Mouse Models and the Evolutionary Developmental Biology - DASH
The use of models in biology
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Transcript of The use of models in biology
The use of models in biology
Bas Kooijman
Afdeling Theoretische Biologie
Vrije Universiteit Amsterdam
http://www.bio.vu.nl/thb/
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/