The use of models in DEB research Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Nijmegen, 2005/02/23

Contents DEB theory introduction Scales in space & time Empirical cycle Dimensions Plasticity in parameters Stochastic vs deteriministic Dynamical systems Nijmegen, 2005/02/23

Dynamic Energy Budget theory for metabolic organisation Uptake of substrates (nutrients, light, food) by organisms and their use (maintenance, growth, development, reproduction) First principles, quantitative, axiomatic set up Aim: Biological equivalent of Theoretical Physics Primary target: the individual with consequences for sub-organismal organization supra-organismal organization Relationships between levels of organisation Many popular empirical models are special cases of DEB

Empirical special cases of DEB yearauthormodelyearauthormodel 1780Lavoisier multiple regression of heat against mineral fluxes 1950Emerson cube root growth of bacterial colonies 1825Gompertz Survival probability for aging 1951Huggett & Widdas foetal growth 1889Arrhenius temperature dependence of physiological rates 1951Weibull survival probability for aging 1891Huxley allometric growth of body parts 1955Best diffusion limitation of uptake 1902Henri Michaelis--Menten kinetics 1957Smith embryonic respiration 1905Blackman bilinear functional response 1959Leudeking & Piret microbial product formation 1910Hill Cooperative binding 1959Holling hyperbolic functional response 1920Pütter von Bertalanffy growth of individuals 1962Marr & Pirt maintenance in yields of biomass 1927Pearl logistic population growth 1973Droop reserve (cell quota) dynamics 1928Fisher & Tippitt Weibull aging 1974Rahn & Ar water loss in bird eggs 1932Kleiber respiration scales with body weight 3/ Hungate digestion 1932Mayneord cube root growth of tumours 1977Beer & Anderson development of salmonid embryos DEB theory is axiomatic, based on mechanisms not meant to glue empirical models Since many empirical models turn out to be special cases of DEB theory the data behind these models support DEB theory This makes DEB theory very well tested against data

Some DEB pillars life cycle perspective of individual as primary target embryo, juvenile, adult (levels in metabolic organization) life as coupled chemical transformations (reserve & structure) time, energy & mass balances surface area/ volume relationships (spatial structure & transport) homeostasis (stoichiometric constraints via Synthesizing Units) syntrophy (basis for symbioses, evolutionary perspective) intensive/extensive parameters: body size scaling

1-  maturity maintenance maturity offspring maturation reproduction Basic DEB scheme foodfaeces assimilation reserve feeding defecation structure somatic maintenance growth 

molecule cell individual population ecosystem system earth time space Space-time scales When changing the space-time scale, new processes will become important other will become less important Individuals are special because of straightforward energy/mass balances Each process has its characteristic domain of space-time scales

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 Crocodylus johnstoni, Data from Whitehead 1987 yolk embryo

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

Vacancies at VUA-TB PhD 4 yr: 2005/02 – 2009/02 EU-project Modelkey Effects of toxicants on canonical communities Postdoc 2 yr: 2006/02 – 2008/02 EU-project Modelkey Effects of toxicant in food chains PhD 4 yr: 205/06/01 – 2009/06/01 EU-project Nomiracle Toxicity of mixtures of compounds

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