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Author M. Matz is a PLOS ONE Editorial Board member. This does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.

Conceived and designed the experiments: MVM. Performed the experiments: RMW. Analyzed the data: MVM JGS. Contributed reagents/materials/analysis tools: MVM. Wrote the paper: MVM JGS.

Model-based analysis of data from quantitative reverse-transcription PCR (qRT-PCR) is potentially more powerful and versatile than traditional methods. Yet existing model-based approaches cannot properly deal with the higher sampling variances associated with low-abundant targets, nor do they provide a natural way to incorporate assumptions about the stability of control genes directly into the model-fitting process.

In our method, raw qPCR data are represented as molecule counts, and described using generalized linear mixed models under Poisson-lognormal error. A Markov Chain Monte Carlo (MCMC) algorithm is used to sample from the joint posterior distribution over all model parameters, thereby estimating the effects of all experimental factors on the expression of every gene. The Poisson-based model allows for the correct specification of the mean-variance relationship of the PCR amplification process, and can also glean information from instances of no amplification (zero counts). Our method is very flexible with respect to control genes: any prior knowledge about the expected degree of their stability can be directly incorporated into the model. Yet the method provides sensible answers without such assumptions, or even in the complete absence of control genes. We also present a natural Bayesian analogue of the “classic” analysis, which uses standard data pre-processing steps (logarithmic transformation and multi-gene normalization) but estimates all gene expression changes jointly within a single model. The new methods are considerably more flexible and powerful than the standard delta-delta Ct analysis based on pairwise t-tests.

Our methodology expands the applicability of the relative-quantification analysis protocol all the way to the lowest-abundance targets, and provides a novel opportunity to analyze qRT-PCR data without making any assumptions concerning target stability. These procedures have been implemented as the MCMC.qpcr package in R.

Real-time quantitative PCR ^{−ΔΔCT} (“delta-delta Ct”) method ^{−ΔΔCT} equation with a four-story formula incorporating the efficiencies of PCR for control and target gene

The need to base qPCR analysis on dilution series data made relative quantification practically equivalent to the other flavor of qRT-PCR analysis, absolute quantification

We would draw an analogy with the literature on data analysis for DNA microarrays. Here, it has been repeatedly argued that the joint analysis of the whole dataset is more appropriate. Such an approach can borrow information across multiple genes, improve the precision of gene-specific estimates, and properly account for complex experiment designs, as well as both biological and technical replication

Despite the attractiveness of this approach, three major issues remained unresolved. First, the approach of Steibel et al. disregards heteroscedasticity, or the increase in sampling variance at the lower end of target abundances. In this respect it is similar to essentially all existing qPCR analysis pipelines: the statistical model ignores the discrete nature of the amplification process. This heteroscedasticity arises because qRT-PCR is fully capable of amplifying just a few target molecules within each trial

Here, the mixed-modeling approach of Steibel et al is extended to account for all these issues. First, a generalized linear mixed model based on the Poisson-lognormal distribution replaces the original Gaussian model. This properly handles zero counts, as well as the shot-noise variance associated with low-abundant targets. Second, the model fitting process involves a Bayesian MCMC sampling scheme, and can directly incorporate information about control genes in the form of priors. Our implementation of the method leverages the MCMCglmm package in R

The dataset that was chosen for re-analysis addressed the effects of heat-light stress and recovery in a reef-building coral

This dataset is interesting from the analytical standpoint because of three reasons. First, one of the main effects of interest is the interaction term, Condition:Timepoint, describing the gene regulation in coral fragments that were first stressed and then allowed to recover. Evaluation of the interaction term necessitates the use of linear models or ANOVA rather than non-parametric methods or pairwise t-tests

The standard practice for qRT-PCR analysis is to analyze each gene individually, using control genes to estimate the required normalization factors. In contrast, we build a hierarchical model that can be used to jointly estimate the effects of experimental treatments on the expression of all genes. Under such an approach, control genes can sharpen estimates of model parameters, but are not strictly necessary, as all normalization happens within the model.

In this respect, our approach is similar to that proposed by Steibel et al _{q}_{q}

Our approach differs in that we directly model the initial copy number using generalized linear mixed models (GLMMs) with Poisson-lognormal errors

A third advantage of our model is that it naturally accounts for the so-called “shot noise” that arises from the discrete nature of PCR amplification. Shot noise refers to Poisson-like fluctuations that become discernible when the number of target molecules is small enough so that such fluctuations are the dominant source of variability after signal amplification. To see why this occurs for weak signals, observe that if the true number of target molecules in a sample is Poisson distributed, then the absolute magnitude of shot-noise variation grows like the square root of the expected number of molecules. This is much slower than linear growth, meaning that the relative contribution of shot noise decreases and the signal-to-shot-noise ratio increases as the expected number of counts gets larger. This explains why shot noise is more frequently observed when amplifying samples with very few target molecules, as we illustrate experimentally below.

In describing our model, we use the following subscript conventions:

Thus _{gijkr}_{gijkr}

There is no closed-form expression for the density of the Poisson-lognormal, but it may be interpreted as a mixture of Poissons. Specifically, suppose that

Then the marginal distribution of

A notable feature of the Poisson-lognormal model is its overdispersion relative to the Poisson: if

This is important for adequately describing the technical variability of qPCR measurements, which need not match the strict mean-variance relationship implied by the Poisson distribution. Observe that in the limit as the log-variance goes to 0, the model becomes Poisson.

For the purpose of model-fitting, we appeal to (1) and re-write the original model in an equivalent hierarchical form, which facilitates computation via Markov-chain Monte Carlo:

The

Our model for log-rate term

We describe each component of the model in more detail, along with the priors used for the random effects.

_{g}

_{ig}

_{k}

_{jg}

_{kg}

To make sure we are understood not only by statisticians but also by qRT-PCR practitioners, below we explain in more colloquial terms how the model (2) is constructed for our motivating example.

The model has a single response variable, the transcript count, whose rate is modeled on a log-linear scale. The most basic explanatory variable in the model is ‘gene’, which corresponds to the term _{g}_{ig}

Even though this model specification seems to contain all the terms we want to estimate, we must take care of other important sources of variation that, while being of no real interest to us, must be taken into account to ensure that the model is accurate and powerful. The most important of these is the random effect of the biological replicate (i.e., an individual RNA sample), accounting for the variation in quality and/or quantity of biological material among samples, which corresponds to the term _{k}

Note that, since the variation in cDNA quality and/or quantity affects all genes in a sample in the same way, this random factor is not gene-specific. The introduction of this random factor into the qRT-PCR model was perhaps the most important innovation in the model of Steibel et al

The experimental design might have involved additional “grouping factors” that are not directly related to the experimental treatments being studied but still might be responsible for a considerable proportion of variation and must be accounted for to achieve more accurate predictions. These factors, if present, would correspond to the term _{jg}

Once again, the model is flexible in the number of grouping factors that could be included.

The two remaining terms that we still need to add are both error terms, accounting for the residual variation that remained unexplained. The first one is specified as a random factor and reflects the unexplained differences between biological replicates (samples), corresponds to the term _{kg}

Finally, the remaining unexplained variation would be due to the differences between technical replicates, reflecting the precision of the qPCR instrument used. This term corresponds to the error term

It is important to note that separating variances due to

For fixed factors involving genes that are not designated as control genes, a diffuse normal prior is used, with mean = 0 and very large variance (10^{8})

In addition to restricting the mean change of the control genes in response to fixed factors, we might also wish to restrict their variances due to gene-specific random effects, _{jg}_{kg}

One notable advantage of the MCMC-based approach is that point estimates and credible intervals for any modeled effects can be easily calculated based on the parameter values sampled by the Markov chain. A credible interval is a Bayesian analogue of the confidence interval in frequentist statistics, and is defined as an interval that, with a specified posterior probability (e.g. 0.95), contains the true value of the parameter. Pairwise differences between conditions characterized by various factor combinations can also be computed, along with their credible intervals. This is useful for situations when factors have more than two levels. In addition to the fixed effects, credible intervals of variance components can be similarly examined; however, it must be remembered that interval estimates for variance components are robust only for data sets with many replications at the corresponding level of the model hierarchy.

The question of interest in qPCR analysis is whether a specific treatment had an effect on a specific gene. The posterior distribution for _{ig}_{ig}

Although it is less natural to do so under the Bayesian paradigm, one may also use posterior tail areas to construct a procedure that behaves very much like a classical significance test. Specifically, define the two-sided Bayes tail area _{ig}_{ig}_{MCMC}_{ig}

For large MCMC samples, our definition of _{ig}

There are at least two reasons why proceeding in this manner yields a sensible, though not exact, test. First, the Bernstein–von Mises theorem implies that, under quite general conditions, the joint posterior distribution behaves asymptotically like a multivariate normal distribution centered at the maximum-likelihood estimate, and with inverse covariance matrix given by the Fisher information matrix

While this asymptotic guarantee may be cold comfort for researchers with modest sample sizes, it should be emphasized that even purely classical analyses of generalized linear mixed models yield significance tests that are valid only asymptotically (e.g.

Second, the key feature of a p-value is that it has a uniform distribution under the null hypothesis. We conducted a simulation study to check whether this fact holds for _{ig}_{ig}

The central procedure in our method is the transformation of raw Cq values into molecule counts. In principle, this can be achieved using absolute quantification curves

rounded to integer. (3)

To directly estimate

(a–c) Examples of amplification of different gene targets from a series of four-fold template dilutions with six-fold technical replication. The red line is the linear regression across 6–7 most concentrated dilutions where the increase in variance was not yet pronounced. The amplification efficiency (

The

The mis-specification of

The points are posterior means, the 95% credible intervals are denoted as dashed lines connecting upper and lower interval limits across genes, to better visualize changes in their width. (a) Comparison of the results based on formula-approximated

(a) Plot of lognormal residuals against predicted values to test for linearity. (b) Scale-location plot to test for homoscedasticity. A good fit is corroborated by the lack of pronounced mean trend in these two plots (red lines). (c) Plot of quantiles of standardized lognormal residuals against theoretical quantiles of the normal distribution. Red diagonal corresponds to the exact match. (d) Probabilities of experimental Poisson residuals plotted against their theoretical probabilities for one of the MCMC samples. All MCMC samples show the same nearly perfect fit to the Poisson expectations.

The points are posterior means, the 95% credible intervals are denoted as dashed lines connecting upper and lower interval limits across genes. (a) Comparison between naïve model (no control genes specified) and two informed models, with one (

Good performance of the naïve model, not relying on any control gene information, may not be too surprising for a dataset that contains many genes demonstrating various expression patterns, but will it work when genes are few and their expression patterns are unbalanced? To explore this issue, a smaller dataset was extracted from the coral stress data containing only four genes: a control gene

(a, c): Three heat shock protein genes plus one control gene (

Some datasets may not conform to the assumption of our model (2) that the variation in template loading between samples, _{k}_{k}

We call this model “classic” since it is based on earlier developments

For the coral stress dataset, the “classic” model generated virtually identical point-estimates of fold changes as the full Bayesian models (

(a) Point-estimates and credible intervals for the effects of stress in coral dataset inferred by the informed model (allowing control genes

The p-values based on Bayesian z-test agreed well with _{ig}^{corrected} = _{2}(

(a) Correspondence between p-values based on posterior tail areas (horizontal axis) and z-test based p-values (vertical axis) for the stress and recovery effects in the coral dataset. (b–d) Frequency distribution of z-test based p-values obtained using naïve (b), informed (c) and “classic” (d) models from datasets simulated under null hypothesis. The fraction of simulated p-values that are less than 0.05 is given above each plot.

(a) Effects of stress and recovery under naïve model. It can be seen that recovery gene regulation is basically a mirror image of stress response. (b) Transcript abundances of selected genes (see legend) across conditions of interest. The points are posterior means, the whiskers denote 95% credible intervals.

(a) The match between the fold-changes inferred in the current reanalysis and previously reported changes, for naïve, informed and “classic” models. (b) Correspondence between p-values under the naïve, informed, and “classic” models (see legend on panel a) and the previously reported p-values. Points above the line indicate higher power (lower p-values) of the new models. The reanalysis p-values were derived by the Bayesian z-test.

To see how the new methods compares to the classic delta-delta-Ct procedure

(a) Fold-changes inferred by the naïve, informed and “classic” models plotted against fold-changes derived by the delta-delta-Ct analysis with a single control gene. (b) Comparison of p-values derived by the Bayesian z-test from naïve, informed, and “classic” models (see legend on panel a) with p-values obtained by pairwise t-tests within the delta-delta-Ct pipeline. On both panels, the line denotes 1∶1 correspondence. On panel b, points above the line indicate higher power (lower p-values) of the new models.

The purpose of

This model is designed to be universally applicable in qRT-PCR. It properly handles the full range of target abundances down to the individual molecule level, derives information from no-amplification trials, and provides a possibility to specify control genes with the desired degree of confidence. In the same time, it is fully flexible in terms of experimental designs that it can accommodate, and would handle any number of fixed and random effects and their interactions.

The “classic” model represents a viable alternative to the full Bayesian analysis especially for the cases when the quantity and/or quality of the RNA samples varies systematically across experimental conditions, and when the expression levels are not too low (i.e., the majority of Cq values are below 30). The model is considerably more powerful than previously used procedures based on the same principles of data processing (

When the targets' abundances remain relatively high and there is little concern about shot noise, it is possible avoid making assumptions concerning

The models described here might be ranked according to their increasing power, manifested as narrowing credible intervals, in the order naïve – informed – ”classic”/fixed. It is encouraging that, at least for the two qRT-PCR datasets that we examined here, all models gave very similar point-estimates of gene expression changes when applied to the same data (

From a practical qRT-PCR standpoint, arguably the most attractive feature of the full Bayesian methodology is its robustness with respect to the number and even the very presence of control genes among the analyzed targets. Even when control genes are not specified as priors (naïve model), the model still can successfully discount the variation due to different amounts of template across samples (

Even when naïve model analysis is intended, it would be prudent to keep one or two presumably stable genes among analyzed targets: such genes could serve as indicators that the model performs reasonably, in addition to diagnostic plots (

Even though our fixed model makes assumptions that are hardly ever realistic in analysis of gene expression

The MCMC.qpcr package accompanying this paper (

The qPCR analysis of over-extended dilution series (

(TAR.GZ)

(PDF)

We are grateful to Joshua Beckham (UT Austin) for providing the raw data for benchmarking the new method against delta-delta Ct procedure. We are also indebted to the R linear mixed model discussion community (