The use of models in biology

1. 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

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 = 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

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 = k0exp{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 = 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

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 • Problem: • 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 CO2 HCO3- 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 = 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)

19. 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/