1. The use of models in biology
Bas Kooijman
Afdeling Theoretische Biologie
Vrije Universiteit Amsterdam
Eindhoven, 2003/02/15

2. 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)

3. Empirical cycle

4. 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

5. 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!

6. 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

7. Allometric functions Two curves fitted: a L2 + b L3
with a = μl h-1 mm-2
b = μl h-1 mm-3
a Lb
with a = μl h-1 mm-2.437
b = 2.437
O2 consumption, μl/h
Length, mm

8. 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 = k0 exp{1 – T0 /T}, with k0 = exp{a – 1}
Difference with allometric model:
no reference value required to solve dimension problem

9. Arrhenius relationship
ln pop growth rate, h-1
r1 = h-1
T1 = K
TH = K
TL = K
TA = K
TAL = K
TAH = K
103/T, K-1
103/TH
103/TL

10. 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

11. Biodegradation of compounds
n-th order model
Monod model
scaled conc.
scaled conc.
scaled time
scaled time

12. 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)

13. 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 errors
Alternatives:
deterministic systems with stochastic inputs
differences in parameter values between individuals
parameter estimation methods become very complex

14. 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)

15. 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

16. Embryonic development
O2 consumption, ml/h
weight, g
time, d
time, d ;
 : scaled time
l : scaled length
e : scaled reserve density
g : energy investment ratio
::

17. 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
CO HCO CO2 ingestion only
No maintenance, full excretion
79.5 h-1
0.73 h-1

18. C,N,P-limitation Nannochloropsis gaditana in sea water
half-saturation parameters
KC = mM for uptake of CO2
KN = mM for uptake of NO3
KP = mM for uptake of PO4
max. specific uptake rate parameters
jCm = mM/OD.h, spec uptake of CO2
jNm = mM/OD.h, spec uptake of NO3
jPm = mM/OD.h, spec uptake of PO4
reserve turnover rate
kE = h-1
yield coefficients
yCV = mM/OD, from C-res. to structure
yNV = mM/OD, from N-res. to structure
yPV = mM/OD, from P-res. to structure
carbon species exchange rate (fixed)
kBC = h-1 from HCO3- to CO2
kCB = h-1 from CO2 to HCO3-
initial conditions (fixed)
HCO3- (0) = mM, initial HCO3- concentration
CO2(0) = 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) = initial biomass (free)

19. Further reading Basic methods of theoretical biology
freely downloadable document on methods
Data-base with examples, exercises under construction
Dynamic Energy Budget theory