Efficiency of DNA replication in the polymerase chain reaction

Abstract

A detailed quantitative kinetic model for the polymerase chain reaction (PCR) is developed, which allows us to predict the probability of replication of a DNA molecule in terms of the physical parameters involved in the system. The important issue of the determination of the number of PCR cycles during which this probability can be considered to be a constant is solved within the framework of the model. New phenomena of multimodality and scaling behavior in the distribution of the number of molecules after a given number of PCR cycles are presented. The relevance of the model for quantitative PCR is discussed, and a novel quantitative PCR technique is proposed.

Figure 1

Probability of replication p(t) as a function of time t, for different template lengths N (in number of nucleotides without including the primers) arising from a numerical simulation of Eqs. 1, with parameters: κ₁ = 10⁹ M⁻¹·s⁻¹, κ₂ = 10⁻² s⁻¹, κ₃ = 10⁷ M⁻¹·s⁻¹, κ₄ = 10⁻³ s⁻¹, κ₅ = 10⁷ M⁻¹·s⁻¹, κ₆ = 15 s⁻¹, κ₇ = 10⁹ M⁻¹·s⁻¹, κ₈ = 10⁻¹ s⁻¹, [pr](0) = 10⁻⁶ M, [n](0) = 10⁻⁵ M, [q](0) = 10⁻⁶ M, [ss](0) = 10⁻¹¹ M. The square and arrowheads indicate, respectively, the mean rise-time and rise-time width as predicted by the simplified model of Eqs. 3.

Figure 2

Efficiency p_k as a function of the cycle number k, for different polymerization times. The length of the template is N = 45; the other parameters are as in Fig. 1.

Figure 3

(a) The pdf of the number of molecules after n = 10 cycles and M₀ = 1 of a branching process with constant efficiency p, in log–log scale. Notice the multimodality for p = 0.9, and the power law regimes (straight lines). (b) Same as in a for M₀ = 50 (linear scale). The multimodality has disappeared even for p = 0.99.

Figure 4

(a) The invariant manifold that crosses z = 1 tangent to the unit disk, of the map z_n+1 = (1 − p)z_n + pz_n² (with z in the complex plane), for p = 0.9. It is parametrized by θ_∞,1(ω). In the same plot the curve parametrized by θ_15,1(ω) is shown; it cannot be resolved from the invariant manifold. (b) Absolute value of θ_15,1(ω) (solid line) and the predicted power law (broken line).