Contents

## Basic Methods in Theoretical Biology Empirical cycle 1.1 Assumptions summarize insight 1.1 Model: definition & aims 1.1 Model properties 1.1

Walker, Jim, Features Editor has reference to this Academic Journal, PHwiki organized this Journal Basic Methods in Theoretical Biology 1 Methodology 2 Mathematical toolkit 3 Models as long as processes 4 Model-based statistics http://www.bio.vu.nl/thb/course/tb/tb.pdf Empirical cycle 1.1 Assumptions summarize insight 1.1 task of research: make all assumptions explicit these should fully specify subsequent model as long as mulations assumptions: interface between experimentalist theoretician discrepancy model predictions measurements: identify which assumption needs replacement models that give wrong predictions can be very useful to increase insight structure list of assumptions to replacebility (mind consistency!)

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Model: definition & aims 1.1 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) observations/measurements: require interpretation, so involve assumptions best strategy: be as explicitly as possible in assumptions Model properties 1.1 language errors: mathematical, dimensions, conservation laws properties: generic (with respect to application) realistic (precision; consistency with data) simple (math. analysis, aid in thinking) complex models are easy to make, difficult to test simple models that capture essence are difficult to make plasticity in parameters (support, testability) ideals: assumptions as long as mechanisms (coherence, consistency) distinction action variables vs measured quantities need as long as core in addition to auxiliary theory Modelling 1 1.1 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)

Modelling 2 1.1 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 as long as mechanisms (coherence, consistency) distinction action variables/meausered quantities core/auxiliary theory Presumptions Laws 11.1 Laws Theories Hypotheses Presumptions decrease in demonstrated support amount of support is always limited Proofs only exist in mathematics role of abstract concepts 0 large facts general theories no predictions possible predictions possible Theories Models 1.1 Theory: set of coherent in addition to consistent assumptions from which models can be derived as long as particular situations Models may or may not represent theories it depends on the assumptions on which they are based If a model itself is the assumption, it is only a description if it is inconsistent with data, in addition to must be rejected, you have nothing If a model that represents a theory must be rejected, a systematic search can start to assumptions that need replacement Unrealistic models can be very useful in guiding research to improve assumptions (= insight) Many models dont need to be tested against data because they fail more important consistency tests Testability of models/theories comes in gradations

Auxiliary theory 1.1 Quantities that are easy to measure (e.g. respiration, body weight) have contributions as long as m several processes they are not suitable as variables in explenatory models Variables in explenatory models are not directly measurable we need auxiliary theory to link core theory to measurements St in addition to ard DEB model: isomorph with 1 reserve & 1 structure that feeds on 1 type of food Measurements typically involve interpretations, models 1.1 Given: the air temperature in this room is 19 degrees Celsius Used equipment: mercury thermometer Assumption: the room has a temperature (spatially homogeneous) Actual measurement: height of mercury column Height of the mercury column temperature: model! How realistic is this model What if the temperature is changing Task: make assumptions explicit in addition to be aware of them Question: what is calibration Complex models 1.1 hardly contribute to insight hardly allow parameter estimation hardly allow falsification Avoid complexity by delineating modules linking modules in simple ways estimate parameters of modules only

Causation 1.1 Cause in addition to effect sequences can work in chains A B C But are problematic in networks A B C Framework of dynamic systems allow as long as holistic approach Dimension rules 1.2 quantities left in addition to right of = must have equal dimensions + in addition to only defined as long as quantities with same dimension ratios 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, in addition to pH1 pH2 dont 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 in addition to x Distinguish dimensions in addition to units! Models with dimension problems 1.2 Allometric model: y = a W b y: some quantity a: proportionality constant W: body weight b: allometric parameter in (2/3, 1) Usual as long as m ln y = ln a + b ln W Alternative as long as m: y = y0 (W/W0 )b, with y0 = a W0b Alternative model: y = a L2 + b L3, where L W1/3 Freundlichs 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 as long as m: C = C0 (c/c0 )1/n, with C0 = kc01/n Alternative model: C = 2C0 c(c0+c)-1 (Langmuirs model) Problem: No natural reference values W0 , c0 Values of y0 , C0 depend on the arbitrary choice

Egg development time 1.2 Bottrell, H. H., Duncan, A., Gliwicz, Z. M. , Grygierek, E., Herzig, A., Hillbricht-Ilkowska, A., Kurasawa, H. Larsson, P., Weglenska, T. 1976 A review of some problems in zooplankton production studies. Norw. J. Zool. 24: 419-456 Space-time scales 1.3 molecule cell individual population ecosystem system earth time space When changing the space-time scale, new processes will become important other will become less important Models with many variables & parameters hardly contribute to insight Each process has its characteristic domain of space-time scales Problematic research areas 1.3 Small time scale combined with large spatial scale Large time scale combined with small spatial scale Reason: likely to involve models with large number of variables in addition to parameters Such models rarely contribute to new insight due to uncertainties in as long as mulation in addition to parameter values

Different models can fit equally well 1.5 Length, mm O2 consumption, l/h Two curves fitted: a L2 + b L3 with a = 0.0336 l h-1 mm-2 b = 0.01845 l h-1 mm-3 a Lb with a = 0.0156 l h-1 mm-2.437 b = 2.437 Plasticity in parameters 1.7 If plasticity of shapes of y(xa) is large as function of a: little problems in estimating value of a from {xi,yi}i (small confidence intervals) little support from data as long as underlying assumptions (if data were different: other parameter value results, but still a good fit, so no rejection of assumption) A model can fit data well as long as wrong reasons Biodegradation of compounds 1.7 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

Biodegradation of compounds 1.7 n-th order model Monod model scaled time scaled time scaled conc. scaled conc. Verification falsification 1.9 Verification cannot work because different models can fit data equally well Falsification cannot work because models are idealized simplifications of reality All models are wrong, but some are useful Support works to some extend Usefulness works but depends on context (aim of model) a model without context is meaningless Model without dimension problem 1.2 Arrhenius model: ln k = a T0 /T k: some rate T: absolute temperature a: parameter T0: Arrhenius temperature Alternative as long as m: k = k0 exp{1 T0 /T}, with k0 = exp{a 1} Difference with allometric model: no reference value required to solve dimension problem

Central limit theorems 2.6 The sum of n independent identically (i.i.) distributed r in addition to om variables becomes normally distributed as long as increasing n. The sum of n independent point processes tends to behave as a Poisson process as long as increasing n. Number of events in a time interval is i.i. Poisson distributed Time intervals between subsequent events is i.i. exponentially distributed Sums of r in addition to om variables 2.6 Exponential prob dens Poisson prob Normal probability density 2.6

Dynamic systems 3.2 Defined by simultaneous behaviour of input, state variable, output Supply systems: input + state variables output Dem in addition to systems: input state variables + output Real systems: mixtures between supply & dem in addition to systems Constraints: mass, energy balance equations State variables: span a state space behaviour: usually set of odes 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 Statistics 4.1 Deals with estimation of parameter values, in addition to 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) Stochastic vs deterministic models 4.1 Only stochastic models can be tested against experimental data St in addition to ard way to extend deterministic model to stochastic one: regression model: y(x a,b, ) = f(xa,b, ) + e, with e N(0,2) Originates from physics, where e st in addition to s as long as 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

Profile likelihood 4.6 large sample approximation 95% conf interval Comparison of models 4.6 Akaike In as long as mation Criterion as long as sample size n in addition to K parameters in the case of a regression model You can compare goodness of fit of different models to the same data but statistics will not help you to choose between the models Confidence intervals 4.6 time, d length, mm 95% conf intervals excludes point 4 includes point 4

## Walker, Jim Features Editor

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