Environ. Sci. Technol. 1904. 18, 713-718

Conney, A. H.; Burns, J. J. Science (Washington, D.C.) 1972, 178, 576. O’Brien, R. D. In “Insecticides-Actions and Metabolism”; O’Brien, R. D., Ed.; Academic Press: New York, NY, 1967; p 209. Malins, D. C.; Collier, T. K. Aquat. Toxicol. 1981,1, 257. Koller, L. D.; Zinkl, J. G. Am. J. Pathol. 1973, 70 (3), 363. Nishizumi, M. Arch. Environ. Health 1970, 21, 620. Hinton, D. E.; Klaunig, J. E.; Lipsky, M. M. Mar. Fish. Rev. 1978, 40, 47. Kimbrough, R. D.; Linder, R. E.; Gaines, T. C. Arch. En- viron. Health 1972,25, 354. Iwasaki, T.; Dempo, K.; Kaneko, A.; Onoe, T. Gann 1972, 63, 21. Domingo, E. 0.; Warren, K. S.; Stenger, R. J. Am. J. Pathol. 1967, 51 , 307. Taper, H. S.; Bannasch, P. Enuiron. J. Cancer 1979, 15, 189. Williams, G. W. Biochim. Biophys. Acta 1980, 605, 167. Tsuda, H.; Hagiwara, A.; Shibata, M.; Masato, 0.; Ito, N. J. Natl. Cancer Inst. 1982, 69 (6). 1383.

Beliles, R. P. In “The Basic Science of Poisons”; Casaratt, L. J.; Doull, F., Eds.; Macmillan Publishing Co.: New York, 1975; p 454. Aoki, K.; Matsudaira, H. J. Natl. Cancer Inst. 1977,59 (6)) 1747. Pliss, G. B.; Khudoley, V. V. J. Natl. Cancer Inst. 1975, 55 (l), 129. Hendricks, J. D.; Scanlon, R. A.; Williams, J. L.; Sinnhuber, R. 0.; Grieco, M. P. J. Natl. Cancer Inst. 1980,64 (6), 1511. Schultz, M. E.; Schultz, R. J. Environ. Res. 1982,27,337. Roubal, W. T. In “Progress in the Chemistry of Fats and Other Lipids”; Holman, R. T., Ed.; Pergamon Press: London, 1972; Vol. XIII, Part 2, pp 61-87. Malins, D. C.; Myers, M. S.; Roubal, W. T. Environ. Sci. Technol. 1983, 17, 679. Malins, D. C.; Myers, M. S.; MacLeod, W. D.; Roubal, W. T., presented at the Second International Symposium on Responses of Marine Organisms to Pollutants, Woods Hole Oceanographic Institution, Woods Hole, MA, 1983.

Jones, G.; Butler, W. H. In “Mouse Hepatic Neoplasia; Butler, W. H.; Newberne, P. M., Eds.; Elsevier Science Publishing Co.: Amsterdam, 1975; p 21.

Received for review December 12,1983. Accepted March 16,1984. A major portion of this work was funded by the Ocean Assess- ment Division, NOAA, Seattle, WA.

Ecotoxicological Assays with Algae: Weibull Dose-Response Curves

Erik R. Christensen*+ and Nlels Nyholmt

Department of Civil Engineering, University of Wisconsin-Mliwaukee, Milwaukee, Wisconsin 53201, and Water Quality Institute, DK-2970 Hoersholm, Denmark

w A newly proposed Weibull model is compared to the probit and logit models for the growth rate of algae as a function of the concentration of metallic and organic toxicants. Four batch assays with Selenastrum capri- cornutum exposed to potassium dichromate and copper and with Scenedesmus suspicatus affected by 3,5-di- chlorophenol and potassium dichromate are carried out. Literature data for four other experiments are also exam- ined. The data fitting is based on linear transformations of the three models using proper weighting. We find that the Weibull model is a t least as appropriate as the probit and logit models. The effective concentrations giving 10 and 90% growth rate reduction, EClO and EC90, are generally lowest for the Weibull model. The slopes (0.5 to 3-5) in the Weibull and logit transformations may be interpreted as the number of toxicant molecules reacting per active receptor of the organism.

Introduction Microorganisms play a key role in aquatic ecology. Algae

and bacteria are at the bottom of the aquatic food chain, and their activity and growth are, therefore, necessary for the adequate functioning of ecosystems. Furthermore, most sewage treatment plants rely on microorganisms so that their activity is of primary importance for proper wastewater treatment.

Toxicants such as heavy metals or organic herbicides can severely inhibit the beneficial growth of microorganisms, both in natural waters and in wastewater treatment plants. This is the basis for the considerable recent interest in ecotoxicological studies relating to microorganisms (1-3).

‘Department of Civil Engineering, University of Wisconsin-

*Water Quality Institute. Milwaukee.

A desirable goal for research on the growth of mi- croorganisms under toxicant stress is to develop and verify mathematical models for their growth as a function of dose (concentration) of the toxicant, that is, dose-response models. Such models are useful for environmental risk assessment, for establishing regulations, and as an aid in elucidating mechanisms of toxic action. The response parameter can, for example, be organism yield or growth rate. In the case of algal assays used in connection with entrophication studies to identify limiting nutrients, the maximum algal yield is a natural response parameter (4). The reason is that there is a direct quantitative relation- ship between the amount of the limiting nutrient in the growth medium and the resulting maximal biomass. However, such a relationship does not exist for toxicants. Their action is better described by the generally delaying effect on growth, that is, through their influence on the growth rate of the microorganisms (1, 2).

Models for microbial growth have dealt mainly with substrate limitation. A prime example is the Monod model which relates growth rate to extracellular substrate con- centration (5). Efforts to model toxicant-affected microbial growth have been comparatively limited. Recent work includes Wong et al. (3) and Rachlin et al. (2). The latter authors fitted their experimental results to the probit model (6). Other dose-response relationships that may prove useful include the logistic (logit) model (7) and a newly proposed Weibull model (8, 9).

The objective of the present work is to compare the Weibull model to the well-known probit and logit models for the growth of unicellular algae under toxicant stress. The response parameter is growth rate. Proper weighting to be used with linear transformations of these models will be developed. We consider single species and several metallic and organic toxicants. Emphasis is placed not only on the concentration giving 50% reduction in growth

0013-936X/84/0918-0713$01.50/0 0 1984 American Chemical Society Environ. Sci. Technol., Vol. 18, No. 9, 1984 713

rate (EC50) but also on the whole range of toxic concen- trations including those for which the growth rate reduc- tion is 10 and 90% (EC10 and EC90). The model pro- ducing the best fit will be the one that is most appropriate for environmental risk assessment and for the establish- ment of water quality criteria. In addition, it can provide clues to the mechanisms of toxic action. While the focus here is on algae, we expect the results to be applicable to microbial growth in general.

Materials and Methods Algal Assays. A total of eight groups of experiments

were considered (Table I). The experiments are batch assays in which the growth rate is determined from the initial (0-3 days) near-exponential part of the growth curve. The calculation of growth rate is based on the total algal cell volume which is closely related to the total biomass and which is considered the fundamental measure of growth. The experimental procedures for cases 4,6,7, and 8 were carried out as described below. In the other cases growth data from the literature were analyzed. Details of culturing methods pertaining to this category are given in the following references: case 1 (10); case 2 (11); cases 3 and 5 (12).

The experimental procedures followed essentially either the EPA bottle test (4 ) or a newly proposed IS0 draft standard (13). The algal cultures were unialgal but not axenic and were grown in 125-mL glass Erlenmeyer flasks containing 50 mL of medium. The flasks were stoppered with foam rubber plugs (nontoxic) and agitated on a rotary shaker. Incubation temperature was 24-26 "C. Continu- ous illumination was provided from a rack of fluorescent tubes of either "cool-white" or "daylight" color, horizontally arranged above the shaker. Light intensity was measured with a Weston illumination meter, Model 756. Water for growth media was obtained fresh from a Millipore Super Q system. The media were not autoclaved. Biomass measurements were carried out with a Model ZBI coulter counter equipped with a mean cell volume (MCV) calcu- lator using a 100-pm aperture tube. Inocula for the ex- periments were derived from cultures propagated under the test conditions. Experimental conditions which dif- fered between experiments were the following:

(1) Copper toxicity experiments were with Selenastrum (case 8). The procedure was as in ref 4. Light intensity was 4.3 klx, and shaking rate was 100 rpm. The medium contains 300 bg L-l disodium ethylenediaminetetraacetate dihydrate (Na2EDTA.2H20) as chelating constituent and has an initial pH of 7.5. For the experimental conditions selected, pH increases to a maximum of about 8.5 in the control cultures after 3 days of exponential growth and then drops. The pH variation is less in toxicant-affected cultures with slower growth rates. Experiments were conducted using inocula from 3-day-old (exponentially growing) cultures. The inoculated cell density was 1.00 x IO4 cells mL-l or 0.18 mg dry weight (dwt) L-l.

(2) Experiments with Scenedesmus (cases 6,7) were part of a round robin test of a proposed IS0 draft standard method (13). The proposed growth medium is nitrogen limited (2 mg of nitrogen L-l) and contains 150 mg L-' NaHC03 which results in an initial pH of 8.5. EDTA is used as a chelator in a concentration of 100 pg L-' Na2EDTA.2H20. Light intensity was 6.9 klx and the shaking rate 130 rpm. During the test, pH increased to a maximum of 8.9 in the control cultures. The inoculated cell density was 1.00 X lo4 cells mL-l or 0.17 mg dwt L-l.

(3) The experiment with Selenastrum and potassium dichromate (case 4) was carried out as above, except that the medium was made phosphate limited (18 pg of phos-

714 Envlron. Scl. Technol., Vol. 18, No. 9, 1984

e a "

i V

Table 11. Comparison of the Weibull Transformation with the Probit and Logit Transformations

type Weibull probit

logit

transformationn

u = Ink + 7 In z (A = Ink) Y = (Y + p log z

I = 8 + 4 In z

relative inhibition (0 - l), P (P = 1 - Q)

1 - ex ( kzn) = 1 - exp(-e') Jl;6tA'[l/(2~)1/2] exp(-t2/2) dt =

1/(1 + e-%*) = 1/(1 + e-')

relative growth rate (0 - l), Q exp(-kzq) = exp(-eu) S L t 8 1,,[1/(2~)'/~1 exp(-t2/2) dt =

1/(1 + e%+) = 1/(1 + e') (1/2)[1 + erf[(Y - 5)/21/2]] (1/2)[1 - erf[(Y- 5)/21/2]]

z is a toxicant concentration; a, 6, k, 7, 8, and 4 are constants.

Table 111. Classification of Models for the Relative Growth Rate of Microorganisms under Nutrient Limitation and Comparison with Complementary Models for the Growth Rate of Microorganisms under Toxicant Stress

complementary ref modelo (nutrient) model (toxicant)

Q = 1/(1 + e%) Q = 1/(1 + e%+)

Monod (5) P = S/(K, + S) Moser (14) P = Sm/(Kx + Sm) Teissier (15), P = 1 - exp(-S/K,) Q = exp(-kz)

"S = nutrient concentration; m, K,, K,, and K. are constants.

phorus L-l) and the bicarbonate concentration was reduced to 50 mg L-l NaHC03 (equilibrium pH 8.1). The inocu- lated cell density was 1.0 X lo3 cells mL-l or 0.014 mg dwt L-l. Due to the low biomass prevailing, pH remained virtually unchanged.

DoseResponse Models. Linear transformations of the three models are given in Table 11. In this table, u (Weibull units), Y (probits), and 1 (logits) are linear transforms, and t is an integration variable. The trans- forms are linear functions of the logarithm of the toxicant concentration.

Models for the growth rate of microorganisms limited by a nutrient are complementary to some of the models considered here for the growth rate under toxicant stress (Table 111). This is to say that the relative growth rate Q for toxicant-affected growth is equal to one minus the relative growth rate P for substrate limited growth when the nutrient concentration S is replaced by the toxicant concentration z. The Monod equation is complementary to the logit model for 4 = 1 and 0 = -In K,. The Moser equation is complementary to the general logit model with 4 = m and 0 = -In K,. Furthermore, the equations by Teissier and Shelef et al. correspond to the Weibull model for q = 1 and k = l/Ka.

The probit, logit, and Weibull models must be consid- ered mainly empirical, although some theoretical basis may be claimed. Argumenh for the probit model are based on the often found appropriateness of the log-normal dis- tribution function in biological systems. The logit model which resembles the probit model over a large range in response is complementary to the Moser equation which, in turn, is identical with the Hill equation for enzyme kinetics (27). Thus, the interpretation of 4 may be the average number of toxicant molecules reacting per active receptor of the organism. As we shall see by the following argument, an analogous interpretation of q in the Weibull model is possible. Consider n toxicant molecules A re- acting with a receptor R:

nA + R F? RA, (1) Assuming equilibrium with a stability constant K, we have

Shelef et al. (16)

The meaning of the growth rate p is essentially the number of cell divisions completed per time unit. The probability

that a cell division will be blocked because of an increase in toxicant concentration from z to z + dz can, therefore, be written as -dp/p. The basic assumption is now that this probability is proportional to the increase in the concentration of blocked receptors d[RA,]:

(3)

where c is a constant. For a large number of receptors, [R] may be considered constant, and integration of eq 3 yields

(4) where z = [A], po is the growth rate for z = 0, p is the growth rate for the actual concentration z of the toxicant, and K = cK[R] is a constant. Note that eq 4 is the Weibull model for the relative growth rate when n is allowed to assume a noninteger value q.

The Weibull, probit, and logit models are fitted to ex- perimental data for the growth of algae by using the linear transformations in Table 11. The statistical weighting is assumed to be constant, and the functional weighting is the reciprocal of the variance caused by the transformation. For example, for the Weibull model (Table 11), we have

u = In (-ln Q) (5) and

du/dQ = 1/(Q In Q) (6)

(7)

where w is the weighting and the variances ou2 and uQ2 are given by u,2 = (du)' and uQ2 = (dQ)2. Because of the assumption of constant statistical weighting, we may drop l/aQ2 from the above equation. We obtain then the fol- lowing weighting expressions: Weibull

-dp / p = cd [ RA,]

Q = p / p 0 = exp(-kzn)

or

w = l / u , 2 = (Q In Q)'(1/uQ2)

w, = (Q, In QJ2

w, = exp[-(P, - 5)']

w, = [&,(I - QJI '

(8)

probit

(9)

(10) where Q, = actual relative growth rate in trial i and P, = probit of Q,.

The procedure for weighted linear least-squares fitting will be illustrated by using the Weibull model as an ex- ample. The linear expression is

u = A + q X (X = In z ) (11)

logit

From ref 18 we obtained intercept

A = (l/O)[(Cw,u,)(Cw,X,') - (Cw,X,)(Cw,u,X,)] (12) slope

t = (1/~)[(C~,)(Cw,uJ,) - (CwiX,)(Cw,u,)l (13)

Environ. Sci. Technol., Vol. 18, No. 9, 1984 715

variances

(SA)' = [l/(Dv)l[(CwiXi2)[Cwi[ui - ( A + xi)]']] (14)

(h12 = [l/(Dv)l[(Cwi)[C~i[~i - ( A + sXJI~II (15) where N = number of trials, v = N - 2 = degrees of free- dom, Xi and ui = actual values of the transforms, and D = (Cwi)(CwiXt) - (CwiXi)'. All summations are from i = 1 to i = N . The fits are compared on the basis of x2 tests:

X' = C(Qj - qiI2 (16) where qi is the calculated relative growth rate. Thus, as required, the goodness-of-fit test is carried out on the untransformed data.

The above weighted linear least-squares scheme can also be derived by determining A and q such that x 2 has a minimum. Following the expressions in Table I1 we have

Qi = exp(-eui) (17)

q; = exp(-eui') (18) where u;' = A + ?Xi is the calculated Weibull unit in trial i. Let us assume that the fit is reasonably good, meaning that ui = ui'+ Aui where Aui is small. Thus, we may write

(19)

(20) which is identical with the weighted sum implicit in the above least-squares method. Since we are in effect min- imizing x 2 , it is reasonable to use the resultant x' as a measure of goodness-of-fit. In other cases, the Kolmogo- rov-Smirnov test may be more appropriate.

For a chosen level of response uo corresponding to, for example, 10, 50, or 90% reduction in growth rate, the effective concentration zo, _e.g., EC10, EC50, or EC90, is calculated from zo = exp(Xo), where

(21)

The 95% confidence upper limit zu and lower limit zL are then determined as follows (6, 19):

Qi - qi = exp(-eui) - exp(-eui') N -Aui exp(ui - cui) = AuiQi In Qi

The expression for x2 (eq.16) becomes

x 2 = C(Qi In Qi)'[ui - ( A + vXJl2

20 = ( l / ~ ) b o - A ) 2.0

Ta 1.5 B

f 1 . 0 -

where X = Cw,Xi(Cw,, g = t2s2/(q2Sxx), t = t statistic = t(v,p), p = probability in t distribution = 5%, s2 = (1/ v)[Cwi[uc - ( A + ~ X i ) l ~ l , and Sxx = ( l / C W i ) [ ( C W i ) . (CwiXi2) - (CwixJ2I. Results

The results of the eight algal assays are shown in Table I. The eight cases have been arranged in ascending order of the magnitude of the steepness parameter q of the Weibull model. For example, Scenedesmus quadricauda with cetyltrimethylammonium chloride has a small slope (0.467), whereas Selenastrum capricornutum with copper shows a steep, almost thresholdlike slope (3.08). A com- parison of x2 values shows that the Weibull model provides the best fit in cases 1,3,5, and 6, the probit model in cases

718 Envlron. Sci. Technol., Vol. 18, No. 9, 1984

- finla"= 1.81 day.' c

-

2.0 1 Seienaslrum ~ i p i i ~ ~ r n u i ~ m

I S 0 -1medium cam 3

c_

f 1.0 -

0.5 - P I L

1. w / '

0003 0 0 1 0 0 3 0 1 0 3 1 3 10 30

K2Cr20, concentration rng1.l

Figure 1. Growth rate vs. K,Cr,O, concentration for Selenastrum capricornutum in IS04 medium (72). The experimental points are fltted to the Weibull (W), probit (P), and logit (L) models.

Scenedesmus rur~ ics lus

ISO. I medium

case 8

9

D C P . C O ~ C ~ ~ I ~ ~ ~ I O ~ . mg1.l

Flgure 2. Growth rate vs. 3,5-dichlorophenol concentrations for Scenedesmus suspicatus in ISO-I medium. The experimental points are fitted to the Weibull (W), probit (P), and logit (L) models.

'. a \ R W I l l i i l l l l 1 1 1 1 ( 1 1 1 1 8 8 / 8 8

0003 001 0 0 3 0 1 0 3 1 3 10 30

CU-EOnEe"llaliDn rngi.1

Figure 3. Growth rate vs. Cu concentration for Selenastrum capri- cornutum in standard algal assay medium (SAAM). The experimental points are fitted to the Weibull (W), probit (P), and logit (L) models.

2,4, and 7, and the logit model in case 8. The interpre- tation of q and 4 described above suggests that the number of toxicant molecules per receptor ranges from 0.5 in case 1 to 3-5 in case 8.

Absolute growth rates vs. toxicant concentration are plotted for cases 3, 6, and 8 in Figures 1-3. The three model curves, based on weighted linear least-squares fits, are also shown in these figures. Since the relative response must be between 0 and 1, experimental relative responses above or equal to 1 have been assigned a value of 0.999, whereas zero response has been assigned a value of 0.001.

Table IV. Equivalent Concentrations and 95% Confidence Limits (mg L-I) for Algal Growth Experimentsn

case alga toxicant modelb EClO EC50 EC90 3 Selenastrum K2Cr207 W 0.0268 (0.00688, 0.0565) 0.662 (0.483, 0.959) 5.11 (2.78, 15.2)

capricornutum P 0.0333 (0.00793, 0.0686) 0.666 0.433, 1.25) 13.3 (4.72, 114) L 0.0309 (0.00729, 0.0645) 0.660 (0.445, 1.15) 14.1 (5.18, 107)

6 Scenedesmus 3,5-dichloro- W 0.572 (0.155, 1.05) 5.32 (3.96, 7.83) 22.1 (13.0, 67.7) suspicatus phenol P 0.641 (0.151, 1.15) 5.87 (4.03, 11.9) 53.8 (21.0, 634)

L 0.546 (0.101, 1.04) 6.10 (4.19, 12.7) 68.1 (24.6, 1094) 8 Selenastrum c u W 0.0263 (0.0175, 0.0323) 0.0485 (0.0426, 0.0535) 0.0717 (0.0641, 0.0861)

capricornutum P 0.0315 (0.0270, 0.0348) 0.0476 (0.0445, 0.0512) 0.0721 (0.0648, 0.0849) L 0.0317 (0.0276, 0.0347) 0.0476 (0.0449, 0.0506) 0.0715 (0.0647, 0.0833)

'Response parameter is relative growth rate. W, Weibull; P, probit; L, logit.

From the previously given weighting formulas (eq 8-10), it may be verified that this is equivalent to assigning little weight to these points. This has also been verified by performing the calculations without these points.

From Figures 1 and 2, it is seen that the Weibull model better reflects the experimentally found abrupt decreases in response at high toxicant concentrations. The thresh- oldlike nature of the response is evident for Selenastrum capricornutum with copper (Figure 3) compared to Sele- nastrum capricornutum with potassium dichromate (Figure 1) and Scenedesmus suspicatus with 3,5-di- chlorophenol (Figure 2).

Equivalent concentrations for 10, 50, and 90% growth inhibition (EC10, EC50, and EC90) pertaining to the above cases are listed in Table IV. When the experimental points are rather evenly distributed over the response in- terval, as in cases 3 and 8 (Figures 1 and 3), the EC50 values are nearly equal. For example, in case 3 we find the values 0.662, 0.666, and 0.660 mg L-' for the three models. It is clear that EClO and EC90 are less reliable than EC50, and this is probably a major reason for the common usage of EC50 for comparison and extrapolation. However, because the slopes of the dose-response curves can vary considerably, more complete information about the toxicological response is obtained when EClO and EC90 are specified as well. This is fully appropriate as long as confidence levels are given for all three levels.

The difference between the upper and lower 95 % con- fidence limits is least for the model that provides the best fit. A case is point is no. 8 for EC50 where these differences are 0.0057,0.0067, and 0.0109 mg L-I for the logit, probit, and Weibull models, respectively. This sequence is in accordance with the hierarchy of models based on goodness of fit (Table I).

A further observation for cases where the experimental points are evenly distributed over the response interval is that EClO and EC90 tend to be the lowest for the Weibull model. In case 3, these values are 0.0268 and 5.11 mg L-l, respectively, compared to considerably higher values for the two other models. The ecotoxicological implication is that the Weibull model, compared to the probit and logit models, predicts a low upper response threshold and at the same time a high response at low concentration.

The fact that EClO is lowest for the Weibull model is significant since this limit often is considered a threshold level for ecological risk management. Reporting of threshold effect levels is now required in many test pro- tocols from the Organization for Economic Cooperation and Development (OECD) and the International Stand- ards Organization (ISO).

Discussion In addition to the three models described above, there

are more empirical methods that can prove useful under certain circumstances. These include, for example, the

moving-average method for the determination of EC10, EC50, and EC90 (6) or simple linear least-squares fitting of a portion of the dose-response curve. However, the advantages of the Weibull, probit, and logit models, and other smooth functions with adequate asymptotic prop- erties and with some theoretical basis, are that they de- scribe the whole feasible concentration range and that they are more general and as such may be used also for mod- eling toxic effects in ecosystems.

Some of the growth rate models considered here for toxicants are complementary to well-known models for substrate limitation. Such a relationship exists between a version of the logit model and the Monod equation, between the general logit model and the Moser equation, and between a version of the Weibull model and Teissier's equation. The general Weibull model can be considered complementary to an extension of Teissier's equation analogous to the generalization of the Monod into the Moser equation (Table 111).

The linear transformations of the Weibull, probit, and logit models are analogous (Table 11) such that linear leastisquares fitting can be carried out in the same general way. While the functional weighting is straightforward, the statistical weighting is more problematic. For quantal assays with a tolerance distribution for individual organ- isms, this weighting is binomial. When the response pa- rameter is based on a counting process without dilutions, as in the 14C method for evaluation of photosynthesis in- hibition, the statistical weighting should be of the Poisson type. However, for growth rate determinations, there is no a priori statistical weighing. A relative growth rate is estimated from biomass measurements performed on toxicant-affected cultures and control cultures. If organ- isms are counted electronically, dilutions are frequently introduced to avoid coincidence losses. This and other factors such as evaluation of the growth rate by averaging over the first few days of growth make it difficult to es- timate the statistical weighting. On the basis of experience, and for convenience, we have assumed that the relative growth rate has a constant statistical weighting. Although proper for relative growth rates above 0.2, this weighting obviously underestimates the weight attached to very small relative growth rates, since the standard deviation ap- proaches zero as the growth rate becomes very small.

The question of statistical weighting has a direct im- plication for the calculation of x2 values. The general expression for this statistic is

where pi and ui are the average and standard deviations, respectively, for the ith observation and xi is the actual response observed. N is the number of trials. Since there are two model parameters, the number of degrees of

Environ. Sci. Technoi., Voi. 18, No. 9, 1984 717

freedom is N - 2. Because we are primarily concerned with relative and not absolute goodness of fit, gi has been as- signed a value of 1. Thus, the listed x2 values are correct except for a constant factor. The utility of the x2 values lies, therefore, in the comparison of different models with respect to their fit to a given set of experimental data.

Responses of zero or one can systematically be taken into account by using standard procedures for maximum like- lihood estimation (6). This method was developed for binomial statistical weighting, for which it can be adapted directly to the three linear transformations in Table 11. It would be desirable to have similar maximum likelihood procedures for other types of statistical weighting such as those of the constant type considered here.

Conclusions (1) The Weibull model provides fits to growth rate for

algae exposed to toxicants which are generally at least as good as fits based on the probit and logit models. This is a significant result since the Weibull model does not appear to have been used for such a purpose previously and is simple to apply. It should, therefore, be useful for the modeling of toxic effects in aquatic ecosystems.

(2) We have developed appropriate weighting to be used with linear transformations of the three models. This weighting is different from the one relevant for quantal assays with macroorganisms.

(3) The toxicant concentration giving 10% growth rate reduction, EC10, is generally lowest for the Weibull model. This is important because EClO is often considered a threshold level in environmental risk management. Thus, if the EClO value for the Weibull model is not used, the derived water quality criterion may be too lenient. The concentration giving 90% growth rate reduction, EC90, is also generally lowest for the Weibull model. This is rel- evant for the control of algal growth in, for example, swimming pools.

(4) The Weibull, probit, and logit models are mainly empirical. We have, however, advanced a heuristic argu- ment showing that the slope 9 in the Weibull transfor- mation may be interpreted as the number of toxicant molecules reacting per active receptor of the organism. The slope # in the logit model has the same meaning based on the Hill equation for enzyme kinetics. This suggests that the number of toxicant molecules per receptor, in our study, ranges from 0.5 for Scenedesmus quadricauda af- fected by cetyltrimethylammonium chloride to 3-5 for Selenastrum capricornutum exposed to copper.

Acknowledgments

The assistance of D. F. Fox and C.-Y. Chen in computer programming is gratefully acknowledged. The work done by N.N. was carried out during a sabbatical visit to the U.S. EPA Environmental Research Laboratory, Corvallis, OR.

Registry No. K2Cr207, 7778-50-9; KC103, 3811-04-9; Cu, 7440-50-8; Pb, 7439-92-1; 3,5-dichlorophenol, 591-35-5; cetyltri- methylammonium chloride, 112-02-7.

Literature Cited Calamari, D.; Chiaudani, G.; Vighi, M. SGOMSEC Work- shop, Roma, Instituto di Ricerca Sulle Acque, CNR, Milano, Italy, July 12-16, 1982. Rachlin, J. W.; Jensen, T. E.; Warkentine, B. 16th Annual Conference on Trace Substances in Environmental Health, University of Missouri, Columbia, May 31-June 3, 1982. Wong, P. T. S.; Chau, Y. K.; Patel, D. In “Advances in Environmental Science and Technology: Special Volume on Aquatic Toxicology”; Wiley: New York, 1983. Miller, W. E.; Greene, J. C.; Shiroyama, T. “The Sele- nastrum capricornutum Printz Algal Assay Bottle Test Experimental Design, Application, and Data Interpretation Protocol”; U.S. Environmental Protection Agency: Envi- ronmental Research Laboratory, Corvallis, Oregon, 1978;

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Damgaard, B. M.; Nyholm, N.; Petersen, J. M. “Comparative Ring-Test with Microorganisms”; Water Quality Institute: Hoersholm, Denmark, 1983. Christensen, E. R. Doctoral Dissertation, University of California, Dissertation Abstracts International, 38/09, 4369B, 1977. Damgaard, B. M. Nyholm, N. “Toxicity-Algae, 2nd IS0 Ring-Test”; Water Quality Institute: Hoersholm, Denmark,

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EPA-600/9-78-018.

5 8 1 ~ .

1981.

Received for review January 23, 1984. Accepted April 9, 1984. This work was supported by U.S. National Science Foundation Grant CEE8103650 to E.R.C. and by the Danish Council of Technical and Scientific Research.

718 Environ. Sci. Technot., Vot. 18, No. 9, 1984