Abstract
BackgroundParalog reduction, the loss of duplicate genes after whole genome duplication (WGD) is a pervasive process. Whether this loss proceeds gene by gene or through deletion of multi-gene DNA segments is controversial, as is the question of fractionation bias, namely whether one homeologous chromosome is more vulnerable to gene deletion than the other.ResultsAs a null hypothesis, we first assume deletion events, on either homeolog, excise a geometrically distributed number of genes with unknown mean μ, and a number r of these events overlap to produce deleted runs of length l. There is a fractionation bias 0 ≤ ϕ ≤ 1 for deletions to fall on one homeolog rather than the other. The parameter r is a random variable with distribution π(·). We simulate the distribution of run lengths l, as well as the underlying π(·), as a function of μ, ϕ and θ, the proportion of remaining genes in duplicate form. We show how sampling l allows us to estimate μ and ϕ. The main part of this work is the derivation of a deterministic recurrence to calculate each π(r) as a function of μ, ϕ and θ.ConclusionsThe recurrence for π provides a deeper mathematical understanding of fractionation process than simulations. The parameters μ and ϕ can be estimated based on run lengths of single-copy regions.
Highlights
Paralog reduction, the loss of duplicate genes after whole genome duplication (WGD) is a pervasive process
We can see that the number of deletion events contributing to a run is somewhat dependent on μ when half of the the sequence has been deleted, but is strongly dependent when 90% has been deleted
We have developed a model for the fractionation process based on deletion events excising a geometricallydistributed number of contiguous paralogs from either one of a pair of homeologous chromosomes
Summary
Simulations to determine π We carried out simulations on an interval of Z of length 100,000. The probability pCii that a deletion event, on either chromosome, does not touch the run of deletions on the left, does touch or overlap the run of deletions on the right entirely on the same chromosome (homeolog), but does not extend over the entire run of undeleted terms beyond that is, for i Î {1, 2}:. Events of type Diif, with i ≠ f, turn two deleted runs with r and s events, respectively, with the latter containing terms from both chromosomes, into a single run with r + s + 1 events. The probability pEiff that a deletion event touches the run of deletions on the left of the run of undeleted terms and touches or overlaps the run of deletions on the right, all on the same chromosome, but does not extend over the entire run of undeleted terms beyond that is: pEiff φiφf τ. Extensive testing of various cutoff values has indicated such errors to be negligible in our implementation
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