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

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[ [1][2][3]  Furthermore, we show that the sample mean C t 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 µ x and µ y are the corresponding sample mean C t values then µ x > µ y .The order reverses because less initial template requires more cycles of PCR to amplify.We utilize ordering as a convenient and natural device to quantify the role of replicates 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 template.
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 C t values had the smallest variance above 10 4 initial copies.Most published standard curves focus on this range [2].Few studies have analyzed issues of variability and robustness below this range.We show that below 10 4 initial copies the probability of misclassification of the initial template concentration given a C t value grows rapidly and saturates near a half.The dispersion in the C t value distributions and the rise in misclassification correlate with an independent measure of the thermal wear of the TAQ polymerase enzyme.
Driven by the observed broadening of the C t value distributions below a thousand initial copies, and inspired by elegant methods that sidestep the issues created by the dynamics of exponential growth [14][15][16][17][18], we examined a format for single molecule detection utilizing 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.

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 restriction 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 Biosystems 7900HT thermocycler and the SDS 2.3 software.The C t value threshold was set at 5.0 RFU (Relative Fluorescence Units) for all samples.The DNA concentrations of concentrated stocks were measured using a Nano-Drop 100 spectrophotometer prior to use.Subsequent dilutions were performed using sterile, nuclease free water from Ambion # AM9937.The following thermo-cycling program was used.

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 replicates 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 supermix 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 reaction mix was loaded into a well of the 384 well plates.

TAQ Polymerase Pre-Wear Assay
The PCR supermix was prepared as described above, but without template DNA.Steps 1 and 2 of the PCR process were executed following which samples were pre-worn by thermocycling the supermix as described in steps 3 through 6 above.Samples were pre-worn from 5 to 40 cycles.10 8 copies of initial template DNA were Copyright © 2010 SciRes.JBiSE added to the preworn enzyme with subsequent resumption of cycling.An efficiency was calculated by averaging the derivative over the resultant amplification curve.

Statistical Analysis of C t Distributions
The sample mean C t values for each initial template concentration 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 C t value distributions may be skewed by outliers, we also considered the median as a measure of central tendency.The median C t values were compared pair-wise using a bootstrap test that has been shown to outperform all reasonable alternative methods [22].
, of the mean/ median C t values against x = log(n), the log of the number of initial copies of template, a relative error was calculated from the quantiles of the C t value distributions as follows.Allow x h Q and x l Q to be high and low quantile values chosen from the C t value distribution generated by initial log template x.Since the C t value generally 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: Let U represent the universe of possible C t 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 probability of misclassifying an observed C t value given a known class label.Suppose that we draw a C t value from a given class, how likely is it to find that value in any of the other classes?The mean misclassification probability is estimated from the C t value distributions corresponding to different initial template concentrations according to the following formula.
is the conditional probability of finding the C t value i, given the initial template concentration x, and the 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 probability of misclassi-fication.The conditional probabilities are estimated from the measured C t value frequency distributions.

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.

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: 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 Mathematica 6.03 (Wolfram Research), and the notebooks are available upon request.

RESULTS
The C t value data are summarized in Figure 1.The figure shows that above 10 4 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.

Mean and Median C t values
While the C t value distributions below 10 4 copies of initial 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 10 4 initial copies the data distributions seen in Figure 1 appear  2.
The C t va because we riances, stan

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Relative err e standard curve rresponding to limits were cal n Subsection 2.   It remains an open problem to determine the conditional probability with which PCR can amplify above threshold in 40 cycles from a single strand of DNA.While it is currently impossible to enumerate individual 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 multiwell PCR plate.
In this way, we are currently exploring the relationship between amplification and DNA copy number.
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Using a multiwell plate format we measured hundreds of replicates to produce C t value distributions.Using standard and novel statistical techniques we analyze the C t value distributions and demonstrate that the sample mean and/or median C t values are statistically significantly distinguishable over ten orders of magnitude.

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JBiSErobust 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.