Modelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency

We propose a stochastic modelling of the PCR amplification process by a size-dependent branching process starting as a supercritical Bienaymé-Galton-Watson transient phase and then having a saturation near-critical size-dependent phase. This model allows us to estimate the probability of replication of a DNA molecule at each cycle of a single PCR trajectory with a very good accuracy.

Unaccounted uncertainty from qPCR efficiency estimates entails uncontrolled false positive rates

Background

Accurate adjustment for the amplification efficiency (AE) is an important part of real-time quantitative polymerase chain reaction (qPCR) experiments. The most commonly used correction strategy is to estimate the AE by dilution experiments and use this as a plug-in when efficiency correcting the ΔΔC_q. Currently, it is recommended to determine the AE with high precision as this plug-in approach does not account for the AE uncertainty, implicitly assuming an infinitely precise AE estimate. Determining the AE with such precision, however, requires tedious laboratory work and vast amounts of biological material. Violation of the assumption leads to overly optimistic standard errors of the ΔΔC_q, confidence intervals, and p-values which ultimately increase the type I error rate beyond the expected significance level. As qPCR is often used for validation it should be a high priority to account for the uncertainty of the AE estimate and thereby properly bounding the type I error rate and achieve the desired significance level.

Results

We suggest and benchmark different methods to obtain the standard error of the efficiency adjusted ΔΔC_q using the statistical delta method, Monte Carlo integration, or bootstrapping. Our suggested methods are founded in a linear mixed effects model (LMM) framework, but the problem and ideas apply in all qPCR experiments. The methods and impact of the AE uncertainty are illustrated in three qPCR applications and a simulation study. In addition, we validate findings suggesting that MGST1 is differentially expressed between high and low abundance culture initiating cells in multiple myeloma and that microRNA-127 is differentially expressed between testicular and nodal lymphomas.

Conclusions

We conclude, that the commonly used efficiency corrected quantities disregard the uncertainty of the AE, which can drastically impact the standard error and lead to increased false positive rates. Our suggestions show that it is possible to easily perform statistical inference of ΔΔC_q, whilst properly accounting for the AE uncertainty and better controlling the false positive rate.

Computational analysis of stochastic heterogeneity in PCR amplification efficiency revealed by single molecule barcoding

The polymerase chain reaction (PCR) is one of the most widely used techniques in molecular biology. In combination with High Throughput Sequencing (HTS), PCR is widely used to quantify transcript abundance for RNA-seq, and in the context of analysis of T and B cell receptor repertoires. In this study, we combine DNA barcoding with HTS to quantify PCR output from individual target molecules. We develop computational tools that simulate both the PCR branching process itself, and the subsequent subsampling which typically occurs during HTS sequencing. We explore the influence of different types of heterogeneity on sequencing output, and compare them to experimental results where the efficiency of amplification is measured by barcodes uniquely identifying each molecule of starting template. Our results demonstrate that the PCR process introduces substantial amplification heterogeneity, independent of primer sequence and bulk experimental conditions. This heterogeneity can be attributed both to inherited differences between different template DNA molecules, and the inherent stochasticity of the PCR process. The results demonstrate that PCR heterogeneity arises even when reaction and substrate conditions are kept as constant as possible, and therefore single molecule barcoding is essential in order to derive reproducible quantitative results from any protocol combining PCR with HTS.

Random variation and concentration effects in PCR

Even though the efficiency of the polymerase chain reaction (PCR) reaction decreases, analyses are made in terms of Galton-Watson processes, or simple deterministic models with constant replication probability (efficiency). Recently, Schnell and Mendoza have suggested that the form of the efficiency, can be derived from enzyme kinetics. This results in the sequence of molecules numbers forming a stochastic process with the properties of a branching process with population size dependence, which is supercritical, but has a mean reproduction number that approaches one. Such processes display ultimate linear growth, after an initial exponential phase, as is the case in PCR. It is also shown that the resulting stochastic process for a large Michaelis-Menten constant behaves like the deterministic sequence x(n) arising by iterations of the function f(x)=x+x/(1+x).