The polymerase chain reaction: A stochastic model, methods of quantification, and applications to HIV
Abstract
This thesis is concerned with the development of the polymerase chain reaction (PCR) as an accurate and reliable measure of specific DNA copy number. This development is motivated by the need to quantify the number of copies of HIV in infected cells. In particular the extent of HIV infection, in terms of proviral load, can be determined by using PCR, leading to more accurate evaluation of drug treatments. Successful completion of this development requires answering the following questions: (i) Can PCR be used to quantify low copy number of specific DNA? (ii) What mathematical and statistical models give the most consistent and reproducable measure of DNA copy number? (iii) What conditions affect assay variability? As these questions involve aspects of several disciplines, the thesis is presented in four parts: (I) the assay, (II) the mathematics, (III) the models, and finally (IV) the applications. The first section includes a complete description of the assay. As PCR mimics natural mechanisms of DNA replication, this section also includes descriptions of DNA structure and of cellular DNA replication. The second section contains the background material and presentation of new developments in the mathematics and statistics needed for the modeling of the assay. The assay is modeled as a branching process, and various aspects of the reaction dictate different types of branching processes. As a result, three types of processes are presented, classic Galton-Watson, generation-dependent, and population-size-dependent. These models lead to quantification procedures involving weighted linear regression and inverse prediction. In addition, new material is presented for the development of comparison methods and confidence intervals in this setting. The third section contains the actual modeling of the reaction through the three different types of branching processes mentioned. Complete characterizations of the distributions of the processes are derived for two of the models, from which new parametric statistical tests for the quantification of DNA, in particular of HIV, are developed. For the third model, simulations are used to explore the process and its moments. This model necessarily leads to a submodel reminiscent to that found in stochastic epidemics. The final section illustrates the application of methods developed to data from HIV-infected patients.
This paper has been withdrawn.