Overcoming the Inflexibility of Most-Probable-Number Procedures
Paul Woomer,* James Bennett and Russell Yost
ABSTRACT
cation compared to the enumeration of microbial
populations in diagnostic culture media. In general, plant
infection procedures are replicated between two and four
times and with base dilution ratios of 2, 4, 5, and 10
(Brockwell, 1963; Brockwell et al., 1975; Fisher and Yates,
1943; Vincent, 1970).
Experiments using MPN methods are currently restricted
to published MPN tables (Table 1), These tables form an
incomplete matrix of base dilution ratio and replicate
number that do not allow for variations such as increasing
accuracy through increasing the replication number (i.e.,
from 2 to 3 or from 5 to 6) or by decreasing the base
dilution ratios (i.e., from 4 to 3 or from 2 to 1.5). The
original tables of Fisher and Yates (1943) from which the
tables of Vincent (1970) were derived, offer greater variety
in replicate numbers but the solutions require additional
calculations in order to yield an MPN estimate. In fact,
tables which present the population estimates for MPN data,
based on the dilution ratio and the replicate number, are less
available than are the published descriptions of either the
confidence factors (Cochran, 1950) or the probability of the
experimental results (Scott and Porter, 1986; Stevens,
1958). In order to improve versatility of the MPN
technique, we have developed a computer program which
generates MPN tables for a wide range of dilution ratios at
selected replicate levels, and which solves for an MPN
estimate for many base dilution ratios and replicate
numbers.
The design of most-probable-number (MPN) experiments is restricted by the
availability and completeness of tables for certain dilution ratio and replicate
number combinations. The results presented in this paper are from a computer
program, the Most Probable Number Enumeration System (MPNES), which
generates solutions and confidence limits (P = 0.05) for population estimates of
MPN data. Tables generated by MPNES agreed with existing tables yet MPNES
was able to generate other tables and discrete solutions for design combinations
that have not been published. The program was also used to generate population
estimates from fractional base dilution ratios. The MPNES increases the
accuracy of MPN estimates by decreasing the base dilution ratio, by increasing
the number of replicates per dilution or by correcting for constantly inaccurate
diluent volumes. The MPNES program also adjusts for inoculation volume and
initial dilution ratios. In this way, MPN experiments can be designed to better
measure the organism of interest rather than have the design dictated by
published tables.
THE MPN TECHNIQUE is widely used to enumerate
microorganisms in foods, water and soil (Brockwell, 1963;
deMan, 1975; Halvorson and Ziegler, 1933; West and
Coleman, 1986; Woodward, 1957). This method is most
important when a single type of microorganism must be
measured within a heterogenous microbial population or
when direct enumeration or plate counting are difficult. The
MPN method does not rely on direct counts of single cells
or colonies, but on the statistical probability of diluting to
extinction (Alexander, 1982).
Standard methods have been developed for some
applications of the MPN procedure (American Public
Health Assoc., 1985). In general, for microbiological
applications, serial dilutions are applied to 5 or 10
replicates of some selective medium (American Public
Health Assoc., 1985; deMan, 1975) because increases in
replicate number result in narrower confidence intervals
than less replicated MPN procedures (Cochran, 1950).
In the case of the plant-infection technique for counting
rhizobia, serial dilutions are applied to the root systems of
plant units. There are no established standards for the
number of replicates used. The range of microbial densities
varies greatly between samples, preventing standard base
dilution ratios. The need for uniformity of the plant host
results in reduced repliP. Woomer, NifTAL Project, University Hawaii, 1000 Holomua Avenue, Paia,
Hawaii, USA 96779; J. Bennett, Dep. Mathematics, Univ. Hawaii, Honolulu,
Hawaii, USA 96817; R. Yost, Dep. Agronomy and Soil Science, Univ. Hawaii,
Honolulu, Hawaii, USA 96822. This work was supported by the NSF Grant
BSR-8516822
and
the
U.S.
AID
Cooperative
Agreement
D.AN-4177-A-00-603500 (NifTAL Project). Journal Series no. 3320 of the
Hawaii Inst. Tropical Agric. and Human Resour. Received 1 I July 1988.
*Corresponding author.
Published in Agron. J. 82:349-353 (1990).
349
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AGRONOMY JOURNAL, VOL. 82, MARCH-APRIL 1990
found such that f [(X 1 + X2)/2] - K = 0, and hence a solution has
been found.
Implementation
The above algorithm was used to develop a program that
provides two options for generating MPN estimates. One which
generates tables of MPN data (positive tubes at various dilutions)
and the corresponding MPN estimate, and the other which, given
specific conditions, generates a specific MPN estimate.
The confidence limits (P = 0.05) of MPN estimates were first
calculated by Cochran (1950). The MPNES program calculates
the confidence factor (CF) as
where a = the dilution ratio and n = the number of replicates of
the MPN equation. The upper and lower confidence limits (P =
0.05) are obtained from the confidence factor through
multiplication with and division into the population estimate,
respectively. The MPNES program presents the upper and lower
confidence limits (P = 0.05) for every population estimate.
The algorithm was initially written in IBM Basic and complied
with Microsoft Basic Compiler. The program will run on
IBM-compatible microcomputers with one floppy disk. The
operating system requires MS DOS 2.0 or above and does not
require additional software. Memory requirements are minimal
(less than 256K RAM). Other formats may be accommodated on a
case-by-case basis.
RESULTS AND DISCUSSION
Using the program
Algorithm
The computer solution follows the arguments above and is
based on the following algorithm. First, the computer finds a
number XI such that f(Xl) - K > 0. It then finds a number X2 such
that f(X2) - K < 0. Now f[(X1 + X2)/2] - K is computed. If it
equals 0 the algorithm is finished and a solution (X 1 + X2)/2 has
been found. If it is greater than 0, then (X1 + X2)/2 is substituted
for X1 above, X2 remains X2, and f [(X1 + X 2)/2] - K is again
computed. If it is less than 0, (X1 + X 2)/2 is substituted for X 2, X
1 remains the same, and f [(XI + X2)/2] - K is computed. This
cyclic process starts over again and continues until values of X1
and X 2 are
The program is composed of two parts; one for generating MPN
tables, and the other for computing individual MPNs. For a table,
the user is asked to provide R (the base dilution ratio) and N (the
number of tubes per dilution) via input statements. The combinations of positive tubes (P1-P6) in the table are provided by the
program. The combinations of positive tubes do not include data
statements in which the lowest serial dilution is not entirely
positive (P1 < N). Only the most probable combinations are
provided (deMan, 1975; Scott and Porter, 1986). While the base
dilution ratio (R) may be any number, the number of tubes per
dilution (N) is limited to 2, 3, 4, and 5. The individual MPN format
can be used to obtain MPN estimates for MPN data not included
among the combinations printed in the table.
For an individual MPN, the base dilution ratio (R), number of
tubes per dilution (N), and the numbers of positives per dilution
(P1-P6) are provided by the user in response to prompts (Table 2).
Options are available for correction in inoculation volume (when
other than 1 mL) and/or initial dilution (when different from the
dilution ratio). When all the necessary information is provided, an
MPN for the combination of positive tubes for each dilution is
generated.
Accuracy of the Program
The accuracy of the MPNES program is apparent when the
MPN estimates generated by this program
WOOMER ET AL.: OVERCOMING THE INFLEXIBILITY OF MPN PROCEDURES
are compared to the estimates of other authors (Table 3).
Note that the MPN estimates yielded through the
MPNES program agrees least with the tables in Vincent
(1970) and is more similar to the solutions offered by
Brockwell (1963), Alexander (1982) and deMan (1975).
The tables in Vincent (1970) are constructed through the
mathematics of Fisher and Yates (1943) in which the
sum of the positive experimental units is used to derive a
population estimate regardless of the pattern (the number
of positive results at each dilution step) of the MPN data.
The other authors (Brockwell, 1963; Alexander, 1982;
deMan, 1975) as well as the authors of the MPNES
program rely on the mathematics of Halvorson and
Ziegler (1933) in which a unique population estimate is
assignable to each pattern of MPN data.
Improvements Offered by the Program
The MPNES program offered several advantages over
existing MPN tables. Researchers were able to adjust
base dilution ratios and replicate combinations to meet
specific degrees of accuracy (Tables 4A and 4B).
Infrequent codes which are not found in existing tables
were assigned a population estimate (Table 4C).
Through the ability to generate tables and estimates from
individual code data of fractional base dilution ratios,
adjustments of the base dilution ratio to actual diluent
volumes were conveniently obtained (Table 4D).
Cochran (1950) demonstrates that the confidence
interval of a given MPN procedure is defined by the
base dilution ratio and the replicate number through their
influence on the standard error of the mean. Thus,
certain combinations of base dilution ratio-replicate
numbers allow for greater accuracy through reduction of
the confidence factor. An important ex-
351
ample of increased accuracy is demonstrated in studies of
relatively low numbers of microorganisms in agricultural
(Singleton and Tavares, 1986) or public health (West and
Coleman, 1986) applications using MPN procedures.
More accurate measurement of microbial populations
may lead to improved management of these
microorganisms.
Increasing the Range or Accuracy
The flexibility of the MPNES table format is presented in Tables 5 and 6. Data in these tables illustrate
how MPN estimates of increased accuracy (Table 5) or
increased range of detection (Table 6) were generated.
These experiments require fewer total growth units when
compared to other tables (Vincent, 1970). The MPN
procedure represented by Table 5 requires 18
experimental units yet has a confidence interval similar
to a fourfold dilution table with four replicates (Vincent,
1970).
The MPNES program produced unique MPN tables
with extremely wide ranges that required few growth
units (Table 6). This table has a base ratio of 12.5 with
two replicates for six serial dilutions (total of 12 growth
units) and is able to detect between 6.0 and 2.4 X 106
organisms per gram of sample.
Nonstandard Dilution Ratios
The ability to generate MPN estimates for nonstandard
dilution ratios is important in three regards: it can be
used to further adjust the range of the MPN to the
population densities expected by the researcher (i.e.,
Table 5), fractional decreases in the base dilution ratio
result in increased accuracy of MPN procedures through
reduction of the confidence interval as in Table 413, and
finally, should a researcher discover that the diluent
blanks prepared for a given dilution series
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AGRONOMY JOURNAL, VOL. 82, MARCH-APRIL 1990
are of uniform but incorrect volume, the MPNES program may be used to generate corrected population
estimates for the nonwhole number base dilution ratio.
The importance of this feature is demonstrated in Table
4D where instead of a fivefold dilution series, a
four-and-one-half-fold series is initiated due to insufficient volume of diluent. In the example given, using
the fivefold estimate results in an overestimation of the
population by 70%.
As situations require, researchers may choose to alter
the inoculation volume or to include an initial dilution
that is different from the base dilution ratio. For the
individual MPN format, these modifications to the MPN
procedure may be entered into the calculation of the
MPN estimate. Increases in the inoculation volume and
decreases in initial dilutions (lower than the base
dilution ratio) serve to detect lower microbial
populations, although not with greater accuracy, since
neither the base dilution ratio nor the replicate number
are being altered. Initial dilutions also serve to center the
range of transition of the MPN data within the serial
dilution series, allowing the microorganisms in
the sample to be diluted to extinction [when all replicates
of the final dilution(s) are entirely negative]. It is
important that the sample be diluted to extinction since
these treatments are the controls for the MPN
experiment. The MPNES program generates MPN estimates for nonstandard initial dilutions without additional calculations. The accuracy of this MPNES option is compared to other sources (Table 7) where
changes in the MPN estimate due to an initial dilution or
inoculation volume are either incorporated into the MPN
table (Brockwell, et al., 1975) or require additional
calculations (Vincent, 1970).
Limitations of the Program
The MPNES table format was able to generate MPN
tables for two, three, four, and five replicates. The range
of base dilution ratios for which the tables accurately
generate MPN estimates was between 1 and 14.5. The
MPNES individual MPN format could generate
population estimates for any positive whole number of
replicates. Population estimates were generated by
entering nonwbole numbers of replicates but these
estimates are meaningless.
Criteria to accept or reject specific MPN data are
described by Stevens (1958) and Scott and Porter (1986)
and applied by MPN researchers (Brockwell, 1963;
Woomer, et al., 1988). The range of transition (Stevens,
1958) is the number of dilution steps between the first,
not entirely positive and the last, not entirely negative
dilution level. Extended ranges of
WOOMER ET AL.: OVERCOMING THE INFLEXIBILITY OF MPN PROCEDURES
transition indicate lower probability data or failure of
technique. Other authors (deMan, 1975; Woodward,
1957) have assigned probabilities to the occurrence of
specific patterns of MPN data. Certain data of low
probability are acceptable when these occur infrequently
and may be demonstrated as acceptable by the range of
transition. Under these circumstances, the MPNES
program is useful in assigning population estimates to
codes which are not included in published MPN tables
(Brockwell, 1963; Brockwell et al., 1975; deMan, 1975;
Woodward, 1957). This feature must be used with
caution. For example, the MPN data given in Table 4C
does not warrant an MPN population estimate because
the extended range of transition of 4 is extremely
improbable (P = 0.0036). This is indicative of a failure in
experimental technique (Stevens, 1958).
In conclusion, the MPNES program was shown to be
an improvement from existing MPN tables because the
options available through this program allowed
researchers greater flexibility. Restraint must be exercised in application of this program because MPN
estimates can now be derived from MPN data that are
very improbable (deMan, 1975; Halvorson and Ziegler,
1933) and may be the result of failure in meeting the
principle assumptions underlying the MPN procedure
(Stevens, 1958). Researchers must be aware that
MPNES is not a solution to poor experimental results,
but a tool in designing improved MPN experiments.
ACKNOWLEDGMENTS
The authors appreciably acknowledge technical assistance by
Mr. Joe Rourke and Mr. Zhi-Cheng Li in development of the
MPNES program and thank Ms. Susan Hiraoka for manuscript
preparation.
353
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