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In biology, the concept of genome doubling is usually expressed as tetraploidiza-tion or autotetraploidization, and the both the doubled genome and its doubling descendant are called tetraploid, even though, generally, the descendants soon undergo a process called (re-)diploidization and function as normal diploids, still carrying a full complement of duplicate markers that evolve independently of each other. Though unambiguous in biological context, implicit in this terminology are many assumptions that are not pertinent to our study. In the yeast data we study here, for example, Saccharomyces cerevisiae exists during most of its life cycle as a haploid, only sometimes as a diploid, while Candida glabrata exists uniquely as a haploid.

In our considerations, the key aspect of genome doubling is the global duplication of chromosomes and markers at the moment of doubling. Ploidy is not relevant in that in any organism that reproduces by meiosis or even by mitosis, the order of the markers on any of the haploid components (e.g., maternal versus paternal chromosomes) is essentially identical. There may be different alleles, or other local differences, but the order is basically invariant. Ongoing variation and evolution at the level of chromosomal structure in an individual or species are considered negligible in comparison with the major rearrangements that exist between genomes separated on an evolutionary time scale.

Although this paper is about polyploidy, then, we will rely largely on terminology independent of ploidy: genome doubling, doubling descendant, unduplicated genomes, genome halving.

The marker complement of a genome may also double by another process, allotetraploidization, or fusion of two different genomes, a kind of hybridization that is probably at least as important biologically as the doubling of a single genome we focus on in this paper. We do not consider this process here, for three reasons. One is our interest in exploring the essential difficulty in the mathematics of doubling, namely the complete ambiguity as to which set of duplicate markers were together in each of the two copies of the original genome. For hybrid doubled genomes, DNA sequence evidence from related but unduplicated genomes can generally resolve this ambiguity [5]. Second, hybrids require reticulate phylogenies which, though of interest themselves, constitute an unwanted layer of difficulty that we wish to keep separate at this stage. Finally, some of the most interesting doubling events (outside the plant kingdom), such as the ones hypothesized in the "2R" model of early vertebrate evolution or the well-established doubling in the ancestor of budding yeasts Saccharomyces cere-visiae and Candida glabrata, which furnish the empirical example for this paper, are usually treated as doubling of a single genome.

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