We have investigated the utility of curves used to represent concentration (or dose) plotted against response (or effect). We applied crude survival data from our laboratory to demonstrate that a known hyperbolic curve compares favourably with
the conventionally used sigmoidal regression in robustness, while avoiding problems associated with the use of transforms

Cridland, J.S.

Witten, G.Q.

South African Journal of Science

English

Cape Town University OpenUCT

302 South African Journal of Science 99, July/August 2003 News & ViewsReceptor simulation and the shape ofthe concentration–response curve andits congenersJ. Stephen Cridlanda* and Gareth WittenbWE HAVE INVESTIGATED THE UTILITY OFcurves used to represent concentration(or dose) plotted against response (oreffect). We applied crude survival data fromour laboratory to demonstrate that a knownhyperbolic curve compares favourably withthe conventionally used sigmoidal regressionin robustness, while avoiding problems asso-ciated with the use of transforms.IntroductionWe consider the response of a biologicalsystem, in which the response (or effect)originates from a set of receptors forwhich the promoter or agonist is endoge-nous and the inhibitor or antagonistbeing studied is exogenous. Plotting re-sponse against antagonist concentrationgives rise to graphs where the formermight be, for instance, viability or survivalor some clinical parameter such as bloodpressure, any of them decreasing non-linearly with increased dose or concentra-tion.Such curves may be approximated byequations so that a descriptive factorcan be calculated, such as the EC50 (effec-tive concentration for 50% maximal re-sponse). In general, it can be shownmathematically that artifacts are lesslikely the fewer the constants (or transfor-mations) that are used in these regressionequations. The reductio ad absurdum of ig-noring this rule can be demonstrated byusing the same number of constants asthere are points in a set of data: the fit isperfect, the total variance is zero, but thevalues of the constants are without practi-cal meaning.While agonist concentrations may notremain static during experiments, and thepreparation of the antagonist, especially ifit is a crude plant extract, may containagonists, partial agonists, more than oneantagonist, promoters, antioxidants andso on, it ought still be safer to choose asimple general descriptive equationderived from relevant theory. Gaddumoriginated such an equation by combin-ing the Law of Mass Action and the Clarkhypothesis that relates the effect tothe proportion of receptors occupied.He later recommended what has nowbecome the convention, to plot response(between 0% and 100%) against the loga-rithm of the concentration of antagonist,giving rise to a sigmoidal (S-shaped)curve in reverse.1 This is visually satisfy-ing (compare two different ways of show-ing the same data set: Figs 1 and 2) but hasled, in part, to the mythical concept of the‘threshold concentration’ that is merely adistortion arising from logarithmicallytransforming the data. The EC50 and itsattached statistical scatter is calculatedusing one of a number of semi-empiricalmethods, of which the best is probablystill inspection with the naked eye.This article presents a more rigorousapproach prompted by the needs ofstudents for a sound and robust computerprogram to fit data obtained by testingherbal extracts on malaria parasitesgrown in vitro.For simplicity ’s sake, the equationrelating response to log10 (concentration)will be referred to as the ‘sigmoid’; thedecaying hyperbola [Equation (1)] will becalled the ‘hyperbola’. Confidence limitsis taken to mean 95% confidence limits ofthe corresponding EC50 value through-out.MethodThe first step taken was to write a simu-lation of the interaction between recep-tors, agonists and antagonists, moleculeby molecule, and to examine the meanresponse arising from the former to vary-ing concentrations of the latter, using theClark hypothesis (effect is proportionalto the proportion of receptors occupiedby agonist molecules), the Avogadronumber (for number of molecules in agiven volume) and the Einstein-Smoluchowski equation (for root-mean-square distance travelled by molecules inthermal motion). Each receptor is treatedindependently. Account is taken of therandom position, movement and orienta-tion of active molecules in relation to thereceptors; thus it was shown that theroot-mean-square deviation in responseis theoretically independent of concentra-tion, that is, all concentration/effect datashould be equally weighted. The programwas used to generate ‘ideal’ data to iden-tify candidate regression equations.In the event, the Gaddum equation wasfound to fit data generated by the recep-tor simulation described. For antagonists,it can be expressed in the form:E = k1/(k2 + [Z]) (1)where E is the effect (however expressed)and [Z] is the concentration of antagonist.This is a falling (decaying) hyperbola withk1 and k2 (the EC50) as constants.ResultsNote that Equation (1) is nonparametricand must be solved by repeated iterationuntil the variance is minimized. Theseemingly more obvious approach, tolinearize it by conversion to1/E = k1/k2 + [Z]/k2, (2)then regressing 1/E on [Z] for data sets Aand D in Table 1, gave rise to EC50 values(133 and 45 mg/l, respectively) that wereunrealistically high. This is a consequenceof distorting the relative weighting of thepoints by using an hyperbolic transfor-mation.As stated, the main objective of thisstudy was to produce a robust regressionaDivision of Pharmacology, Faculty of Health Sciences,University of Cape Town, Observatory 7925, South Africa.bSchool of Pure and Applied Mathematics, University ofCape Town, Private Bag, Rondebosch 7701, South Africa.*Author for correspondence.E-mail: scridlan@uctgsh1.uct.ac.za Fig. 1. Hyperbolic regression. Fig. 2. Sigmoidal regression.News & Views South African Journal of Science 99, July/August 2003 303equation, solidly founded in theory andproviding plausible EC50 values with their95% confidence limits. To this end, 45 setsof crude data were selected from mea-surements of viability taken using visiblelight absorptiometry on the conventional96-well plate used to screen herbal prepa-rations for antimalarial activity. The statis-tical scatter within each data set was oftenhigh. The sets were selected consecu-tively from a laboratory notebook exceptthat the test sets summarized in Table 1were included. Each set of data wasregressed upon the equations for the‘hyperbola’ and for the ‘sigmoid’ usingcomputer programs minimizing the un-weighted total sum-of-squares differencebetween the points actually measuredand those derived by calculation. Plausi-bility of EC50 values is difficult to identifyexcept by visual inspection of the crudedata. To be more precise, one can excludeall sets for which no EC50 values and/ortheir confidence limits can be calculated.A confidence range overlapping zerowould be included in this class as itimplies that there is a chance that the EC50does not exist. A further refinement is toexclude, arbitrarily but realistically, setsfor which the EC50 values are insuffi-ciently precise because their upper confi-dence limits are more than twice as large.An example would be the ‘sigmoid’regression for data set A in Table 1.Three sets of data could not be fitted atall, that is, no EC50 values nor confidencelimits could be calculated using eitherequation. For the ‘hyperbola’, the EC50could be calculated for a further sevensets but the confidence limits overlappedzero. For the remaining sets, upper confi-dence limits were all satisfactorily small.EC50 values plausible by inspection andtheir confidence limits could therefore becalculated for 35/45 sets (78%). For the‘sigmoid’, a further three sets could not befitted at all. EC50 values but not their confi-dence limits could be calculated for 10sets. For five sets, both EC50 and confi-dence limits could be calculated but theupper confidence limit was too large. Ad-equate means and confidence limits,then, could only be calculated for 24/45sets (53%). These results do not apply uni-versally but it seems clear that the regres-sion of concentrat ion-responseobservations on to the decaying hyper-bola of Equation (1) is robust. Observationsuggests that confidence limits (where ac-ceptable) are similar for both curves.We thank the creators of the GraphPad Prism v.2.01graphical and statistical package used for thesigmoidal regression and plotting of graphs, theauthors of Statistica v.6 used for the hyperbolic regres-sion, and also the students who provided the motiva-tion for this project.1. Burgen A.S.V. and Mitchell J.F. (eds) (1985). InGaddum’s Pharmacology, 9th edn, chap. 1, p. 5.Oxford Medical Publications, Oxford.Table 1. Comparison of fit of concentration-effect data fitted to a hyperbola (hyp) with thesame data fitted to a sigmoid (sig).Data set Curve form Mean EC50 (mg/l) 95% confidence limitsA Hyp 72 31–113Sig* 58 24–164B Hyp 20 12–29Sig 20 11–37C Hyp 9 5–14Sig 10 5–18D Hyp 6 3–10Sig 7 5–9*‘Sigmoid’ incomplete.                   

Receptor simulation and the shape of the concentration-response curve and its congeners

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Receptor simulation and the shape of the concentration-response curve and its congeners

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