The use of models in biology

Bas Kooijman

Afdeling Theoretische Biologie

Vrije Universiteit Amsterdam

http://www.bio.vu.nl/thb/

Bas@bio.vu.nl

Eindhoven, 2003/02/15

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

time, d time, d

wei

ght,

g

O2 c

onsu

mpt

ion,

ml/

h

l

ege

d

d

ge

legl

d

d

3

332 l

d

dbalO

::

: scaled timel : scaled lengthe : scaled reserve densityg : energy investment ratio

;

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

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/