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From 2012.igem.org

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Modeling the evolution of a population during MAGE

Figure 1. Schematic of model.

The distribution of specific mutations in MAGE is a stochastic process that we model as as a function of each oligo's predicted efficiency of allelic replacement (which can be estimated in E. coli as discussed in "Programming cells by multiplex genome engineering and accelerated evolution," Wang & Isaacs et al 2009), assuming that each mutation event is binary and exclusive. Then a population after c cycles is a weighted sum of n Bernoulli trials, each zero if the oligo does not mutate its target 'i' and otherwise equal to the number r of mutations it induces. Given efficiencies of allelic replacement p, this probability mass function becomes:

In doing this, we have derived a more general form of the binomial distribution. Computing this PMF involves solving the subset sum problem, but we optimized our algorithm to avoid slowdowns by using a recursive formula (Wadyicki, Shah et al. 1973) for the occasional, simpler case when all oligos carry the same number of mutations, and in other cases a branched, dynammic programming algorithm (Horowitz and Sahni 1974).

Survey for off-target binding sites

Not all MAGE-induced mutations will be at the intended sites; to identify likely unintended mutations, we scripted a search of the genome using BLAST+ to find subsequences with four base pairs or more matching oligos in the MAGE oligo pool, and estimates the change in Gibbs energy likely upon hybridization at each such off-target pairing, using the UNAFold software package.

Figures generated by our MAGE model

Both of these scripts will be bundled into an cloud-based tool for genomic engineering (unpublished work).