pop growth rate, h-1

r1 = 1.94 h-1

T1 = 310 K

TH = 318 K

TL = 293 K

TA = 4370 K

TAL = 20110 K

TAH = 69490 K

103/T, K-1

103/TH

103/TL

Biodegradation of compounds

n-th order model

Monod model

;

;

X : conc. of compound, X0 : X at time 0

t : time k : degradation rate

n : order K : saturation constant

Biodegradation of compounds

n-th order model

Monod model

scaled conc.

scaled conc.

scaled time

scaled time

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 eN(0,2)
- Originates from physics, where e stands for measurement error
- Problem:
- deviations from model are frequently not measurement errors
- Alternatives:
- deterministic systems with stochastic inputs
- differences in parameter values between individuals
- Problem:
- parameter estimation methods become very complex

Statistics

- Deals 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 systems

Defined by simultaneous behaviour of

input, state variable, output

Supply systems: input + state variables output

Demand systems input state variables + output

Real systems: mixtures between supply & demand systems

Constraints: mass, energy balance equations

State variables: span a state space

behaviour: usually set of ode’s with parameters

Trajectory: map of behaviour state vars in state space

Parameters:

constant, functions of time, functions of modifying variables

compound parameters: functions of parameters

Embryonic development 3.7.1

Crocodylus johnstoni,

Data from Whitehead 1987

embryo

yolk

O2 consumption, ml/h

weight, g

time, d

time, d

: scaled time

l : scaled length

e: scaled reserve density

g: energy investment ratio

;

C,N,P-limitation

N,P reductions

N reductions

P reductions

Nannochloropsis gaditana

(Eugstimatophyta) in sea water

Data from Carmen Garrido Perez

Reductions by factor 1/3

starting from 24.7 mM NO3, 1.99 mM PO4

CO2 HCO3- CO2 ingestion only

No maintenance, full excretion

79.5 h-1

0.73 h-1

C,N,P-limitation

Nannochloropsis gaditana in sea water

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)

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/

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