Christine Jacob

Theoretical uncertainty of measurements using quantitative polymerase chain reaction

Current quantitative polymerase chain reaction (PCR) protocols are only indicative of the quantity of a target sequence relative to a standard, because no means of estimating the amplification rate is yet available. The variability of PCR performed on isolated cells has already been reported by several authors, but it could not be extensively studied, because of lack of a system for doing kinetic data acquisition and of statistical methods suitable for analyzing this type of data. We used the branching process theory to simulate and analyze quantitative kinetic PCR data. We computed the probability distribution of the offspring of a single molecule. We demonstrated that the rate of amplication has a severe influence on the shape of this distribution. For high values of the amplification rate, the distribution has several maxima of probability. A single amplification trajectory is used to estimate the initial copy number of the target sequence as well as its confidence interval, provided that the amplification is done over more than 20 cycles. The consequence of possible molecular fluctuations in the early stage of amplification is that small copy numbers result in relatively larger intervals than large initial copy numbers. The confidence interval amplitude is the theoretical uncertainty of measurements using quantitative PCR. We expect these results to be applicable to the data produced by the next generation of thermocyclers for quantitative applications.

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.

Statistical Estimations of PCR Amplification Rates

Quantitative applications of the Polymerase Chain Reaction (PCR), also known as Quantitative-PCR (Q-PCR) are intended either to determine the number of copies of a given nucleic acid sequence, or more generally, to determine the relative abundance of two sequences. Current methods to determine exact numbers of molecules overcome the difficulty of determining the amplification rate by assuming identical amplification rates for a target DNA sequence and a standard of known quantity introduced into the experiment design, so that only the ratio of amplified products need be determined. Violations of the hypothesis of identical amplification rates for two sequences will result in a systematic bias in the experimental results that underestimates or overestimates the initial copy numbers. Acquisition of kinetic PCR data was pioneered by Higuchi et al. (1992, 1993), and commercial instruments have been available since early 1996. Kinetic data provide a new way to determine the amplification rate, and we can foresee that their availability will rekindle interest in the algorithms used to compute the initial quantities of DNA sequences. Analysis of kinetic PCR patterns will soon make its way into the family of recipes that have been in use for some years in this field. This chapter provides evidence that a statistical analysis of the amplification rate is critical to ensuring a reliable estimate of the initial copy number.

PREDICTION OF SEQUENTIALLY OPTIMAL RNA SECONDARY STRUCTURES

A rigorous mathematical modeling of the RNA sequential folding process during transcription is proposed. It is based, at each transcription step, on a homogeneous markovian jump process, the state space of which is the set of structures constructible on the part of the RNA already transcribed. A theoretical formula permitting the computation of the structures probabilities at the end of the RNA transcription is derived. Successive approximations, aimed at reducing the size of the state space, permit the design of a prediction algorithm. The algorithm is tested on some structural RNAs (tRNA, 5S, 16S, hammerhead, ...), results are discussed and possible improvements are proposed.

Stochastic theories of the activated complex and the activated collision: The RNA example

We propose a common rigorous foundation to the classical collision theory and that of the classical activated complex. Based on the notion of activated complex, this foundation relies on a stochastic approach showing up the different influence factors of a chemical reaction. The thermodynamic formulation is obtained here by assuming the exponential statistical distribution within each stable chemical state. The general model we obtain yields two stochastic formulations called the stochastic transition state theory (denoted STST) and the stochastic activated collision theory (denoted SACT) respectively, depending on whether the rate of the reaction is of the same scale as, either the rate of passing over the potential energy barrier, for the STST, or the rate of reaching the activated complex state (which is a generalization of the collision rate), for the SACT. The modeling is first done in the case of a closed small system undergoing stochastic changes of chemical state and then, by extension, in the fra...

Branching Processes: Their Role in Epidemiology

Branching processes are stochastic individual-based processes leading consequently to a bottom-up approach. In addition, since the state variables are random integer variables (representing population sizes), the extinction occurs at random finite time on the extinction set, thus leading to fine and realistic predictions. Starting from the simplest and well-known single-type Bienaymé-Galton-Watson branching process that was used by several authors for approximating the beginning of an epidemic, we then present a general branching model with age and population dependent individual transitions. However contrary to the classical Bienaymé-Galton-Watson or asymptotically Bienaymé-Galton-Watson setting, where the asymptotic behavior of the process, as time tends to infinity, is well understood, the asymptotic behavior of this general process is a new question. Here we give some solutions for dealing with this problem depending on whether the initial population size is large or small, and whether the disease is rare or non-rare when the initial population size is large.

Bayesian Estimation for Quantification by Real-time Polymerase Chain Reaction Under a Branching Process Model of the DNA Molecules Amplification Process

The aim of Quantitative Polymerase Chain Reaction is to determine the initial amount X 0 of specific nucleic acids from an observed trajectory of the amplification process, the amplification being achieved through successive replication cycles. This process depends on the efficiency {p n } n of replication of the DNA molecules, p n being the probability that a molecule will duplicate at replication cycle n. Assuming p n = p for all n, Bayesian estimators of the unknown parameter θ = (p, X 0) are constructed by Markov Chain Monte Carlo methods under a Bienaymé-Galton-Watson branching model of the amplification process. The Bayesian approach takes into account some prior information on the parameter. Relying on simulated data, the proposed Bayesian estimators and their credibility sets are shown to be quite accurate.

Probability distribution of the chemical states of a closed system and thermodynamic law of mass action from kinetics: The RNA example

We show some thermodynamic results concerning chemical kinetics, either from statistical physics, under the strong assumption (EXS) of the statistical exponential distribution of the physical states, or from kinetics. In that frame, the reaction rates are modeled according to the stochastic activation collision theory, under the weaker assumption (EXW) of the statistical exponential distribution not on the whole space but only within each structure; moreover a symmetry condition (SY) concerning the local properties of the rate constants is assumed. The Boltzmann distribution and an extension of the more general result of the thermodynamic law of mass action are proved. Moreover, we show that the assumptions (EXS) and (EXW) are equivalent, under (SY). At last, we deal with the stochastic example of the RNA structure for which we calculate the temporal probabilities of the structures by means of a jump process.

Saturation Effects in Population Dynamics: Use Branching Processes or Dynamical Systems?

This chapter deals with the behavior of a branching population undergoing saturation effects when it becomes too large. We study in particular the limits of the prediction given in the setting of the deterministic dynamical system related to the stochastic branching process modeling the evolution of the population. We also generalize the usual Markovian branching processes of order one to size-dependent branching processes that may have a longer memory and give conditions leading to an almost sure extinction of the process while the dynamical system is persistent. The notion of reproductive rate is explained and generalized. We give some examples, in particular the amplification process in the polymerase chain reaction (PCR).

Risk analysis of Transmissible Spongiform Encephalopathies in animals: state-of-the-art

The Bovine Spongiform Encephalopathy (BSE) crisis of the last two decades has shown that proper interaction of risk assessment, risk management and risk communication is essential. Mathematical models and risk assessments have been used as a basis for BSE risk management options and much of the legislation regarding the control and eradication of BSE. Much uncertainty regarding important input parameters remains a major constraint in risk assessment. Uncertainty is one of the most critical and most difficult aspects of communication of risks about Transmissible Spongiform Encephalopathies (TSEs). Nevertheless, the decline in the BSE epidemic in the UK and most European countries demonstrates that management has been, for the most part, successful. Literature pertaining to the three inter-related facets of risk analysis: risk assessment, risk management and risk communication of TSEs of animal origin was reviewed and used to describe the state-of-the-art of risk analysis for TSEs.