**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:

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:

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):

^{n} Y_{0} exp (Y_{0}))

where Y and Y_{0} 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:

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 we^{w} = x. Fritsch, F. N., Shafer, R. E., and Crowley, W. P.; Comm.
ACM **16**: 123-124 (1973)