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Journal of Microbiological Methods 107 (2014) 214–221
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Journal of Microbiological Methods
journalhomepage:www.elsevier.com/locate/jmicmeth
Estimation method for serial dilution experiments
a, b
⁎
Avishai Ben-David , Charles E. Davidson
a RDECOM,EdgewoodChemicalBiologicalCenter,Aberdeen Proving Ground, MD21010, USA
b Science and Technology Corporation, Belcamp, MD 21017, USA
article info abstract
Article history: Titration of microorganisms ininfectious or environmental samples is a corner stone of quantitative microbiology.
Received 9 May 2014 Asimplemethodispresentedtoestimatethemicrobialcounts obtained with theserial dilution technique for
Received in revised form 29 August 2014 microorganismsthatcangrowonbacteriologicalmediaanddevelopintoacolony.Thenumber(concentration)
Accepted30August2014 of viable microbial organismsisestimated from a single dilution plate (assay) without a need for replicate plates.
Available online 7 September 2014 Ourmethodselectsthebestagarplatewithwhichtoestimatethemicrobialcounts,andtakesintoaccount the
Keywords: colony size and plate area that both contribute to the likelihood of miscounting the number of colonies on a
Titration plate. The estimate of the optimal count given by our method can be used to narrow the search for the best
Serial dilution (optimal) dilution plate and saves time. The required inputs are the plate size, the microbial colony size, and the
Viable bacterial counts serial dilutionfactors.Theproposedapproachshowsrelativeaccuracywellwithin±0.1log fromdataproduced
10
Agarplates bycomputersimulations.Themethodmaintainsthisaccuracyeveninthepresenceofdilutionerrorsofupto10%
Toonumeroustocount(TNTC) (for both the aliquot and diluent volumes), microbial counts between 104and 1012 colony-forming units, dilution
Density (concentration) of microorganisms ratios from 2 to 100, and plate size to colony size ratios between 6.25 to 200.
Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction Givenanunknownsamplewhichcontainsn colonyformingunits
0
(CFUs),aseriesofJdilutionsaremadesequentiallyeachwithadilution
−1
Quantitative estimation of the number of viable microorganisms in factor α. From each of the J dilutions a fraction α is taken and spread
p
bacteriological samples has been a mainstay of the microbiological (plated) on an agar plate (assay) where colonies are counted. Thus, in
laboratory for more than one-hundred years, since Koch first described generaltherearetwodilutionfactors:αandα .Forexample,α=10in-
p
the technique (Koch, 1883). Serial dilution techniques are routinely dicates a 10-fold dilution, e.g., by diluting successively 0.1 ml of sample
used in hospitals, public health, virology, immunology, microbiology, into0.9mlofmedia;andα =1meansthattheentirevolumeisspread
p
pharmaceutical industry, and food protection (American Public Health, (plated) on the agar plate. For an experiment with a larger dilution
2005; Hollinger, 1993; Taswell, 1984; Lin and Stephenson, 1998) for factor α , multiple plates may be spread at the same dilution stage. For
p
microorganisms that can grow on bacteriological media and develop example, α = 20 represent a 5% plating of the dilution, and thus up
p
into colonies. A list of bacteria that are viable but nonculturable to 20 replicates could be created. At each dilution the true number of
(VBNC),thedetectionofsuchmicroorganisms,andtheprocessofresus- −j −1 ^
colonies is nj = n0α αp and the estimated number is nj.The
citation of cells from VBNC state are addressed by Oliver (2005, 2010). estimated quantities are denoted with a “hat” (estimated quantities
In the work presented here it is assumed that the microorganisms are can be measured quantities, or quantities that are derived from mea-
culturable. sured or sampled quantities); symbols without a “hat” denote
Theobjectiveoftheserialdilutionmethodistoestimatetheconcen- true quantities (also known as population values in statistics) that do
tration (number of colonies, organisms, bacteria, or viruses) of an not contain any sampling or measurement error. In this work both n
j
unknownsamplebycountingthenumberofcolonies cultured from andn are“counts”, i.e., number of colonies. Knowing the aliquot vol-
0
serial dilutions of the sample, and then back track the measured counts ume, one can easily convert counts to concentration (for example
to the unknown concentration. CFU/ml).
Theimportanceofserialdilutionandcolonycountingisreflectedby
thenumberofstandardoperatingproceduresandregulatoryguidelines
describingthismethodology.Inalloftheseguidelinestheoptimalnum-
Abbreviations:TE,totalerror;RE,relativeerror;MPN,mostprobablenumber;TNTC,too ^
numeroustocount;VBNC,viablebutnonculturable;CFU,colonyformingunit. ber(nj)ofcoloniestobecountedhasbeenreported(ParkandWilliams,
⁎ Correspondingauthor.Tel.:+14104366631;fax:+14104361120. 1905; Wilson, 1922; Jennison and Wadsworth, 1940; Tomasiewicz
E-mail addresses: avishai.bendavid@us.army.mil (A. Ben-David), et al., 1980; FDA, 2001; Goldman and Green, 2008)as40–400,
charles.e.davidson2.ctr@us.army.mil (C.E. Davidson). 200–400, 100–400,25–250,30–300.Itis interesting to note that these
http://dx.doi.org/10.1016/j.mimet.2014.08.023
0167-7012/PublishedbyElsevierB.V.This isanopenaccess articleunder theCCBY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
A. Ben-David, C.E. Davidson / Journal of Microbiological Methods 107 (2014) 214–221 215
references do not specify the area in which the colonies grow, nor the
diameterof theparticular organism assayed. The result is that titration
andcountingcolonies is done within a range that may be inadequate,
and may introduce considerable error. In our work these parameters
are addressed.
Themainchallengeinserialdilution experiments is the estimation
^
of the undiluted microorganisms counts n0 from the measured nj.
Therearetwocompetingprocesses(Tomasiewiczetal.,1980)thataf-
fecttheaccuracyoftheestimation:samplingerrorsandcountingerrors.
Samplingerrorsarecausedbythestatisticalfluctuationsof thepopula-
tion. For example, when sampling an average of 100 colonies, the fluc-
pffiffiffiffiffiffiffiffiffi
tuations in the number of the population are expected to be 100
when the sampling process is governed by a Poisson probability
(Poisson and Binomial distributions are often used in statistical analysis Fig. 1. The concept of δ is given with Poisson and a shifted-Poisson probability density
j
^ ^
to describe the dilution process (Hedges, 2002; Myers et al., 1994)) functionsforthetruen,andforthecounted(measured)n ,respectively. n istheobserved
j j j
wherethestandarddeviationequalssquare-rootofthemean;therelative (counted) number of colonies on plate j of the serial dilution process, for which the true
pffiffiffiffiffiffiffiffiffi numberof colonies is nj with a mean μj. Due to uncounted colonies in the measuring
error (ratio of the standard deviation to the mean) is 100=100 ¼ 0:1. process (i.e., merging of colonies that are counted as one colony due to overcrowding),
Thus, the larger the sample size is, the smaller the relative sampling ^
nj bnj.
error; hence, one would like to use a dilution plate with the largest num-
^
ber nj (i.e., the least diluted sample, j → 1). However, as the number of
colonies increases, counting error is introduced due to the high probabil-
^ ^
for n is the same as that for n , and therefore the variance Var n
ity of two (or more) colonies to merge (due to overcrowding) and j j j
becomeindistinguishable, and be erroneously counted as one colony. 2
¼Var n ¼En−μ . E(⋅) is the expectation operator (i.e., an av-
Anoptimum(a“sweetspot”)betweenthesetwoprocesses(sampling j j j
^
andcounting error) needs to be found for using the optimal dilution nj erageprocess),andμ isthemeanofn.Thereisnotenoughdatatocom-
j j
(i.e., the optimal jth plate) with which to estimate n . Cells can grow puteavariancewiththeexpectationoperator(becauseonlyoneplate
0
into colonies in various ways. Wilson (1922) states that when two cells ^ −j −1
with one value of n for a dilution α α is available). Therefore, we
j p
are placed very close together only one cell will develop, and when two 2
defineameasureof“spread”givenbyV =(n −μ) ,thatiscomputed
cells are situated at a distance from each other both cells may grow and n j j
fromasinglevalueofn.The“spread”canbesolvedforμ byμ ¼n −
then fuse into one colony. Either way, the end result is the appearance pffiffiffiffiffiffi j j j j
V . The distance δ b μ (see Fig. 1), and thus there must be a constant
of one colony which causes counting error. j j j pffiffiffiffiffiffi
cb1suchthatδ ¼cμ ¼cn −c V .Becausewedonotwanttomodify
Estimation of bacterial densities from the most probable number j j j n
theindividualn values (with c), we instead construct δ with a con-
(MPN)method(Cochran,1950)requiresmultiplereplicatesofthe jth j pffiffiffiffiffiffi j pffiffiffiffiffiffi
dilution plate, and analyzes the frequency of plates with zero colonies stant k N 1whereδj ¼ nj−k Vn,andkonlymodifiesthespread Vn.
instead of using counts directly. MPN is often used to measure microbes Ournotionof“spread”isasubstituteforthenotionofvariance(thatwe
in milk, water and food (Blodgett, 2010). The MPN method (Cochran, cannot compute) and is weak; hence, no harm is done in adjusting it
1950)“is of low precision, as is to be expected from a method that does withafudgefactork.Thefactorkisafunctionofthegeometryofthese-
not use direct counts. Large number of samples [replicate agar plates] rialdilutionexperiment.Atthispointitsufficestostatethatkexists(the
mustbetakenateachdilution if a really precise result is wanted”.In numericalvalueofkisaddressedlater).WiththePoissonanddisplaced
our work we seek a method where the counts from a single plate are ^ ^
Poisson assumptions for nj and nj,wesetVn ¼ V^ ¼ nj,leadingto
pffiffiffiffiffi n
used to estimate bacterial concentrations. δ ¼n −k n. Wedon'thaveaccesston,andthereforewereplacethe
j j j j
A simple method to estimate the number of colonies n in the ^
0 unknownpopulationvaluen withthemeasuredn .Thisisinspiredby
j j
^
unknown sample from the counted number of colonies nj at the jth the principle that is often used in signal processing of replacing un-
assay is presented. Our method is easy to implement. The method known population parameters with maximum-likelihood estimates
selects the optimal count (i.e., a best single plate) with which n is
0 (as is done, for example, in the generalized likelihood ratio test
estimated. There are only a few inputs needed: the incubation plate (Scharf and Friedlander, 1994)).
size, the microbial colony size, and the dilution factors (α and α ).
p Withanundeterminedk(forthemoment)amodel-estimateofthe
Thedilution error (although present in the serial dilution experiment) ^
true δ by δ is given by
is not an input. The relative accuracy of our method is well within qffiffiffiffiffi
±0.1log (i.e., within 100% error) which is much better than the 8 9
10 >^ ^ ^ >
>δj ¼nj−k nj >
> >
common requirement of ± 0.5 log (i.e., within 500% error) that is < qffiffiffiffiffi =
10 ^
^ : ð1Þ
often regarded as accepted accuracy in many biological experiments. >δj N 0 for 1bkb nj>
> qffiffiffiffiffi >
> >
:^ ^ ;
2. Material and methods δj ¼ 0 for k ≥ nj
^
^ The inequalities in Eq. (1) are necessary to ensure δ to be a
The measured (counted) number of colonies nj is related to the true
^ non-negative quantity less than the value of μ.Eq.(1) implies that the
number of colonies n by n ¼ n þδ where δ is a bias that accounts
j j j j j
for uncounted colonies due to the merging (overcrowding) of nearby ^ ^ 2 ^
counting error is negligible δ→0 whennjbk .Givenδj inEq.(1)
^
colonies, and thus, n ≥ n . The challenge is to obtain an estimate of δ
j j j weproceedtoestimaten by
^ 0
from a single measurement nj of the jth Petri dish. The challenge is
met in an ad-hoc manner. The following assumptions are made: (i)
Thetruen (whennocoloniesaremiscounted)isdescribedbyaPoisson
j ^ ^ ^ j
n ¼ n þδ α α : ð2Þ
probabilityforwhichthevarianceequalsthemean(Forbesetal.,2011), 0 j j p
Eq.(2)producesanestimateofn bycorrectingthecountfortheex-
^ 0
(ii) Theprobabilitydensityfunctionfornj(withtheeffectofmergingof
^
nearbycolonies)isadisplacedversion(byδ)ofthePoissondistribution pectednumberofmissedcoloniesδj,andbymultiplyingbythetotaldi-
j
j −j
for n,seeFig. 1. Hence the variance of the probability density function lutionfactorsα α (toreversetheeffectoftheserialdilutionα andthe
j p
216 A. Ben-David, C.E. Davidson / Journal of Microbiological Methods 107 (2014) 214–221
−1 ^
plating dilution α in n →n).Thequestionremains“whichjthplate
p 0 j whereTEisgiveninEq.(3),δj is givenEq. (1),andthevalueofk(with
^
to use for nj?” The rule of thumb that is commonly used (Tomasiewicz ^
whichδjiscomputed)isgivenlaterinEq.(5).Theestimateduncertainty
^ ^
et al., 1980) is to choose a plate with 30 ≤ nj ≤ 300 or 25 ≤ nj ≤ 250. with Eq. (4) underestimates the total error (discussed in Section 3).
^
Withourmethodnj ischosenasfollows. ^
Relative error as a function of n is shown in Fig. 2 for a nominal 10-
The standard definition of total error (also known as mean squared fold dilution ratio (α = 10) and for 100% plating (α =1)asafunction
error, see p. 330 in Casella and Berger, 2002) for an estimated p
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q of colony size (2 to 10 mm diameter) on a 10 cm diameter agar plate.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hi ^
^ ^ 2 ^ ^ 2 The decreasing part of the curves (e.g., nb100 for dcolony =3mm)in
quantity n is given by E ðÞn −n ¼ varðÞn þ½EðÞn −n ¼
0 0 0 0 0 0 the figure represents the effect of the decrease of the sampling error
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^
j ^2 2 as a function of increased n, whereas the increasing portion of the
^ ^
α αp n þδ þ δ −δ where½EðÞn −n is the error due to the
j j j j 0 0 ^
curves(e.g.,fornN100,d =3mm)representstheeffectofcounting
colony
bias of the estimates. The total error combines the effects of precision ^
error that increases with n. The location of the minimum of the curves
^ dependsonthetrade-off between sampling and counting errors. The
(via the varianceoftheestimatedn )andaccuracy(viathebiasinesti-
0
^ shapeoftheerrorcurveshowsthatasthecolonysizeincreasesitisad-
mating n ). In computing total error, one performs many Monte Carlo
0
simulations (where n , δ, and all other parameters are known) for the vantageous to select a plate with smaller number of colonies and that
0 j
dilutionprocess,constructsestimatesforn ,andcomputesthevariance the optimal counts (for a minimum error) is contained in a narrower
0
andthemeanfromalltheestimatesofn .Thisiswhatwedolaterin range of counts. Observing the behavior of the analytical TE curve as a
0 j
Section3(Results)whenwevalidateourestimationmethod.However, function of k and comparing to the expected behavior of the minimum
^ j location (as a function of sampling error and colony size) allows us to
the true δj is not known in a real scenario (only nj, α ,andαp are
^ determine a value for k.
known), and thus we resort to the estimated δj with our model as is If the total dilution factor αα isdecreased,thenmoredilutionssteps
given in Eq. (1) to construct an estimator of total error. Our total error p
is givenasasumofthesamplingerrorandthecountingerror.Thesam- willbenecessarytoobtainacountableplate,andthesamplingerrorata
^
pling error which is due to the variance of the sampled n on a plate is given count n will increase (due to compound error of the multiple
j steps), thereby pushing the location of the minimum of the total error
^ ^ ^ ^
given by var nj ¼ var nj þδj ¼ nj þδj (usingthePoisson distribu- curve to larger values (to mitigate the compound error). Thus, it is ad-
^ vantageous to sacrifice an increase of counting error to mitigate the
tionfornj ¼ nj þδjwhosePoissonintensityparameterλisestimat-
^ ^ fact that sampling error increased by seeking a serial dilution plate
edasλ¼En ¼n þδ),andthecountingerroronaplatethatisdue
j j j 6
with more colonies. For example, consider a scenario with n0 =10
2 2
^ ^
to the residual bias given by δj−δj which we approximate as δj andassumethatn =100istheoptimalcountinaserialdilutionexper-
j
(becausethetrueδ isnotknown).Ourtotalerror(TE)modelisgivenby iment with α =10andαp = 1. Four sequential dilutions (j =4)
j
are needed in order to obtain nj ≅ 100. For α = 2, thirteen dilutions
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 (j = 13) are needed to obtain n ≅ 100 and the compounded sampling
> 2 2 > j
TE ¼ ðÞsampling error þðÞcounting error
> j > error is larger. Thus, the optimal count for α = 2 will occur at a value
> >
> rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >
> >
< j 2 = larger than 100.
^
¼ααp Var nj þδj : ð3Þ If the colony size decreases relative to the plate size, it is less likely
> rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >
> >
> >
> j 2 > ^
> ^ ^ > for two colonies to merge, and counting error at a given count n will
: ^ ;
¼ααp nj þδj þδj have decreased, again pushing the location of the minimum of the
total error curve to larger values (i.e., we are willing to sacrifice some
Thetotal error in estimating n originates from sampling error (the ofthebenefitofthereducedcountingerrorinordertoreducesampling
0 error by seeking a serial dilution plate with more colonies).
j
variance of the true n that is transformed by αα to the space of n )
j p 0 Since the location of the minimum of TE increases with k (see
^ j
and the counting error due to δj (also transformed to the n0 space). Eq. (6), below), k should be inversely proportional to relative colony
Weconstruct a table for TEj values for each serial dilution plate, and size, and inversely proportional to the combined dilution factor αα .
^ p
choose the n value that produces the smallest TE (example is shown
j j
^
later in Table 1). With this optimal n , n is estimated with Eq. (2).The
j 0
predicted total error in Eq. (3) underestimates the truth (to be shown
2 2
^ ^ ^
in the results section) because usually δj−δj N δj.Notethatδj in
qffiffiffiffiffi
^ ^
Eq. (1) cannot exceed the value nj− nj (which occurs when k =1),
^
whereasδ −δ isunbounded,becausethetrueδ ≤n isonlybounded
j j j j
^ ^
bynj,wherenj≥nj (Fig. 1). Furthermore, δj issettozeroinEq.(1)
qffiffiffiffiffi
^
for k≥ n whereasthetrueδ onlyapproaches zerowhenthecolony
j j
^
size approaches zero (shown later in Eq. (8)). Hence, usually δj N δj.
^ →0 because αj
Note that TE becomes very large as the counts nj
j ^
increases rapidly (α ∝1=nj, small counts occur at large j values with
hugedilution).
^
Theestimateduncertainty(error)forthesolutionn (Eq.(2)andits
0
relative error (RE), are given by Fig. 2. Relative error curve in percent (Eq. (4)) as a function of measured number of colo-
^
niesnonaplatefordifferentvaluesofcolonysize(diameter).Theexperimentalcondition
8 9 for the TE curve is d =10cm(diameter),dilutionratiosα=10,andα =1.Theop-
plate p
> ^ j > ^
>^ ^ > timalmeasurednthatminimizesthetotalerroristheonethatisclosesttotheminimumof
n ¼ n þδ α α TE
> 0 j j p j >
> rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ^ ^
> > thecurve(e.g., around n≅100 for d =3mm).Eq.(6)predictsthevalueofnatthe
< = colony
2 ^
^ ^ ^ minimumofthetotal error curve (e.g., n ¼ 101 for dcolony = 3mm). Microbial count
^ TE nj þδj þδj ð4Þ
> Δn0 j > ^
> > in the range 25bnb170is expected to produce an estimate of n0 within 20% error
>RE¼ ¼ ¼ >
> >
> ^ ^ > for d =3mm.Thetypicallycitedoptimalrangeof25–250countsispredicted
: n n ^ ^ ; colony
0 0 nj þδj
byourmodelforacolonyofdcolony= 2.5mm(alsowithin20%error).
A. Ben-David, C.E. Davidson / Journal of Microbiological Methods 107 (2014) 214–221 217
Therefore, the previously undetermined value of k in (Eq. (1))is
set to
81bk¼3k 9
> experiment >
< sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =
1 1 area 1 d ð5Þ
> bk ¼ plate ≅ plate >
:3 experiment αα area αα d ;
p colony p colony
wheretheproportionally constant is set to be 3 (the value of 3 gave
us good empirical results in our extensive testing, especially for
ααp =10),andtheinequality 1 bkexperiment is to ensure that k N 1
3
so that δj b μj (see Fig. 1). In most experiments the colony sizes are
mono-dispersedandhaveacircularshape,hencearatioofdiameters
(plate to colony) can be used for k.
^
Weseekthelocationnoftheminimumofthetotalerrorcurvefor
^ ^
TEðÞn;k in Eq. (3),whereδj and k are given by Eqs. (1) and (5).
Although this is a constrained minimization problem subject to k ≤ ^
pffiffiffi Fig. 3. Optimalcounts n onaplate(thatminimizesthetotalerrorinEq.(3))asafunction
^
n, the constraint does not need to be enforced, and it is sufficient of the ratio d /d in a serial dilution process with 10-fold dilution (α = 10) and
plate colony
to minimize Eq. (3) ignoring the constraint. 100% plating (αp = 1). The likelihood of miscounting colonies due to overcrowding
^ ^ increases as d /d decreases.
j n n plate colony
FromEq.(2)weuseα αp ¼ 0 ≅ 0 (i.e., we neglect δ in the de-
^ ^ j
^ n
njþδj j
nominator) in Eq. (3) to obtain a continuous function of the measured d
^ ^ TheinverseofEq.(6)givesanestimateoftheratio plate asafunction
nj. We then, by taking a derivative with respect to nj and solving for d
colony
^ ^
the derivative to be zero, obtain the location n of the minimum of the of optimal n (i.e., the counts that minimizes TE) on an agar plate, as
total error curve. This approximation gives us a nice compact solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
withlessthan13countserror(whenδjisnotneglected)inthelocation d αα pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 ^
plate p ^ ^2 ^ 9þ3 4n −20nþ9
^ ¼ pffiffiffi 4n−4þ2 4n −20nþ9þ : ð7Þ
of optimalnforallpossiblek N1(lessthan4countserrorfork N2,and ^
d 6 2 n
^ colony
lessthan10%errorfork N4).Oursolutionfortheoptimalnthatproduce
a minimumtotalerrorisgivenby, Thesizeofacolony(d )increasesasafunctionoftime(through
colony
8 2 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 growth)ontheagarplate.Letusassumethatonewantstocountagiven
> 2 6 8 >
> 4þk þk þ 16þ8k þ2k þk > ^ ^
>^ > numberofcolonies n onaplate(say,n ¼ 50).Theminimumtotalerror
^ 2 > colony
>n≅k for kN4:5 > d
> > plate
:1bk¼3k ; isfies the ratio d that is predicted with Eq. (7). Thus, if the rate of
experiment colony
growthofaparticularorganismisknown(say,τmm/day),andapartic-
^
^ ularnisdesiredapriori,thetimeforincubationintheserialdilutionex-
Thesolutionisonlyafunctionofk.Eq.(6)showsthattheoptimaln
2 perimentcanbepredictedbyd /τ where d is computed with
^ colony colony
increases with k (as was discussed earlier). For k N 4.5, n≅k is within ^ ^
5% of the analytic solution given by Eq. (6), and within 15% of the Eq. (7) for a given desired value of n. Thus, if a plate with n ¼ 50 is
morecomplicatedsolutionthatcanbederivedusingthecorrectexpres- observed in the serial dilution experiment it is assured to produce
j ^ minimumtotalerror.
sionforα α .Theestimateoftheoptimalcountncanbeusedtonarrow
p ^ ^
Theestimation algorithm for n from measured n on a single agar
^ 0
the search for the best dilution plate nj and save time. plate is summarized by the following six steps (knowledge of the
^
In principle TEðÞn;k (Eq. (3)) can be solved for a given value of error in sampling the aliquot and diluent volumes is not required):
^ ^ ^
TE = const to produce a range n bnbn for which the total error
min max qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
area d
^ ^ 1 plate 1 plate
is bounded (i.e., n and n are given by the intersection between
min max Step 1: Compute k ¼ ≅ (see Eq. (5))
experiment ααp areacolony ααp d
the curves given in Fig. 2 and a horizontal line at a constant relative colony
error level). This range can be obtained pictorially from Fig. 2 for the spe- for theserialdilutionexperiment,whereαisthedilutionfactor
(e.g., for 10-fold sequential dilution, α =10),α is the plating
cific serial dilution experiment (i.e., α =10,α =1,d =10cm),for p
p plate dilution ratio when a fraction of the dilution volume is placed
^
other serial dilution parameters the figure REðÞn must be redrawn onanagarplate(e.g.,when5%ofthedilutionvolumeisplated,
^ ^ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(since k is a function of α, αp, dcolony, dplate — see Eq. (5) — and δðÞn α =20),andtheratioofplatesizeto colonysize areaplate ≅
p area
is a function of k through Eq. (1)). For example, in Fig. 2,countsin colony
d
^ plate 1
therange25bnb170isexpectedtoproduceanestimateofn0within d is estimated from the agar plate. If kexperiment N 3, proceed
20%error for d =3mm.Thetypicallycitedoptimal range of colony
colony to step 2. Our algorithm is only valid for k =3k N1.
25–250countsonaplatethatisusedtoestimaten ispredictedby experiment
0 ^
our model(for d =10cmandthecommonlyused10-folddilution Step 2: Count the number of colonies nj on the J serial dilution plates.
plate ^ ^
d Therelevant plates are those with nj that are close to n (given
with αp = 1) for a colony size of dcolony =2.5mm(i.e., plate ¼ 40,
dcolony with Eq. (6), and in Fig. 3 for the standard dilution α =10
^ ^
k = 12) to within error bound of 20% in estimating n .InFig. 3 we andα =1).EstimatingthebestnwithEq.(6)mayeliminate
0 p
^
showthebestnonaplate(thatminimizethetotalerrorinEq.(3))as the need to count irrelevant serial dilution plates that are
a function of the experimental scenario for a given ratio of plate size ^
clearly far from the optimal n and save time.
to colony size (d /d ) as is predicted by Eq. (6) for 10-fold dilu- ^
plate colony Step3: Compute δ withEq.(1).NotethatkinEq.(1)isk=3k .
tion(α=10)andplatingdilutionα =1(i.e.,platingtheentiredilution j experiment
p ^ 2
Thus,fornj b9k themodelpredictsnegligiblecounting
^ experiment
volume).Thefigureshowsthattheoptimalcountnincreaseswithd /
plate ^
d , wherethelikelihoodofmiscountingcoloniesduetooverlapping error and thus δj→0.
colony Step4: ComputetotalerrorTE foreachplatewithEq.(3).Selectthejth
andmergingtogetherincreasesasd /d decreases. Fig. 3 can be j
plate colony plate for which the error is the smallest. For this optimal value
used as a quick guideline for selecting the best available agar plate
^ ^
fromwhichtoestimaten . of j we now have n and δ . This agar plate is defined as the
0 j j
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