STAGE: EARLY DRAFT
DOCUMENT TYPE: Definition Document
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.
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 .