2007-09-06

Population

Two groups have the same population if to each individual in one group correspond exactly one in the other regarding his genotype. It says the two groups are clons.

For illustration is considerate a specie folowing a single philogenetic tree like bacteria. From a groupe {e1,e2} there are among others the following possible phylogenies:
a: {e1,e2}->1a{e1}->2a{}
b: {e1,e2}->1b{e1,e1}->2b{e1}
c: {e1,e2}->1c{e1,e2}->2c{e1}
d: {e1,e2}->1d{e1,e2,e2}->2d{e2}

The group 1a{e1} 2b{e1} and 2c{e1} have the same population P.

This chapter is about to determine the probability of each possible groups that can orginate from a initial group. The group does not split so that only the reproduction behaviour inside the group are considered. The variable is the birth-rate distribution attributed to each individual. The birth-rate depend on the group he is but it does not dependant from the actualy number of child of any other individual. This can also be say this way; if the actual number of descendant of a individual is known that does not change the birth-rate for any other individual. In connect with the chapter phylogenetic trees a specie that follow a simple phylogenetic trees the individual and the element of the trees are same. For species that does not follow a simple phylogenetic tree the individual is constructed which elements of simple phylogenetic trees.

Some convention:
In a set an individual is represented with e. e' is the underset with all individual having the same gens than e.
1(y,p1,..,px) is a function equal to 1 if p1+..+px=y else equal to 0 while the value p1,..,px are integer equal or greater than 0 and equal or less than y.
S[p1,..,px] 1(y,p1,..,px)*dp1*..*dpx is the sum of every factor 1(y,p1,..,px)*dp1*..*dpx when p1,..,px goes thrue all possible arrangements when given each index p1,..,px a value from 0 to y.

To begin the calculation detail for the single phylogenetic tree probability over one generation.

To each individual is associate the probability distribution of his descendants: d0,d1,.. . d0 is the probability that he has no child, d1 one and so one. Like every distribution d0+d1+d2+..=1. The sum 0*d0+1*d1+2*d +..=v is the birth-rate. Over 1 means that the number of his descendants statisticaly rises and under that it sinks. The maximal size of the next population is the add up of the maximal number of possible descendant of each individual. The minimal possibly number of the next population is the sum of the minimal number of possible descendant. If for every individual d0 is not null it is the empty set and the population can become extinct. The maximal possibly descendance for every individual is s. From a first population with n individual with m different phenotyps one forms all the possible set from n*s to 0 individual from a set of m individual with different genotyp while each individual can occurs several times. Each set e' with y individual has the probability to come from a group e' with x element of:
S[p1,..,px] 1(y,p1,..,px)*dp1*..*dpx.
The value for all underset e' making up the next group are then multiplicate. It is the probability that this particular group originate.

It is possible to give each individual from a group e' a different birth-rate distribution. The calculation proceeding is the same but it is then useful to give the factor of dp1*..*dpx an index; d1p1*..*dxpx. It remains that the probabiltiy that a given population comes from an other only depend on the probability distribution every individual.

To determine the probability of population over several generation the path procedure can be apply. All the element on each path are multiplicate and then all this value are added together.

The formulization is about the probability of particular groups over several generation and for other phylogenetic trees.

Formalization

eijkt is an individual. t his Generation, k his population, j his descent or genotyp, i an index for the identification of each individual from the same Population, Generation and descent.
e'jkt identify the underset of individual with the same descent. N(e'jk) is the number of individual in this underset.
P the Population, G the Groupe. P[..] is the product and S[..] the sum symbol. P can also be used like this P2(||) to symbolise a disjunctive relation (of set of 2 elements whith the relation ||) or like this 2P for all couples of P without couples with identical elements.

If 2 groupes have the same Individuales as for their descendences they belong to the same population: G in P
eijlkt is an individual when his apertenance to a groupe has to be consider.
e´jlkt the corresponding underset.

d=d0,d1,.. and d0+d1+..=1	; d0 is the probality to have 0 child, d1 one and so one.
v(d)=0*d0+1*d1+2*d2+..	; the birth-rate.
\/ p1,..,px >= 0 and =<y 1(y,p1,..,px)=1 if p1+..+px=y or x=y=0 else 0	; the definition of the function 1(..). \/ means for all.

The index l is used to identify groups generated from different groups.

	; from the example when Pk={e1} for k=1; Pk={e1,e2,e2} for k=2
Glkt=[i,j]{..,eijlkt,..}	; Gb12=2b{e1}, Gc12=c2{e1},Gd12={}
G'lkt=[j]{..,e'jlkt,..}	; G'd21=1d{e1',e2'}
Pkt=[i,j]{..,eijkt,..}	; P12={2b{e1},2c{e1}}
P'kt=[j]{..,e'jkt,..}	; P'21={1d{e1',e2'}},P'11={1a{e1'}}
P'k=[t]{..,P'kt,..}	; P'1={1a{e1'},2b{e1'},2c{e1'}}

A population founded on a single philogenetic tree.

d(eijkt)=d0ijkt,d1ijkt,..=d=dijkt
v(dijkt)=v(d(eijkt))=v=vijkt
w((Pk1t1,Pk2t2))=wk1t1k2t2
v(e'jk1t,e'jk2t+1)=vjk1k2t=S[p1,..,px : x=N(e'jk1)] 1(y=N(e'jk2),p1,..,px)*dp1ijkt*..*dpxijkt
wk1tk2t+1 = P[j] vjk1k2t

wkat1kbt2 =
S[k0,k1,..,kt2-t1 : k0=ka; k2..kt2-t1-1 >= 0; kt2-t1=kb]P[t : 0<=t<t2-t1]P[j]vjktkt+1t1+t
=
S[k0,k1,..,kt2-t1 : k0=ka; k2..kt2-t1-1 >= 0; kt2-t1=kb]P[t : 0<=t<t2-t1]P[j]S[p1,..,px : x=N(e'jkt)] 1(y=N(e'jkt+1),p1,..,px)*dp1ijktt*..*dpxijktt

wkat1kbt2 is the probality that from the population ka at the generation t1 comes the population kb at the generation t2.
example

A population founded on a double philogenetic tree.

2Pkt=[i1, j1, i2, j2; i1/=i2 or j1/=j2]{..,{ei1j1kt,ei2j2kt},..}
{ei1j1kt,ei2j2kt}=cpkt=ci1j1i2j2kt with i1/=i2 or j1/=j2
2Pt=[k]{..,2Pkt,..}
d(ci1j1i2j2kt)=d0i1j1i2j2kt,d1i1j1i2j2kt,..=d=dpkt=di1j1i2j2kt
v(di1j1i2j2kt)=v=vpkt=vi1j1i2j2kt

w((2Pk1t1,Pk2t2))=wk1t1k2t2

A population founded on a dual philogenetic tree.

PP=P2(||)
PPkt=[q]{..,{ei1j1kt,ei2j2kt},..}={..,pq,..}
2PPkt=[q1,q2; q1/=q2]{..,{pq1,pq2},..}

A population founded on a asymetric dual philogenetic tree.

P=PX and PY={..,xi1j1kt,..} and {..,yi2j2kt,..}
PP=P(P2(||))
PPkt=[q]{..,{xi1j1kt,xi2j2kt},..}and{..,{xi1j1kt,yi1j1kt},..}= {..,pxxq,..} and {..,pxyq,..}
2PPkt=[q1,q2; q1/=q2]{..,{pxxq1,pxyq2},..}

next not revise.

A population grunded on a multiple dual philogenetic tree

PPs=P(P2(||))
PPkt=[q]{..,{es0i1j1kt,es0i2j2kt},..}and{..,{es1i1j1kt,es1i2j2kt},..}and..=
{..,ps0q,..}and{..,ps1q,..}and..
2PPkt=[q1,q2; q1/=q2]{..,{ps0q1,ps0q2},..}{..,{ps1q1,ps1q2},..}..

A population grunded on a multiple asymetric dual philogenetic tree.

P=PX and PY={..,xi1j1kt,..} and {..,yi2j2kt,..} and {..,es0ijkt,..} and ..
PPs={PX and PY}(P2(||)) and P(P2(||))
PPkt=[q]{..,{xi1j1kt,xi2j2kt},..}and{..,{xi1j1kt,yi1j1kt},..}= {..,pxxq,..} and {..,pxyq,..}
2PPkt=[q1,q2; q1/=q2]{..,{pxxq1,pxyq2},..}