My very old 'analytical' USS evolution model

I found my old USS evolution model in the files with my 1999 NIH grant proposal. It's not a computer simulation of evolution but analysis of equations describing an equilibrium. It uses only very simple algebra, so calling it an 'analytical' model is probably giving it more credit than it deserves. The introductory text gives an excellent description of the background, so here it is:

This model starts with the following assumptions:

1. H. influenzae cells have a preexisting DNA uptake system that preferentially transports fragments containing a USS0 (the 9bp core: 5'AAGTGCGGT). Fragments with imperfect (USS1) sites are not favoured.
2. Fragments of H. influenzae DNA are frequently brought into cells by this system.
3. Once inside the cell these fragments recombine with and replace homologous regions of the chromosome.
4. The DNA in the cells' environment comes from lysed cells ('donors') having the same distribution of USS0 and USS1 sites as the cells that are taking up DNA (the 'recipients').
5. Random mutation acts on both USS0 and USS1 sites to destroy and create new USS.

How these assumptions cause USS0 to accumulate:
Because the DNA uptake mechanism is biased in favour of fragments containing USS0 sites, any donor DNA fragment containing a new mutation that has created a USS0 will be taken up more efficiently than the wildtype version of the fragment. Similarly, donor fragments containing new mutations that have eliminated a USS0 will be taken up less efficiently than their wildtype counterparts. Consequently the DNA fragments within cells will be enriched for USSs relative to the external DNA, and recombination between these fragments and the resident chromosome will increase the number of genomic USS0s more frequently than it will decrease it. The bias will similarly affect the fate of new mutations arising within the recipient cell; mutations removing a USS0 will often be replaced by donor fragments carrying the wildtype USS0, whereas mutations creating new USS0s in the recipient will less frequently be replaced. This biased gene conversion can thus both compensate for mutational loss and amplify mutational gain of USS0s, and will cause them to be maintained at a higher equilibrium number than would be expected for a random sequence of the same base composition.

How the model works:
This model addresses processes acting within a single genome. The model uses the observed equilibrium numbers of perfect and imperfect USS sites to derive an equation relating transformation frequency to the bias of the DNA uptake system. This equation tells us the values that the transformation frequency and uptake bias parameters would have to take in order to be responsible for the maintenance of perfect and imperfect USS at the high equilibrium ratio that we observe.

The model then defines its terms ('variables'? 'parameters'?). It assumes that the frequencies of USS0 and USS1 observed in the real H. influenzae genome represent an equilibrium between forces that create USS0s and forces that convert them into USS1s. It then derives an equation relating the bias of the DNA uptake machinery (enrichment of USS0 over USS1) to the transformation frequency (the probability that a USS site will be replaced a fragment brought in by the uptake machinery in each generation). Conveniently, the mutation rate drops out of the equation.

Again quoting from what I wrote ten years ago:

What the final equation means: This equation tells us how frequent transformation must be (T), given a specified bias B of the DNA uptake system in favor of the USS, in order to fully account for the observed numbers of USS0 and USS1 sites in the Rd genome. The range of values is given below.












This is a pleasingly tidy result. It makes good biological sense, and the values are not unreasonable. Estimates of the actual bias vary, no doubt partly because they have been determined using USSs with different flanking sequences, but are usually between 10 and 100. We have no good estimate of actual transformation frequencies for H. influenzae in its natural environment, but if cells grow slowly, and frequently take up DNA as food then an average of 1% transformation per generation seems plausible, and even 10% not impossible.

I'm not very sure that this work is sound. The mutation rate dropping out is a bit suspicious, and I'm not sure how to interpret the transformation frequency when there are no other terms depending on generations (such as mutation rate).