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Multitype PCR branching processes

P. Chigansky, F. Klebaner, M. Mrksa, S. Sagitov

Abstract

To model amplification Polymerase Chain Reaction (PCR) techniques targeting DNA sequences of several types, we introduce a multitype PCR branching process as a generalized version of the Michaelis-Menten-based branching process model introduced in Jagers-Klebaner, 2003. We establish two limit theorems extending the results of Chigansky-Jagers-Klebaner, 2018 to the multitype case.

Multitype PCR branching processes

Abstract

To model amplification Polymerase Chain Reaction (PCR) techniques targeting DNA sequences of several types, we introduce a multitype PCR branching process as a generalized version of the Michaelis-Menten-based branching process model introduced in Jagers-Klebaner, 2003. We establish two limit theorems extending the results of Chigansky-Jagers-Klebaner, 2018 to the multitype case.

Paper Structure

This paper contains 11 sections, 10 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Multitype PCR branching process
  3. Main results
  4. Illustrations of Theorems \ref{['mainthm']} and \ref{['thm']}
  5. Proof of Theorem \ref{['mainthm']}
  6. Proof of Theorem \ref{['thm']}
  7. Properties of $\bar{F}$
  8. Invertibility.
  9. Regularity and growth
  10. Iterates
  11. Scaling limits

Key Result

Theorem 3.1

Consider the MPCR branching process $\bar{Z}(n)$. Given that parameters viq and the initial vector $\bar{Z}(0)$ are fixed, assume that parameter $K\in\mathbb (0,\infty)$ varies in such a way that Then putting $\kappa=\log_{b_1}K$, we have where $W_1,\ldots,W_d$ are independent random values defined by branchlim and $W_0=W_1+\ldots+W_{d_0}.$

Figures (3)

  • Figure 2: An illustration of Theorem \ref{['mainthm']} in the case $d=2$ with $b_1=1.9$, $b_2=1.2$ and $\kappa=25$ based on 200 simulations.
  • Figure 3: The joint distribution of $H(W_1)$ and $W_2G_2(W_1)$ in the case $d=2$ with $b_1=1.9$, $b_2=1.2$.
  • Figure 4: An illustration of Theorem \ref{['thm']} in the case $d=5$, $b_1=b_2=b_3=b_4=b_5=1.9$, and $K=b_1^{-29}$. Here the directly simulated values $Z_i(26)/K$ are matched against the limiting values $V_i=W_iW^{-1}f^{(-3)}(H(W))$.

Theorems & Definitions (19)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 5.1
  • proof
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • ...and 9 more