Journal of Plankton Research Vol.17 no.6 pp.1391-1394, 1995

LETTER TO THE EDITOR

The fate of comparisons of models in temperature-dependentgrowth of copepods: A reply to the comment by McLaren.

Jose Ma Blanco1, Francisco Guerrero2 and Valeriano Rodriguez1

1 Departamento de Ecologia, Universidad de Mdlaga, Campus de Teatinos, 29071Mdlaga and 2Departamento de Biologta Animal, Vegetal y Ecologia, Facultad deCiencias Experimentales, 23071 Jain, Spain

We thank McLaren (1995) for his careful and detailed analysis of our paper(Guerrero et al., 1994) even though we do not agree with much of his criticism.We have structured our discussion of the principal points of disagreement in theframework of the several objectives of the different models describing the closedependence of growth (in Copepods and any poikilothermic organism) ontemperature to:

(1) predict growth rates;(2) characterise the response by using parameters that can be compared in

different situations;(3) study the physiological response.

(1) Prediction of growth rates. The problem is mainly mathematical. We have aset of values of growth-rates or growth-times recorded at different temperaturesand from these try to predict the growth-rates or growth-times at othertemperatures within/outside the temperature range studied (interpolationextrapolation). We need an easily manageable equation with a robust fit andwhich does not need a large amount of data to give convergent results. Tauti'sequation (Guerrero et al., 1994) fulfils these requirements and both Guerrero etal. (1994) and McLaren (1995) show that it gives a good fit in all the cases givenin these two works. It appears to be the best model when the data available areinsufficient for other types of equations, especially those with three parameters,like that of Belehrddek.

The fit of three-parameter equations necessarily requires numerical reiterationso that the final solution depends on the initial exploration conditions and eventhe computer search algorithm. Yes, normally, this problem does diminish themore data one has, but the extra effort needed to obtain a large amount of data iscostly in time and money. Moreover, the size of data sets for those organismsthat live in a narrow range of temperatures will usually be small and, for thisreason alone, two-parameter equations are generally better, provided, of course,that they predict satisfactorily. Conditions being equal, Tauti's equation will givebetter fits than other two-parameter equations. Is the extra effort and expensethat the use of three-parameter equations cause the researchers really worthwhile, particularly when their need for more data is because their fits are morecomplicated?

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When one needs to predict data outside the range of experimental dataavailable, one must choose the equation more carefully. Normally, there is littleneed to extrapolate ambiental temperature values for extreme environmentalsituations of an organism (very cold or very warm) because in these cases, oneusually has to take into account additional factors.

(2) Equation parameters. The parameters of the resultant equation can servemoreover to compare the responses of similar species. In this sense, it has alwaysbeen easier to interpret B£lehradek's equation than exponential equations, butonly when one BeHehrddek parameter is fixed, otherwise the free fit of the threeparameters will have very wide ranges that are impossible to compare (forexample, McLaren, 1995, gives a values that vary in three orders of magnitude).

As Belehradek's equation only serves well when one parameter is fixed, whichparameter should be fixed? Guerrero et al. (1994) discussed the nuisance offixing To since this parameter is associated with the growth temperature of theorganism in question. This consideration led to the observation by Guerrero etal. (1994) that the three parameters of BSlehradek's equation have one degree offreedom more than necessary for its intended purpose. Generally, b is fixed togive useful relationships between a and 7o for similar species, usually a valueclose to 2 is assigned. But, just as one expects the value of To to be different foreach species, the response curve or 'shape' can also be different for each species.It is easy to see that if one assigns to b a value other than 2 one obtains otherrelationships between the parameters. Why choose b~2? Each author has hisown idea of the appropriate value and if each has a different one it is obviousthat meaningful comparisons between their works will be impossible.

(3) Physiological response. To analyse effectively the physiological response totemperature the model employed should be biologically significant and, in thissense, only Arrhenius' equation is justified. However, one of the requirements ofthe Arrhenius model is that the activation-energy of the process under studyshould remain constant and this means that the enzymes involved are workingwithin their range of optimum temperatures. Consequently, it is not surprisingthat outside this range exponential equations like those of Arrhenius and Tautiare poor predictors. Above a certain temperature all enzymes denature and thisdenaturing increases enormously the activation-energy. Nevertheless, when wetake into account the change of the system, the application of these sameexponential laws explained the denaturing of enzymes and associated processes.This means that equations calculated to fit values within the normal physiologicalrange are inappropriate for studying the growth inhibition caused by hightemperatures. To find an equation that might reproduce the full range ofbehaviour of an organism from abnormally low environmental temperatures toabnormally high ones is a purely mathematical quest that has no biologicalsignificance or meaningful application.

Consequently, Arrhenius' equation conserves all its biological significancewithin the normal physiological range of temperatures of the species. Tauti'sequation also does this because it was derived directly from Arrhenius' equation

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Letter to the Editor

(see Appendix 1) and it assumes that the value of Qx0 remains constant withinthe range of physiological temperatures. In spite of McLaren's allegation that'the use of the Celsius scale in the Tauti equation destroys any theoretical basisfor its use', the Tauti equation has the great advantage that one can use degreesCelsius because the temperature term denotes a temperature-difference not anabsolute value (see Appendix 1). Yes, when we use Arrhenius' equation we haveto use the absolute scale, but not in Tauti's equation.

The biological significance of Belehrddek's equation is seriously damagedwhen one fixes one of the three parameters because the original theoreticalliberties of each of the other two parameters are also restricted. Nevertheless,this restricted equation can still be used to identify qualitively the differentpatterns of the different responses to temperature of the copepod species understudy. The interpretation of To as the 'biological zero temperature' (below whichlife-processes are arrested) is perfectly valid as an impartial interpretation of theBelehrddek model (see Guerrero et al., 1994). Normally, this parameter showsnegative values when the value of b is fixed at b~2, that is to say it absurdlypredicts functioning processes at temperatures well below freezing. If, however,we take into account the observations made above about the working of theequations and we recognize that this is a case of extrapolation to an abnormallylow range of temperatures, it is nothing to worry about. If one accepts that thenegative values have no importance when they fall outside the physiologicalrange, for the same reason one has to accept that one just cannot comparetheoretical results for these extremes of temperature.

In conclusion, Tauti's equation is mathematically and statistically better, its fitis easy, it needs only a small amount of data, and it predicts effectively. In thesame conditions, Belehrddek's equation is only manageable when one fixes oneparameter and this then introduces an arbitrary and subjective bias into theanalytical process. This distortion can make the results uncomparable eventhough the shape of the equation identifies more directly with biologicalparameters than that of Tauti's equation.

ReferencesGuerrero^F., BlancoJ.M° and Rodriguez,V. (1994) Temperature-dependent development in marine

copepods: a comparative analysis of models. /. Plankton Res., 16, 95-103.

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Appendix 1

Derivation of Tauti equation from Arrhenius equationArrhenius-Van't Hoff equation:

dT = dmt-blf (1)

where dT is the rate of growth at temperature T, dm is the maximum growth rateand b is a constant that depends on the activation energy. Then

(2)G l 0 = =dT

and, in the same manner

reordering

if we take k = {T^+ 10)/T, we have

dT = dTiQwk<!^r (5)

that is Tauti equation,

dr = A ec r (6)

where

-kll k

A = dTlQio '» and c = — In Q10

If we assume Qi0 as constant (true in biological systems) then A and c will beconstant, since k remains constant when the temperatures T and T\ (thetemperature of reference for (2io) are quite close, as it is usual in these studies.Note that in equation (5) we have a difference of temperatures that it is the sameyet in Kelvin or Celsius scale, and the effect on equation (6) will affect only to A,being easy to calculate the effect.

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