Quality control for quantitative PCR based on amplification compatibility test
Ales Tichopad a,b,c,*, Tzachi Bar b
, Ladislav Pecen c
, Robert R Kitchen d
, Mikael Kubista c,e
, Michael W. Pfaffl a,e
a
Technical University Munich, Physiology Weihenstephan, Weihenstephaner Berg 3, 85354 Freising-Weihenstephan, Germany
b
Labonnet Ltd. Itamar St. 10/1, 52531 Ramat-Gan, Israel
c
Institute of Biotechnology AS CR, Lb building, 2nd floor, Videnska 1083, 142 20 Prague 4, Czech Republic
d
School of Physics, University of Edinburgh, 10 Crichton Street, Edinburgh, EH8 9AB, United Kingdom
e
TATAA Biocenter AB, Odinsgatan 28, 411 03 Göteborg, Sweden
a r t i c l e i n f o
Article history:
Accepted 22 January 2010
Available online 28 January 2010
Keywords:
Quantitative PCR
DNA quantification
Z-score
Quality control
Kineret
Mahalanobis distance
Pattern recognition
Inhibition
Amplification efficiency
Kinetic outlier detection
a b s t r a c t
Quantitative qPCR is a routinely used method for the accurate quantification of nucleic acids. Yet it may
generate erroneous results if the amplification process is obscured by inhibition or generation of aberrant
side-products such as primer dimers. Several methods have been established to control for pre-processing
performance that rely on the introduction of a co-amplified reference sequence, however there is
currently no method to allow for reliable control of the amplification process without directly modifying
the sample mix. Herein we present a statistical approach based on multivariate analysis of the amplification
response data generated in real-time. The amplification trajectory in its most resolved and
dynamic phase is fitted with a suitable model. Two parameters of this model, related to amplification
efficiency, are then used for calculation of the Z-score statistics. Each studied sample is compared to a
predefined reference set of reactions, typically calibration reactions. A probabilistic decision for each individual
Z-score is then used to identify the majority of inhibited reactions in our experiments. We compare
this approach to univariate methods using only the sample specific amplification efficiency as reporter of
the compatibility. We demonstrate improved identification performance using the multivariate approach
compared to the univariate approach. Finally we stress that the performance of the amplification compatibility
test as a quality control procedure depends on the quality of the reference set.
Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction
The quantitative polymerase chain reaction (qPCR) is a method
to quantify a selected polynucleotide sequence by amplifying its
initial concentration to a level at which an accurate detection can
be made [1–3]. Fifteen years after its invention it is the experiment
design, proper data analysis, and data quality assurance rather
than the instrumental performance and chemistry that pose the
major challenges in acquiring valid biological inference [4–6].
PCR amplifies the targeted nucleic acid in the sample and this
amplification is considered exponential in its most progressive
phase [7–11]. The fundamental improvement from a qualitative
PCR technique to a quantitative approach was facilitated by the
inclusion of a fluorescence emitting reporter into the reaction
mix, whose fluorescence can be monitored throughout the reaction
progress to reflect the increasing concentration of the nucleic acid
as the reaction progresses [12–17]. The signal emitted by interaction
of the signalling agent with the DNA is monitored at least once
per cycle and the cycle number at which the signal reaches a
threshold level that is significantly above background is defined
as the cycle of quantification (Cq). This threshold is set based on
qualified decisions or a computing procedure [7,18]. The amplification
kinetics can be visualised in a plot of signal intensity versus
PCR cycle number and the full plot of all signal readings has a sigmoid
character, provided enough cycles are included. The traditional
assumption underlying PCR is that it is a chain reaction
that progresses in a fashion close to perfect doubling after the completion
of each cycle. That is, every target DNA molecule is used as
a template for its complementary copy within one cycle of the
reaction. A further common assumption is that the reporter fluorescence
is proportional to the amount of target DNA present,
despite the fact that the reporter dye concentration is constant
during the reaction while the DNA concentration, and thus the
DNA to dye ratio, changes several orders of magnitude.
Several methods have been published that describe the computation
of amplification efficiency from the region of the reaction
trajectory considered close to exponential [8,11,19–23]. This region
is usually selected somewhere between the departure from
the noisy background and the entry into the plateau phase and typically
contains some 3–10 cycle readings. An exponential model is
fit to the selected data to generate an estimate of the amplification
1046-2023/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.ymeth.2010.01.028
* Corresponding author. Address: Technical University Munich, Physiology
Weihenstephan, Weihenstephaner Berg 3, Freising-Weihenstephan 85354, Germany.
Fax: +49 8161 71 4204.
E-mail address: ale@tichopad.de (A. Tichopad).
Methods 50 (2010) 308–312
Contents lists available at ScienceDirect
Methods
journal homepage: www.elsevier.com/locate/ymeth
efficiency (EPCR) or, alternatively, this region can be log-transformed
prior to fitting a linear model [19–23]; although both approaches
assume that the amplification is exponential. The
selection of this exponential region is somewhat arbitrary and
has the potential to profoundly influence the efficiency estimate
(Table 1). Also, the use of exponential models to describe the reaction
kinetics is a simplification of the true nature of the reaction
however sigmoid models with compensation for saturation may
prove more appropriate for quantification [9,19,24–26]; such models
can be fitted with better precision, particularly around the midpoint
of inflection.
Quality assurance and quality control of the data produced by
qPCR is essential to obtain valid biological results [27]. Due to
the nonlinear character of the amplification process even minor
initial disturbances due, for example, to the presence of inhibitors
or formation of primer may be amplified to a large error in the final
quantification. As it is meaningful to compare samples that have
been analysed using the same qPCR assay [28–30] it is essential
that they should have compatible kinetics. Comparable kinetics
can be validated by either introducing an internal control sequence
to be co-amplified with the target sequence [31–33], the SPUD assay
[34], or by analysing the amplification trajectory of each individual
sample and comparing it with defined references [20,35–
37]. While the internal amplification control may negatively interact
with the target sequence and bias the results, the SPUD assay
does not control for amplification errors such as undisclosed generation
of primer dimers. Further, both methods are laborious
and costly.
Recently developed amplification kinetics compatibility tests
are inexpensive (no consumables needed), fast, and they allow
for discrimination between valid and invalid samples [20,35–37].
To date, all of these methods have been based on comparing the
exponential amplification efficiency of the individual test sample
with that of defined reference samples, which are typically the calibration
samples used for the quantification of absolute copy
numbers.
In our experience, the calculated single sample efficiency from
univariate analysis is not a robust enough estimate (Table 1) to enable
the reliable identification of deviant samples. Herein we present
a modified approach to the detection of deviant reactions that
introduces a multivariate test to compare selected geometric properties,
rather than relying on the amplification efficiency alone. By
excluding deviant reactions from an experiment, the overall experimental
effect recognition is improved together with the precision
and accuracy. The comparison of the multivariate approach with
the univariate approaches, based on estimating EPCR, is the focus
of this paper.
2. Description of method
Two experiments were performed in order to evaluate the performance
of the multivariate kinetics outlier detection; one with
well controlled effect of the inhibition by primer competition
[38] and the second with inhibition by tannic acid, which is a naturally
occurring PCR inhibitor [35,39–41]. In each case, three different
assays were evaluated.
2.1. Primer competimers in controlled assay
Analysis of a selected locus of the beta-actin gene was performed
with PCR using the DyNamo SYBR green qPCR kit (F-
410L) and primer concentrations of 250 nM in a total reaction volume
of 20 ll using the ABI 7300 thermal cycler. Six standard
curves based on purified PCR products, were constructed using five
DNA concentrations, each analysed with three qPCR replicates (all
together 6 Â 5 Â 3 = 90 reactions). The range of concentrations was
102
–107
copies. A No Template Control (NTC) samples was included
for each standard curve. The PCR were inhibited by adding
primer competimers at 0% (reference), 1%, 2%, 4%, 8%, and 16% concentration
of the total amount of the forward primer (mr-BetAct_F)
to the six standard curves. The competimer had the same sequence
as the forward primer but was modified at the 30
-end such that it
could not be extended. By competing with the normal primers for
the same target sequences the competimer reduces PCR efficiency
[38]. The total molar concentration of normal primer plus the competimer
was the same (250 nM) in all standard curves. The qPCR
cycling conditions were: 50 °C for 20
, 90 °C for 100
, 40 cycles of
95 °C for 1500
followed by 60 °C for 10
. Dissociation curve profile
was: 95 °C for 1500
, 60 °C for 10
and automatic rap rate to 95 °C.
2.2. Effect of tannin on different sequences
Three qPCR assays were performed using DyNamo SYBR green
qPCR kit (F-410L) and primer concentration of 250 nM in a total
reaction volume of 20 ll using the RotorGene 6000 thermal cycler.
Beta-actine, IGF1 and Histon3 transcripts were assayed independently
in singleplex reactions. Two standard curves were produced
from the same cDNA stock solution, one without inhibitor and one
with 2.0 ng tannic acid added per 15 ll reaction mix at each dilution
(Fig. 1A). Each standard curve consisted of 5-fold dilutions (1-,
5-, 25-, 125-, and 625-fold) in triplicates (total 15 reactions). The
initial target cDNA concentrations were unknown. For non-inhibited
assays the standard curves had r2
> 0.98 for all three genes.
2.3. Statistical analysis
2.3.1. The univariate approach
The univariate analysis of the amplification kinetics was performed
in the same way as described in Bar et al. [20]. Briefly,
the fluorescence background was removed by subtracting the
arithmetic average of the five lowest fluorescence readings from
all data points in the amplification curve and four data points just
above a manually set threshold were used for analysis. These were
fitted with the exponential model to estimate the amplification
efficiency, EPCR [8]. The EPCR values were then compared with the
average of those EPCR values of the reference set [36].
2.3.2. The multivariate approach
The multivariate approach differs from the univariate in that
information from several amplification curves is used to identify
outlying reactions. In this approach, for each curve the maximum
of the first derivative (t1) and the maximum of the second derivative
(t2) [18] were calculated for a region of eight data points, six
below the inflection point (IP) of the response curve and two
above. These points were smoothed using a symmetric sigmoid
function [25,26]. These two vectors, t1 and t2, provide an effective
means of assessing the dynamics and progression of the reaction.
When plotted in a scatter plot with axes t1 and t2 we find the typTable
1
Discrepancy between methods for amplification efficiency estimation.
Tichopad
et al. [8]
Ramakers
et al. [21]
Peirson et al.
[22]
Wilhelm
et al. [23]
Liu and Saint
[19]
DE SD DE SD DE SD DE SD DE SD
0.44 0.076 0.26 0.102 0.24 0.118 0.31 0.076 0.33 0.071
A standard curve with five dilution steps and three replicates at each dilution steps
was constructed and overall amplification efficiency was calculated from its slope
as Estd = 10À1/slope
À 1. Five Eindividual estimation methods for individual samples
were then employed and the SD and mean Eindividual were calculated. The Eindividual
was eventually compared with Estd and DE was calculated as DE ¼ Estd À Eindividual.
A. Tichopad et al. / Methods 50 (2010) 308–312 309
ical Pearson correlation coefficient to be of the order of r > 0.6. Outliers
are defined as data points that exceed a defined threshold in a
combined measure, in this case the Z-score, obtained from a transformation
of the t1 and t2.
Assuming that n variables (X1; . . . ; Xn) are approximately normally
distributed, Z statistics are calculated as follows
Z ¼ X2
1 þ . . . þ X2
n ð1Þ
where Z is approximately v2
(chi-square) distributed with n degrees
of freedom: Z $ v2
(n). The critical percentile for the given n degrees
of freedom can be obtained from tabulated v2
(n) distribution or calculated
from its cumulative distribution function. The linear model
describing the relationship between t1 and t2 is simply
t2 ¼ t1 Á b þ a þ s ð2Þ
where b and a are the linear coefficients and s is the residual error
term that is independent of t2. If standard samples are available
they are used to produce the t1 and t2 pairs for the modelling. If
standards are not available regression is performed over all observations.
From the modelling s is estimated
s ¼ t2 À bt2 ð3Þ
t1 and s are transformed to standard normal distribution by subtracting
the mean and dividing by the standard deviation:
t1norm ¼ ðt1 À t1Þ=rt1
ð4Þ
snorm ¼ ðs À sÞ=rs ð5Þ
where t1 and s are the means and rt1
and rs are the standard
deviations.
snorm and t1norm are finally used in the calculation of Z statistics.
Z ¼ t2
1 þ . . . þ s2
ð6Þ
Kinetic outliers are identified based on the 95th percentile of the v2
distribution for two degrees of freedom (5.991) and the 99th percentile
(9.210) is further used to distinguish weak and strong outliers.
As positive outliers (samples for which the amplification
efficiency is noticeably above average) may occur in an experiment
together with samples with suppressed performance, one may decide
that only inhibited samples shall be considered invalid. Hence,
only those samples that have a negative Z-score and either negative
t1norm , and/or negative snorm values shall be considered as outliers to
assure that only truly inhibited samples are excluded. This may be
considered a one-tailed statistical test. This calculation was implemented
as a module of the Kineret software (Fig. 1B).
3. Concluding remarks
3.1. The assay performance
All assays at all dilutions and degrees of inhibition generated
valid amplification curves with clearly defined exponential trajectory
that reaches plateau phase. The linearity of the log-transformed
standard curve had r2
> 0.98 in all non-inhibited assays.
Melt curve analyses evidenced formation of specific amplification
products. NTC samples generated responses that reached threshold
very late due to primer–dimer formation, and those were readily
identified by the melt curve analysis.
3.2. Effect of inhibition on Cq
To study the effect of inhibition the Cqs for the replicates in
each dilution curve were averaged. The differences between the
Cq values of the inhibited step and non-inhibited reactions at each
dilution were calculated. The null hypothesis, that these differences
are zero (i.e. there is no inhibition), is tested by one-tailed
t-test. The corresponding significant probability achieved for the
test with a = 0.05 is p < 0.1, provided the difference is negative.
Only inhibition with 8% and 16% competimers could be considered
significant based on the t-test. The effect of 1%, 2%, and 4% competimers
was not significant (Table 2).
3.3. Amplification compatibility
Test samples were compared to the reference set using the Zscore
to detect outliers (Fig. 2A). The test identified 44 of the set
Fig. 1. Reactions inhibited with tannin and their retrieval with Kineret software. (A) The blue curves present 15 reference reactions and the red curves present 15 reactions
produced from the same DNA stocks as the reference with 2.0 ng tannic acid added per 15 ll reaction mix. (B) Screenshot of the Kineret software with the Z-score calculated.
Red color of sample positions on the 96 well-plate format indicates Z > 9.21, critical value for 99% confidence level. Twelve out of 15 inhibited samples (C1 to D7) were
retrieved based on the calculated Z-score.
310 A. Tichopad et al. / Methods 50 (2010) 308–312
of 75 inhibited reactions as deviant. All NTC samples were also
identified as outliers by the test, showing that this method is also
suitable to distinguish negative and positive samples. The number
of identified deviant samples varied with the strength of inhibition.
At 8% and 16% competimer inhibitions, 73% and 100%, respectively,
of the samples were correctly identified as outliers. At 1%, 2%, and
8% competimer inhibitions, 40%, 13%, and 13%, respectively, of the
samples were identified as outliers (Table 3). This result is encouraging,
since the measured Cq values at this low degree of inhibition
were not significantly affected (Table 2). One sample was identified
as outlier in the reference set. The retrieval rates in the experiment
with tannin inhibition showed reliable performance for the Z-score
with a success rate between 67% and 100% (Table 4). The number
of outliers found in the reference set of the beta-actine, IGF1, and
Histon3 assays were 1, 0, and 1, respectively.
The approach based on estimating EPCR to identify deviant samples
showed less reliable retrieval, with less than 33% in both experiments
(Tables 3 and 4). It is obvious from Fig. 2B that the spread of
the estimated a EPCR values is too large to be a reliable indicator.
In conclusion, when a set of reference samples is available that
can be used to define the performance of a reaction, we show that
qPCR with incompatible kinetics can reliably be identified. Calibration
reactions used in absolute quantification may serve as ideal
standards if they are performed with the same template and comparable
sample matrix. In cases where such reference samples are
not available, the entire set of reactions may be used for calibration
and outliers are identified using suitable metrics such as, for example,
the Z-score. In such cases more robust procedures may be of
advantage, such as the ‘‘leave-one-out” classification, i.e. sequentially
removing one sample and testing it against the others. Another
approach is the repeated excluding of outliers and
redefining the reference (Fig. 3). An implementation of this powerful
multivariate method of outlier detection among qPCR samples
is available in the Kineret software from Labonnet Ltd.
(www.kineretsoftware.com).
A generalised solution for n > 1 kinetics parameters is to calculate
the n-dimensional Mahalanobis distance [42]. The Mahalanobis
distance is the uncorrected sum of squares of the principal
component scores calculated from the center of the reference data
set. Also other multivariate approaches may be employed such as
the Kohonen self-organising networks, principal component analysis,
and support vector machines. Our work so far suggests that
obtaining two traces from the amplification kinetics is sufficient
to represent the amplification kinetics well enough; fitting the response
curve to a model of higher dimensionality does not improve
the stringency appreciably, on the contrary, there is risk data are
over-fitted. We could also get reasonably good results simply by
using the slope of the sigmoid curve calculated at the point of
inflection combined with the plateau height. These parameters
can be obtained by fitting a sigmoid model to the data [25,26].
We also suggest that kinetics outliers are validated based on their
Cq values, ensuring that the outlier detection is not overly sensitive
and does not discriminate against minor deviations in Cq. This can
happen if the reference set is highly homogeneous with low spread
as obtained when using multiple replicates of the same sample as a
Fig. 2. Comparison of bivariate Z-score with the univariate amplification efficiency E in detection of inhibited samples. (A) The bivariate Z-score shows increased values in the
8%, 16% inhibition as well as in the NTC reactions. (B) The EPCR shows too large spread of values to reliable distinguish inhibited groups from the reference.
Table 2
Effect of the inhibition on the Cq value.
DNA conc. Differences from reference as DCq by inhibition
strength
1% 2% 4% 8% 16%
xÃ10000 0.05 À0.29 À0.223 À0.257 À0.363
xÃ1000 0.233 0.327 0.143 À0.077 À0.307
xÃ100 0.213 0.017 À0.073 À0.303 À0.47
xÃ10 À0.23 À0.173 À0.457 À0.753 À0.737
x NA NA NA NA NA
p of t-test (H0: Dif. < 0) 0.58 0.84 0.31 0.09 0.02
NA indicates that not enough observations were obtained to perform statistical test.
Table 3
Retrieval of samples inhibited by competimers by the multivariate and univariate
test.
Multivariate (Z) 1% 2% 4% 8% 16% NTC
N/total 6/15 2/15 2/15 11/15 15/15 6/6
Retrieval [%] 40 13 13 73 100 100
Univariate (E)
N/total 4/15 5/15 2/15 1/15 2/15 2/6
Retrieval [%] 27 33 13 7 13 13
The bivariate Z-score and the amplification efficiency E were separately used to
discriminate between the reference set and the inhibited sets.
Table 4
Retrieval of samples inhibited by tannin by the multivariate test and univariate test.
Multivariate (Z) ACTB H3 IGF
N/total 12/15 15/15 10/15
Retrieval [%] 80 100 67
Univariate (E)
N/total 1/15 5/15 2/15
Retrieval [%] 7 33 13
The bivariate Z-score and the amplification efficiency E were separately used to
discriminate between the reference set and the inhibited sets.
A. Tichopad et al. / Methods 50 (2010) 308–312 311
reference set. A good strategy is to use a reference set with similar
spread in concentrations as expected for the test samples.
Finally, the precision of the kinetic outlier detection depends on
the size of the reference set. Our preliminary work suggests that at
least 10 reference measurements should be available for every
assay.
Acknowledgment
This work was supported in part by grants GAAV IAA500970904
and GAAV IAA500520809 to Institute of Biotechnology AS CR.
Labonnet Ltd. is the developer of the Kineret software.
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Fig. 3. Two dimensional 95% confidence interval. Both dimensions are normalized to mean = 0 and SD = 1. Only the reference set and the set with 16% inhibition by primer
competimers are shown to maintain clarity. This method presents another alternative to Z-score calculation. (A) The entire reference dataset is taken. (B) Reference set
cleaning was performed by exclusion of two extreme observations, resulting in stronger distinction from the incompatibility pattern.
312 A. Tichopad et al. / Methods 50 (2010) 308–312