The use of models in biology Bas Kooijman Afdeling Theoretische Biologie Vrije Universiteit Amsterdam 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 pH 1 – pH 2 ? 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 = y 0 (W/W 0 ) b, with y 0 = a W 0 b Alternative model: y = a L 2 + b L 3, where L  W 1/3 Freundlich’s model: C = k c 1/n C: density of compound in soil k: proportionality constant c: concentration in liquid n: parameter in (1.4, 5) Alternative form: C = C 0 (c/c 0 ) 1/n, with C 0 = kc 0 1/n Alternative model: C = 2C 0 c(c 0 +c) -1 (Langmuir’s model) Problem: No natural reference values W 0, c 0 Values of y 0, C 0 depend on the arbitrary choice

Allometric functions Length, mm O 2 consumption, μl/h Two curves fitted: a L 2 + b L 3 with a = μl h -1 mm -2 b = μl h -1 mm -3 a L b with a = μl h -1 mm b = 2.437

Model without dimension problem Arrhenius model: ln k = a – T 0 /T k: some rate T: absolute temperature a: parameter T 0 : Arrhenius temperature Alternative form: k = k 0 exp{1 – T 0 /T}, with k 0 = exp{a – 1} Difference with allometric model: no reference value required to solve dimension problem

Arrhenius relationship 10 3 /T, K -1 ln pop growth rate, h /T H 10 3 /T L r 1 = 1.94 h -1 T 1 = 310 K T H = 318 K T L = 293 K T A = 4370 K T AL = K T AH = K

Biodegradation of compounds n-th order modelMonod model ; ; X : conc. of compound, X 0 : X at time 0 t : time k : degradation rate n : order K : saturation constant

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

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 {x i,y i } 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 errors Alternatives: deterministic systems with stochastic inputs differences in parameter values between individuals Problem: parameter estimation methods become very complex

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)

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

Embryonic development time, d weight, g O 2 consumption, ml/h ::::  : scaled time l : scaled length e : scaled reserve density g : energy investment ratio ;

C,N,P-limitation Nannochloropsis gaditana (Eugstimatophyta) in sea water Data from Carmen Garrido Perez Reductions by factor 1/3 starting from 24.7 mM NO 3, 1.99 mM PO 4 CO 2 HCO 3 - CO 2 ingestion only No maintenance, full excretion N,P reductionsN reductions P reductions 79.5 h h -1

C,N,P-limitation half-saturation parameters K C = mM for uptake of CO 2 K N = mM for uptake of NO 3 K P = mM for uptake of PO 4 max. specific uptake rate parameters j Cm = mM/OD.h, spec uptake of CO 2 j Nm = mM/OD.h, spec uptake of NO 3 j Pm = mM/OD.h, spec uptake of PO 4 reserve turnover rate k E = h -1 yield coefficients y CV = mM/OD, from C-res. to structure y NV = mM/OD, from N-res. to structure y PV = mM/OD, from P-res. to structure carbon species exchange rate (fixed) k BC = h -1 from HCO 3 - to CO 2 k CB = 79.5 h -1 from CO 2 to HCO 3 - initial conditions (fixed) HCO 3 - (0) = mM, initial HCO 3 - concentration CO 2 (0) = mM, initial CO 2 concentration m C (0) = j Cm / k E mM/OD, initial C-reserve density m N (0) = j Nm / k E mM/OD, initial N-reserve density m P (0) = j Pm / k E mM/OD, initial P-reserve density OD(0) = initial biomass (free) Nannochloropsis gaditana in sea water

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