The ratio of dNTP to ddNTP used is critical for the success of Sanger sequencing since it determines the distribution of DNA fragment lengths produced. Despite this, I have been unable to find a detailed mathematical treatment of how this ratio affects the results.

**Thus, in the article I will explore the mathematical relationship between the dNTP to ddNTP ratio and the distribution of fragment lengths, and develop an equation to predict the optimum ratio under specified conditions.**The accuracy of this model depends on the validity of the assumptions which I have made to develop it:

- The reaction goes fully to completion, all dNTP and ddNTP are used up, all strands being synthesized are terminated.
- The kinetics for the incorporation of a dNTP or a ddNTP are identical, both processes have identical rate constants.
- The DNA sample contains roughly the same amounts of guanosine, cytidine, adenosine, and thymidine, thus each nucleotide type will have the same optimum ratio of dNTP to ddNTP.
- The DNA sample being sequenced is long such that the fraction of fragments produced spanning the entire sequence is small.
- The separation from gel electrophoresis is ideal.
- No complicating secondary structure of DNA exists during sequencing.

We can define p as the fraction of nucleotides triphosphates derivatives which are normal dNTP:

Note that f(n) is always maximum at n=1 (because p<1) and each term is successively smaller than the last. Thus for any distribution the number of detectable fragments will be equal to the length of the longest detectable fragment (n*). This length has the lowest frequency which is still above or equal to the detection limit: