Statistical Inference for Quantitative Polymerase Chain Reaction Using a Hidden Markov Model: A Bayesian Approach

Quantitative Polymerase Chain Reaction (Q-PCR) aims at determining the initial quantity of specific nucleic acids from the observation of the number of amplified DNA molecules. The most widely used technology to monitor the number of DNA molecules as they replicate is based on fluorescence chemistry. Considering this measurement technique, the observation of DNA amplification by PCR contains intrinsically two kinds of variability. On the one hand, the number of replicated DNA molecules is random, and on the other hand, the measurement of the fluorescence emitted by the DNA molecules is collected with some random error. Relying on a stochastic model of these two types of variability, we aim at providing estimators of the parameters arising in the proposed model, and, more specifically, of the initial amount of molecules. The theory of branching processes is classically used to model the evolution of the number of DNA molecules at each replication cycle. The model is a binary splitting Galton-Watson branching process. Its unknown parameters are the initial number of DNA molecules and the reaction efficiency of PCR, which is defined as the probability of replication of a DNA molecule. The number of DNA molecules is indirectly observed through noisy fluorescence measurements resulting in a so-called Hidden Markov Model. We aim at inference of the parameters of the underlying branching process, and the parameters of the noise from the fluorescence measurements in a Bayesian framework. Using simulations and experimental data, we investigate the performance of the Bayesian estimators obtained by Markov Chain Monte Carlo methods.