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
(Gam,Mate,Par,Fert)
with components
Gam,
the set of underlying gametes,
Mate,
the set of zygotes formed by the fusion of egg gametes and sperm
gametes,
Par,
a mapping from child gametes to parent zygotes, and
Fert,
a mapping from zygotes to fertilization time.
For convenience, given a gametic genealogy,
Gam0
denotes the set of egg gametes,
Gam1
denotes the set of sperm gametes, and
Mate∗
denotes the mapping from gametes to the zygotes they formed during
fertilization.
Formally, a gametic genealogy must satisfy the
following conditions.
Mate⊂Gam0×Gam1
where Gam0∩Gam1=∅,
Gam0∪Gam1=Gam
and Mate
forms a one-to-one mapping between Gam0
and Gam1.
Par
is a function C↦Mate,
where C
is a subset of Gam
representing child gametes.
Fert
is a function Mate↦R
such that for all child gametes g∈domPar,
Fert(Mate∗(g))>Fert(Par(g))
Note that domPar
denotes the domain of Par,
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
(Loc,G,Lin)
where
Loc
is the set of all genomic locations,
G
is a gametic genealogy (Gam,Mate,Par,Fert),
and
Lin
is a function Loc×Gam↦2Gam
mapping a genomic position in a gamete to the set of gametes that
transmitted genetic information to that position.
For every location ℓ∈Loc
and gamete g∈Gam,
Lin(ℓ,g)
is the lineage ending at gamete g
via locus ℓ
and it must satisfy the condition Lin(ℓ,g)={g}∪Lin(ℓ,Par(g)i) for either i=0 or i=1
when g∈domPar,
otherwise Lin(ℓ,g)={g}.
Example mathematical application
Given a sample of gametes S,
define the genomic locations reached by an ancestral gamete as
RS(g):={ℓ∈Loc:∃g′∈S(g∈Lin(ℓ,g′))}
We conjecture that the set {RS(g):g∈Gam}
is the set of haplotype blocks defined in
[1].
References