(Jennison, 1937). This lack of agreement may be overcome, at

Transcription

1 EVALUATION OF THE ERRORS INVOLVED IN ESTIMATING BACTERIAL NUMBERS BY THE PLATING METHOD' MARSHALL W. JENNISON AND GEORGE P. WADSWORTH Department of Biology and Public Health, and Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts Received for publication August 1, 1939 The plate count method for estimating bacterial populations is satisfactory for many comparative purposes if relative rather than absolute numbers of cells are wanted, although in some cases, because of clumping, plate counts may not bear a constant relation to total counts even during the logarithmic growth phase (Jennison, 1937). This lack of agreement may be overcome, at least with some organisms, by proper shaking to break up clumps of cells (Ziegler and Halvorson, 1935).2 Considering the plating method per se, the total error of the mean plate count of a given dilution of cells is chiefly made up of two rather distinct sources of deviations: (a) the distribution or sampling error, sometimes inaccurately called the counting error, (i.e., variation in number of colonies, due to sampling, between replicate plates of the given dilution), and (b), the dilution error, (i.e., the errors of pipetting involved in reaching the given dilution). For purposes of discussion, it is assumed that optimum conditions are provided for the growth of the organisms. It is customary to measure the reliability of the plate count 1 Contribution No. 159 from the Department of Biology and Public Health, Massachusetts Institute of Technology. 2Recently, Riker and Baldwin (Phytopathology, 29, , 1939) have pointed out that the problem of coalescence of surface colonies on plates has apparently not been treated quantitatively in bacteriology, but may have an application both to plate counts per se and as regards their relation to total counts. A brief mathematical study estimates the probability that the growth from original bacterial loci may coalesce as the colonies develop. Practical evaluation appears difficult. 389

2 390 MARSHALL W. JENNISON AND GEORGE P. WADSWORTH by calculating only the distribution error, and assuming that the dilution error is small, constant, and unimportant. We shall show, however, that at best this dilution error is of about the same order of magnitude as the distribution error, and is, therefore, equally deserving of consideration in arriving at the total error of plate counts. Furthermore, the dilution error increases with higher dilutions, whereas the distribution error does not. Obviously, one must take into account both sources of variation in evaluating the total error, as, for example, in a problem involving significance of differences, in which the same dilution might be employed. The size of the distribution error depends upon the number of replicate plates counted, within limits, other conditions being the same. To obtain a small distribution error, a dilution giving the proper number of colonies per plate ( ) for enumeration must be available, in order to minimize errors of crowding and of sampling (Wilson, 1922), and a sufficient number of replicate plates (3-5) to give a precise mean must be used (Wilson and Kullmann, 1931). So far as evaluating the distribution error is concerned, this is usually done by calculating the standard deviation of the mean (standard error) of the replicate plates, assuming that the variation between such plates is that of random samples. Under good experimental conditions, this coefficient of variation will average 4-45 per cent (Jennison, 1937). In order to test whether observed variations between replicate plates are due to chance or to technique, the x2 ("chi square") test may be used (Wilson and Kullmann, 1931). The calculated values of X2 will be distributed in a known manner if the replicate samples are from a Poisson series, that is, if their variation is that of random samples from the same population. Fisher, Thornton, and MacKenzie (1922), and Fisher (1938), have shown that a Poisson distribution is obtained in parallel plate counts made under standardized experimental conditions. Both the x2 test and calculation of the. standard error of replicate plates apply only to a given dilution; they do not account for errors involved in arriving at that dilution.

3 ESTIMATING BACTERIAL NUMBERS BY PLATING METHOD 391 The size of the dilution error will depend upon errors in volume of dilution blanks, the variation in delivery of pipettes, and upon the number of dilutions made. We shall assume that dilutions are made in powers of 10, using 9 ml. and 99 ml. dilution blanks and 1 ml. pipettes. With the same percentage error in 99 ml. and 9 ml. blanks, it is, of course, better to use the former in preference to the latter, since fewer are required to reach a given dilution. The average error in volume of blanks, and in delivery of pipettes under experimental conditions must be known. PURPOSE It is the purpose of this paper to derive an expression for evaluating the dilution error, and to show how, having an estimate of both the dilution error and distribution error, the most probable total error in the plate count may be determined. A useful table of dilution errors is included. The distribution error, dilution error, and total error are conveniently expressed as percentage standard deviations. The expression for the dilution error is derived as follows: Let x = 1 ml. with a standard deviation of delivery of a ml., y = 9 ml. with a standard deviation of volume of b ml., and z = 99 ml. with a standard deviation of volume of c ml., where a, b, and c are standard deviations of a series of pipettes and blanks. Thus x + y = u ml. has a standard deviation of -/a2 + b2 ml., and x + z = v ml. has a standard deviation of Va2 + c2 ml. Also, let m = number of 9 ml. blanks used in making a series of dilutions, and n = number of 99 ml. blanks used in making a series of dilutions. In making dilutions, we add x ml. of original bacterial suspension to y or to z sterile blanks. If we add x to z, we get v; then the fraction of the original number of bacteria in a unit x volume (x) of the first z blank is -. If the process is repeated, using another z blank, the fraction of original bacteria pipetted

4 392 MARSHALL W. JENNISON AND GEORGE P. WADSWORTH into this second z blank is x - V or A, and the fraction of the original V amount now in a unit volume of the second z blank is]-. If the V2 dilution process is repeated n times, the fraction of the original Zn number in a unit volume of the last blank will be -. Now we Vs will dilute this latter suspension in y ml. blanks, where x + y = u. The amount of this new suspension (-) pipetted into the first Xn Xn+l y blank (to give u) =-* x or -, and the fraction of the original XX+1 suspension now in a unit volume (x) of the first y blank is - -. If this is repeated m times, the fraction of original bacteria in a unit volume of the last y blank is f-. This latter expression gives the fraction of the original bacterial suspension in a unit volume (x) of the last dilution blank after a series of dilutions with various combinations of m and n blanks. xm+n Let, the dilution of bacteria, = F. To obtain the error in F, we must first take total differential of F, or df = (m + n)e+n-' dx - e+" du- d+n umvn UnV"+l U"'t+1Vn Then, the total error in F (i.e., the dilution error) equals the square root of the sum of the squares of the separateterms. Since the error in x is dx, and is also a (expressed as standard deviation in ml.) dx2 = a2. Also du2 = (a2 + b2), and dv2 = (a2 + c2). Then the approximate change A F (dilution error as standard deviation), in F, due to deviations in x, u and v is: (1) AF + Znm+n /a s+ )2 + n2(ao + b2) +m(ac +C2) Since x2 and x"+n are always 1, and u = 10 and v = 100, we can

5 ESTIMATING BACTERIAL NUMBERS BY PLATING METHOD 393 substitute in the equation. Also we can put umvn (which indicates the number of times that the original suspension isdiluted) in the numerator, giving: (2) AF = V-n /a2(m + n)2 + n2(a2 + b2) + m2(a2 + c2) 'V 1 (10)2 (100)2 The numerator outside of the square root, when expressed as 10 with a negative exponent, is the actual dilution of the original suspension, e.g., = The actual number of bacteria in aunit volume (x) of the last dilutionblank of a series is therefore u-mv-s of the original. If we start with K bacteria per ml. originally, then the number of organisms after a series of dilutions is K(u-mV-n), with a standard deviation of K(u"v -n),4a2(m + n)2 + n2(a2 + b2) +m2(a2 + C2) organisms, or, expressing the number with plus or minus its standard deviation, (3) K(i-iV-n) [1 + (M + n)2 + n2(a2 + b2) m2(a2 + c2)1 + L1,4/'a2(m J bacteria. For comparative purposes it is convenient to express the dilution error (AF) in percentage standard deviation, which is therefore given by: Percentage dilution n2(a2 + b2) m2(a2 + C2) (4) error = /a2(m + n) This generalized equation gives the most probable percentage error, as standard deviation, of any series of dilutions, with any size average values (in ml.) of a, b and c, and with any combinations m and n of dilution blanks. For most work, however, equation 4 may be simplified. The absolute size of a2, in comparison with b2 and c2 is usually so small that for practical purposes a2 may be omitted from the last two terms of equation 4, giving as a working equation: Percentage dilution error = a (m + n)2 + n 1b0 mc2

6 394 MARSHALL W. JENNISON AND GEORGE P. WADSWORTH1 Experimentally, with a careful technique and good 1 ml. pipettes (tolerance =0.01 ml.), the standard deviation of delivery of a series of such pipettes may be kept to 0.01 ml. Also, the standard deviation of the contents of 9 ml. blanks may be kept to 0.1 ml., and of 99 ml. blanks to 1.0 ml. Then b = 10 a, and c = 100 a. Substituting in equation 5, factoring a out of the square root, and cancelling, we get: (6) Percentage dilution error = A V(m + n)2 + m2 + n2 It is to be noted that equation 6 is applicable only under the conditions specified, except that if all deviations are twice, three times, etc., as large as those specified, the results from equation 6 may be multiplied by 2, 3, etc., respectively, to obtain the new error. The conditions specified above (all deviations ±1 per cent) are perhaps the best that obtain in actual practice. Often, the deviations in the pipettes and in the blanks will be of different sizes, and will be larger than 1 per cent, to which other cases equation 5 applies. In any event, the magnitude of one's own experimental deviations must first be determined in order to evaluate the dilution error. In table 1, based on equations 5 and 6, are calculated values of the dilution error, as percentage standard deviations, corresponding to various deviations in pipettes and in dilution blanks. This table covers the range of pipette and dilution blank deviations usually encountered in work susceptible to statistical analysis, except that fractional percentage deviations in pipettes and blanks are not included. For deviations in pipettes and blanks other than those in the table, equation 5 must be used for calculating the dilution error. All possible combinations of 9 ml. and 99 ml. blanks for obtaining a given dilution are shown, and it is obvious that for the smallest dilution error, it is necessary to use 99 ml. blanks in preference to the 9 ml. size when the percentage deviations of the two sizes are the same, since fewer of the larger size are required. By definition, a, b and c refer, respectively, to standard deviations in milliliter of 1 ml. pipettes, 9 ml. blanks, and 99 ml. blanks. Finally, the best estimate of the total error of the plate count is given by the square root of the sum of the squares of the dilu-

7 ESTIMATING BACTERIAL NUMBERS BY PLATING METHOD 3g5 TABLE 1 Dilution errors of the plating method, corresponding to various combinations of deviations in pipettes and dilution blanks DILU- STANDARD DEVIATIONS OF PIPETTES AND BLNKS, IN MILULITlBS 1 ml. pipettes (a), first line across; 9 ml. blanks (b), second line; COMBINA- ~~~~~~99 ml. blank (c), third line TIONS OF ml. ml. ml. ml. ml. ml. ml. ml. ml. ml. ml. ml. ml. ml. BLANKS TION FOR THUZ GIVEN DILUTION s DILUTION ERROR, AS (PLUS OR MINUS) PERCENTAGE STANDARD DEVIATION,. -..~~pe per pa Per Pa pw PtPW pw pa per per per pe cent cent cent cent cent t cent cent cent cent cent cent ct per cent , , , 3-9, , , , , , , , ,

8 396 MARSHALL W. JENNISON AND GEORGE P. WADSWORTH tion error and the distribution error, all being conveniently expressed in percentage standard deviation, according to the equation :3 (7) Percentage total error = i,/(percentage distribution error)2 + Vy (percentage dilution error)2 In equation 7, the dilution error is based upon the standard deviations of a series of pipettes and dilution blanks, while the distribution error is computed from the standard deviation of the mean (standard error) of replicate plates. In each case we are using the best estimate of the error. EXAMPLE Assume that we have triplicate plates of a 1:1,000,000 (10-6) dilution of bacteria, which dilution gives in this case the proper number of colonies for enumeration. Three 99 ml. blanks were used in making the dilution. The distribution error is determined by calculating the standard deviation of the mean (standard error) of the three plates, and expressed as a percentage (coefficient of variation) is found to be i5 per cent of the mean. Let us say also that the standard deviation of pipettes (a) was 0.01 ml., and of 99 ml. dilution blanks (c) 1.0 ml. Then, entering table 1, we find opposite dilution 10-6 and 3-99 blanks, and in the first (or second) column under pipette and blank deviations, a dilution error of 44.2 per cent. (With no 9 ml. blanks, there is, of course, no b value.) Then from equation 7, the total percentage error (as standard deviation) of the plate count is ± /(5)2 X (4.2) per cent. Since by definition a standard deviation indicates a probability of 0.68, our standard deviation of ±6.6 per cent means that the chances are about 7 in 10 that the actual value of the plate count is within ±6.6 per cent of the value found experimentally. a This equation is statistically only approximate, since the percentage errors of the two terms are referred to different quantities. The percentage error of counting is referred to the estimate of the actual value of the bacterial counts, and the error of dilution referred to the actual number of bacteria instead of the estimated number. For practical purposes the difference will be negligible if both are referred to the same quantity, and calculation is much simplified.

9 ESTIMATING BACTERIAL NUMBERS BY PLATING METHOD 397 SUMMARY The dilution error and the distribution error are the chief sources of variation accounting for the total error involved in estimating bacterial numbers by the plating method. An expression is derived for evaluating the dilution error, deviations in pipettes and dilution blanks being known. A convenient table of dilution errors, as percentage standard deviations, is given, covering the range of dilutions ordinarily employed, for all combinations of dilution blanks and for various deviations in pipettes and blanks. The usual method for estimating the distribution error is indicated, and a formula given for calculating the total error from the distribution error and the dilution error. REFERENCES FISHER, R. A Statistical Methods for Research Workers. 7th Ed. Edinburgh, Oliver and Boyd. FISHER, R. A., THORNTON, H. G., AND MACKENZIE, W. A The accuracy of the plating method of estimating the density of bacterial populations: with special reference to the use of Thornton's agar medium with soil samples. Ann. Applied Biol., 9, JENNISON, M. W Relations between plate counts and direct microscopic counts of E8cherichia coli during the logarithmic growth period. J. Bact., 33, WILSON, G. S The proportion of viable bacteria in young cultures, with special reference to the technique employed in counting. J. Bact., 7, WILSON, P. W., AND KULLMANN, E A statistical inquiry into methods for estimating numbers of Rhizobia. J. Bact., 22, ZIEGLER, N. R., AND HALVORSON, H Application of statistics to problems in bacteriology. IV. Experimental comparison of the dilution method, the plate count, and the direct count for the determination of bacterial populations. J. Bact., 29,