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Abstract
STAGE: EARLY DRAFT
DOCUMENT TYPE: Definition Document
OBJECTIVES
Formal mathematical definition to be used in stochastic process model under development.
Precise mathematical definition for clarification of certain concepts discussed in the population genetics literature.
Gametic genealogy
A gametic genealogy is a convenient mathematical formalism of the genealogy of a population from the perspective of gametes. Mathematically, it is a quadruple with components
, the set of underlying gametes,
, the set of zygotes formed by the fusion of egg gametes and sperm gametes,
, a mapping from child gametes to parent zygotes, and
, a mapping from zygotes to fertilization time.
For convenience, given a gametic genealogy,
denotes the set of egg gametes,
denotes the set of sperm gametes, and
denotes the mapping from gametes to the zygotes they formed during fertilization.
Formally, a gametic genealogy must satisfy the following conditions.
where , and forms a one-to-one mapping between and .
is a function , where is a subset of representing child gametes.
is a function such that for all child gametes ,
Note that denotes the domain of , that is, the set of child gametes.
Gametic lineage space
A gametic lineage space is a mathematical formalism representing the lines of transmission of genetic information via gametes of a population over time. It is a triplet where
is the set of all genomic locations,
is a gametic genealogy , and
is a function mapping a genomic position in a gamete to the set of gametes that transmitted genetic information to that position.
For every location and gamete , is the lineage ending at gamete via locus and it must satisfy the condition when , otherwise .
Example mathematical application
Given a sample of gametes , define the genomic locations reached by an ancestral gamete as
We conjecture that the set is the set of haplotype blocks defined in [1].