A Quantitative Model of PCR

The essence of the model is defined by equation (33) in [1], where the efficiency of amplification (at any time during the amplification process) is stated to be:

(33)e = 1 -[ET] / ([E] + [ET])

Here [E] is the concentration of free enzyme (polymerase) and [ET] is the concentration of enzyme-template complex. Since the term ([E] + [ET]) is the total enzyme concentration, the expression [ET] / ([E] + [ET]) is the fraction of enzyme in a complex, which is the efficiency of enzyme use. Thus (33) is 1 - (efficiency of enzyme use). However, what we are really interested in is the efficiency of substrate (template) use. This would be:

e = [T] / ([T] + [ET])

While e of Schnell and Mendoza doesn't make sense to me, it is at least correlated with the efficiency of substrate use. In early cycles of PCR, enzyme is in excess, thus its efficiency is much less than 1, thus the Schnell and Mendoza efficiency is high. In later cycles, most enzyme is bound to substrate, thus its efficiency is close to 1, thus the Schnell and Mendoza efficiency is low.

Schnell and Mendoza [2] gave an expression for the reduced product concentration at amplification cycle n using the Omega function, W(x):

(14)Y = W ( 2n Y0 exp (Y0))

where Y and Y0 are the reduced concentrations of product at cycle n, and at the start of the PCR respectively. Reduced concentrations are raw concentrations divided by Km, the dissociation constant for enzyme-substrate complex, and the only adjustable parameter of the model.

W(x) is the Omega function [3], which is defined by the equation:

W(x) exp (W(x)) = x

1. Enzymological considerations for a theoretical description of the Quantitative Competitive Polymerase Chain Reaction (QC-PCR). S. Schnell and C. Mendoza; J theor. Biol. 184: 433-440 (1997)

2. Theoretical description of the polymerase chain reaction. S. Schnell and C. Mendoza; J. theor. Biol. 188: 313-318 (1997)

3. Algorithm 443: Solution to the transcendental equatiohn wew = x. Fritsch, F. N., Shafer, R. E., and Crowley, W. P.; Comm. ACM 16: 123-124 (1973)