The Annals of Probability

Immortal branching Markov processes: Averaging properties and PCR applications

Didier Piau

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The immortal branching Markov process (iBMP) is a modification of the usual branching model, in which each particle of generation n is counted, in addition to its offspring, as a member of generation $n+1$, its state being unchanged. When the number of offspring is Bernoulli, iBMP accounts, for instance, for the variability of the biological sequences that are produced by polymerase chain reactions (PCRs). This variability is due to the mutations and to the incomplete replications that affect the PCR. Estimators of PCR mutation rate and efficiency have been proposed that are based, in particular, on the mean empirical law $\eta_n$ of the mutations of a sequence. Unfortunately, $\eta_n$ is not analytically tractable. However, the infinite-population limit $\eta^*_n$ of $\eta_n$ is easily characterized in the two following, biologically relevant, cases. The Markovian kernel describes a homogeneous random walk, either on the integers or on some finite Cartesian product of a finite set. In the PCR context, this corresponds to infinite or finite targets, respectively. In this paper, we provide bounds of the discrepancy between $\eta_n$ and $\eta^*_n$ in these two cases. As a consequence, iBMP exhibits a strong averaging effect, even for surprisingly small starting populations. The bounds are explicit functions of the offspring law, the Markovian kernel, the number of steps n, the size of the initial population and, in the finite-target case, the size of the target. They concern every moment and, what might be less expected, the histogram itself. In the finite-target case, some of the bounds undergo a phase transition at an explicit value of the mutation rate per site and per cycle. We use precise estimates of the harmonic means of classical nondecreasing branching processes, whose proofs are included in the Appendix.

Article information

Ann. Probab., Volume 32, Number 1A (2004), 337-364.

First available in Project Euclid: 4 March 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D20: Protein sequences, DNA sequences
Secondary: 60K37: Processes in random environments 60J85: Applications of branching processes [See also 92Dxx]

Markov branching process Galton--Watson process mean field approximation polymerase chain reaction (PCR) error-prone PCR mutation rate DNA amplification estimation


Piau, Didier. Immortal branching Markov processes: Averaging properties and PCR applications. Ann. Probab. 32 (2004), no. 1A, 337--364. doi:10.1214/aop/1078415838.

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  • Athreya, K. B. (1994). Large deviation rates for branching processes I. Single type case. Ann. Appl. Probab. 4 779--790.
  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • Biggins, J. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25--37.
  • Brunnert, M., Müller, O. and Urfer, W. (2000). Genetical and statistical aspects of polymerase chain reactions. Technical Report 6/2000, Univ. Dortmund.
  • Grey, D. (1980). A new look at convergence of branching processes. Ann. Probab. 8 377--380.
  • Hayashi, K. (1990). Mutations induced during the polymerase chain reaction. Technique 2 216--217.
  • Heyde, C. C. and Brown, B. M. (1971). An invariance principle and some convergence rate results for branching processes. Z. Wahrsch. Verw. Gebiete 20 271--278.
  • Jacobs, G., Tscholl, E., Sek, A., Pfreundschuh, M., Daus, H. and Trümper, L. (1999). Enrichment polymerase chain reaction for the detection of Ki-ras mutations: Relevance of Taq polymerase error rate, initial DNA copy number, and reaction conditions on the emergence of false-positive mutant bands. J. Cancer Res. Clin. Oncol. 125 395--401.
  • Joffe, A. (1993). A new martingale in branching random walk. Ann. Appl. Probab. 3 1145--1150.
  • Krawczak, M., Reiss, J., Schmidtke, J. and Rosler, U. (1989). Polymerase chain reaction: Replication errors and reliability of gene diagnosis. Nucleic Acids Res. 17 2197--2201.
  • Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (K. B. Athreya and P. Jagers, eds.) 217--221. Springer, New York.
  • Maruyama, I. N. (1990). Estimation of errors in the polymerase chain reaction. Technique 2 302--303.
  • Ney, P. E. and Vidyashankar, A. N. (2001). Harmonic moments and large deviation rates for supercritical branching processes. Technical Report Stat2001-5, Univ. Georgia.
  • Pakes, A. G. (1975). Non-parametric estimation in the Galton--Watson process. Math. Biosci. 26 1--18.
  • Piau, D. (2001). Processus de branchement en champ moyen et réaction PCR. Adv. in Appl. Probab. 33 391--403.
  • Piau, D. (2002). Mutation--replication statistics for polymerase chain reactions. J. Comput. Biol. 9 831--847.
  • Reiss, J., Krawczak, M., Schlösser, M., Wagner, M. and Cooper, D. N. (1990). The effect of replication errors on the mismatch analysis of PCR-amplified DNA. Nucleic Acids Res. 18 973--978.
  • Saiki, R. K., Gelfand, D. H., Stoffel, S., Scharf, S. J., Higuchi, R., Horn, G. T., Mullis, K. B. and Erlich, H. A. (1988). Primer-directed enzymatic amplification of DNA with a thermostable DNA polymerase. Science 239 487--491.
  • Sun, F. (1995). The polymerase chain reaction and branching processes. J. Comput. Biol. 2 63--86.
  • Wang, D., Zhao, C., Cheng, R. and Sun, F. (2000). Estimation of the mutation rate during error-prone polymerase chain reaction. J. Comput. Biol. 7 143--158.
  • Weiss, G. and von Haeseler, A. (1995). Modeling the polymerase chain reaction. J. Comput. Biol. 2 49--61.
  • Weiss, G. and von Haeseler, A. (1997). A coalescent approach to the polymerase chain reaction. Nucleic Acids Res. 25 3082--3087.