A mathematical analysis of in vitro molecular selection-amplification

We construct a mathematical model for in vitro molecular selection with amplification. Using DNA-protein binding as the illustrative example, we obtain an expression for the probability that a randomly selected molecule from the final in vitro selection products is a molecule with the highest binding affinity. Experiments of this type have been reported for several examples of DNA binding proteins. Our study requires a model of the DNA-protein binding constant between DNA molecules and the target protein. The relationship between binding constants and selection probabilities is presented under simplifying but reasonable assumptions. From our analysis, we find that for successful in vitro selection experiments there should be a certain relationship between the number of polymerase chain reaction cycles and the concentration of free protein. The results obtained should be widely applicable to a variety of selection-amplification procedures.