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)

Empirical cycle

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/