Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-27T03:47:11.669Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency

Part of: Applications Parametric inference Markov processes

Published online by Cambridge University Press:  01 July 2016

N. Lalam ,
C. Jacob  and
P. Jagers
N. Lalam*
Affiliation:
Institut National de la Recherche Agronomique, Jouy-en-Josas
C. Jacob*
Affiliation:
Institut National de la Recherche Agronomique, Jouy-en-Josas
P. Jagers*
Affiliation:
Chalmers University of Technology, Göteborg
*
∗ Current address: Eurandom, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: lalam@eurandom.tue.nl
∗∗ Postal address: INRA, Laboratoire de Biométrie, 78352 Jouy-en-Josas Cedex, France. Email address: cj@banian.jouy.inra.fr
∗∗∗ Chalmers University of Technology, S-412 96 Göteborg, Sweden. Email address: jagers@math.chalmers.se
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Polymerase chain reactionsize-dependent branchingkinetic modelconditional least squares

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62F12: Asymptotic properties of estimators
Secondary: 62P10: Applications to biology and medical sciences
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 602 - 615
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Dacunha-Castelle, D. and Duflo, M. (1982). Probabilités et Statistiques, Tome 1. Masson, Paris.Google Scholar
[2] Ferré, F., (ed.) (1998). Gene Quantification. Birkhäuser, Boston.CrossRefGoogle Scholar
[3] Gilliland, G., Perrin, S., Blanchard, K. and Bunn, H. F. (1990). Analysis of cytokine mRNA and DNA: detection and quantitation by competitive polymerase chain reaction. Proc. Nat. Acad. Sci. USA 87, 2725–2729.CrossRefGoogle ScholarPubMed
[4] Higuchi, R., Dollinger, G., Walsh, P. S. and Griffith, R. (1992). Simultaneous amplification and detection of specific DNA sequences. Biotechnology 10, 413–417.Google Scholar
[5] Jacob, C. and Peccoud, J. (1998). Estimation of the parameters of a branching process from migrating binomial observations. Adv. Appl. Prob. 30, 948–967.CrossRefGoogle Scholar
[6] Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
[7] Jagers, P. and Klebaner, F. C. (2003). Random variation and concentration effects in PCR. J. Theoret. Biol. 224, 299–304.CrossRefGoogle ScholarPubMed
[8] Kersting, G. (1990). Some properties of stochastic difference equations. In Stochastic Modelling in Biology, ed. Tautu, P., World Scientific, Singapore, pp. 328–339.Google Scholar
[9] Kimura, B., Kawasaki, S., Nakano, H. and Fujii, T. (2001). Rapid, quantitative PCR monitoring of growth of clostridium botulinum type E in modified-atmosphere-packaged fish. Appl. Environ. Microbiol. 67, 206–216.Google Scholar
[10] 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.Google Scholar
[11] Lalam, N. and Jacob, C. (2003). Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process. Adv. Appl. Prob. 36, 582–601.CrossRefGoogle Scholar
[12] Lalam, N. and Jacob, C. (2003). Modelling the PCR amplification process with size-dependent branching processes and estimation of the efficiency. Tech. Rep., Applied Mathematics and Informatics, INRA, Jouy-en-Josas.Google Scholar
[13] Mackay, I. M., Arden, K. E. and Nitsche, A. (2002). Real-time PCR in virology. Nucleic Acids Res. 30, 1292–1305.CrossRefGoogle ScholarPubMed
[14] Mullis, K. B. and Faloona, F. (1987). Specific synthesis of DNA in vitro via a polymerase-catalysed chain reaction. Methods Enzymol. 155, 335–350.Google Scholar
[15] Mullis, K. B., Ferré, F. and Gibbs, R. A. (1994). The Polymerase Chain Reaction. Birkhäuser, Boston.Google Scholar
[16] Nedelman, J., Heagerty, P. and Lawrence, C. (1992). Quantitative PCR: procedures and precision. Bull. Math. Biol. 54, 477–502.Google Scholar
[17] Peccoud, J. and Jacob, C. (1996). Theoretical uncertainty of measurements using quantitative polymerase chain reaction. Biophys. J. 71, 101–108.CrossRefGoogle ScholarPubMed
[18] Peccoud, J. and Jacob, C. (1998). Statistical estimations of PCR amplification rates. In Gene Quantification, ed. Ferré, F., Birkhäuser, Boston.Google Scholar
[19] Piau, D. (2001). Processus de branchement et champ moyen. Adv. Appl. Prob. 33, 391–403.CrossRefGoogle Scholar
[20] Raeymakers, L. (1995). A commentary on the practical applications of competitive PCR. Genome Res. 5, 91–94.CrossRefGoogle Scholar
[21] Saiki, R. K. et al. (1988). Primer-directed enzymatic amplification of DNA with a thermostable DNA polymerase. Science 239, 487–491.Google Scholar
[22] Schnell, S. and Mendoza, C. (1997). Enzymological considerations for a theoretical description of the quantitative competitive polymerase chain reaction. J. Theoret. Biol. 184, 433–440.CrossRefGoogle ScholarPubMed
[23] Stolovitzky, G. and Cecchi, G. (1996). Efficiency of DNA replication in the polymerase chain reaction. Biophysics 93, 12947–12952.Google Scholar
[24] Sun, G. (1995). The PCR and branching processes. J. Comput. Biol. 2, 63–86.Google Scholar
[25] Vandenbroucke, I. I., Vandesempele, J., De Paepe, A. and Messiaen, L. (2001). Quantification of splice variants using real-time PCR. Nucleic Acids Res. 29, e68.CrossRefGoogle ScholarPubMed
[26] Weiss, G. and Von Haeseler, A. (1995). Modeling the PCR. J. Comput. Biol. 2, 49–61.CrossRefGoogle Scholar