An analysis of quantitative PCR reliability through replicates using the Ct method

J. Biomedica doi: 10.4236/jb

Published On

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Chris C. St

1_{Bioprocess R} 2_{Department o} 3_{Department o}

Email: erik.bo Received 6 Fe

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There is con suring nucle tions. The m thod is the r analyzed wit (Ct) method. performed h of initial con a concentrat resultant Ct dard and no riability/relia supports th replicates, th tically distin ten orders o tion. As exp grow as the demonstrate confound qu tion at low t that a miscl 3000 initial c tion region thermal wea vide data th detection str and plate fi classification becomes unr Keywords:M Molecule Co Replicates an

1. INTRO

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number of in e that these v uantitative cla template conc lassification t copies of temp

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ge dramatical observation a ection thresho ror. But varia om the statist ver replicates eplicates. In p

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460 C. C. Stowers et al. / J. Biomedical Science and Engineering 3 (2010) 459-469

Copyright © 2010 SciRes. JBiSE

Using a multiwell plate format we measured hundreds of replicates to produce Ct value distributions. Using

standard and novel statistical techniques we analyze the

Ct value distributions and demonstrate that the sample

mean and/or median Ct values are statistically

signifi-cantly distinguishable over ten orders of magnitude. Furthermore, we show that the sample mean Ct values

are reliably ordered according to the initial concentration of template. In other words, if x and y are initial template

concentrations with x < y and µx and µy are the

corres-ponding sample mean Ct values then µx > µy. The order

reverses because less initial template requires more cycles of PCR to amplify. We utilize ordering as a con-venient and natural device to quantify the role of repli-cates on reliability. We ask and answer the following question: Given an unlabeled dilution series how many replicates are required to reliably order the tubes? We find that the answer depends on the range of initial tem-plate.

A focus of this work was to cover as broad a range of initial conditions as possible with the same experimental format. We observed that the mean and/or median Ct

values had the smallest variance above 104 _{initial copies.}

Most published standard curves focus on this range [2]. Few studies have analyzed issues of variability and ro-bustness below this range. We show that below 104

ini-tial copies the probability of misclassification of the ini-tial template concentration given a Ctvalue grows

ra-pidly and saturates near a half. The dispersion in the Ct

value distributions and the rise in misclassification cor-relate with an independent measure of the thermal wear of the TAQ polymerase enzyme.

Driven by the observed broadening of the Ct value

distributions below a thousand initial copies, and in-spired by elegant methods that sidestep the issues created by the dynamics of exponential growth [14-18], we examined a format for single molecule detection uti-lizing an endpoint analysis and the statistical properties of well mixing and plate filling. We present data that such an assay is accurate where the real time method becomes unreliable.

2. MATERIALS AND METHODS

2.1. PCR

Rt-PCR results were generated using linearized double stranded EC3 plasmid DNA containing the ybdO gene. The plasmid was linearized by digestion with the restric-tion enzyme BamH1 prior to PCR. The following primer sequences were used.

1) Forward: 5’-AAT TAT TCT AAA ACC AGC GTG TC-3’

2) Reverse: 5’-TTT GGG ATT GAA TCA CTG TTT C-3’

The PCR supermix was prepared as described in [19], with the exception that we used Qiagen HotStarTaq Cat

# 203203, Roche dNTPs Cat# 13583000, DMSO Sigma # D8418 at 2%, and Sybr Green (Sigma # 86205) at 5-times the recommended concentration. Primers were used at a concentration of 1 µM. All samples were run

on the 384 well plate platform using an Applied Biosys-tems 7900HT thermocycler and the SDS 2.3 software. The Ct value threshold was set at 5.0 RFU (Relative

Fluorescence Units) for all samples. The DNA concen-trations of concentrated stocks were measured using a Nano-Drop 100 spectrophotometer prior to use. Subse-quent dilutions were performed using sterile, nuclease free water from Ambion # AM9937. The following thermo-cycling program was used.

1) 2 min at 50°C Initial Warmup Phase; 2) 15 min at 95°C Initial TAQ Activation Step; 3) 1 min at 95°C DNA Denaturation;

4) 1 min at 50°C Primer Annealing; 5) 1 min at 72°C DNA Extension;

6) 0.25 min at 80°C Fluorescence Measurement; 7) Repeat Steps 3-6 forty times.

2.2. Preparation of Identical Replicates

To ensure uniformity in the face of pipetting error the PCR supermix was prepared in well-mixed batches in a 14-mL conical tube. Each sample consisted of 184 rep-licates and 8 negative controls, requiring exactly half of a 384 well plate. All of the components except for DNA were loaded into a 14 mL conical tube in the following order: 800 µL PCR buffer, 5.6 mL of nuclease free water,

160 µL DMSO, 320 µL MgCl2 (Qiagen Cat #

124113012), 160 µL of a primer mix (a 1:1 mix of the

forward and reverse primer stored at a concentration of 50 M each), 160 µL Sybr Green (100X stored in DMSO),

160 µL of dNTPs, and lastly 80 µL of Taq polymerase.

We have noticed that the order at which these are added affects the reproducibility of the assay. The mixture was vortexed at high speed for 1 minute. 335 µL of supermix

was then removed to be used as a negative control and placed into a 1 mL eppendorf tube and 25 µL of water

was added. This mixture was then briefly vortexed to ensure well mixing. The remaining 7.105 mL of super-mix was then split equally four ways into 2 mL cryostat tubes, and 134 µL of water plus the amount of desired

DNA was added to each cryostat tube. Each tube was then briefly vortexed. For each reaction contained within a single well of the plate, 10 µL of the respective

reac-tion mix was loaded into a well of the 384 well plates.

2.3. TAQ Polymerase Pre-Wear Assay

C. C. Stowers et al. / J. Biomedical Science and Engineering 3 (2010) 459-4 461

Copyright © 2010 SciRes. JBiSE

added to the preworn enzyme with subsequent resump-tion of cycling. An efficiency was calculated by averag-ing the derivative over the resultant amplification curve.

2.4. Statistical Analysis of Ct Distributions

The sample mean Ct values for each initial template con-

centration were compared pairwise using a permutation test that is asymptotically valid and robust in situations where the distributions are not necessarily normal and/or the ratio of the variances is unknown, indicating that a t-test is not supported [20,21]. The test statistic T [20], measures the difference in mean rank of the samples within their union, scaled by a consistent estimator of their variance. Because the Ct value distributions may be

skewed by outliers, we also considered the median as a measure of central tendency. The median Ctvalues were

compared pair-wise using a bootstrap test that has been shown to outperform all reasonable alternative methods [22].

Given a linear regression, ymxb, of the mean/

median Ct values against x = log(n), the log of the

num-ber of initial copies of template, a relative error was cal-culated from the quantiles of the Ct value distributions as

follows. Allow x h

Q and x l

Q to be high and low

quan-tile values chosen from the Ct value distribution

gener-ated by initial log template x. Since the Ct value

gener-ally increases with decreasing amount of initial template the slope m of the regression line(s) is negative. Thus,

the difference in the predicted amount of initial template DNA from the distributions divided by the input amount is given as:

x

m b Q m b

Q_lx _hx

n n

10 10 10(  ) _ ( ) 



(1)

Let U represent the universe of possible Ct values, and

let T stand for the collection of possible initial template

concentrations. The initial template concentrations are thought of as the class labels. We consider the probabil-ity of misclassifying an observed Ctvalue given a known

class label. Suppose that we draw a Ctvalue from a

giv-en class, how likely is it to find that value in any of the other classes? The mean misclassification probability is estimated from the Ct value distributions corresponding

to different initial template concentrations according to the following formula.



U i

x \ T | i P x | i P x

P( ) ( ) ( ) (2)

where P(i|x) is the conditional probability of finding the Ct value i, given the initial template concentration x,

and the P(i|T\x) is the conditional probability of observing that same value given any initial template concentration other than x. The later conditional prob-ability is interpreted as the probprob-ability of misclassi-

fication. The conditional probabilities are estimated from the measured Ct value frequency distributions.

2.5. Plate Filling with Microbeads

Experiments were performed using 20µm latex beads

from Beckman Coulter (#PN6602798) using flat bottom 96 well plates from Becton Dickinson Labware. 96 well plates were used in place of 384 well plates for ease of microscopic analysis. Various dilutions of beads were prepared using a Beckman Multisizer Coulter Counter 2. 25 µL of each dilution was loaded into each well of the

96 well plate. The number of beads in each well was counted with a Nikon TE-2000 microscope.

2.6. Plate Filling Simulations

The statistics of the plate filling stochastic process were modeled using Monte-Carlo simulation. For instance, the expected number of empty wells in a 96 well plate was estimated by simulation using the following function:

Table[Mean[ Table[Length[

Complement[Range[96],

RandomInteger[Range[96], m]]], {10000}]], {m, 1, 600}]

Here m is the number of molecules being plated from a well mixed solution. The mean is estimated from 10,000 realizations. The standard deviation is computed by replacing the function Mean by Standard Deviation. A graph of these functions is shown in Figure 9. All

simulations and analysis were carried out in Mathemati-ca 6.03 (Wolfram Research), and the notebooks are available upon request.

3. RESULTS

The Ct value data are summarized in Figure 1. The

fig-ure shows that above 104 _{copies the data are distributed}

about the median with smaller variance than those below. Outliers exist across all the data, mostly trending upward of the median, indicative of reactions lagging behind the pack. The data show the distributions as collected across ten orders of magnitude in initial template.

3.1. Mean and Median Ct values

While the Ct value distributions below 104 copies of

ini-tial template DNA are broad and noisy, the sample means and medians form a monotone increasing series when stratified according to initial template. These data are shown in Table 1. The means and medians are very

nearly equal in all cases and the data distributions appear unimodal.

At initial template concentrations larger than 104

462

Copyright ©

F o r d

to identify d no hypothesi template con not within a another. Wh count the me a level of sig ception of th cally differen result of the in Table 2.

The Ctva

because we riances, stan

© 2010 Sci

Table 1. Samp Observe that th

Distribution N

Log Copies T

Replicates

Mean Ct Valu

Median Ct Va

Figure 1. Summ of DNA amplif run at each initi dots. Outliers ar

istinguishable is test appears ncentrations it a fraction of hen all of the eans and medi gnificance gre he smallest tw

nt at the 3/10 significance t

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C. C. Stowers

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C. C. Stowers

Res.

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10) 459-469

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close to the stripe, indica error, misclas

3.4. Rank O

In the first s sample mean data are stat This was the plate concen stand how few

In this sec Suppose that in a set of tub a correctly ca taining the scrambled as that the amou series. The q PCR runs are

Ct values cor

given level o is one level o closely relate

Conditione Monte-Carlo we draw k r

their sample

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fraction of po sample mean of k. The resu

© 2010 Sci

regression li ating the con ssification and

Ordering wi

section we sum n and median C

tistically disti case using al ntration. Doub w data are req ction we imag

an investigato bes, with no er alibrated pipett dilution serie s to order, but unt of DNA w question is to e required such rrectly order, a of confidence?

of quantitation ed pre-requisite ed on our data simulation. F replicates unif

mean . Th The resulting s alues and if thi

x, we score thi

ositive draws t ns, from a tota ults are shown

Figure 7. rank order

C. C. Stowers

Res.

ine and well cordance betw d pre-wear dat

ith Replicate

mmarized the

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inguishable an ll 184 replicat

tless it is of quired to make gine the follo or has produce rror other than

teman. Howev es are unlabe t not contents. within the tubes determine ho h that the resul and hence labe

We imagine n removed from

e.

we can answe From each Ct

formly at rand his mimics an set are sorted is order agrees is trial positive that resulted in l of 20,000 dr in Figures 7a

Reliability of r ring the initial t

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within the e ween the rela ta.

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findings that mputed from all nd rank orde tes per initial t interest to un e this same clai owing experim ed a serial dilu n that coming f ver, the tubes c eled and bec

It is unequiv s form a mono ow many repli ltant sample m el, the tubes wi

that rank orde m inversion, b

er this question value distribu dom and com n experiment w d according to t

s with that of t e. We compute n correctly ord raws at each v and8.

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Copyright

theory and e solutions of methods. Uti DNA from th ical bead cou endpoint ana pendent expe for PCR wer

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© 2010 Sci

Figure results results

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experiment for 20 micron la ilizing PCR to hose that are n unting. Figure

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SIONS

PCR plate rea al time PCR

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that using le template con are statistica ponding with have shown gressed over

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copies of in independent tion. Indepen enzyme expe in the corresp

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fication analy to high. The perhaps mor ror analysis

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Figure cules sp theory in

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stems from t e distribution

C. C. Stowers

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10. The numbe pread over a 38 n blue. The red

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interpret whi re quantitative ts an alterna nverse proble onvert a Ct va

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function of the umber of empt PCR endpoint d

n the variance n the data dist es are require ility? Rank or nt and meanin nk ordering si mber of replic tial template. a points provi or more repli ion for the sam heartening res ates are requir rce. For samp tions it may b

thod using th lls than to con These data, an monstrated tha n with statistic me PCR capab NA samples ra wever the dat ndreds of initi antitation. We ernative meth unting. In this alysis shows si We have des

ule counting ve demonstrat t the expecte

neering 3 (201

number of DN ty wells calcula data.

e of the data d tributions tea d in practice f rdering of the ngful device f imulations wit cates required Below the tr ide better than icates are sugg

me degree of sult for the fo red precisely ples with sma e more accura he expected n nsider replicat

nd the work at the use of cal replicates ble of quanti anging from u ta presented a

al copies real e and other gr hods for singl s regard, a p ignificant prom cribed a deco into plate fill ted through s ed number of

10) 459-469

NA mole-ated from

distributions. ach about how

for resolution e sample mean

for exploring th our data su d depends on ransition regi n 90% rank ac gested below rank accuracy ollowing reaso where the sa all initial temp

ate to conside number of (u es.

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renders the p itative analys upwards of 1 above show th

time PCR is roups have be le molecule d process involv

mise. omposition o

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What lessons w many repli-n arepli-nd reprodu-ns is a conve-this question. uggest that the the range of on individual ccuracy, while the transition y. But this is a on: More rep-ample may be plate concen-er an endpoint un) amplified

ups [24] have d in conjunc-process of real sis for initial 04 _molecules.

hat for tens to unreliable for een exploring detection and ving endpoint

of single mo-lification. We d experiment ed wells is a

C. C. Stowers et al. / J. Biomedical Science and Engineering 3 (2010) 459-4 469

Copyright © 2010 SciRes. JBiSE

robust statistic on which to base an inverse problem or a rigorous hypothesis test to count small numbers of single molecules. The observed linear scaling between expected and perceived DNA concentration indicates that amplification by PCR is directly related to plate

filling.

It remains an open problem to determine the condi-tional probability with which PCR can amplify above threshold in 40 cycles from a single strand of DNA. While it is currently impossible to enumerate individ-ual molecules or particles smaller than a nanometer, it is straightforward to count macroscopic objects such as latex beads or single yeast cells with a Coulter counter. Haploid yeast cells provide a convenient and verifiable means to deliver single copies of Bacillus subtilis genes such as ybdO into the wells of a multi- well PCR plate.

In this way, we are currently exploring the relation-ship between amplification and DNA copy number.

5. ACKNOWLEDGEMENTS

This work was partially supported through NSF-DMS 0443855, NSF ECS 0601528, NIH EB009235, and the short-lived W. M. Keck Foun-dation Grant#062014.

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