Micro-evolution according to
the Poisson distribution[1]
Summary
You expect
the genetic changes in a population are described as random or non random and
thus selective, but this appears not to be the case. Random genetic changes are
nearly always described as genetic drift with changes in the heterozygosis, but
the gene frequencies and so the random genetic changes are not to be reduced
from the heterozygosis. So it is tried to develop a uniform theory with the
random expected genetic change as the neutral theory and the zero hypothesis
for the selection. For the procreation and gene transfer from the individual in
a large, relative unlimited population the Poisson distribution is the
obvious method for calculating the
random expected genetic change. Yet also in small populations this distribution
appears accurate and well applicable. The Poisson distribution appears very
flexible in the sense that the parameters determining the intensity can
describe well the small populations and the population dynamics (change in size
and allele selection). By means of the parameters you can also find indications
for the complicated effect of the selection in the procreation on the allele
transfer. Even with a very simple application of this theory there are
immediate indications that the selection by people has been suddenly stopped
with the entrance of the modern society.
The relevancy in the appearance of the mutations for the dynamics of the
organism and the species is checked. By making simple distributions the transfer
of the alleles through the generations is followed in coherence with the random
variation in the effective procreation. These distributions are superposed over
some generations. This superposition of the distributions is possible by
working systematically with the Poisson distribution, but it is very laborious.
It appears than that the accumulation of the distributions over many
generations can easily be calculated, only for the exponential part of the
Poisson distribution The result of it the P0, the extinction of the neutral
alleles is essential for the random theory. The calculations of the extinction
with the exponential recurrence formula is also possible with population
specific parameters determining the cumulated exponential intensity are interesting.
They give information over the random neutral path and the non random
selection. This is showed by describing the decay of the alleles over many
generation in tables for populations, large and small, increasing and
decreasing in size, with and without selection, with and without inbreeding.
The random path of larger quantities of the alleles and thus allele frequencies
is also described. Important is however that these simple distributions do
describe primarily the decay of the absolute quantities, the “quanta”, of the
offspring and the alleles. In literature investigation later I found that these
extinction is also described plain by Motoo Kimura, but he reduced them not in
this basic way. There is some evidence that the conclusion of Kimura’s and others ware: the calculations of the
extinctions are not relevant, because they are not applicable in a limited
population. Nowise this, the extinction is essential in a logic consistent
theory.
Points of
attention are:
The neutral
theory as the zero hypothesis for the genetic selection concerns exclusively
the direct changes in numbers or frequencies of alleles and/or descendants of
individuals as described here. Changes in the heterozygosis are an indirect
basis for the neutral theory.
The direct
neutral change or allele extinction is also described in the limited and small
population and gives here besides the genetic drift extra information over the
genetic changes.
Selection
always is the result of non random differences in the parities, the offspring
of the individuals.
Preface
After my study medicine and some years later the training in the
epidemiology in tuberculosis and similar things until 1976 I did lose all the
tangible connections with the organisations of science and research. Moreover I
do not anymore practice the profession of a doctor already since my 40th.
Now I am 61 and I looked after my children for a long time, while my wife was
working. My curiosity and interest in different fields and the increasing quantity of spare time results in
a number of hobby-studies. Magazines, books and later on the internet did
provide me afterwards plentiful in information The study of the topic
evolutionary biology was one of my most favourites. No official training or
studies were followed in this. My study activities consist of collection here
and there interesting data and than thinking about it endless with my super
critical dialectical customs, or you may can call it also addiction. Anything
you read than is denied and that negation follows a laborious constructed
meaning, but that meaning again is denied, etc. So the negation of the negation
in order to find at last the all synthesis, the logic, the unity, the truth in
which anything is participating. This in principle is the rational method of
Spinoza, which was described later on by Hegel. By these negations points of
view mostly are not simply accepted or rejected, because often a synthesis is
possible so than the and/or is the best solution. Study in this way is an
endless ruminating, destructing and constructing of meanings and theories.
Using these radical dialectics you do not need a teacher, but it can result in
a stomach ulcer, for with this method of negation you are not a nice teacher
for yourself. The advantage of these primary negations is that it makes you
independent of other people like teachers and authors. Their information it is
not followed and taken over, but negated. So I tried to be no man’s follower and an open minded searcher to the
uniform principles.
Publication by internet is for me the most convenient way to share my ideas
about the evolution with other people. Perhaps it can help starting discussions
and deepening studies to the stirring, interesting and in many aspects so
important topics of the evolution. Furthermore I do hope I also will be able to
publish here more ideas about the macro-evolution and the
religious-philosophical aspects of the evolution as a logical integral.
Some principles of the
evolution theory
The evolution of the organisms and their species still is somewhat
disputed. It is nevertheless without doubt that anything we can observe is
changing and that nothing is able to remain the same forever. If the changes in
this are irreversible there always is a development or evolution. So the
a-priori statement is plausible: the
changeable and evolutionary characteristic of the nature, living and not
living, is nothing more than its very existence in the time or properly the
space-time. There are indeed observations that confirm the evolution.
Already in 1859 Charles Darwin did describe and prove that the similarities and
differences between the yet living species and the died out fossils indicate
evolution, as well as the frame of the embryo’s. Although there now is much
more knowledge and we can fill now probably a bookshelf of more than
Micro-evolutionary principles
Within a population is a knock-out competition between the different
allelic variations on the gene loci. By two factors all the gene variations or
alleles are not transferred through the generations of descents and so the numbers
and frequencies of the alleles will increase or decrease in the following
generations. These 2 factors are:
1st the distribution
of the reproduction. The individual organisms of the parent generation F0 do
have different numbers of effective descendents, that reach adultness and are
able to reproduce themselves.
2nd The endowment of the alleles to the effective
descendants.
The organisms of the F0 that have been able to reproduce effectively in
this way, will pass at most the same, but on the average fewer different
alleles to their total offspring than they do have themselves. Even if all
parents should have an equal number of offspring they will pass different parts
of their genetic variations to the next generations or otherwise not. There are
many mutations and often an individual has a number of seldom mutations. Also
are many mutations seldom and are they in small frequencies in large
populations, or a total species. However the absolute numbers of seldom
mutations are large in large populations, of in the whole species. One percent
of 10^8 yet ever is 10^6. So it is obvious a-priori that seldom mutations
practically never will vanish in large populations, unless they are ultimate
seldom and occur in immeasurable small frequencies, or are very unfavourable.
In this it also is obvious that alleles will be practically never be fixed in
large homogeneous populations. In small populations seldom alleles do have
small absolute numbers and by this they can vanish or increase in number and
sometimes be fixed in small populations. One percent of 100 yet only is 1. That
this is a-priori at random to be expected may appear from the following:
Pose a bag with 100 marbles. They have a number of different colours,
some colours are singular, some occur on 2 marbles, some on 3 or more. The
marbles all are drawn under replace. The results of the total turn of 100
drawings under replace are recorded and a new bag is composed, so that the
colours of the marbles are distributed following these results. It than appears
that the composition has been changed: Some colours have been disappeared and
some colours that were singular in the first bag now are present in twofold or more. At the
second turn, starting from the results of the first bag again the composition
changes evidently. If these drawing turns are ever repeated more and more
colours will disappear (extinction) and ultimately after a big number of turns
only one colour will remain in the bag (fixation). The same experiment can be
executed as well with the help of a computer in a bag with 10^8 marbles, in
which some colours are present on 10^6 marbles or on two or more times 10^6. It
will be evident that in this bag the composition hardly will change in the
drawing turns; 10^6 may become 9.10^5, but not easy 2.10^6 and practical never
0. So the frequencies will hardly change here and can at most fluctuate
somewhat in the turns. Yet is the change, that a singular allele (marble) is
not drawn and will disappear in a population (bag) of 10^8, nearly equal to
that in a population (bag) of 100 and so the change that all the 10^6 alleles
will disappear, is practically zero. This is in principle the model of the
random or neutral genetic change in a population. Essential in this however is
that the non random genetic change, the selection, comes upon to this as a
parameter of the chance distributions. As well in the case of selection are
these drawing turns valid in the model, but the drawings than are not ‘honest’.
In the selection for instance the red
marbles will have a smaller chance to be drawn and the green ones a larger
chance than at random, because the red ‘marbles’ are unfavourable alleles and
the green ones are favourable for the survival and the reproduction of the
individuals. This is the essence of the
micro-evolution that is elaborated here further.
Genetic Drift
This process by which the alleles
will vanish or settle totally in a close population with limited size is called
in literature genetic drift. So the allele frequencies always become 0 (in extinction)
or 1 (in fixation) and after a longer period this also occurs in larger
populations. The heterozygosis and thus the genetic variation within a
population is getting smaller and smaller by this genetic drift. By the genetic
drift arise ultimately a population that is genetic total identical, which is
of course also total homozygote if there were no mutations. Theoretically the
population becomes even identical exclusive by descent, after it was already a
long time homozygote and identical in general occurring alleles, but in
practice this event is not likely because the population will dye out before.
From the binomial distribution Sewell
Wright deduced there is a decrease in the heterozygosis [2] by
the drift with the average factor (1- 1/2n) per generation. In this is n the
size of the population and so 2n the number of alleles on the loci in a diploid
population with sexual procreation. This decrease is to be calculated with the
formula H g +1 = H g [ 1 – 1/2n ],
in which Hg is the heterozygosis in generation g. This means for instance that
in a population with 50 animals participating in the procreation is a decrease
of 1% per generation. So this is an important problem for many threatened
species. This decrease does not mean however that such a population will be
total homozygote and genetic identical by descent already after 100
generations. It yet is an exponential decrease; in general the heterozygosis
changes by a factor ℮^-1 = 0,3679 after 2n generations, so the decrease
than is 63,2%. After a x 2n generations the heterozygosis changes with a factor
℮^-a. In this is ℮ the logarithmic base, so ℮ = 2,7183.. This
decrease in the heterozygosis at the genetic drift is based on random
inbreeding. The drift to extinction or fixation of the alleles can be
intuitively a-priori approached in two ways:
1st By the inevitable or random inbreeding in a close
population arises homozygosis, so that the heterozygosis decreases, being its complement.
This process implicates imperatively the vanishing of some alleles and the
increase of their alternatives on the loci until it remains only one, but now
is it not easy to guess how this will happen.
2nd By random sampling there ever is fluctuation of the
numbers of the alleles, but if the decrease goes incidentally to zero there is
no way of return. This makes the curve of the chances for the smaller numbers
asymmetrical. This vanishing of some alleles means the increase of their
alternatives on the loci and so also the increase of the homozygosis and
decrease of heterozygosis. This happens in a population with limited size as
well as in the unlimited population. This vanishing or extinction of the
alleles however is limited in a pool with a limited number of alleles, because
not all the alleles can disappear here. There must remain in the limited pool
one of all the possible variations and in the unlimited pool are infinite
variations and so there will remain nothing. If this happens there is fixation
in the limited pool, with a fixation chance 1/2n.
Which allele will be fixed by the drift and which will vanish is of
course not to be predicted. You can pose the allele with the largest frequency
on the datum date at start should have the greatest probability. The
differences however in the probabilities often are very small, because there
are many events with random fluctuations between the datum date and the real
fixation. In a population of some size it will last a very long time till an
allele is fixed, but the increases and decreases of the allele frequencies can
go fast temporarily. Conditions for the
genetic drift as it is described by the formula: H g +1 = H g [ 1 – 1/2n ] are:
1st The close
population without genetic exchange. 2nd The constant size of the
population. 3rd Random breeding, so no more or no fewer inbreeding than at random. 4th
There is no selection. 5th There arise no new mutations after the
datum date. 6th There is no mating between the generations. 7th
Self-fertilisation is possible, because the individuals are fertile in both
genders.
Criticism on this model, Disadvantages
1st The great drawback
of this formula is: it describes how the heterozygosis
decreases in a close population, but unfortunately not how do allele frequencies
change in populations, as it is sometimes suggested indeed. The in- and
decreases of the frequencies and numbers of the alleles also is not to be
derived from this formula or from this model. Insight in the random and non
random changes however is essential for insight into the micro evolution.
2nd This formula H g +1 = H g [ 1 – 1/2n ] only is valid in very
restricted situations, because of the above called conditions. Further on it
is, I think, disputed if this formula and model fulfil if the population has
more than 2 allelic variations on the locus. If for instance 4 different
alleles a; b; c and d are at start on the locus, the extinction of any of these
alleles should be described as a separate process. Yet the vanishing of the
first allele is not lied with the fixation of the last allele. Evident further
is that a number stochastic processes independent of each other can not be
described as one process with one formula. So this should mean that another
condition for the formula is: there should
be only two allelic variations for the locus in the population.
3rd The formula appears than also not applicable in many real
situations. It can not describe for instance how a new arisen and thus very seldom mutation often disappears
very fast from a large population. Also the fast genetic changes that arise in
populations shortly after they got isolated can not be explained well by this
formula. These fast genetic changes arise
for instance in animals that got isolated in small populations after
people did disturb the ecology of their old life area. If so a mother
population splits into a number of deems there will be initially in these deems
alleles singular, in twofold, in threefold etc and by the small numbers of
these alleles many of them will disappear in a little generations. Also the
stocks in descent of the domestic animals are models of these very isolated
populations, that underwent impressive genetic changes in the course of a
restricted number of generations. The different races of the domestic animals
may origin from source populations of
minimal 50 to about 1000 of animals. According to the formula H g +1 = H g [ 1 –
1/2n ] the
heterozygosis should decrease in these effective populations with ca 1% to ca
0,5‰, per generation, while thus the observation indicates us that the changes
in the genes in these populations must have taken place much faster.
The large advantage of the formula however is that it is simple and
gives good and easy insight in the important aspects of the genetic changes:
the decrease in the heterozygosis and the increase of the homozygosis. This
easy calculation of the heterozygosis in this model, means thus a reduction,
which restricts the flexibility of this model in the different situations. By
this it only is possible to get more specific information with very complicated
further calculations, that than again do not give any more at all the simple
intuitive insight in the biologic events. So it could be useful and is any way
harmless trying to approach this matter in another way with models primary describing
what will happen in general with the allelic variations in a close, limited
population and in the theoretical unlimited population of Hardy and Weinberg.
In search of another model
In this model is started from the generation F1, that is born, or arises
and receives at random alleles from the former generation F0 on a distinct
locus in the genome. The size of the population in generation F0, F1, F2, etc
is constant on n examples. Thus are 2n alleles on the diploid loci, so that the
chance that a distinct allele of F0 comes into the zygote of F1 is 1/2n and the
chance that this allele does not come into the zygote is 1-1/2n. In this way
for all the n zygotes in F1 are ‘drawn’ 2n alleles for the locus from the
generation F0 alleles. Standard should be drawn in this way all the alleles or
gametes of F0 and so should be passed the total set of alleles from generation
F0 to generation F1. This standard event however is in reality as likely as a
long street in a poker game with a lot (2n) of different cards. Always are
drawn a number of alleles two times or more and an accordingly number are not
drawn. We can follow with the aid of a game with marbles or a computer module
of it what are exactly the fortunes of the genes with their potential and real
allelic varieties in a population. We start with a bag of 2n marbles, that all
have a singular number 1; 2; 3; …2n. These numbers represent the separate, in
generation F0 singular alleles, or potential variations of the genes. Further
on the marbles do have colours so that some marbles have the same colour. The
colours indicate the real existent gene variations. 2n marbles are drawn under
replace and the drawings are recorded. After a turn of 2n drawings the contents
of the bag is replaced by the results of the 2n drawings as recorded. So after
the first turn of drawings the first bag, F0, is replaced by the second, F1,
and so on. It than appears from the recordings that already in the first turn a
lot the of numbers on the marbles is not drawn and that many numbers are drawn
2x and some 3x or more. Also the colours did change in number in this way and
some very seldom colours were not drawn. At the second turn from bag F1 is
formed bag F2. Now also many numbers are not drawn, but less than at the turn
from bag F0, because in F1 not all the numbers are singular. If these turns are
ever repeated the singular numbers the singular numbers of F0 will vanish ever
more giving rise to increase of the frequencies of the remaining numbers and
colours. The numbers on the marbles and their colours are fluctuating in the
further turns The numbers and colours will decrease some turns and than again
increase, but if they go to zero no return is possible and it vanishes. By
these vanishing the remaining numbers and colours are ever increasing and by
this the vanishing becomes ever more seldom in the later turns. After a large
number of turns will remain only one colour and later also only one number. If
the bag contains a small number (2n) of marbles this process of fixation goes
very fast and if there are many marbles the fixation is slow and is only
possible after a huge number of turns. In the beginning however will the
singular numbers on the marble vanish in the large bag as fast as in the small
one. If the remaining numbers on the marbles become somewhat larger to about 50
or 100 than they will only very seldom disappear. The colours will be present
in the large bags already at the first turn mostly in numbers of more than 100,
so that they will scarcely disappear from the beginning.
This model of the marble game is a simplification, a reduced principle,
that does not describe the total biological reality. This hazard game model
than also does not intend to describe the total biological reality of the
genetic changes, but only the hazardous events in this. That is why it must
have indeed intrinsically restrictions: The possible gene changes caused by the biological functions are to be excluded in
the model. Unfortunately the model must have also extrinsically restrictions:
For reasons of clarity and survey-ability not all possible extrinsic events can
be included in a model. It is convenient
to describe a model within a standard situation with exclusion, or freezing of
all possible events. Later on some events may be included into the model as a
new parameter. These restrictions are mostly the same as in the general model
of the genetic drift, as they are:
1st The close
population without genetic exchange. 2nd The constant size of the
population. 3rd Random breeding, so no more or fewer inbreeding than
at random. 4th There is no selection. 5th No new
mutations do arise after the datum date. 6th There is no mating
between the generations. 7th Self-fertilisation is possible, because
the individuals are fertile in both genders.
The conditions random mating and no selection are largely or totally
intrinsically, as they are causal lied with the biological functions. The
events that may open the population and will change its size can be both
extrinsic and intrinsic. The arise of mutations is seen mostly as an extrinsic
factor, but this may be disputed. The mating between the generations is
intrinsic. The fertility in the genders and their participation in the
procreation is a biological or intrinsic factor.
Most of these restrictions are described as a parameter further on here.
This is not the case in the mating only within the same generation. The
possibilities of allele transfer in genealogic studies indicate however the
influence of mating between the generations on the allele transfer may be
small, because this mating does not cause inbreeding. It can cause however some
fluctuation of the effective size of the population while the real size remains
constant. Making models and calculations with this mating is difficult. The
measure of the mating between the generations depends from the length of
generation time in relation to the period of fertility and reproduction of the
individuals. Mice can mate with much more generations than people. The random
possibility of self-fertilisation itself in somewhat larger populations is very
small and thus unimportant. If the individuals of the species have separated
genders and can be fertile in only one gender this also is of no influence if
these both genders participate equally in the procreation. The problem however
is that in practical live the genders do not equal participate. The
observations learn us that more individuals of the gender that ‘invests’ the
most in the next generation participate in the procreation than those of the
minor investing gender, thus mostly the masculine. The phenomenon of biological
functions by which is caused this unequal participation of the genders in the
procreation is called sexual selection. It is possible and often practised to
correct for this intrinsic factor in calculating the effective population size
for these cases, but it is disputed if it always is useful to correct in the
biological data to get the random situation.
The potential most important restriction of this model however is that at 8th the transfer of the
alleles from generation Fn to F(n+1) must be one uniform event. In reality
it yet is a composed event. So this
restriction is very important, but it is not generally acknowledged in the
literature I guess. As pointed out before there is in fact a drawing or
distribution of:
1st The reproduction. The individual organisms of the parent
generation F0 draw different numbers of effective descendents, that reach
adultness and are able to reproduce themselves.
2nd The effective descendants can draw different alleles from
their parents and further ancestors.
It is pointed out here further on[3]
that it is possible and in many situations necessary to make a model with
specification of these two drawing events.
Further more it is possible to extend this model by putting more data
into it in order to get more information about the changes of the genes in the
course of time. So the marbles can have besides their number and colour also
other marks by means of which can be reed
for instance which individual is carrier of the allele and which was the
carrier in the former generation. With data like these the genealogy within the
population can be followed and so you can have much more information about
random genotypic distributions, the measure of homozygosis and especially the
important random changing linking as there is between the allele on a distinct
locus and many other loci of the genes of the ancestors. Many interesting
computer models can be made for the study of these problems. Primary however is
this simple reduced model. But besides of the models it is necessary to
describe in algebraic terms what happens at random to the genes and what
happens in essential in the biological reality:
Deduction why the
Poisson-exponential distribution is appropriate.
In a population are n individuals, so 2n diploid alleles are in the
model and they are seen as singular, so that they represent the potential
variations on the loci of the genes in the total population. If the size of the
population remains constant, the chance that one distinct allele is drawn in
one fertilization, so in one descendant, or is transferred from F0 to F1, is
1/2n and its complement, the chance that this allele is not drawn thus is
1-1/2n. This means that this distinct allele is not drawn on the average in 2n
draws, so in one generation (1-1/2n)^2n and so it is than not transferred in
one generation. It appears now that this relation (1-1/2n)^2n converges fast to
1/℮, for if n→∞ becomes (1 - 1/2n)^2n = 1/℮, in this is
℮ the base of the natural logarithm, so ℮=2,7183.. The conversion
goes fast, if the size of the population n=10, it is (19/20)^20=0,3585 so that
the ‘base’ than already is 2,7895, only 2,6% more than ℮. So is 1/℮,
or ℮^-1 the proportion of the singular alleles that is not transferred
from generation F0 to F1. The alleles however only can be transferred in this
way at random in a population with individuals that are fertile in the both
genders. In a population with ½ n
individual of the masculine and ½ n of the feminine gender the alleles are
‘drawn’ or transferred separately for the genders. In both of the genders n
alleles are present and are drawn. At one fertilization one allele is drawn in
both of the genders. In this are the proportions 1/n and 1-1/n transferred
respectively not transferred and so the proportion not transferred is nearly
1/℮ in somewhat larger populations. So is indeed this principle also
valid in separated genders, if the participation in the reproduction is equal. In
somewhat larger populations (n>ca 10) singular alleles are not transferred
and will vanish in the proportion or in the rate 1/℮=0,3679. In the
smallest possible population, if n=1, so in self fertilizing, this rate is (1-
½ )^2=0,25. Further it is obvious in a population with 2n alleles that if the
different alleles are not singular but are present in absolute numbers 1; 2; 3;
or q they are transferred or not
transferred to one descendant in proportions q/2n and 1- q/2n respectively. In
2n drawings, so in one generation they are not transferred in the proportion (1- q/2n)^2n. This is if n→∞ (1/℮)^q=℮^-q. It is evident to
that if the effective size of the population is not constant, but changes by a
factor p and the alleles do occur in the number q, these alleles are
transferred of not transferred with chances, or in proportions pq/2n and 1-
pq/2n respectively. So in one generation are (1- pq/2n)^2n alleles not
transferred. This is for n→∞
℮^-pq. This formula P0 = ℮^-qp is easily to be deducted and is than
also applied in many specialities. If a number of events occur in a period of
time and the events appear ‘memory less’ in general is valid: P0(t) = ℮^-qt. In this is
P(0)t the chance on no observation or hit of any event within period of time t.
The complement of this, Pi(t) = 1 -
℮^-qt, is the exponential distribution. So it is the
chance on one or more ‘events’ ‘arrivals’ or ‘hits’ within period of time t. This
period may be a constant, for instance the time of one generation. The events
or arrivals can be drawn or transferred alleles, if they are transferred memory
less at random. This is the case in this
biologic field if any individual has any moment the same chance on effective
reproduction.
This (negative) exponential distribution is used generally in science. It is a statistical
distribution, but you can see it also as an essential natural law. It also is
supplied for instance in the field of epidemiology. An unfortunately realistic
instance for illustration: a group of 10 young people goes to
In the Application of the Poisson-exponential
distribution in this field are taken as example of events, drawings, etc the ‘arrivals’ of the numbers of descendants
or alleles into the next generation. The starting lemma’s as condition at this
application are the lemma’s of the neutral hypothesis, the hypothesis of the negation
or the zero-hypothesis of the evolution by selection:
1st Any individual has any moment the same chance on
effective reproduction and any allele has any moment the same chance on
transfer.
2nd Differences in reproductive success between parents and
differences in the transfer between alleles are caused by accident.
These a-priori and a-posteriori conditions, which differ only in meaning
concerning the time of observation: before or after the event, are apparently
not present in the actual life of the organisms. The reproduction is yet in
many species seasonal, so that at once are born a number of cubs as a litter.
So the chances on mating and reproduction are apparently not memory less,
because a large part of the year the animals are not for mating disposed and
not fertile as do the plants in most climates. This however concerns the actual
reproduction we can easily observe in nature, but for the study of the genetic changes and the evolution we have to
observe the effective reproduction in the nature which is much more difficult.
In the effective reproduction are counted only the cubs that grow adult and get
a litter themselves. The observation of the effective reproduction of course is
necessary, because the genes and their variations are transferred only via
individuals of the following generations that survive. The effective
reproduction as the balance of birth minus juvenile mortality and infertility is
much more or perhaps total time independent, because the death hazards are
memory less and the juvenile mortality takes mostly a large part of the birth figures.
The above conditions concern for instance the question if an animal that
becomes a grandparent will have after this event the same chances to get
effective descendants than before this. Further on are differences in chances
if smaller scales are concerned which cause fluctuations as is pointed out here
later, but this may be averaged on the larger scale. For instance in an area or
in a period of drought the litters are smaller than in an area or period of
abundance and this may effect even the effective reproduction and so by this it
is fluctuating. On the larger scales however this may be averaged and than the
condition ‘the same chances in any time’ should also be taken larger. The
neutral hypothesis of equal chances and thus random causes for the differences
in the reproductive results and ultimately for the evolution can be tested by
observations of the results of the effective reproduction. Primary are equal
chances on effective reproduction and not equal reproduction. Real equality of
effective reproduction and allele transfer is excluded, because in nature does
not exist something like rationing systems for mating, birth and dying. So not real
or potential existing equality, but equal chances in effective reproduction and
transfer of the genes should be the (negative) basis for the evolutionary
theory.
The Poisson distribution is used here in the explication of the neutral
random theory for the general populations. This because it concerns here at
first the descendants of one person or the transfer of one or a very small
number of alleles in a relative large population. In very small populations as
pointed out later on here the binomial distribution is taken. The Poisson
distribution can be used best in this cases with small numbers in a large space
and it also has great advantages, I think: its flexibility and its simplicity.
The Poisson intensity does describe the average events in time and this intensity,
or ‘λ’ can easily be defined by
parameters as is done here: For the
distribution can be used the formula P(i) = ℮^-λ . λ^i/i! in the standard generation time. In this field the Poisson intensity λ apparently is determined by
some factors, q, or Q, p, r and s, so that λ=qprs
or λ=Qprs. In this is Q the primary quantum the
absolute number of ancestors in the parent generation F0 (which always is
So are explained some basic conditions for the neutral theory, which is
elaborated further in the chapters about the calculations of the Poisson
distributed reproduction and allele transfer. At first however some
observations and practical implications are concerned to show the relevance of
the random neutral theory as the basis of the total evolutionary theory.
Observations
It is possible to examine in what extend the participation in the
effective reproduction and thus the transfer of genetic variations is indeed
Poisson distributed. It is possible but not easy to count numbers of effective descendants
of the living animals and plants in the nature. This is a lot of work, but the
statistic data of the numbers of children people do have in the different
counties are direct available:
Table 1
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Table H2. Distribution of Women 40 to 44 Years Old by
Number of Children Ever Born and Marital Status: Selected Years, 1970 to 2004 |
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Source: U.S. Census Bureau |
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Internet
release date: |
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(leading dots indicate sub-parts) |
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(Years ending in June. Numbers in thousands) |
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Year |
Women 40-44 yr x1000 |
Women by number of children ever born in % |
Children ever born per woman |
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Total |
None |
One |
Two |
Three |
Four |
Five and six |
Seven or more |
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All Marital Classes |
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.1976 |
5684 |
100 |
10,2 |
9,6 |
21,7 |
22,7 |
15,8 |
13,9 |
6,2 |
3,091 |
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Poisson
λ=3,091 |
100 |
4,546 |
14,051 |
21,715 |
22,374 |
17,289 |
16,195 |
3,827 |
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.1982 |
6336 |
100 |
11 |
9,4 |
27,5 |
24,1 |
13,8 |
10,4 |
3,9 |
2,783 |
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Poisson
λ=2,783 |
100 |
6,185 |
17,214 |
23,953 |
22,22 |
15,46 |
12,596 |
2,373 |
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.2004 |
11535 |
100 |
19,3 |
17,4 |
34,5 |
18,1 |
7,4 |
2,9 |
0,5 |
1,895 |
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Poisson
λ=1,895 |
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100 |
15,032 |
28,485 |
26,99 |
17,049 |
8,077 |
4,028 |
0,34 |
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Women Ever Married |
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.1970 |
5815 |
100 |
8,6 |
11,8 |
23,8 |
21,4 |
14,6 |
12,9 |
6,8 |
3,096 |
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Poisson
λ=3,096 |
100 |
4,523 |
14,003 |
21,168 |
22,371 |
17,315 |
16,254 |
3,858 |
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.1976 |
5455 |
100 |
7,5 |
9,6 |
22,4 |
23,4 |
16,4 |
14,4 |
6,3 |
3,19 |
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Poisson
λ=3,190 |
100 |
4,117 |
13,134 |
20,948 |
22,275 |
17,764 |
17,359 |
4,396 |
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.1982 |
6027 |
100 |
7,6 |
9,6 |
28,7 |
25,1 |
14,3 |
10,8 |
4 |
2,885 |
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Poisson
λ=2,885 |
100 |
5,585 |
16,114 |
23,245 |
22,354 |
16,123 |
13,776 |
2,804 |
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.1985 |
6836 |
100 |
8 |
12,9 |
34,2 |
24,1 |
11,4 |
7,4 |
2 |
2,548 |
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Poisson
λ=2,548 |
100 |
7,823 |
19,935 |
25,397 |
21,571 |
13,741 |
9,976 |
1,557 |
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.1988 |
7543 |
100 |
10,2 |
14,7 |
37,3 |
22,1 |
9,5 |
5,2 |
0,9 |
2,28 |
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Poisson
λ=2,280 |
100 |
10,228 |
23,321 |
26,586 |
20,205 |
11,517 |
7,247 |
0,896 |
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.1998 |
9995 |
100 |
13,7 |
18,1 |
38,7 |
19,6 |
6,2 |
3,2 |
0,6 |
2,002 |
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Poisson
λ=2,002 |
100 |
13,506 |
27,04 |
27,067 |
18,063 |
9,04 |
4,828 |
0,456 |
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.2004 |
10036 |
100 |
13,2 |
17,4 |
38 |
19,9 |
8,2 |
2,9 |
0,4 |
2,046 |
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Poisson
λ=2,046 |
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100 |
12,925 |
26,445 |
27,053 |
18,45 |
9,437 |
5,179 |
0,513 |
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The US Census Bureau did collect in her table H2 the data of all the
women from the total American population. The figures of the US Census Bureau
are given here in Table 1 and they
show that particular in the short period from 1980 to 1990 has been a sharp
decrease from about 3 to
We see in Table 1 that the initial
figures of the years ’70 and begin ’80 show some resemblance between the
observation and the expectation following the Poisson distribution. Also in the
group ever married are the figures under no children however larger than
expected, probably because there have been always a minimum number of families
that keep childless by biological infertility of one of the partners. In the
later years the childlessness is more in accordance with the Poisson
distribution and this is an indication for the small difference between the
actual and the effective reproduction especially in the later years. In general
are in the observed figures fewer women with 1 child, more with 2 children and
fewer in the higher parities, although initial were the observed figures the
very high parities larger. These general tendencies are increasing in the years
and all the high parities than have lower figures later on. The aspect than in
the later years of the observed figures in relation to the Poisson is that of a
shift from the extreme values to the average value (=2). So the divergence in
the distribution of the natural parities becomes obvious smaller than in the
distribution of the parities following Poisson. So the genetic differences
between the generations have become smaller the at random. This is evidence for the view there is no (more) selective evolution in
the modern American population. These differences are large and the
divergences are so much smaller than random that it is very unlikely that
figures from the effective reproduction should give another indication. The observed
figures also did differ initially from the Poisson figures. The figures here
are somewhat conflicting: the average values (=3) are about equal, but there
was initial a shift from the moderate to the more extreme values in the observed
distribution. So this is indicative for a little larger divergence and thus is likely
a larger genetic difference between the generations than at random.
The divergence in the observed distributions in relation to the random
Poisson distribution is an important datum, which directly indicates the
changing of the genes and so the evolutionary intensity of a population.
To learn the relevance of this you
should inquire some more characteristics of the population. A population that
is very heterogenic in the reproduction will have a large divergence, but such
a population may be not a real existent social or natural group of individuals
and than its evolutionary intensity is not so relevant. Such a population here
in Table 1b is for instance the population women never
married with 0,88 children per woman on average and a large divergence. The
descendants of this population will of course be genetically different in the
next generation, but that may be not relevant. This population must consist of
women that are real singles and get only a little children and a group that
have about average children and a family life, but they only are not married
for the law and there may be others. So it is necessary to inquire the
composition of the populations and the US Census Bureau gives than also the
figures of the typical American subpopulations as they are called: Whites,
Blacks, Asians and Hispanics of any race. Some of their data are given in Table
1b and I do compare these also with the Poisson distribution. In trying to make
a better comparison and to estimate the evolutionary intensities I used a provisory coefficient in the right
column. This is an instance, this method is not satisfying, I think, but is
must be possible to develop here good exact methods. There may be relative
small differences between the American subpopulations in these aspects of
reproduction and genetic evolution. It is possible that these differences
between the European subpopulations are larger but their data are not, or more
difficult available.
Table
1b
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Table 1. Women by Number of Children Ever Born by
Race, Hispanic Origin, Nativity Status, Marital Status, and Age: June 2004 |
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(Numbers in thousands. For meaning of symbols, see table of
contents.) |
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(leading dots indicate sub-parts) |
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(Column B is in persons, all others are percents) |
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Women by number of children ever born |
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age |
total
x1000 |
Total
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None
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One |
Two |
Three
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Four
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5 and 6
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≥ 7 |
children
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coëf
MP |
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women |
women % |
% |
% |
% |
% |
% |
% |
% |
ever
born |
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ALL
RACES |
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per
woman |
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All
Marital Classes |
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.40
to 44 |
11.535 |
100 |
19,3 |
17,4 |
34,5 |
18,1 |
7,4 |
2,9 |
0,5 |
1,895 |
0,868 |
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Poisson
λ=1,895 |
100 |
15,032 |
28,485 |
26,99 |
17,049 |
8,077 |
4.028 |
0,34 |
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ALL
RACES |
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.Women
Ever Married |
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..40
to 44 |
10.036 |
100 |
13,2 |
17,4 |
38 |
19,9 |
8,2 |
2,9 |
0,4 |
2,046 |
0,723 |
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Poisson
λ=2,046 |
100 |
12,925 |
26,445 |
27,053 |
18,45 |
9,437 |
5,179 |
0,513 |
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ALL
RACES |
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.Women
Never Married |
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40
to 44 |
1.498 |
100 |
59,8 |
17 |
11,2 |
6,2 |
2,1 |
2,9 |
0,9 |
0,88 |
1,973 |
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Poisson
λ=0,88 |
100 |
41,478 |
36,501 |
16,06 |
4,711 |
1,036 |
0,042 |
0,001 |
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WHITE
ONLY |
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.Women
Ever Married |
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40
to 44 |
8289 |
100 |
13,4 |
16,8 |
39,3 |
19,7 |
8 |
2,4 |
0,3 |
2,02 |
0,694 |
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Poisson
λ=2,02 |
100 |
13,266 |
26,796 |
27,064 |
18,223 |
9.203 |
4,97 |
0,478 |
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WHITE
ONLY, NOT HISPANIC |
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.Women
Ever Married |
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40
to 44 |
7206 |
100 |
14,1 |
17,2 |
39,8 |
19,6 |
7,1 |
1,9 |
0,2 |
1,959 |
0,718 |
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Poisson
λ=1,959 |
100 |
14,1 |
27,622 |
27,056 |
17,667 |
8,653 |
4,498 |
0,406 |
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HISPANIC
(of any race) |
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.Women
Ever Married |
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40
to 44 |
1179 |
100 |
8,1 |
14,3 |
36,1 |
20,5 |
14,6 |
5,2 |
1,2 |
2,437 |
0,854 |
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Poisson
λ=2,437 |
100 |
8,742 |
21,305 |
25,96 |
21,088 |
12,848 |
8,8056 |
1,251 |
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BLACK
ONLY |
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.Women
Ever Married |
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40
to 44 |
1.054 |
100 |
12,3 |
20,3 |
27,9 |
24,7 |
9,5 |
4,2 |
1,1 |
2,198 |
0,928 |
|
Poisson
λ=2,198 |
100 |
11,102 |
24,403 |
26,819 |
19,65 |
10,797 |
6,485 |
0,743 |
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ASIAN
ONLY |
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|
.Women
Ever Married |
|
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|
40
to 44 |
470 |
100 |
12,9 |
20,5 |
40,1 |
13,4 |
6,3 |
6,1 |
0,7 |
2,052 |
0,711 |
|
Poisson
λ=2,052 |
100 |
12,848 |
26,364 |
27,049 |
18,502 |
9,491 |
5,227 |
0,519 |
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Source:
U.S. Census Bureau, Current Population Survey, June 2004. |
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A situation different from the data of the US Census Bureau shows the
picture of the historical data on Table
2. This population is describes more detailed at Table 9b. In this population is described concretely the effective
reproduction. Used are data from a genealogy of a family of fishermen and
skippers living in the South-western of the
Tabel 2
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
Poisson,
λ=3,05556 |
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|
0,0471 |
0,14391 |
0,21986 |
0,22393 |
0,17106 |
0,10453 |
0,05324 |
0,02324 |
0,00888 |
0,00301 |
|
populatie
n=72 |
gemiddeld
3,0555 kinderen per ouder |
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|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
0,09722 |
0,194444 |
0,22222 |
0,11111 |
0,06944 |
0,13889 |
0,08333 |
0,06944 |
0 |
0,01389 |
|
7/72 |
14/72 |
16/72 |
8/72 |
5/72 |
10/72 |
6/72 |
5/72 |
0 |
1/72 |
There are many publications about birth and fertility figures in the
various countries. Data about the parities are however much more difficult to
be found and are apparently not collected in most counties, but there are other
countries than the
Figure
2.9--Distribution of Women by Number of Children Born by Age
Random or[4]
selective changes
The premise of random mating, which often is pointed in literature, can
be described in plain English as: any individual has any moment the same chance
on mating. This mating can better be specified to its consequence we are
interested in: the effective reproduction. This random effective reproduction
does exist actually in real existing biological populations, if the condition
equal chances have been fulfilled and if they have not been fulfilled, as is
mostly, the random effective reproduction will exist only potentially in the
population. The dialectical neutral theory starts from the idea: there are
possibly random differences and non-random differences in the reproductive
results individuals have and in the transfer of the individual alleles. It is
possible with the modern technical tool to observe in all kind of levels and
situations in actual populations differences in these both processes: the
effective reproductive results and the transfer of the gene variations
(alleles). As pointed out further here (page 55) reproduction and allele
transfer are causal linked and in the random form they (or it) even exist
uniformly. The observational results are of course the sum of random and
non-random differences. The random differences are well-known, as they are
given by the Poisson distribution. This
also is ascertained in literature. The observed distributions of the effective
offspring thus will mostly differ from the Poisson distribution. This fact thus
should nowise induce the meaning that the statistical distributions are of none
or very limited importance in the inquiry of genetic changes in the
micro-evolution. The random events are of course physical present, but what we
observe on living beings never is the result of hazard alone. Children of
people and animals are not born and do not dye only by accident. The results we
can observe like products of effective reproduction and so gene transfer always
are a combination of hazard and biological action or function. The latter of
this, the organic skill to procreate and survive, can be defined as selection.
So selection than also is the causal unity of the non-random differences, which
is the dialectic complement of hazard in the struggle for life and so has here
a somewhat more specific meaning than in the pure Darwinist sense. Although..,
there is survival of the fittest and there is survival of the luckiest, but only
the survival of the fittest is survival by selection. It ever is important I
think, in philosophical and scientific research to find the essential detail in
the background of the noise of accidental events.
Random transfer or selection
at work
In the data of Table 1 is for instance an about 30% smaller variation
than Poisson-random in the parities in the data, the US Census Bureau collected
in the later years, of women getting children in the years about 1970-
Another picture gives the reproduction of salmons. A population of
salmons is reproducing far in the upper reaches of a distinct river. The
circumstances in the spawning-grounds are rather uniform. The salmons seldom
reproduce more than once in their lives and dye afterwards. The next generation
hatches from the egg and grows in the
river and further in the sea. A small part of the young salmons return later into
the source river as an adult for reproducing and dying. As for concerning the
parent generation the young salmons will apparently behave as real equal luck
‘Poissons’ (Fr: fishes). The parent salmons did yet produce about the same
numbers of fertilized eggs and in this river the young’s became all about the
same change on growing and survival from their parents, that brought them
hitherto. It is all up to the youngsters now. The eventual non-random or
selective differences and so non random allele transfer may appear in the
possible differences in chances to survive and to grow up to adulthood and to
swim at last all the way back to this source river. If some salmons have more
chances on success than others, this will give rise to more variations in the
distribution of the offspring and in the larger genetic differences between the
generations in the population. This possible selection than will also
strengthened by the inbreeding the
salmons will probably have by mating in relative small populations for many
generations in the source rivers.
By birds and many other species again is the situation different. Many species
of animals and even plants will give indeed some form of parent care to their
living children or give them something to survive better. The salmons were
particular in this because they do invest into the offspring already before
mating by swimming up to grounds that are favourable for the survival of the
brood. The real parent care birds give come after mating, brooding and
hatching. The survival of the young generation than depends on the dedication
and the possibilities of the parent(s) and the young itself, so of some more genetic different individuals. The
difference in the success the parents have at the raising of their young’s
often is based on good luck or blind evil. It so can happen that at one year there
is a drought in their living area, so that the parents can find less food for
the young’s than do parents in other living area’s. Such incidental differences
are not important, for the non-random distributions of the population in
somewhat larger scales may stay valid, because in more litters and more
generations the incidental differences will compensate each other. It is
important that the differences do exit systematically as capacities in the care
and in finding the food some parents may have more than other parents. Because
the young’s themselves are present at this care the better caring capacities
can be transferred in two ways through the generations: by the genetic transfer
and by the imitation, because the young’s will imitate their parents behaviour
later when they get litters themselves. This imitation can make some stocks of
birds systematically more successful in survival and reproduction. This however
may happen without regard of the characteristics of some genes. The genes of
these more successful individuals, because of their acting well by imitation
will also be transferred more than at random, although these genes did not
attribute to the success. In this can the transfer of some ill-functioning
genes accidentally be promoted it they occur in smart acting birds. This,
however may cause drawbacks later on, because physical defects in the birds can
develop by the accumulation of less functioning genes. So on the larger scales
the non-random differences are more consistent if the are transferred indeed by
the genes.
Some evolutionary problems in
people
From the beginning some million years ago the human (hominid) species
have had to give - in relation to average animal species - a very intense and
lasting care for their children and this further is increased during their evolution.
They did have than also more abilities in this care than the animals had and
could transfer their abilities more effective through the generations than the
other species. They were able to do this and many other things, because their brain
is very large in relation to individuals of all other species with about the
same bodyweight. This brain makes people capable to use their sense organs more
efficient and that improvement of their sensory perception was very useful for
the people in their care for the children, their defence against predators and
their search for food. The possibilities of the brain however go much further,
we do know now as modern people. Your brain makes also possible to perceive the
things behind the things. This deeper perception however of the causes etc
behind the things was a huge problem for the primitive people in prehistory and
they generally avoided to gather information about the things behind the
things. If they did inquire these or should invent new possibilities this ever
brought themselves and their tribes in large difficulties. Although the brains
of our direct ancestors, homo sapiens and perhaps also of the other hominids as
homo rectus and his later form the homo Neanderthal could work as well as ours
or perhaps even better, those people ware not able to use their brains as we do,
because of their social situation. An important problem is yet that people do observe
the world indeed much deeper than animals do, because much more efficient receivers
are opened for the information from the outside, but people do have the same anxious
cautiousness animals have for survival in dangerous surroundings. This excess
of info about potential dangers makes ancient people, but often also modern
people, very anxious and also very aggressive. The problems are much aggravated
by the consciousness people have of the things behind the things and the
communication about these with each other. People will so, in lack of modern
knowledge, experience various threatening phenomenon’s and see a whole
threatening world as causes for different events. They can be very anxious for
the thunder, for the shining of the moon, for the strange behaviour of other
people being friends of evil ghosts, etc. The destructive consequences of magic
and fear were moreover not the only disadvantages of the peoples brains. In
this matter also important is the interference between the transfer of the
genes - the genetic information - and the transfer of the ‘neuronic’
information through the generations: More than at the birds the problem here
was if your tribe or stock of people have success by their smart solutions they
may gather ill-genes. Drawbacks by this must have happened, but this problem
will be prevented mostly by a smash with the knout on the smart brain that should
not let people behave strange and thus evil. The human evolution has been a
very complicated process, which is only partly unravelled by the scientific
research, I think. The micro-evolution in this, so the genetic transfer through
the homo sapiens generations in about 200000 years, of course is of special
importance. A question that rises is how fast was the evolution in the sapiens
generations, or how large are the genetic differences between the generations.
The problem is that the indications for the answer to this question are
conflicting:
A-priori there is indication for fast evolution within sapiens, because
the non-random or selective differences are made by the biological, social and
whatever functions of the organisms selves. Humans are relatively well equipped
to achieve their targets and do often use aggressive and radical modes to do
so. For the things they can prefer, as are the appearances, it thus must have
gone very fast. So, many people in our part of
Also what we know from the historical and prehistorical data about the
reproduction possibilities in the live of the people is indicative for fast
evolution. The more different possibilities will cause probably more variations
in not modern people than average in most species. You can imagine that when
people became somewhat more knowledge together with increasing social
inequalities more chances for the privileged groups will arise. Also the often
very aggressive tendencies in the social life of people (also in modern!) as in
war making in combination with killing the conquered enemies, burning down
their possessions and violating their wives may cause strong selective
differences, although in the larger scales many of the effects by these
inequalities may be balanced. An important issue however in the historical view
at the human evolution is that the fitness in ancient times was mostly very
different from what fits in our present society and in our present biological
situation. It often was fit to be an aggressive man raging at people of other
ethnicity and violating their women. This now is very criminal behaviour, but
it is no wonder that you can watch this kind of behaviour everyday in the
streets of our cities. These problems were still worse if behaviour was total
inheritable and people could not correct by intelligence for their natural
tendencies. In other situations the historical–evolutionary problems are still
more evident. In the past many alleles have accumulated in the genomes of
people by selection at fitness towards situations that do not exist anymore. Examples
of this are the alleles that make people resistant for specific infectious
diseases that are easily to control now. The sickle cell anaemia is a famous example
of this. There are found many more of these cases and some will never be found
because the infectious agent has been disappeared for a long time. The cause of
the systematic occurrence of this phenomenon is very simple: The pathogen needs
a key to come into some specific cell of the host organism and it is
specialized by genetic selection to use the key, which often is a protein on the
cell wall. If the key does not function the pathogen has a problem, but also
the host organism. The less functioning key at the heterozygote does increase
the resistance, but this does ever mean also a less functioning protein or
total cell, which means non functioning in the case of homozygosis. This
problem is evident in some monogenetic diseases in people as the different
haemoglobin disorders, cystic fibrosis and others. The problem may be much
broader: also many polygenetic inheritable diseases are possibly induced by
genetic selection. Our ancestors did live close together without any form of
hygiene and also got many traumata and this did them so suffer a lot from all
kinds of infections during many ten thousands of generations. No wonder from
the evolutionary view that we now posses a very aggressive immune system that
easily deregulates giving rise to auto immune diseases, in which the immune
system attacks the cell of the own body and also to allergies with the
exaggerated reactions on harmless vectors.
This comes to the conclusions: People may have had a relative fast
evolution. The evolution now suddenly has stopped, but we have definitely not
to worry about this ceasing. We are on the contrary in big troubles by the
genetic variations accumulated by selection in our ancestors. The genes are of
course now not to be removed by natural evolution, also not by sharp artificial
evolution, or as it is called radical eugenics. The results of artificial
evolution is to be seen at the sad genetic state of our house animals. The
races of the house animals have been bred mostly under veterinarian control,
but there are still gigantic genetic problems. There are also very important ethical
objections against eugenics. A free medical advise for a family in specific
cases is of course something else.
Random transfer or sexual
selection
The distribution of descendants and alleles is, as a-priori expected, in
the masculine gender different from the feminine. The masculine and feminine
organisms do reproduce with different properties and these differences will
result in different chances on reproduction and so in a different distributions
of their descendants. The more general differences between feminine and
masculine is not always corresponding with the biological sex even in the field
of reproduction. So it may be better to use other words for describing the
sexual characteristics more general and more typical: yin for the feminine and
yang for the masculine reproduction. In the yin the woman’s part is uniform
within the mother’s part and the pure yin will receive the both parts. The yin
does invest maximal into the child to
create maximal effective numbers of
children. This is attended with few sexual competition and even with
cooperation between the females. The yin will limit to the minimum the
variation in the numbers of children the different females have. A more unequal
distribution of the numbers of the descendants would be unfavourable for the
survival. It yet is not ‘efficient’ human economics would say if the hard work
of motherhood is not borne by all the
females in proportion to ability. This minimal variation in the effective
children is given by the Poisson distribution, if the abilities are equal and
if there is large juvenile mortality. This last condition is mostly present in
nature, but in our modern human populations the infant mortality is very small
and so than the most efficient distribution of the children among the mothers is
with a smaller variation than Poisson. By differences between the females in abilities for the motherhood the yin wants
a larger variety than Poisson and it so creates selection. Particularly in more
intelligent organisms there may be systematic differences in abilities.
In the yang the man’s part is distinct from the father’s part and so in
the pure yang can present only the man’s part. The yang invests minimal in the child for the
possibility to create maximal
numbers of children. This brings much competition between the males and even
with females. In that eternal male-female conflict the ever heard female
argument is: you always are thinking of the one thing and the male argument is:
you cannot make well more than one thing simultaneously. The yang is working on
distance the effectiveness of his procreation (care) is via the yin. The number
of the partners and their possibilities are the results of the yang, whereas
the numbers of the children and their possibilities are the results of the yin.
So the yang has to follow the yin in the minimum variation augmented by the
variation the yang has in the numbers of partners. This extra variation the
yang has in the parities is created by the differences between the male
individuals for mating in competition with each other. These extra variations
are called the sexual selection. The sexual selection generally is present in
nature. The sexual selection however probably is larger in species that do have
large individual differences, as is in intelligent creatures. If there also is
a large social inequality, as it was in historical human societies, sexual
selection can become extreme large. Some men with much power and high
distinction did have a lot of descendants. So nearly everymen in the
neighbourhood descents apparently from the old celebrities as Charlemagne or
Dzengis Khan. It may be obvious that the selective variation by the fatherhood
(or sexual selection) in general is larger and also has other qualities,
because it has been selected on other characteristics (mating ability) than by the selective variation by the
motherhood (care ability). These differences in quality may be of evolutionary importance: systematic selection
on characteristics. In the larger scales may exceed furthermore the female
selection for instance also the quantity of the male selection, if the larger
differences by the male selection in more generations can be more neutralized
in the allele transfer, while the female selection can be more systematic.
So because yin and yang are different biologic functions it is useful to
observe and calculate the male reproduction distinct from the female. This
however is not done in the genetic drift theory; in the literature the distinct
yin and yang selection always are equalized by the formula of the effective
population size. Oh, girls did not I say that you can not do your work well if
you try to fix the different things at the same.
Calculation of the Poisson
distributed reproduction.
Suppose the size of the population is constant on n individuals in the generations F0; F1 and F2. The reproduction
population in study consists of parents of effective children. Only descendants
in the first generation that have become parents themselves are counted as
individuals of the population. So parents with 0 children are parents that did
not become grandparents. There is random reproduction. So any individual of F0
has the same change of 1/n to be the
parent of the new individual of F1 and 1-1/n
to be not the parent. For 2n effective
children[6]
of F1 the change to be not the parent is (1−1/2n)^2n. This is 1/℮
for larger values of n. In this is
℮ the natural base 2,7183.. In the case n=10 this “base” already is 2,7895.. On this account the
distribution of the effective descendents in the next generation is essentially
according to the Poisson-exponential distribution, if the population is not
very small. So the average proportions of the individual organisms in F0 with i descendants in F1 are to be
calculated by substitution in P(i) = ℮^-λ .
λ^i/i! In this the intensity λ can be determined
by parameters, so that λ=Qprs or λ=qprs. In this is Q the primary quantum; q is the general quantum;
p the change in the size of the population; s the theoretical or
virtual selection and r is the replacement factor by neutral population
dynamics. Because the children and further descendents of 1 individual in the
generation of the first parents, the F0, are studied in Table 3 Q=1.
This individual has in many cases (proportions) more than 1 descendant in the
F1, These will than also participate in a number (quantum) >1 participate in
the new parent generation. That is why q is indeed variable and has natural
values as 1;2;3, etc. In this example the population size is constant, so p=1.
By sexual reproduction the individual organisms have on the average 2
descendants in the next generation in neutral population dynamics, so r=2.
There is random mating with equal chances, without selection, so s=1. So
it is obvious that p, s and thus λ can have in
principle any values ≥0.
The average random or Poisson distributed offspring of the individual
organisms of the primary parent population F0 is given in Table 3. If you calculate by substitution in the formula with λ=2 it appears that
the proportion of F0 with 0 descendents in F1 is ℮^-2 = 0,1353.. This
proportion participates active in the reproduction but will have no descendants
by random mating with equal chances. The proportion 2.℮^-2 = 0,2707 of F0
has 1 descendant on the average in F1. The proportion 4/3.℮^-2 = 0,1804 has 3
descendants. 2/3 .℮^-2 = 0,0902
has 4 descendants etc. So the average of 2 descendants is in this way Poisson
distributed and the sum of the descendants, calculated in this way indeed is 0,2707x1
+ 0,2707x2 + 0,1804x3 + 0,0902x4 +… = 2. These are descendants that individuals
in the F0 will have together with their different sexual partners so that the
population keeps the same size.
In the distribution of the descendants of F0 in F1 there is only one
intensity λ of the expected
number of children of F0 in F1. This intensity is in random mating only
determined by the population dynamics, so what is necessary for maintaining of
changing the effective population size. But if we consider the descendants of
F0 in F2, the grandchildren, there must be different intensities. The children
the F0 individual has in F1 determine by their numbers q the expected number of their grandchildren of F0 in F2 and so
determine the λ of the
distribution for the new generation together with any possible changes. So,
because of the different numbers of children in the F1, the distribution of the
descendants of the primary parents, the F0, in the further generations has no uniform intensity. As a symbol for
this variable intensity is used λ*,
so that : λ*=qprs. By
1 descendant in F1 the expected number of descendants in F2 is in neutral
dynamics : λ*=qprs=1x1x2x1=2. By 2
children there are 4 grandchildren on the average, because than q=2, so
that λ*=2x1x2x1=4. By 3
children there are 3x1x2x1=6 descendants in the F2 etc. So it is possible to
calculate the 4 descendants the individual from F0 has on the average in F2 in
a further Poisson distribution. In this would be not right to consider the
originate of the generations F1 and F2 out of F0 straight away as one process
and do so calculating this as a Poisson distribution with intensity 4. That is
not right, because the origin of the F1 and the F2 are two processes all within
its own space of time. In these the individuals of the F1, the parents and not
the grandparents from F0, are concerned in the events, the “arrivals” that give
rise to the F2. The only overlap of the two processes is that individuals in F0
that have no descendents in F1 will also have no descendents in F2.
The proportion 2℮^-2=0,2707 of F0 has one descendant in F1. The
size of the population in F0 and F1 is constant on n. So there are n2℮^-2
individuals coming in this way. You can calculate the proportion of this, so the
way of 1 descendant from F0 to F1 and zero descendants from F1 to F2 (notice
→1→0) by substitution with λ*=2 and i=0, than it
is 2℮^-2 . ℮^-2 = 2℮^-4. So is (→1→1): 2.℮^-2 . 2.℮^-2 = 4 ℮^-4.
(→1→2) is 2.℮^-2 . 2.℮^-2 = 4 ℮^-4.
(→1→3) is 2.℮^-2 . 4/3.
℮^-2 = 8/3 . ℮^-4. (→1→4) is 2.℮^-2. 2/3 .℮^-2 = 4/3 .
℮^-4. (→1→5) is 2.℮^-2 . 4/15 .℮^-2 = 8/15 .
℮^-4. (→1→6) is 2.℮^-2 . 4/45 . ℮^-2 = 8/45 .
℮^-4. (→1→7) is 2.℮^-2 . 8/315 .℮^-2 = 16/315 . ℮^-4,
etc. Simply used is as substitution in the Poisson formula is q=1; r=2; λ*=2 en i=0,of i=1, of i=2, etc and
multiplied with the factor 2℮^-2, the proportion of the one descendant in
the F1.
So has also the proportion 2℮^-2 of F0 two descendants in the F1. These
parents in F0 expect however to get here 4 grandchildren, so q=2; r=2 and λ*=4 and
(→2→0) is 2.℮^-2. ℮^-4=2℮^-6. (→2→1)
is 2.℮^-2. 4.℮^-2 =8.℮^-2. (→2→2) is
2.℮^-2. 8.℮^-2 =16.℮^-2, etc. In these calculations
continually λ*=4 en i=0, i=1, i=2, etc. The total
distribution of the descendants of F0 in F1 and F2 is given in Table 3. The way of descend is showed
with the arrows.
Table 3
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|
Descendants of
F0 in F1 |
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||
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
|
℮^-2 |
2.℮^-2 |
2.℮^-2 |
4/3. ℮^-2 |
2/3 .℮^-2 |
4/15℮^-2 |
4/45℮^-2 |
8/315℮^-2 |
2/315℮^-2 |
|
0,135342 |
0,27067 |
0,27067 |
0,18045 |
0,09022 |
0,03609 |
0,01203 |
0,00344 |
0,00086 |
|
x2/1 |
x2/2 |
x2/3 |
x2/4 |
x2/5 |
x2/6 |
|
|
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|
Descendants
of F0 in F2 |
|
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|
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|||
|
→0→0 |
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|
℮^-2 |
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|
|
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
→1→7 |
→1→8 |
|
2.℮^-4 |
4. ℮^-4 |
4 ℮^-4 |
8/3 ℮^-4 |
4/3 ℮^-4 |
8/15 . ℮^-4 |
8/45 ℮^-4 |
16/315℮^-4 |
4/315℮^-4 |
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
→2→7 |
→2→8 |
|
2℮^-6 |
8℮^-6 |
16℮^-6 |
64/3℮^-6 |
64/3℮^-6 |
256/15℮^-6 |
512/45℮^-6 |
2048/315℮^-6 |
1024/315℮^-6 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
→3→7 |
→3→8 |
|
4/3℮^-8 |
8℮^-8 |
24℮^-8 |
48℮^-8 |
72℮^-8 |
86,4℮^-8 |
86,4℮^-8 |
2592/35℮^-8 |
1944/35℮^-8 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
→4→7 |
→4→8 |
|
2/3℮^-10 |
16/3℮^-10 |
32/3℮^-10 |
512/9℮^-10 |
1024/9℮^-10 |
8192/45℮^-10 |
242,736℮^-10 |
277,401℮^-10 |
277,401℮^-10 |
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
→5→7 |
→5→8 |
|
4/15℮^-12 |
8/3℮^-12 |
40/3℮^-12 |
44,44℮^-12 |
111,11℮^-12 |
222,22℮^-12 |
370,37℮^-12 |
529,10℮^-12 |
661,38℮^-12 |
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
→6→7 |
→6→8 |
|
4/45℮^-14 |
16/15℮^-14 |
6,4℮^-14 |
25,6℮^-14 |
82,29℮^-14 |
184,32℮^-14 |
368,64℮^-14 |
631,95℮^-14 |
947,93℮^-14 |
|
→0→0 |
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|
|
→→ |
→→ |
→7→6 |
→7→7 |
→7→8 |
|
0,135342 |
|
|
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|
265,59℮^-16 |
531,18℮^-16 |
929,57℮^-16 |
|
∑
0 F2 |
∑
1 |
∑
2 |
∑
3 |
∑
4 |
∑
5 |
∑
6 |
∑
7 |
∑
8 |
|
0,042068047 |
0,09604 |
0,12155 |
0,1207 |
0,10737 |
0,09086 |
0,074075 |
0,05832 |
0,04447 |
|
as
℮ functon |
X
1 |
X
2 |
X
3 |
X
4 |
X
5 |
X
6 |
X
7 |
X
8 |
|
[(℮^2^(℮^-2)-1]/℮^2 |
0,9604 |
0,2431 |
0,3621 |
0,42948 |
0,4543 |
0,44445 |
0,40824 |
0,35576 |
|
∑
0 F1+F2 |
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0,177412 |
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|
[(℮^2^(℮^-2)]/℮^2 |
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X
0 |
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|
→9 |
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0,00141093℮^-2 |
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0,00019 |
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|
→1→9 |
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|
→2→9 |
→2→10 |
→2→11 |
|
|
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|
1,4448℮^-6 |
0,5779℮^-6 |
0,2102℮^-6 |
0,07005℮^-6 |
|
|
|
|
|
|
→3→9 |
→3→10 |
→3→11 |
→3→12 |
→3→13 |
→3→14 |
→3→15 |
|
|
|
1296/35℮^-8 |
22,217℮^-8 |
12,118℮^-8 |
6,059℮^-8 |
2,797℮^-8 |
1,199℮^-8 |
0,479℮^-8 |
|
|
|
→4→9 |
→4→10 |
→4→11 |
→4→12 |
→4→13 |
→4→14 |
→4→15 |
→4→16 |
→4→17 |
|
246,579℮^-10 |
197,263℮^-10 |
143,464℮^-10 |
95,643℮^-10 |
58,857℮^-10 |
33,633℮^-10 |
17,937℮^-10 |
8,969℮^-10 |
4,221℮^-10 |
|
→5→9 |
→5→10 |
→5→11 |
→5→12 |
→5→13 |
→5→14 |
→5→15 |
→5→16 |
→5→17 |
|
734,86℮^-12 |
734,86℮^-12 |
668,06℮^-12 |
556,71℮^-12 |
428,24℮^-12 |
305,89℮^-12 |
203,92℮^-12 |
127,45℮^-12 |
74,97℮^-12 |
|
→6→9 |
→6→10 |
→6→11 |
→6→12 |
→6→13 |
→6→14 |
→6→15 |
→6→16 |
→6→17 |
|
1263,91℮^-14 |
1516,69℮^-14 |
1654,57℮^-14 |
1654,57℮^-14 |
1527,30℮^-14 |
1309,11℮^-14 |
1047,29℮^-14 |
785,47℮^-14 |
554,45℮^-14 |
|
→7→9 |
→7→10 |
→7→11 |
→7→12 |
→7→13 |
→7→14 |
→7→15 |
→7→16 |
→7→17 |
|
1446,0℮^-16 |
2024,40℮^-16 |
2576,51℮^-16 |
3005,93℮^-16 |
3237,16℮^-16 |
3237,16℮^-16 |
3021,34℮^-16 |
2643,68℮^-16 |
2172,65℮^-16 |
|
∑
9 |
∑
10 |
∑
11 |
∑
12 |
∑
13 |
∑
14 |
∑
15 |
∑
16 |
∑
17 |
|
0,0329265 |
0,0238454 |
0,0168737 |
0,0116831 |
0.00788 |
0,0052613 |
0,00343908 |
0,00217468 |
0,00135777 |
|
X
9 |
X
10 |
X
11 |
X
12 |
X
13 |
X
14 |
X
15 |
X
16 |
X
17 |
|
0,2963385 |
0,238454 |
0,1856107 |
0,1401972 |
0,10244 |
0,0736582 |
0,0515862 |
0,034795 |
0,02308 |
|
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|
→4→18 |
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|
1,876℮^-10 |
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|
→5→18 |
→5→19 |
→5→20 |
→5→21 |
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|
41,65℮^-12 |
21,92℮^-12 |
10,96℮^-12 |
5,22℮^-12 |
|
|
|
→6→18 |
→6→19 |
→6→20 |
→6→21 |
|
|
|
369,63℮^-14 |
233,45℮^-14 |
140,07℮^-14 |
80,04℮^-14 |
43,66℮^-14 |
22,78℮^-14 |
|
→7→18 |
→7→19 |
→7→20 |
→7→21 |
→7→22 |
→7→23 |
|
1693,33℮^-16 |
1247,72℮^-16 |
873,40℮^-16 |
580,27℮^-16 |
370,54℮^-16 |
225.55℮^-16 |
|
∑
18 |
∑
19 |
∑
20 |
∑
21 |
∑
22 |
∑
23 |
|
0,00083899 |
0,00046922 |
0,000249 |
0,000164 |
0,00008 |
0,00004 |
|
X
18 |
X
19 |
X
20 |
X
21 |
X
22 |
X
23 |
|
0,0151 |
0,00892 |
0,00498 |
0,00344 |
0,00176 |
0,00092 |
In Table 3 are mentioned the
numbers of descendants as a product of ℮ under the field of the arrows.
So under example →3→6 is noticed in
proportions of F0 through 3 descendants in F1 to 6 descendants in F2. Notice
that the numbers of the sums under ∑, so 0,17740 ; 0,09604 ; 0,12155; etc
are proportions of F0 with totally 0; 1; 2; 3; etc descendants in F2. The
further Poisson-like distribution that is given here is thus more asymmetric
than the normal primary Poisson distribution. Of that totals under ∑ only
the sums for 0 descendants can be expressed fully as products of ℮. The
total of all the proportions or change intensities is indeed 1. The sum of the
descendants in F2 to an ancestor in F0, so 0x 0,17740 + 1x 0,09604 + 2x 0,12155
+ … is indeed in total
The Poisson distributed allele
transfer
In Table 4 are described the
fortunes of the allelic variations. So how many of the unique alleles or
possible gene variations are transferred on the average to the next 3 generations
according to the continued Poisson like distributions. Pose an individual has
on a locus the alleles a and b. The change on transfer of allele a by 1
descendant in F1 is 0,5. By 2 descendants, the replacement in neutral
population dynamics, the transfer of allele a is on the average 2x0,5=1. This
the same for allele b. So the parameters for the intensity of the transfer of
one allele to the next generation in a neutral population are Q =1 p=1, s=1 en r=1. The distribution of the alleles
thus is with Poisson intensity λ=1, so that in the F1 the proportion 1/℮=0,368..
has been disappeared, also 1/℮=0,368 occurs singular, the half of this
0,184 occurs in twofold, etc. Also at
the transfer from F1 to F2 and the further generation there will be on the
average 2 descendants and 1 allele in the distribution. To make the distribution
for F2 and F3 you must for all the proportions, those in singular, in twofold
etc from the former generation, separately calculate how their further Poisson
distribution is, as indicated with the arrows. In this are each of these proportions
different distributed with the intensities λ*=q=1; λ*=q=2; λ*=q=3, etc. The total intensity μ of
these 2nd; 3rd and further degrees Poisson distributions
remains
Table 4
|
F0 |
|
|
|
|
|
|
|
|
|
|
Q=1 λ=1 |
|
|
|
|
|
|
|
|
|
|
F1
|
|
|
|
|
|
|
|
|
|
|
1→0 |
1→1 |
1→2 |
1→3 |
1→4 |
1→5 |
1→6 |
1→7 |
1→8 |
1→9 |
|
℮^-1 |
℮^-1 |
1/2.℮^-1 |
1/6.℮^-1 |
1/24.℮^-1 |
1/120.℮^-1 |
1/720.℮^-1 |
1/5040.℮^-1 |
1/40320.℮^-1 |
2,76.10^-6
℮^-1 |
|
F2 |
μ=1 |
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
→0→7 |
→0→8 |
→0→9 |
|
℮^-1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
→1→7 |
→1→8 |
→1→9 |
|
℮^-2 |
℮^-2 |
1/2.℮^-2 |
1/6.℮^-2 |
1/24.℮^-2 |
1/120.℮^-2 |
1/720.℮^-2 |
1/5040℮^-2 |
1/40320.℮^-2 |
2,76
10^-6.℮^-2 |
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
→2→7 |
→2→8 |
→2→9 |
|
1/2.℮^-3 |
℮^-3 |
℮^-3 |
2/3.℮^-3 |
1/3.℮^-3 |
2/15.℮^-3 |
2/45.℮^-3 |
4/315.℮^-3 |
1/315.℮^-3 |
7,06.10^℮^--4 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
→3→7 |
→3→8 |
→3→9 |
|
1/6.℮^-4 |
1/2.℮^-4 |
3/4.℮^-4 |
3/4.℮^-4 |
9/16.℮^-4 |
27/80.℮^-4 |
81/480.℮^-4 |
81/1120.℮^-4 |
243/8960.℮^-4 |
9,04.10^-4℮^-4 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
→4→7 |
→4→8 |
→4→9 |
|
1/24℮^-5 |
1/6.℮^-5 |
1/3.℮^-5 |
4/9.℮^-5 |
4/9.℮^-5 |
16/45.℮^-5 |
32/135.℮^-5 |
128/945.℮^-5 |
64/945.℮^-5 |
0,0301℮^-5 |
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
→5→7 |
→5→8 |
→5→9 |
|
1/120.℮^-6 |
1/24.℮^-6 |
5/48℮^-6 |
25/144.℮^-6 |
125/576.℮^-6 |
125/576.℮^-6 |
0,18084℮^-6 |
0,12618℮^-6 |
0,08073℮^-6 |
0,04485℮^-6 |
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
→6→7 |
→6→8 |
→6→9 |
|
1/720.℮^-7 |
1/120.℮^-7 |
1/40.℮^-7 |
1/20.℮^-7 |
3/40/.℮^-7 |
0,09.℮^-7 |
0,09.℮^-7 |
27/350.℮^-7 |
81/1400.℮^-7 |
0,03857.℮^-7 |
|
→7→0 |
→7→1 |
→7→2 |
→7→3 |
→7→4 |
→7→5 |
→7→6 |
→7→7 |
→7→8 |
→7→9 |
|
1/5040.℮^-8 |
1/720.℮^-8 |
7/1440.℮^-8 |
0,01134.℮^-8 |
0,01985.℮^-8 |
0,02779℮^-8 |
0,03242℮^-8 |
0,03242℮^-8 |
0,02837℮^-8 |
0,02206.℮^-8 |
|
→8→0 |
→8→1 |
→8→2 |
→8→3 |
→8→4 |
→8→5 |
→8→6 |
→8→7 |
→8→8 |
→8→9 |
|
1/40320℮^-9 |
1/5040℮^-9 |
1/1260℮^-9 |
2/945℮^-9 |
0,00433℮^-9 |
0,00677℮^-9 |
0,00903℮^-9 |
0,01032℮^-9 |
0,01031℮^-9 |
0,00917℮^-9 |
|
∑
0 F2 |
∑
1 F2 |
∑
2 F2 |
∑
3 F2 |
∑
4 F2 |
∑
5 F2 |
∑
6 F2 |
∑
7 F2 |
∑
8 F2 |
∑
9 F2 |
|
0,163584164 |
0,195514535 |
0,13372015 |
0,07295863 |
0,036145345 |
0,016973463 |
0,007630948 |
0,003299023 |
0,001378136 |
0,000357693 |
|
as
℮ function |
as
℮ function |
|
|
|
|
|
|
|
|
|
[℮^(
℮^-1) 1]/℮ |
℮^(1/℮-2) |
|
|
|
|
|
|
|
|
|
∑
0 F0 - F2 |
or
℮^(1/℮)/℮^2 |
|
|
|
|
|
|
|
|
|
0,53146305 |
|
|
|
|
|
|
|
|
|
|
≈℮^(1/℮-1) |
|
|
|
|
|
|
|
|
|
|
or
[℮^(℮^-1)]/℮ |
|
|
|
|
|
|
|
|
|
|
F3 |
|
|
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
→0→7 |
→0→8 |
→0→9 |
→0→10 |
|
0,53146305 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
→1→7 |
→1→8 |
→1→9 |
→1→10 |
|
0,07192577 |
0,07192577 |
0,035962888 |
0,011987629 |
0,002996907 |
0,000599381 |
0,000099896 |
1,43E-05 |
1,78E-06 |
1,98E-07 |
1,98E-08 |
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
→2→7 |
→2→8 |
→2→9 |
→2→10 |
|
0,018097054 |
0,036194108 |
0,036194108 |
0,024129405 |
0,012064702 |
0,004825881 |
0,001608271 |
0,000459608 |
0,000114902 |
2,55E-05 |
5,11E-06 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
→3→7 |
→3→8 |
→3→9 |
→3→10 |
|
0,00363234 |
0,010897188 |
0,016345783 |
0,016345783 |
0,012259337 |
0,007355603 |
0,003677801 |
0,001576201 |
0,000591075 |
0,000197025 |
5,91E-05 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
→4→7 |
→4→8 |
→4→9 |
→4→10 |
|
0,000662025 |
0,0026481 |
0,005296201 |
0,007061601 |
0,007061601 |
0,00564928 |
0,003766187 |
0,002152107 |
0,001076054 |
0,000478246 |
0,000191298 |
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
→5→7 |
→5→8 |
→5→9 |
→5→10 |
|
0,000114366 |
0,000578315 |
0,001429579 |
0,002382631 |
0,002978289 |
0,002978289 |
0,002481907 |
0,001772791 |
0,001107994 |
0,000615552 |
0,000307776 |
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
→6→7 |
→6→8 |
→6→9 |
→6→10 |
|
1,89E-05 |
0,000113491 |
0,000340474 |
0,000680948 |
0,001021422 |
0,001225707 |
0,001225707 |
0,001050606 |
0,000787954 |
0,000525303 |
0,000315182 |
|
|
→7→1 |
→7→2 |
→7→3 |
→7→4 |
→7→5 |
→7→6 |
→7→7 |
→7→8 |
→7→9 |
→7→10 |
|
|
2,11E-05 |
7,37E-05 |
0,000171976 |
0,000300958 |
0,00042134 |
0,000491564 |
0,000491564 |
0,000430118 |
0,000334536 |
0,000234175 |
|
|
|
|
→8→3 |
→8→4 |
→8→5 |
→8→6 |
→8→7 |
→8→8 |
→8→9 |
→8→10 |
|
|
|
|
3,95E-05 |
7,89E-05 |
0,000126242 |
0,000168323 |
0,000192369 |
0,000192369 |
0,000170995 |
0,000136796 |
|
|
|
|
|
→9→4 |
→9→5 |
→9→6 |
→9→7 |
→9→8 |
→9→9 |
→9→10 |
|
|
|
|
|
1,21E-05 |
2,17E-05 |
3,26E-05 |
4,19E-05 |
4,71E-05 |
4,71E-05 |
4,24E-05 |
|
∑0
F3 |
∑
1 F3 |
∑
2 F3 |
∑
3 F3 |
∑
4 F3 |
∑
5 F3 |
∑
6 F3 |
∑
7 F3 |
∑
8 F3 |
∑
9 F3 |
∑
10 F3 |
|
0,09445047 |
0,122378031 |
0,095642736 |
0,062799424 |
0,038774185 |
0,023203445 |
0,013552238 |
0,007751407 |
0,004349379 |
0,002394518 |
0,001291877 |
|
∑0
F0-F3 |
|
|
|
|
|
|
|
|
|
|
|
0,625917694 |
|
|
|
|
|
|
|
|
|
∑
11 F3 |
|
≈℮^[℮^(1/℮-1)-1] |
|
|
|
|
|
|
|
|
6,88
E-4 |
|
The number of the events as “arrivals” of descendants and genes in generation
F1 is Poisson distributed with the known primary Poisson distribution. This
means a distribution of the primary quantum Q with the uniform intensity λ
into proportions for the quanta i=0; i=1; i=2; i=3, etc, so that the
distribution results in quanta and proportions. The result of the former
distribution, these proportion can of course be distributed Poisson again. Then
however the proportions must be distributed separately each with its own
intensity λ*=q.μ, so the product of the quantum q of the proportion in the former
generation and μ. In this way
also in the further generations the arrivals of the alleles remain to be
Poisson distributed in the same generation time t and this distribution can be calculated by the same substitution
in the formula P(i) = ℮^-λ*. λ*^i/i! through the
generations. So the proportions of the old generations are always distributed
into the new generations. In this way the 2nd degree Poisson
distribution arises out of the general known primary distribution, the 3rd
degree out of the 2nd degree and the nth degree out of
the (n-1)th degree Poisson
distribution. These further distribution all originate from the normal primary
Poisson distribution with the uniform intensity λ. I do call these 2nd , 3rd and further
degree Poisson distributions, because the same primary quantum Q is distributed
here primary, secondary, tertiary and further. In Table 4 this happens with a
constant total Poisson intensity μ=λ=1 for F(g-1)→Fg. The μ in
this is the intensity in which all the proportions will decrease or increase in
total at the distribution F(g-1)→ Fg. The μ is the proportional
total intensity of the degree g, so that: μ=0x[P(i=0)] + 1x[P(i=1)] + 2x[P(i=2)] + …qx[P(i=q)], in
this is P(i=q) the result of the distribution according to P(i) =
℮^-λ*. λ*^i/i! of degree g-1. This μ of the continued
Poisson distribution is constant in these examples, but the Poisson
distribution of the quanta can also be continued in the next degree with another
intensity. The calculation of a large number of degrees are easily practicable,
I guess, with a computer and the right software, but not in this way.
So there are in the graduated Poisson distributions levels of quantities
and the distributions are from the former to the next level of the quantities.
In this application the levels of the quanta are called the generations F0; F1; F2;..Fg. The degrees of the Poisson
distributions are between these levels or generations, so that degree Gg
distributes the quanta of Fg into those of F(g-1). See on Table 5.
The accumulating exponential
distribution.
The peculiarity of the P(i=0), this is the negation or the complement of
the Poisson event or arrival, the zero-proportion is exponential distributed, according
to P(i=0) = ℮^-λ at
the primary and further Poisson distribution and it is the
complement of the exponential distribution, P(i=n) = 1 - ℮^-λ. The intensity
λ of these exponential distributions is also within the next degrees equal
to the μ of the continued Poisson distribution, of which it is a
part. With λ*=μ.q and
the quanta q the P(i=0) can be calculated with the superposed Poisson
distributions in the way of Table 3 and 4. If you express than the P(i=0) as an
algebraic function of ℮ it just appears that the remaining quantity, so 1-P(i=0) just is negative
exponential distributed through the degrees or generations. So P(i=0) of
generation g simply is ℮^{1-P(i=0)} of the former generation
g-1. The exponential distribution of the non arrival accumulates in this way.
Through the generations is the intensity λ or σ of the
exponential distribution equal to the remaining quantity and decreases, while
the non arrival, the extinction of the allele increases. In this is λ
the intensity of the primary distribution and is σ the accumulated
intensity of the distributions in the further degrees. The superposition of the
exponential part of the Poisson distribution can be calculated in following the
recurrence and this is noticed in -σ(Fg)=ν-
In Table 5 the extinct alleles, the P(∑i=0), shortly P0, is
calculated from the intensities λ,
or σ for the generations F0-F200, starting from λ=μ=1.
Table 5
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
|
λ=1 |
σ=0,6321 |
σ=0,4685 |
σ=0,3741 |
σ=0,3121 |
σ=0,2681 |
σ=0,2352 |
σ=0,2095 |
σ=0,1890 |
σ=0,1723 |
|
P0=0,368 |
P0=0,531 |
P0=0,626 |
P0=0,688 |
P0=0,732 |
P0=0,765 |
P0=0,790 |
P0=0,811 |
P0=0,828 |
P0=0,842 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
|
|
|
|
|
|
|
|
|
|
|
F10 |
F11 |
F12 |
F13 |
F14 |
F15 |
F16 |
F17 |
F18 |
F19 |
|
σ=0,1582 |
σ=0,1464 |
σ=0,1361 |
σ=0,1273 |
σ=0,1195 |
σ=0,1127 |
σ=0,1066 |
σ=0,1011 |
σ=0,0961 |
σ=0,0916 |
|
P0=0,854 |
P0=0,864 |
P0=0,873 |
P0=0,880 |
P0=0,887 |
P0=0,893 |
P0=0,899 |
P0=0,904 |
P0=0,908 |
P0=0,912 |
|
F11 |
F12 |
F13 |
F14 |
F15 |
F16 |
F17 |
F18 |
F19 |
F20 |
|
|
|
|
|
|
|
|
|
|
|
|
F20 |
F21 |
F22 |
F23 |
F24 |
F25 |
F26 |
F27 |
F28 |
F29 |
|
σ=0,0876 |
σ=0,0838 |
σ=0,0804 |
σ=0,0773 |
σ=0,0744 |
σ=0,0716 |
σ=0,0692 |
σ=0,0668 |
σ=0,0646 |
σ=0,0626 |
|
P0=,0916 |
P0=0,919 |
P0=0,923 |
P0=0,926 |
P0=0,928 |
P0=0,931 |
P0=0,933 |
P0=0,935 |
P0=0,937 |
P0=0,939 |
|
F21 |
F22 |
F23 |
F24 |
F25 |
F26 |
F27 |
F28 |
F29 |
F30 |
|
|
|
|
|
|
|
|
|
|
|
|
F30 |
F31 |
F32 |
F33 |
F34 |
F35 |
F36 |
F37 |
F38 |
F39 |
|
σ=0,0607 |
σ=0,0589 |
σ=0,0572 |
σ=0,0556 |
σ=0,0541 |
σ=0,0526 |
σ=0,0513 |
σ=0,0500 |
σ=0,0487 |
σ=0,0476 |
|
P0=0,941 |
P0=0,943 |
P0=0,944 |
P0=0,945 |
P0=0,947 |
P0=0,949 |
P0=0,950 |
P0=0,951 |
P0=0,952 |
P0=0,954 |
|
F31 |
F32 |
F33 |
F34 |
F35 |
F36 |
F37 |
F38 |
F39 |
F40 |
|
|
|
|
|
|
|
|
|
|
|
|
F40 |
F41 |
F42 |
F43 |
F44 |
F45 |
F46 |
F47 |
F48 |
F49 |
|
σ=0,0465 |
σ=0,0454 |
σ=0,0444 |
σ=0,0434 |
σ=0,0425 |
σ=0,0416 |
σ=0,0407 |
σ=0,0399 |
σ=0,0391 |
σ=0,0384 |
|
P0=0,955 |
P0=0,956 |
P0=0,957 |
P0=0,958 |
P0=0,958 |
P0=0,959 |
P0=0,960 |
P0=0,961 |
P0=0,962 |
P0=0,962 |
|
F41 |
F42 |
F43 |
F44 |
F45 |
F46 |
F47 |
F48 |
F49 |
F50 |
|
|
|
|
|
|
|
|
|
|
|
|
F50 |
F51 |
F52 |
F53 |
F54 |
F55 |
F56 |
F57 |
F58 |
F59 |
|
σ=0,0376 |
σ=0,0369 |
σ=0,0363 |
σ=0,0356 |
σ=0,0350 |
σ=0,0344 |
σ=0,0338 |
σ=0,0332 |
σ=0,0327 |
σ=0,0322 |
|
P0=0,963 |
P0=0,964 |
P0=0,964 |
P0=0,965 |
P0=0,966 |
P0=0,966 |
P0=0,967 |
P0=0,967 |
P0=0,968 |
P0=0,968 |
|
F51 |
F52 |
F53 |
F54 |
F55 |
F56 |
F57 |
F58 |
F59 |
F60 |
|
|
|
|
|
|
|
|
|
|
|
|
F60 |
F61 |
F62 |
F63 |
F64 |
F65 |
F66 |
F67 |
F68 |
F69 |
|
σ=0,0317 |
σ=0,0312 |
σ=0,0307 |
σ=0,0302 |
σ=0,0298 |
σ=0,0293 |
σ=0,0289 |
σ=0,0285 |
σ=0,0281 |
σ=0,0277 |
|
P0=0,969 |
P0=0,969 |
P0=0,970 |
P0=0,970 |
P0=0,971 |
P0=0,971 |
P0=0,972 |
P0=0,972 |
P0=0,972 |
P0=0,973 |
|
F61 |
F62 |
F63 |
F64 |
F65 |
F66 |
F67 |
F68 |
F69 |
F70 |
|
|
|
|
|
|
|
|
|
|
|
|
F70 |
F71 |
F72 |
F73 |
F74 |
F75 |
F76 |
F77 |
F78 |
F79 |
|
σ=0,0273 |
σ=0,0269 |
σ=0,0266 |
σ=0,0262 |
σ=0,0259 |
σ=0,0256 |
σ=0,0252 |
σ=0,0249 |
σ=0,0246 |
σ=0,0243 |
|
P0=0,973 |
P0=0,973 |
P0=0,974 |
P0=0,974 |
P0=0,974 |
P0=0,975 |
P0=0,975 |
P0=0,975 |
P0=0,976 |
P0=0,976 |
|
F71 |
F72 |
F73 |
F74 |
F75 |
F76 |
F77 |
F78 |
F79 |
F80 |
|
|
|
|
|
|
|
|
|
|
|
|
F80 |
F81 |
F82 |
F83 |
F84 |
F85 |
F86 |
F87 |
F88 |
F89 |
|
σ=0,0240 |
σ=0,0237 |
σ=0,0235 |
σ=0,0232 |
σ=0,0229 |
σ=0,0227 |
σ=0,0224 |
σ=0,0221 |
σ=0,0219 |
σ=0,0217 |
|
P0=0,976 |
P0=0,977 |
P0=0,977 |
P0=0,977 |
P0=0,977 |
P0=0.978 |
P0=0,978 |
P0=0,978 |
P0=0,978 |
P0=0,979 |
|
F81 |
F82 |
F83 |
F84 |
F85 |
F86 |
F87 |
F88 |
F89 |
F90 |
|
|
|
|
|
|
|
|
|
|
|
|
F90 |
F91 |
F92 |
F93 |
F94 |
F95 |
F96 |
F97 |
F98 |
F99 |
|
σ=o,0214 |
σ=0,0212 |
σ=0,0210 |
σ=0,0208 |
σ=0,0205 |
σ=0,0203 |
σ=0,0201 |
σ=0,0199 |
σ=0,0197 |
σ=0,0195 |
|
P0=0,979 |
P0=0,979 |
P0=0,979 |
P0=0,979 |
P0=0,980 |
P0=0,980 |
P0=0,980 |
P0=0,980 |
P0=0,980 |
P0=0,981 |
|
F91 |
F92 |
F93 |
F94 |
F95 |
F96 |
F97 |
F98 |
F99 |
F100 |
|
|
|
|
|
|
|
|
|
|
|
|
F100 |
F101 |
F102 |
F103 |
F104 |
F105 |
F106 |
F107 |
F108 |
F109 |
|
σ=0,01935 |
σ=0,01917 |
σ=0,01898 |
σ=0,01881 |
σ=0,01863 |
σ=0,01854 |
σ=0,01829 |
σ=0,01812 |
σ=0,01796 |
σ=0,01780 |
|
P0=0,98083 |
P0=0,98102 |
P0=0,98119 |
P0=0,98137 |
P0=0,98154 |
P0=0,98171 |
P0=0,98188 |
P0=0,98204 |
P0=0,98220 |
P0=0,98236 |
|
F101 |
F102 |
F103 |
F104 |
F105 |
F106 |
F107 |
F108 |
F109 |
F110 |
|
|
|
|
|
|
|
|
|
|
|
|
F110 |
F111 |
F112 |
F113 |
F114 |
F115 |
F116 |
F117 |
F118 |
F119 |
|
σ=0,01764 |
σ=0,01749 |
σ=0,01733 |
σ=0,01718 |
σ=0,01704 |
σ=0,01689 |
σ=0,01675 |
σ=0,01661 |
σ=0,01647 |
σ=0,01634 |
|
P0=0.98251 |
P0=0,98267 |
P0=0,98281 |
P0=0,98296 |
P0=0,98311 |
P0=0,98325 |
P0=0,98339 |
P0=0,98352 |
P0=0,98366 |
P0=0,98379 |
|
F111 |
F112 |
F113 |
F114 |
F115 |
F116 |
F117 |
F118 |
F119 |
F120 |
|
|
|
|
|
|
|
|
|
|
|
|
F120 |
F121 |
F122 |
F123 |
F124 |
F125 |
F126 |
F127 |
F128 |
F129 |
|
σ=0,01621 |
σ=0,01608 |
σ=0,01595 |
σ=0,01582 |
σ=0,01570 |
σ=0,01557 |
σ=0,01545 |
σ=0,01533 |
σ=0,01522 |
σ=0,01510 |
|
P0=0,98392 |
P0=0,98405 |
P0=0,98418 |
P0=0,98430 |
P0=0,98443 |
P0=0,98455 |
P0=0,98467 |
P0=0,98478 |
P0=0,98490 |
P0=0,98501 |
|
F121 |
F122 |
F123 |
F124 |
F125 |
F126 |
F127 |
F128 |
F129 |
F130 |
|
|
|
|
|
|
|
|
|
|
|
|
F130 |
F131 |
F132 |
F133 |
F134 |
F135 |
F136 |
F137 |
F138 |
F139 |
|
σ=0,01499 |
σ=0,01488 |
σ=0,01477 |
σ=0,01466 |
σ=0,01455 |
σ=0,01445 |
σ=0,01434 |
σ=0,01424 |
σ=0,01414 |
σ=0,01404 |
|
P0=0,98512 |
P0=0,98523 |
P0=0,98534 |
P0=0,98545 |
P0=0,98555 |
P0=0,98566 |
P0=0,98576 |
P0=0,98586 |
P0=0,98596 |
P0=0,98606 |
|
F131 |
F132 |
F133 |
F134 |
F135 |
F136 |
F137 |
F138 |
F139 |
F140 |
|
|
|
|
|
|
|
|
|
|
|
|
F140 |
F141 |
F142 |
F143 |
F144 |
F145 |
F146 |
F147 |
F148 |
F149 |
|
σ=0,01394 |
σ=0,01385 |
σ=0,01375 |
σ=0,01366 |
σ=0,01356 |
σ=0,01347 |
σ=0,01338 |
σ=0,01329 |
σ=0,01320 |
σ=0,01312 |
|
P0=0,98615 |
P0=0,98625 |
P0=0,98634 |
P0=0,98644 |
P0=0,98653 |
P0=0,98662 |
P0=0,98671 |
P0=0,98680 |
P0=0,98688 |
P0=0,98697 |
|
F141 |
F142 |
F143 |
F144 |
F145 |
F146 |
F147 |
F148 |
F149 |
F150 |
|
|
|
|
|
|
|
|
|
|
|
|
F150 |
F151 |
F152 |
F153 |
F154 |
F155 |
F156 |
F157 |
F158 |
F159 |
|
σ=0,01303 |
σ=0,01295 |
σ=0,01286 |
σ=0,01278 |
σ=0,01270 |
σ=0,01262 |
σ=0,01254 |
σ=0,01246 |
σ=0,01238 |
σ=0,01230 |
|
P0=0,98705 |
P0=0,98714 |
P0=0,98722 |
P0=0,98730 |
P0=0,98738 |
P0=0,98746 |
P0=0,98754 |
P0=0,98762 |
P0=0,98769 |
P0=0,98777 |
|
F151 |
F152 |
F153 |
F154 |
F155 |
F156 |
F157 |
F158 |
F159 |
F160 |
|
|
|
|
|
|
|
|
|
|
|
|
F160 |
F161 |
F162 |
F163 |
F164 |
F165 |
F166 |
F167 |
F168 |
F169 |
|
σ=0,01223 |
σ=0,01216 |
σ=0,01208 |
σ=0,01201 |
σ=0,01194 |
σ=0,01187 |
σ=0,01180 |
σ=0,01173 |
σ=0,01166 |
σ=0,01159 |
|
P0=0,98784 |
P0=0,98792 |
P0=0,98799 |
P0=0,98806 |
P0=0,98813 |
P0=0,98820 |
P0=0,98827 |
P0=0,98834 |
P0=0,98841 |
P0=0,98847 |
|
F161 |
F162 |
F163 |
F164 |
F165 |
F166 |
F167 |
F168 |
F169 |
F170 |
|
|
|
|
|
|
|
|
|
|
|
|
F170 |
F171 |
F172 |
F173 |
F174 |
F175 |
F176 |
F177 |
F178 |
F179 |
|
σ=0,01153 |
σ=0,01146 |
σ=0,01139 |
σ=0,01133 |
σ=0,01126 |
σ=0,01120 |
σ=0,01114 |
σ=0,01108 |
σ=0,01102 |
σ=0,01096 |
|
P0=0,98854 |
P0=0,98861 |
P0=0,98867 |
P0=0,98873 |
P0=0,98880 |
P0=0,98886 |
P0=0,98892 |
P0=0,98898 |
P0=0,98904 |
P0=0,98910 |
|
F171 |
F172 |
F173 |
F174 |
F175 |
F176 |
F177 |
F178 |
F179 |
F180 |
|
|
|
|
|
|
|
|
|
|
|
|
F180 |
F181 |
F182 |
F183 |
F184 |
F185 |
F186 |
F187 |
F188 |
F189 |
|
σ=0,01090 |
σ=0,01084 |
σ=0,01078 |
σ=0,01072 |
σ=0,01066 |
σ=0,01061 |
σ=0,01055 |
σ=0,01050 |
σ=0,01044 |
σ=0,01039 |
|
P0=0,98916 |
P0=0,98922 |
P0=0,98928 |
P0=0,98934 |
P0=0,98939 |
P0=0,98945 |
P0=0,98950 |
P0=0,98956 |
P0=0,98961 |
P0=0,98967 |
|
F181 |
F182 |
F183 |
F184 |
F185 |
F186 |
F187 |
F188 |
F189 |
F190 |
|
|
|
|
|
|
|
|
|
|
|
|
F190 |
F191 |
F192 |
F193 |
F194 |
F195 |
F196 |
F197 |
F198 |
F199 |
|
σ=0,01033 |
σ=0,01028 |
σ=0,01023 |
σ=0,01018 |
σ=0,01012 |
σ=0,01007 |
σ=0,01002 |
σ=0,00997 |
σ=0,00992 |
σ=0,00987 |
|
P0=0,98972 |
P0=0,98977 |
P0=0,98982 |
P0=0,98987 |
P0=0,98993 |
P0=0,98998 |
P0=0,99003 |
P0=0,99008 |
P0=0,99013 |
P0=0,99018 |
|
F191 |
F192 |
F193 |
F194 |
F195 |
F196 |
F197 |
F198 |
F199 |
F200 |
|
|
|
|
|
|
|
|
|
|
|
If the population dynamics are not neutral.
If the population dynamics[7]
are not neutral, because p≠1 or/and s≠1, the
population size or the number of the alleles will change with a constant factor μ=p.s. The population
size and the alleles will increase exponential, if μ>1 and will decrease exponential, if μ<
Table 5b
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F4 |
F9 |
F14 |
F19 |
F49 |
F99 |
F199 |
F299 |
F399 |
F∞ |
|
μ=0,97 |
σ=0,28263 |
σ=0,1436 |
σ=0,0917 |
σ=0,0645 |
σ=0,0156 |
σ=0,0028 |
σ=0,000 |
σ=0,000 |
σ=0,000 |
σ=0,000 |
|
P0=0,37908 |
P0=0,75380 |
P0=0,8662 |
P0=0,9124 |
P0=0,9375 |
P0=0,9845 |
P0=0,9972 |
P0=1 |
P0=1 |
P0=1 |
P0=1 |
|
F1 |
F5 |
F10 |
F15 |
F20 |
F50 |
F100 |
F200 |
F300 |
F400 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F4 |
F9 |
F19 |
F49 |
F99 |
F199 |
F299 |
F399 |
F499 |
F∞ |
|
μ=1 |
σ=0,3121 |
σ=0,1723 |
σ=0,0916 |
σ=0,0384 |
σ=0,0195 |
σ=0,00987 |
σ=0,00652 |
σ=0,00491 |
σ=0,00391 |
σ=0 |
|
P0=0,368 |
P0=0,732 |
P0=0,842 |
P0=0,912 |
P0=0,962 |
P0=0,981 |
P0=0,99018 |
P0=0,9935 |
P0=0,9951 |
P0=0,9961 |
P0=1 |
|
F1 |
F5 |
F10 |
F20 |
F50 |
F100 |
F200 |
F300 |
F400 |
F500 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F4 |
F9 |
F19 |
F49 |
F99 |
F199 |
F299 |
F399 |
F499 |
F∞ |
|
μ=1,01 |
σ=0,32215 |
σ=0,18241 |
σ=0,10197 |
σ=0,04919 |
σ=0,03117 |
σ=0,02298 |
σ=0,02096 |
σ=0,02029 |
σ=0,02006 |
σ=0,01993 |
|
P0=0,36422 |
P0=0,72459 |
P0=0,83326 |
P0=0,90306 |
P0=0,95200 |
P0=0,96931 |
P0=0,97728 |
P0=0,97926 |
P0=0,97991 |
P0=0,98014 |
P0=0,98027 |
|
F1 |
F5 |
F10 |
F20 |
F50 |
F100 |
F200 |
F300 |
F400 |
F500 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F4 |
F9 |
F19 |
F49 |
F99 |
F199 |
F299 |
F399 |
F499 |
F∞ |
|
μ=1,03 |
σ=0,34270 |
σ=0,20364 |
σ=0,0,12451 |
σ=0,07544 |
σ=0,06238 |
σ=0,05955 |
σ=0,05941 |
σ=0,05941 |
σ=0,05941 |
σ=0,05941 |
|
P0=0,35701 |
P0=0,70985 |
P0=0,81576 |
P0=0,88293 |
P0=0,92734 |
P0=0,93953 |
P0=0,94219 |
P0=0,94232 |
P0=0,94232 |
P0=0,94232 |
P0=0,94232 |
|
F1 |
F5 |
F10 |
F20 |
F50 |
F100 |
F200 |
F300 |
F400 |
F500 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F4 |
F9 |
F14 |
F19 |
F29 |
F49 |
F99 |
F199 |
F499 |
F∞ |
|
μ=1,05 |
σ=0,36380 |
σ=0,22605 |
σ=0,178533 |
σ=0,14942 |
σ=0,12412 |
σ=0,10655 |
σ=0,09901 |
σ=0,09838 |
σ=0,09838 |
σ=0,09838 |
|
P0=0,34994 |
P0=0,69503 |
P0=0,79768 |
P0=0,83918 |
P0=0,86121 |
P0=0,88327 |
P0=0,89893 |
P0=0,90573 |
P0=0,90630 |
P0=0,90630 |
P0=0,90630 |
|
F1 |
F5 |
F10 |
F15 |
F20 |
F30 |
F50 |
F100 |
F200 |
F500 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F4 |
F9 |
F14 |
F19 |
F29 |
F49 |
F99 |
F199 |
F499 |
F∞ |
|
μ=1,1 |
σ=0,41874 |
σ=0,28672 |
σ=0,24130 |
σ=0,22020 |
σ=0,20283 |
σ=0,19496 |
σ=0,19374 |
σ=0,19374 |
σ=0,19374 |
σ=0,19374 |
|
P0=0,33287 |
P0=0,65787 |
P0=0,75072 |
P0=0,78560 |
P0=0,80236 |
P0=0,81642 |
P0=0,82287 |
P0=0,82387 |
P0=0,82387 |
P0=0,82387 |
P0=0,82387 |
|
F1 |
F5 |
F10 |
F15 |
F20 |
F30 |
F50 |
F100 |
F200 |
F500 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F2 |
F5 |
F10 |
F15 |
F19 |
F29 |
F49 |
F99 |
F499 |
F∞ |
|
μ=1,5 |
σ=1,03226 |
σ=0,90753 |
σ=0,87729 |
σ=0,87450 |
σ=0,87426 |
σ=0,87421 |
σ=0,87421 |
σ=0,87421 |
σ=0,87421 |
σ=0,87421 |
|
P0=0,22313 |
P0=0,35620 |
P0=0,40352 |
P0=0,41591 |
P0=0,41707 |
P0=0,41717 |
P0=0,41719 |
P0=0,41719 |
P0=0,41719 |
P0=0,41719 |
P0=0,41719 |
|
F1 |
F3 |
F6 |
F11 |
F16 |
F20 |
F30 |
F50 |
F100 |
F500 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F8 |
F12 |
F16 |
F∞ |
|
μ=2 |
σ=1,72933 |
σ=1,64519 |
σ=1,61405 |
σ=1,60184 |
σ=1,59695 |
σ=1,59497 |
σ=1,59385 |
σ=1,59363 |
σ=1,59363 |
σ=1,59363 |
|
P0=0,13534 |
P0=0,17740 |
P0=0,19298 |
P0=0,19908 |
P0=0,20153 |
P0=0,20251 |
P0=0,20291 |
P0=0,20314 |
P0=0,20319 |
P0=0,20319 |
P0=0,20319 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F9 |
F13 |
F17 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
The primary quantity Q.
The allele a or the ancestor in F0 has
as the primary quantity Q. If Q is not 1 but 2; 3 or 1000, there are 2; 3 or
1000 distributions of the quantity 1, which is in the calculation the same as
one distribution of the quantities 2;3 or
With the quantitative Poisson distribution you
can calculate the superposition of the degrees in the distributions for
different values of Q and μ. That is a tough job when you
have only a calculator. I did make only a modest begin for Q=2 and μ=1 in 2 generations and only for the
P(∑i=0) in 3 generations. This is shown on Table 7. If you will
study the extinction you can use only the quantitative part of the Poisson
distribution, the exponential one with is accumulation through the generations.
Table 7
|
F0
|
|
|
|
|
|
|
|
|
|
|
F0→F1;
Q=2; λ=2 |
|
|
|
|
|
|
|
|
|
|
2→0 |
2→1 |
2→2 |
2→3 |
2→4 |
2→5 |
2→6 |
2→7 |
2→8 |
2→9 |
|
℮^-2 |
2.℮^-2 |
2.℮^-2 |
4/3. ℮^-2 |
2/3 .℮^-2 |
4/15℮^-2 |
4/45℮^-2 |
8/315℮^-2 |
2/315℮^-2 |
0,00141093℮^-2 |
|
0,135342 |
0,27067 |
0,27067 |
0,18045 |
0,09022 |
0,03609 |
0,01203 |
0,00344 |
0,00086 |
0,00019 |
|
i=0
F1 |
i=1
F1 |
i=2
F1 |
i=3
F1 |
i=4
F1 |
i=5
F1 |
i=6
F1 |
i=7
F1 |
i=8
F1 |
i=9
F1 |
|
F1→F2;
Q=2; μ=1 |
|
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
→0→7 |
→0→8 |
→0→9 |
|
0,135342 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
→1→7 |
→1→8 |
→1→9 |
|
2.℮^-3 |
2.℮^-3 |
℮^-3 |
1/3.℮^-3 |
1/12.℮^-3 |
1/60.℮^-3 |
1/360.℮^-3 |
1/2540.℮^-3 |
1/20160.℮^-3 |
5,5115E6.℮^-3 |
|
2→0 |
2→1 |
2→2 |
2→3 |
2→4 |
2→5 |
2→6 |
2→7 |
2→8 |
2→9 |
|
2.℮^-4 |
4. ℮^-4 |
4 ℮^-4 |
8/3 ℮^-4 |
4/3 ℮^-4 |
8/15 . ℮^-4 |
8/45 ℮^-4 |
16/315℮^-4 |
4/315℮^-4 |
0.0028219.℮^-4 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
→3→7 |
→3→8 |
→3→9 |
|
4/3.℮^-5 |
4.℮^-5 |
6.℮^-5 |
6.℮^-5 |
4,5.℮^-5 |
2,7.℮^-5 |
1,35.℮^-5 |
0,578571.℮^-5 |
0,21696.℮^-5 |
0,072321.℮^-5 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
→4→7 |
→4→8 |
→4→9 |
|
2/3.℮^-6 |
8/3.℮^-6 |
16/3.℮^-6 |
64/9.℮^-6 |
64/9.℮^-6 |
256/45.℮^-6 |
512/135.℮^-6 |
2048/945.℮^-6 |
1,0836.℮^-6 |
0.48156.℮^-6 |
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
→5→7 |
→5→8 |
→5→9 |
|
4/15.℮^-7 |
4/3.℮^-7 |
10/3.℮^-7 |
50/9.℮^-7 |
125/18.℮^-7 |
125/18.℮^-7 |
625/108.℮^-7 |
4,1336.℮^-7 |
2,5835.℮^-7 |
1,43528.℮^-7 |
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
→6→7 |
→6→8 |
→6→9 |
|
4/45℮^-8 |
8/15.℮^-8 |
1,6.℮^-8 |
3,2.℮^-8 |
4,8.℮^-8 |
5,76.℮^-8 |
5,76.℮^-8 |
4,93714.℮^-8 |
3,70286.℮^-8 |
2,46857.℮^-8 |
|
→7→0 |
→7→1 |
→7→2 |
→7→3 |
→7→4 |
→7→5 |
→7→6 |
→7→7 |
→7→8 |
→7→9 |
|
8/315℮^-9 |
8/45.℮^-9 |
28/45.℮^-9 |
196/135.℮^-9 |
343/135.℮^-9 |
2401/675.℮^-9 |
4,14988.℮^-9 |
4,14988.℮^-9 |
3,63114,℮^-9 |
2,82422.℮^-9 |
|
→8→0 |
→8→1 |
→8→2 |
→8→3 |
→8→4 |
→8→5 |
→8→6 |
→8→7 |
→8→8 |
→8→9 |
|
2/315.℮^-10 |
16/315.℮^-10 |
64/315.℮^-10 |
512/945.℮^-10 |
1,0836.℮^-10 |
1,73376.℮^-10 |
2,31168.℮^-10 |
2,64191.℮^-10 |
2,64191.℮^-10 |
2,34837.℮^-10 |
|
∑i=0
F2 |
∑i=1
F2 |
∑i=2
F2 |
∑i=3
F2 |
∑i=4
F2 |
∑i=5
F2 |
∑1=6
F2 |
∑
i=7 F2 |
∑i=8
F2 |
∑i=9
F2 |
|
0,147118255 |
0,207817487 |
0,180359676 |
0,129835019 |
0,084822699 |
0,051674392 |
0,029717982 |
0,0162779 |
0.00854897 |
0,004325 |
|
∑i=0
F0-F2 |
|
|
|
|
|
|
|
|
|
|
0,282453538 |
|
|
|
|
|
|
|
|
|
|
als
℮ function |
|
|
|
|
|
|
|
|
|
|
≈℮^(2/℮-2) |
|
|
|
|
|
|
|
|
|
|
F2→F3;
Q=2; μ=1 |
|
|
|
|
|
|
|
|
|
|
→0→0 |
|
|
|
|
|
|
|
|
|
|
0,282453538 |
|
|
|
0,02170.℮^-5 |
0,00592.℮^-5 |
|
|
|
|
|
→1→0 |
|
|
|
→4→10 |
→4→11 |
→4→12 |
→4→13 |
|
|
|
0,07645178 |
|
|
|
0,192640.℮^-6 |
0,07005.℮^-6 |
0,02335.℮^-6 |
0,00718.℮^-6 |
|
|
|
2→0 |
|
|
|
→5→10 |
→5→11 |
→5→12 |
→5→13 |
|
|
|
0,024409027 |
|
|
|
0,71764.℮^-7 |
0,3262.℮^-7 |
0,13592.℮^-7 |
0,05228.℮^-7 |
|
|
|
→3→0 |
|
|
|
→6→10 |
→6→11 |
→6→12 |
|
|
|
|
0,006464105 |
|
|
|
1,48114.℮^-8 |
0,8079.℮^-8 |
0,40395.℮^-8 |
|
|
|
|
→4→0 |
|
|
|
|
|
|
|
|
|
|
0,000883308 |
|
|
|
1,9767.℮^-9 |
1,25806.℮^-9 |
0,73387.℮^-9 |
|
|
|
|
→5→0 |
|
|
|
|
|
|
|
|
|
|
0,000348179 |
|
|
|
1,8787.℮^-10 |
1,36632.℮^-10 |
0,91088.℮^-10 |
|
|
|
|
→6→0 |
|
|
|
∑i=10
F2 |
∑i=11
F2 |
∑i=12
F2 |
∑i=13
F2 |
|
|
|
0,000073664 |
|
|
|
0,00210423 |
0,00099929 |
0,00044925 |
0,00006547 |
|
|
|
→7→0 |
|
|
|
|
|
|
|
|
|
|
0,000014835 |
|
|
|
|
|
|
|
|
|
|
→8→0 |
|
|
|
|
|
|
|
|
|
|
0,000002868 |
|
|
|
|
|
|
|
|
|
|
∑1=0
F3 |
|
|
|
|
|
|
|
|
|
|
0,108647765 |
|
|
|
|
|
|
|
|
|
|
∑i=0
F0-F3 |
|
|
|
|
|
|
|
|
|
|
0,391101303 |
|
|
|
|
|
|
|
|
|
|
≈℮^[2℮^(1/℮-1)-2] |
|
|
|
|
|
|
|
|
|
|
≈2.℮^(1/℮-2) |
|
|
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|
The Poisson-exponential distribution continued in degrees is not only a
law of the small numbers, because by the endless iteration also the large
numbers come into the picture. The primary distribution of for instance Q=λ=100 is nearly symmetrical. The P(i=0) is yet very
small, ℮^-100 and that is why this distribution is not to be discerned
from a normal distribution. However if the distribution of the Q=100 is
continued in the further degrees the P(i=o) will increase gradually. Such a
large large superposed distribution is to be calculated with a computer and the
right software. The calculation of the extinction only is much easier: In the
data for the Table 5 was the not rounded value of σ(F198) = 9,922274724.10^-3, so the P(0) for F199 =
℮^-0,009922274724 = 0,990126788. Following σ(g)=℮^{Qν(g-1)-Q} this x100 and afterwards -100 is -0,987321136.
The is the accumulated intensity σ for the 200th degree in
which Q=100 and λ=μ=1. So the P(0) on the 200th level here is ℮^-0,987321136
= 0,372573428. So the 100 alleles, present at the parent generation F0 all are
extinct in 37,3% of the cases at the 200th generation. In this way
it also to be calculated from the data that if Q=10 the σ of the F200 is
0,0982 so that the 10 alleles are extinct in 90,65% of the cases. So the trend
is that after 2Q generations the total extinction is in a little more than
℮^-1 or 37% of the cases and after 20Q generations the extinction is in
more than℮^-0,1 or more than 90% of the cases total. So if there are 1000
alleles, not much in a large population, the extinction up to more than 90%
should last as many as 20000 generations. This is not quite sure as far as
these calculations, but evident is that for any positive small value of x │[(℮^-x)
– 1]│ < │x│ is valid, so that the σ(Fg) of table 5
converges to 0 by increasing g and any value of Q.
The exponential
extinction in the Hardy-Weinberg population.
Because single and very rare alleles can disappear in one generation
they are evidently not in equilibrium in the Hardy-Weinberg population. Also
large but finite quantities are not in equilibrium in the infinite large
population and will ultimately disappear from it in a stochastically process.
In an infinite large population the frequencies and quantities of the alleles
are only constant if the quantities them selves are also infinite large. The
limited quantities always will fluctuate by random sampling. If someone
describes allele frequencies in an
infinite H-W population, the absolute
quantities of the alleles must also be infinite large in the H-W
population. The frequencies of alleles with distinct finite quantities would of
course always be zero in an infinite large population. So in allele frequencies
in the H-W population, as often described in the literature, it always concerns
infinite quantities. There are nevertheless also finite absolute quantities in
an infinite large H-W population and there are stochastic processes describing
the changes of these quantities. In this way is a quantity of for instance 10^6
is small in relation to ∞ and the superposed Poisson-exponential
distribution shows how this quantity disappears ultimately.
If the population size or the selection fluctuates
around a constant.
It generally occurs that populations on
account of the ecologic equilibriums are about constant in size at the long
term. By all kinds of changes in the life surroundings as droughts and
epidemics the size of the population often fluctuates as well. Also the
selection on fitness fluctuates mostly together to the fluctuating size.
Favourable or fit properties are fitting in distinct life surroundings. So for
instance in a time of great drought with low and more specific food offer other
properties are favourable than in a time of abundance and so are in the course
of the time always different circumstances that distinct the measure of the
fitness of the genes and their combinations. For instance the varying occurring
of different species of predators and competitors results also that sometimes
these and than again the other properties are favourable for the survival. The
selection also fluctuates by the linkage of the genes. Somewhat nearby on the
same chromosome localized genes are transferred linked during many generations
and are often only after long time independent of each other by the
recombination. The linked genes are passed through the generations as bigger or
smaller DNA fragments and the resultant of the different selective properties
of all the genetic varieties (alleles) within such a fragment determines of
course the selection factor for all the genes the fragment consists of. By the
cutting and fixing of the recombination there are fluctuating linkages and so
there is fluctuating selection on the distinct alleles. This means that alleles
being selectively neutral on the basis of their own properties are ever lifted
by their neighbours to positive and negative and alleles being positive them
selves transfer temporally with neutral, negative or strong positive selection.
The fluctuation in the selection of the alleles occurs generally and is so of
great importance.
The accumulating distribution and extinction
of the alleles is also fluctuating selection and population dynamics easily to
be found with –σ(Fg)=μν-μ. In this is μ=ps, so that μ has alternating values by the fluctuating and
the average value of it here is pointed out as λ= μ average. The extinction of the alleles for generation Fg than is ∑p(i=0)=℮^-σ for (Fg-1→Fg).
The particularity of the recurrence is in this to be taken into account:
the events should be calculated recurrent in the time. If the population
changed in the first generation with the factor 0,5 and in the second with the
factor 2; in the third again with 0,5 and in the forth 2, etc than the real
historical chronology is turned. So at first −σF0=(2x0)-2=-2 and afterwards −σF1=0,5(℮^−2)−0,5=−0,4323
than −σF2=2℮^[0,5(℮^-2)-0,5]-2=−0,7020
and at last −σF3=0,5℮^{2℮^[0,5(℮^−2)-0,5]-2}-0,5=−0.2522.
The P0 values of the extinction than are to be found with P(0)F1=℮^(2x0−2)=0,1353 and P(0)F2=℮^[0,5(℮^−2)−0,5]=0,6490,
etc. It is evident that in the calculations in this way the P0 values at
the distribution to the uneven generations →F1; →F3, etc as showed on Table 8a in italics, are not the real values
of the extinction for this generations. The calculation for this extinction
again is turned. The extinction to the uneven generations can be found from the
calculation of the values found in this way at the reverse phase. So the false
values in italics μ1=1 and μ2=0,5 on Table 8a are the right values for the extinction to the uneven
levels for μ1=0,5 en μ2=1. This is showed on Table 8b and c. On these tables it
is evident that the extinction in the increasing phases are smaller than on the
phases with deceasing population dynamics. In Table 8d are the accumulating
intensities σ also shown like on Table 5. On the Table 8d
are reproduced only the extinction values to the even levels, so after the
total binary phases. The extinction values at the fluctuation between μ1 and μ2 can be compared with the average extinction according to the constant λ.
The proof for the correctness of these estimations
can be found by the superposing and the full elaboration of the Poisson
distributions. The superposed distributions for μ=2 and μ=0,5 are
elaborated in a very restrictive measure in Table 11. The bold printed
values in table 8 b and c are affirmed by Table 11. This
by itself is definitely not a general deduced proof, but it is possible to
produce unlimited much evidence, inductively and experimentally, by elaborating
these distributions. Besides it may be easy for a mathematician to find the
exact deductive proves by infinitesimal calculus, but I do work in a more
philosophical way: In this view it is a consequence of the qualitative property
of the exponential distribution. The exponential distribution according to P(i=0)
= ℮^-λ and
P(i=n) = 1 - ℮^-λ is in principle a
distribution of the quantity 1, to be hit or not to be hit, so that the
intensity λ should be always λ=1. The number n distributions of the
quantity and intensity λ=1 gives in the calculations however the same
result as 1 distribution with λ=n
at the normal primary exponential distribution and this is much easier to work
with. In the accumulated exponential distribution of the further degrees the principle
difference between n distributions of
λ=1 and 1 distribution of
λ=n of also of practical importance. This induces a perpetual
turning of the intensities as a consequence of the dialectical negation. That
is why the distribution on the one side with the intensity μ2 and on the
other hand the exponential summation of the primary quanta Q, which is here
μ1 as the intensity and also the result of the former distribution,
alternate each other continuous in the fluctuation. In this is, as in the primary
quant Q, at first is to be calculated the distribution with μ2 and
afterwards follows the summation of μ1 of these distributions. In
this elaboration are calculated only the real values after the total binary
phase of the fluctuation, because the values for the uneven levels in Table
8a are the distributions of the even levels before (and without) the
summations. The complement of this, the alternate phase than appears to give
the right data for the uneven levels This again is the dialectic mirror.
So by the fluctuation between the intensity
μ1
and μ2 in
the extinction of the neutral allele (λ=1) there is no change in
the point of convergence, so that the extinction remains complete. However if
the fluctuation begins with a decrease in the population the extinction
proceeds faster and if it begins with an increase, thus in the rising phase, it
proceeds slower than the constant extinction. The point of convergence of the
extinction and so the chance on fixation however is changed by the theoretical
positive allele (λ>1, because ps>1). The change of
fixation is smaller by start in the sinking phase and larger by start in the
rising phase than in the constant population. So new mutations have a larger
change to be absorbed by the population if they initiate during a phase with
increasing population or with temporal positive selection. The problem is
however that we often do not know which phase is primary. Sometimes we have an
impression of it: natural populations do have some decided size in the ecologic
equilibrium, which than is repeatedly broken by “catastrophes”. This kind of
bottleneck situations do indicate lower minima in the population size and thus
on a start in the sinking phase. This distinction is more difficult by the
fluctuating selection and at the non poissonic parities the phases are in fact
at the same time. So than we do have in fact than 2 or more points of
convergence and fixation by the fluctuation cause by the non poissonic
parities.
In general a population grows exponential,
with a constant factor, until limitations come from internal (physiological) or external ( ecological) factors. The
increase is decelerating by this and the population comes at last in
equilibrium with its limitations. That is why the growing curve of for instance
a bacterial colony in the laboratory has a s-form aspect. In the nature there
are yet often changes in the external factors, occurring as disasters like
droughts, inundations, epidemics etc. The growth of the population often is
snapped off in the phase with the constant growing factor and the population
decreases than sharply in short time. The curve of the turning size of the
population has by this probably often a sewing teeth aspect. Starting from the
minimal population size the points are upwards with two dual phases like this: 
. The sew must be turned
horizontal and vertical if you start from the maximum population size, like 
One dual phase with start from the
about average population size looks like 
. This is the most probable situation at the varying population size in
the actual nature. However the transfer of the descendants and alleles is
decided by the smallest population size and so the minimal population size with
and start in the
rising phase is the real starting point for the theory of the neutral
population dynamics. In Table 8e the extinction is calculated in an
unlimited population with fluctuation μ1=100 and μ2=0,01. At the top of Table 8e are again the unreal calculation
values in italics. In the first two columns of
Table 8e this fluctuation is symmetric. In this the increase and
decrease with factors respectively μ1=100 and μ2=0,01 are each within one generation, so there is fluctuation around the
average λ=1. In the following 11 columns of Table 8e the increase is
divided over 10 generations each with μ1=1,585 so that the
total increase also is ς1=1,585^10≈100. The fluctuations in the allele transfer as a result
of the variations in the population size do have probably often this sewing
teeth aspect of Table 8e. Fluctuations by variations in the selection
will have mostly a more symmetrical aspect. The asymmetrical fluctuations are
important because the population has an increased capacity by it to absorb
mutations in the prolonged rising phases. This also is shown on Table 8e,
compared with the constant extinction of Table 5.
The importance of the fluctuation is getting
even more important owing to non Poisson distributed reproduction. As pointed
out the parities often are not Poisson distributed this is partly or total
caused by the non random selection. These larger varieties in the parities can
however also (partly or total) be caused by random differences. These random
differences in the parities can be approximated as extinction with fluctuating
intensity. So the expectation is that the fluctuation of the intensity occurs
very general in natural population by a combination of different causes. The
selection introduces itself as the resultant of a lot of fluctuations and
variations.
Tabel 8a
|
F0 λ=1 |
F1; P0= |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
μ1=2 P0=0 |
|
0,455236288 |
|
0,620721229 |
|
0,70787681 |
|
0,76201347 |
|
0,799017493 |
|
μ2=0,5 |
0,606530659 |
|
0,761563398 |
|
0,827257401 |
|
0,864104477 |
|
0,88781378 |
|
|
F0 λ=1 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
μ1=0,5 |
0,135335283 |
|
0,495586824 |
|
0,640291569 |
|
0,719487576 |
|
0,769719626 |
|
|
μ2=2 P0=0 |
|
0,648993642 |
|
0,777084185 |
|
0,835391989 |
|
0,869135523 |
|
0,891241194 |
Tabel 8b
|
F0 λ=1 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
μ1=2 |
0,135335283 |
|
0,495586824 |
|
0,640291569 |
|
0,719487576 |
|
0,769719626 |
|
|
μ2=0,5
P0=0 |
|
0,455236288 |
|
0,620721229 |
|
0,70787681 |
|
0,76201347 |
|
0,799017493 |
|
F11 |
F12 |
F13 |
F14 |
F15 |
F16 |
F17 |
F18 |
F19 |
F20 |
F21 |
|
0,804513437 |
|
0,830077154 |
|
0,849672275 |
|
0,865180565 |
|
0,877765529 |
|
0,888185996 |
|
|
0,825955852 |
|
0,846465125 |
|
0,862612599 |
|
0,875662272 |
|
0,886431324 |
|
|
F22 |
F23 |
F24 |
F25 |
F26 |
F27 |
F28 |
F29 |
F30 |
F31 |
F32 |
|
|
0,896958347 |
|
0,904446376 |
|
0,910913877 |
|
0,916556861 |
|
0,921524046 |
|
|
0,895471941 |
|
0,903170885 |
|
0,909807246 |
|
0,91558754 |
|
0,9206679 |
|
0,925168586 |
Tabel 8c
|
F0 λ=1 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
μ1=0,5
P0=0 |
|
0,648993642 |
|
0,777084185 |
|
0,835391989 |
|
0,869135523 |
|
0,891241194 |
|
μ2=2 |
0,606530659 |
|
0,761563398 |
|
0,827257401 |
|
0,864104477 |
|
0,88781378 |
|
|
F11 |
F12 |
F13 |
F14 |
F15 |
F16 |
F17 |
F18 |
F19 |
F20 |
F21 |
|
|
0,906881687 |
|
0,918547718 |
|
0,927591476 |
|
0,934812114 |
|
0,940712948 |
|
|
0,904393022 |
|
0,916656861 |
|
0,926105205 |
|
0,933612601 |
|
0,939724187 |
|
0,944797804 |
|
F22 |
F23 |
F24 |
F25 |
F26 |
F27 |
F28 |
F29 |
F30 |
F31 |
F32 |
|
0,945627073 |
|
0,949783869 |
|
0,953346538 |
|
0,956434414 |
|
0,959136796 |
|
0,961521862 |
|
|
0,949078249 |
|
0,95273874 |
|
0,95590535 |
|
0,958672053 |
|
0,961110348 |
|
Tabel 8d
|
|
|
|
|
|
|
|
|
|
|
|
|
F1 |
F3 |
F5 |
F7 |
F9 |
F11 |
F13 |
F15 |
F17 |
F19 |
λ=1 |
|
σ=0,6321 |
σ=0,3741 |
σ=0,2681 |
σ=0,2095 |
σ=0,1723 |
σ=0,1464 |
σ=0,1273 |
σ=0,1127 |
σ=0,1011 |
σ=0,0916 |
μ1=1 |
|
P0=0,531 |
P0=0,688 |
P0=0,765 |
P0=0,811 |
P0=0,842 |
P0=0,864 |
P0=0,880 |
P0=0,893 |
P0=0,904 |
P0=0,912 |
μ2=1 |
|
F2 |
F4 |
F6 |
F8 |
F10 |
F12 |
F14 |
F16 |
F18 |
F20 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F1 |
F3 |
F5 |
F7 |
F9 |
F11 |
F13 |
F15 |
F17 |
F19 |
|
|
σ=0,4323 |
σ=0,2522 |
σ=0,1799 |
σ=0,1403 |
σ=0,1151 |
σ=0,0977 |
σ=0,0850 |
σ=0,0752 |
σ=0,0674 |
σ=0,0611 |
λ=1 |
|
P0=0,649
|
P0=0,777 |
P0=0,835 |
P0=0,869 |
P0=0,891 |
P0=0,907 |
P0=0,919 |
P0=0,928 |
P0=0,935 |
P0=0,941 |
μ1=0,5 |
|
F2 |
F4 |
F6 |
F8 |
F10 |
F12 |
F14 |
F16 |
F18 |
F20 |
μ2=2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
F1 |
F3 |
F5 |
F7 |
F9 |
F11 |
F13 |
F15 |
F17 |
F19 |
|
|
σ=0,7869 |
σ=0,4768 |
σ=0,3455 |
σ=0,2718 |
σ=0,2244 |
σ=0,1913 |
σ=0,1667 |
σ=0,1478 |
σ=0,1328 |
σ=0,1206 |
λ=1 |
|
P0=0,455 |
P0=0,621 |
P0=0,708 |
P0=762 |
P0=0,799 |
P0=0,826 |
P0=0,846 |
P0=0,863 |
P0=0,876 |
P0=0,886 |
μ1=2 |
|
F2 |
F4 |
F6 |
F8 |
F10 |
F12 |
F14 |
F16 |
F18 |
F20 |
μ2=0,5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F1 |
F3 |
F5 |
F7 |
F9 |
F11 |
F13 |
F15 |
F17 |
F19 |
|
|
σ=1,165 |
σ=0,9657 |
σ=0,9075 |
σ=0,8869 |
σ=0,8791 |
σ=0,8761 |
σ=0,8749 |
σ=0,8745 |
σ=0,8743 |
σ=0,8743 |
λ=1,5 |
|
P0=0,311 |
P0=0,381 |
P0=0,404 |
P0=0,412 |
P0=0,415 |
P0=0,416 |
P0=0,417 |
P0=0,417 |
P0=0,417 |
P0=0,417 |
μ1=1,5 |
|
F2 |
F4 |
F6 |
F8 |
F10 |
F12 |
F14 |
F16 |
F18 |
F20 |
μ2=1,5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
F1 |
F3 |
F5 |
F7 |
F9 |
F11 |
F13 |
F15 |
F17 |
F19 |
|
|
σ=1,5829 |
σ=1,3469 |
σ=1,2777 |
σ=1,2535 |
σ=1,2445 |
σ=1,2411 |
σ=1,2398 |
σ=1,2393 |
σ=1.2392 |
σ=1,2391 |
λ=1,5 |
|
P0=0,205 |
P0=0,260 |
P0=0,279 |
P0=0,286 |
P0=0,288 |
P0=0,289 |
P0=0,289 |
P0=0,290 |
P0=0,290 |
P0=0,290 |
μ1=0,75 |
|
F2 |
F4 |
F6 |
F8 |
F10 |
F12 |
F14 |
F16 |
F18 |
F20 |
μ2=3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
F1 |
F3 |
F5 |
F7 |
F9 |
F11 |
F13 |
F15 |
F17 |
F19 |
|
|
σ=0,7127 |
σ=0,5874 |
σ=0,5522 |
σ=0,5400 |
σ=0,5355 |
σ=0,5338 |
σ=0,5332 |
σ=0,5329 |
σ=0,5328 |
σ=0,5328 |
λ=1,5 |
|
P0=0,490 |
P0=0,556 |
P0=0,576 |
P0=0,583 |
P0=0,585 |
P0=0,586 |
P0=0,587 |
P0=0,587 |
P0=0,587 |
P0=0,587 |
μ1=3 |
|
F2 |
F4 |
F6 |
F8 |
F10 |
F12 |
F14 |
F16 |
F18 |
F20 |
μ2=0,75 |
|
|
|
|
|
|
|
|
|
|
|
|
Tabel 8e
|
F0 |
F1 |
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F11 |
|
σ=0,01 |
σ=0,995 |
|
σ=0,01 |
σ=0,0158 |
σ=0,0248 |
σ=0,0388 |
σ=0,0604 |
σ=0,0928 |
σ=0,1405 |
σ=0,2078 |
σ=0,2974 |
σ=0,4077 |
σ=0,5307 |
|
P0=0,990 |
PO=0,368 |
|
P0=0,990 |
P0=0,984 |
P0=0976 |
P0=0,962 |
P0=0,941 |
P0=0,911 |
P0=0,869 |
P0=0,812 |
P0=0,743 |
P0=0,665 |
P0=0,588 |
|
F1 |
F2 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
F12 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F11 |
|
μ1=100 |
μ2=0,01 |
|
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ1=1,585 |
μ2=0,01 |
|
σ=100 |
σ=0,995 |
|
σ=1.585 |
σ=1,260 |
σ=1,135 |
σ=1,076 |
σ=1,044 |
σ=1,027 |
σ=1.018 |
σ=0,012 |
σ=1,009 |
σ=1,007 |
σ=0,5307 |
|
P0=0,000 |
PO=0,368 |
|
P0=0,205 |
P0=0,284 |
P0=0,321 |
P0=0,341 |
P0=0,351 |
P0=0,358 |
P0=0,361 |
P0=0,363 |
P0=0,365 |
P0=0,365 |
P0=0,588 |
|
F1 |
F2 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
F12 |
Discussion about
Table 4 and the specified distributions.
Table 4 shows the transfer of the alleles to the next
generation as one uniform process, but there may be two processes in the
biologic reality. There is yet one distribution of the numbers of offspring
among the parents, the parities, and the second distribution is the endowment (0
or 1) of the alleles from the parent to the individual offspring. Table 9 describes
the random Poisson distributed parities with the average of 2 children and
superposed over this is the endowment of the alleles to the real children. The
distribution of the alleles of the heterozygote parent of F0, with genotype ab,
is calculated with the binomium (a+b)^q. By means of the arrows the
superposition of the binomium is easily to understand. For instance: ¼ of the
parents with 2 children does not transfer the allele a to F1, ½ transfers the
allele a singular and ¼ transfers the allele a in twofold. The sums of the
columns are products of ℮ on behave of converging and it appears that the
total distribution of the alleles over the offspring is a Poisson distribution
with intensity the half of the intensity of the distribution of only the
offspring. The sum of one divided by all the faculties, so 1+1+1/2+1/6+1/24.. etc
is equal to ℮, so that
℮^−2(1+1+1/2+1/6+1/24.. etc)=℮^-1. If the series
begins with ½ and it goes further on with 1/2x1/1; x1/2; x1/3, etc. than the
sum is ½℮. If the series begins with 1/b, the sum is 1/b x ℮. More
convenient notations are ∑ 1/n!=℮ and b∑(1/n!)=b℮. So in the general the random
allele transfer can be considered as one uniform process. In special situations
however it may be better to be described as two processes. In table 9a
is the same distribution of the parities as in table 9, but in this
population is inbreeding with f=0,2.
The change that the parent pass any of their alleles to the next generation
respectively do not pass their alleles is in out breeding populations equal on 0,5
and 0,5. Because the alleles of the partners are in this inbreeding population
in 20% of the cases identical, these chances here are respectively 0,6 and 0,4.
Yet the child receives besides the Mendelian 0,5 of allele a from its father
here also 0,5x0,2=0,1 of allele a from its mother, so in total 0,6 of the cases
allele a is transferred to the child and this is true for allele b as well. So
the Poisson distribution is in table 9a superposed by an asymmetrical
binomium based on (0,6+0,4)^q. The superposition is marked by the arrows and it
appears now indeed that this asymmetrical binomium over the Poisson
distribution with λ=2 results in a Poisson distribution with λ=1,2.
This is also the case in general for other values of f according to the converging series b∑(1/n!)=b℮.
You should possibly pose the statement Table
9a describes also the situation of selection for the allele that is
selective advantageous with the factor s=1,2. This however is not true
in the light of the physical biological events, because selective differences
in the endowment of the alleles to the real individuals in the reproduction
population are impossible. The children always receive the Mendelian
half of the alleles from their parents, so the endowment (0 or 1) of the
alleles to the existent individuals is always at random. The only source of
selective or non-random differences is the reproductive distribution of the
parities. The real existing reproductive selection will cause many fluctuations
and variation in the allele transfer, that result ultimately in the allele
selection. How the allele selection is effected from the physical observable
reproduction selection here is described only superficially and partly on the
tables. The allele selection is nevertheless described in the classical theory
as in Table 5b and 9a, but it seems better to call this the
theoretical or virtual allele selection.
Table 9
|
|
|
|
|
|
|
|
|
|
|
|
|
Descendants of F0 in F1 |
|
|
|
|
|
|
|
|||
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
|
℮^-2 |
2.℮^-2 |
2.℮^-2 |
4/3. ℮^-2 |
2/3 .℮^-2 |
4/15℮^-2 |
4/45℮^-2 |
8/315℮^-2 |
2/315℮^-2 |
4/2835℮^-2 |
|
|
0,135342 |
0,27067 |
0,27067 |
0,18045 |
0,09022 |
0,03609 |
0,01203 |
0,00344 |
0,00086 |
0,00019 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
→0n→0a |
→0n→1a |
→0n→2a |
→0n→3a |
→0n→4a |
→0n→5a |
→0n→6a |
→0n→7a |
→0n→8a |
→0n→9a |
|
|
℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→1n→0a |
→1n→1a |
→1n→2a |
→1n→3a |
→1n→4a |
→1n→5a |
→1n→6a |
→1n→7a |
→1n→8a |
→1n→9a |
|
|
℮^-2 |
℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→2n→0a |
→2n→1a |
→2n→2a |
→2n→3a |
→2n→4a |
→2n→5a |
→2n→6a |
→2n→7a |
→2n→8a |
→2n→9a |
|
|
1/2.℮^-2 |
℮^-2 |
1/2.℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→3n→0a |
→3n→1a |
→3n→2a |
→3n→3a |
→3n→4a |
→3n→5a |
→3n→6a |
→3n→7a |
→3n→8a |
→3n→9a |
|
|
1/6.℮^-2 |
1/2.℮^-2 |
1/2.℮^-2 |
1/6.℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→4n→0a |
→4n→1a |
→4n→2a |
→4n→3a |
→4n→4a |
→4n→5a |
→4n→6a |
→4n→7a |
→4n→8a |
→4n→9a |
|
|
1/24.℮^-2 |
1/6.℮^-2 |
1/4.℮^-2 |
1/6.℮^-2 |
1/24.℮^-2 |
0 |
0 |
0 |
0 |
0 |
|
|
→5n→0a |
→5n→1a |
→5n→2a |
→5n→3a |
→5n→4a |
→5n→5a |
→5n→6a |
→5n→7a |
→5n→8a |
→5n→9a |
|
|
1/120.℮^-2 |
1/24.℮^-2 |
1/12.℮^-2 |
1/12.℮^-2 |
1/24.℮^-2 |
1/120.℮^-2 |
0 |
0 |
0 |
0 |
|
|
→6n→0a |
→6n→1a |
→6n→2a |
→6n→3a |
→6n→4a |
→6n→5a |
→6n→6a |
→6n→7a |
→6n→8a |
→6n→9a |
|
|
1/720℮^-2 |
1/120℮^-2 |
1/48℮^-2 |
1/36.℮^-2 |
1/48.℮^-2 |
1/120℮^-2 |
1/720℮^-2 |
0 |
0 |
0 |
|
|
→7n→0a |
→7n→1a |
→7n→2a |
→7n→3a |
→7n→4a |
→7n→5a |
→7n→6a |
→7n→7a |
→7n→8a |
→7n→9a |
|
|
1/5040℮^-2 |
1/720℮^-2 |
1/240℮^-2 |
1/144℮^-2 |
1/144℮^-2 |
1/240℮^-2 |
1/720℮^-2 |
1/5040℮^-2 |
0 |
0 |
|
|
→8n→0a |
→8n→1a |
→8n→2a |
→8n→3a |
→8n→4a |
→8n→5a |
→8n→6a |
→8n→7a |
→8n→8a |
→8n→9a |
|
|
1/40320℮^-2 |
1/5040℮^-2 |
1/1440℮^-2 |
1/720℮^-2 |
1/576℮^-2 |
1/720℮^-2 |
1/1440℮^-2 |
1/5040℮^-2 |
1/40320℮^-2 |
0 |
|
|
→9n→0a |
→9n→1a |
→9n→2a |
→9n→3a |
→9n→4a |
→9n→5a |
→9n→6a |
→9n→7a |
→9n→8a |
→9n→9a |
|
|
1/362880℮^-2 |
1/40320℮^-2 |
1/10080℮^-2 |
1/4320℮^-2 |
1/2880℮^-2 |
1/2880℮^-2 |
1/4320℮^-2 |
1/10080℮^-2 |
1/40320℮^-2 |
1/362880℮^-2 |
|
|
+… |
+... |
+... |
+... |
+... |
+... |
+... |
+... |
+... |
+... |
|
|
∑ 0a |
∑
1a |
∑
2a |
∑
3a |
∑
4a |
∑
5a |
∑
6a |
∑
7a |
∑ 8a |
∑
9a |
|
|
℮^-1 |
℮^-1 |
1/2℮^-1 |
1/6℮^-1 |
1/24℮^-1 |
1/120℮^-1 |
1/720℮^-1 |
1/5040℮^-1 |
1/40320℮^-1 |
1/362880℮^-1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Table 9a
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
|
℮^-2 |
2.℮^-2 |
2.℮^-2 |
4/3. ℮^-2 |
2/3 .℮^-2 |
4/15℮^-2 |
4/45℮^-2 |
8/315℮^-2 |
2/315℮^-2 |
4/2835℮^-2 |
|
|
0,135342 |
0,27067 |
0,27067 |
0,18045 |
0,09022 |
0,03609 |
0,01203 |
0,00344 |
0,00086 |
0,00019 |
λ=2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
→0n→0a |
→0n→1a |
→0n→2a |
→0n→3a |
→0n→4a |
→0n→5a |
→0n→6a |
→0n→7a |
→0n→8a |
→0n→9a |
|
|
℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→1n→0a |
→1n→1a |
→1n→2a |
→1n→3a |
→1n→4a |
→1n→5a |
→1n→6a |
→1n→7a |
→1n→8a |
→1n→9a |
|
|
0,8℮^-2 |
1,2℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→2n→0a |
→2n→1a |
→2n→2a |
→2n→3a |
→2n→4a |
→2n→5a |
→2n→6a |
→2n→7a |
→2n→8a |
→2n→9a |
|
|
0,32℮^-2 |
0,96℮^-2 |
0,72℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→3n→0a |
→3n→1a |
→3n→2a |
→3n→3a |
→3n→4a |
→3n→5a |
→3n→6a |
→3n→7a |
→3n→8a |
→3n→9a |
|
|
0,085333.℮^-2 |
0,384.℮^-2 |
0,576.℮^-2 |
0,288.℮^-2 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
→4n→0a |
→4n→1a |
→4n→2a |
→4n→3a |
→4n→4a |
→4n→5a |
→4n→6a |
→4n→7a |
→4n→8a |
→4n→9a |
|
|
0,017067.℮^-2 |
0,1024.℮^-2 |
0,2304.℮^-2 |
0,2304.℮^-2 |
0,0864.℮^-2 |
0 |
0 |
0 |
0 |
0 |
|
|
→5n→0a |
→5n→1a |
→5n→2a |
→5n→3a |
→5n→4a |
→5n→5a |
→5n→6a |
→5n→7a |
→5n→8a |
→5n→9a |
|
|
0,002731.℮^-2 |
0,02048.℮^-2 |
0,06144.℮^-2 |
0,09216.℮^-2 |
0,06912.℮^-2 |
0,020736.℮^-2 |
0 |
0 |
0 |
0 |
|
|
→6n→0a |
→6n→1a |
→6n→2a |
→6n→3a |
→6n→4a |
→6n→5a |
→6n→6a |
→6n→7a |
→6n→8a |
→6n→9a |
|
|
0.000364.℮^-2 |
0,0032768℮^-2 |
0,012288.℮^-2 |
0,024576.℮^-2 |
0,027648.℮^-2 |
0,016589.℮^-2 |
0,004147.℮^-2 |
0 |
0 |
0 |
|
|
→7n→0a |
→7n→1a |
→7n→2a |
→7n→3a |
→7n→4a |
→7n→5a |
→7n→6a |
→7n→7a |
→7n→8a |
→7n→9a |
|
|
4,161E-5.℮^-2 |
0,000437.℮^-2 |
0,001966.℮^-2 |
0,004915℮^-2 |
0,007373℮^-2 |
0,006636℮^-2 |
0,003318℮^-2 |
0,000711℮^-2 |
0 |
0 |
|
|
→8n→0a |
→8n→1a |
→8n→2a |
→8n→3a |
→8n→4a |
→8n→5a |
→8n→6a |
→8n→7a |
→8n→8a |
→8n→9a |
|
|
4,16E-6.℮^-2 |
4,993E-5.℮^-2 |
2,621E-4.℮^-2 |
7,864E-4℮^-2 |
0,001475.℮^-2 |
0,001769℮^-2 |
0,001327.℮^-2 |
5,688E-4.℮^-2 |
1,066E-4.℮^-2 |
0 |
|
|
→9n→0a |
→9n→1a |
→9n→2a |
→9n→3a |
→9n→4a |
→9n→5a |
→9n→6a |
→9n→7a |
→9n→8a |
→9n→9a |
|
|
3,699E-7.℮^-2 |
4,438E-6℮^-2 |
2,996E-5℮^-2 |
1,04877E-4℮^-2 |
2,3593E-4℮^-2 |
3,53894E-4℮^-2 |
3,53894E-4℮^-2 |
2,28773E-4℮^-2 |
8,5314E-5℮^-2 |
1,4219E-5℮^-2 |
|
|
+… |
+... |
+... |
+... |
+... |
+... |
+... |
+... |
+... |
+... |
|
|
∑ 0a |
∑
1a |
∑
2a |
∑
3a |
∑
4a |
∑
5a |
∑
6a |
∑
7a |
∑ 8a |
∑
9a |
|
|
2,2255756℮^-2 |
2,6706482℮^-2 |
1,602386℮^-2 |
0,6409423℮^-2 |
0,19223393℮^-2 |
0,0460839℮^-2 |
0,0091509℮^-2 |
0,0001509℮^-2 |
0,0001919℮^-2 |
1,4219E-5℮^-2 |
|
|
0,3011989 |
0,3614329 |
0,2168594 |
0,086721 |
0,026016 |
0,0062368 |
0,0012384 |
0.000204163 |
2,597E-05 |
1,92E-06 |
population |
|
0,30119 |
0,36143 |
0,21686 |
0,08674 |
0,02602 |
0,00625 |
0,00125 |
0,00021 |
3,20E-05 |
4,30E-06 |
λ=1,2 |
|
|
|
|
|
|
|
|
|
|
|
|
Superposition binomium on the non random reproduction
Also if the reproduction is
not exact Poisson distributed, as is mostly the case in the practice of the
nature because the changes are not total equal and the events are not totally
at random in the time, the calculations with the specified distributions of Table 9 should be used instead of the
uniform distribution of Table 4. The
endowment of the alleles (0 or 1) always is total at random, but the
distribution of the reproduction - the parities over the parents - may be non
random. As pointed out the non Poisson, or non random, distribution of the
reproduction in natural populations can
be observed and measured. If you want information over the transfer and
distribution of the alleles in these populations you can superpose the binomium
of the endowment of the alleles over the observed non Poisson distributed
reproduction. It than is to be expected that the allele distribution is more
like the Poisson distribution, because of the random endowment of the alleles.
In the Tables 9b and 9c is the distribution of the
reproduction according to the measurement at first compared with the Poisson
distribution. Further is the endowment of the alleles posed on these observed
natural distributions. Table 9b
concerns a population with a variety in the distribution of the reproduction
smaller than the Poisson distribution does. In this is an excess in the
parities with about the average number of children of the population and there
is a shortage or equal in the extremes on both
sides in relation to the Poisson distribution. The data of Table 9b are taken from the date of the US
Census bureau. These are data of 7,2 million women from the American subpopulation:
all the English speaking white women in the age of 40-44 years in 2004. So
women that have accomplished their families to 99%. But they are data from the actual
reproduction. In Table 9c the
population has distribution with yet a larger variation than the Poisson
does. Here is a shortage in the about
average values and an excess in the extremes on both sides. This is a
population of 72 parents that lived in the past in the
Table 9b
|
WHITE
ONLY, NOT HISPANIC |
Women
Ever Married |
n
= 7,2 E6 |
|
|
|
|
|||
|
Numbers of children par woman 40-44 jr |
Average number of children par woman is 1,959 |
|
|
|
|||||
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
|
0,141 |
0,172 |
0,398 |
0,196 |
0,071 |
0,01425 |
0,00475 |
0,00156 |
0,00044 |
|
|
Poisson
λ=1,959 |
|
|
|
|
|
|
|
λ=1,959 |
|
|
0,141 |
0,27622 |
0,27056 |
0,17667 |
0,08653 |
0,04498 |
0,01107 |
0,0031 |
0,00076 |
|
|
→0n→0a |
→0n→1a |
→0n→2a |
→0n→3a |
→0n→4a |
→0n→5a |
→0n→6a |
→0n→7a |
→0n→8a |
|
|
0,141 |
|
|
|
|
|
|
|
|
|
|
→1n→0a |
→1n→1a |
→1n→2a |
→1n→3a |
→1n→4a |
→1n→5a |
→1n→6a |
→1n→7a |
→1n→8a |
|
|
0,086 |
0,086 |
|
|
|
|
|
|
|
|
|
→2n→0a |
→2n→1a |
→2n→2a |
→2n→3a |
→2n→4a |
→2n→5a |
→2n→6a |
→2n→7a |
→2n→8a |
|
|
0,0995 |
0,199 |
0,0995 |
|
|
|
|
|
|
|
|
→3n→0a |
→3n→1a |
→3n→2a |
→3n→3a |
→3n→4a |
→3n→5a |
→3n→6a |
→3n→7a |
→3n→8a |
|
|
0,0245 |
0,0735 |
0,0735 |
0,0245 |
|
|
|
|
|
|
|
→4n→0a |
→4n→1a |
→4n→2a |
→4n→3a |
→4n→4a |
→4n→5a |
→4n→6a |
→4n→7a |
→4n→8a |
|
|
0,0044375 |
0,01775 |
0,026625 |
0,01775 |
0,0044375 |
|
|
|
|
|
|
→5n→0a |
→5n→1a |
→5n→2a |
→5n→3a |
→5n→4a |
→5n→5a |
→5n→6a |
→5n→7a |
→5n→8a |
|
|
0,00045968 |
0,002298 |
0,0045968 |
0,00459677 |
0,00229839 |
0,00045968 |
|
|
|
|
|
→6n→0a |
→6n→1a |
→6n→2a |
→6n→3a |
→6n→4a |
→6n→5a |
→6n→6a |
→6n→7a |
→6n→8a |
|
|
0,0000742 |
0,00044531 |
0,00111328 |
0,0014844 |
0,00111328 |
0,00044531 |
0,0000742 |
|
|
|
|
→7n→0a |
→7n→1a |
→7n→2a |
→7n→3a |
→7n→4a |
→7n→5a |
→7n→6a |
→7n→7a |
→7n→8a |
|
|
1,21875E-05 |
0,00008531 |
0,0002559 |
0,00042656 |
0,00042656 |
0,0002559 |
0,00008531 |
1,21875E-05 |
|
|
|
→8n→0a |
→8n→1a |
→8n→2a |
→8n→3a |
→8n→4a |
→8n→5a |
→8n→6a |
→8n→7a |
→8n→8a |
|
|
0,00000172 |
0,00001375 |
0,000048125 |
0,00009625 |
0,00012031 |
0,00009625 |
0,000048125 |
0,00001375 |
0,00000172 |
|
|
∑ 0a |
∑
1a |
∑
2a |
∑
3a |
∑
4a |
∑
5a |
∑
6a |
∑
7a |
∑ 8a |
|
|
0,351485 |
0,379092 |
0,205639 |
0,048853 |
0,008396 |
0,0012571 |
0,0002076 |
0,00002594 |
0,00000172 |
∑p=0,994957 |
|
Poisson
λ=0,9759 |
|
|
|
|
|
|
|
∑a=0,978226 |
|
|
0,3755 |
0,3664 |
0,1795 |
0,0584 |
0,0142 |
0,00278 |
0,00045 |
0,00006 |
0,000008 |
|
|
|
|
|
|
|
|
|
|
|
|
Table 9c
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
Poisson,
λ=3,05556 |
|
|
|
|
|
|
|
|
|
|
0,0471 |
0,14391 |
0,21986 |
0,22393 |
0,17106 |
0,10453 |
0,05324 |
0,02324 |
0,00888 |
0,00301 |
|
population
n=72 |
Average
3,0555 children per parent |
|
|
|
|
||||
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
0,09722 |
0,194444 |
0,22222 |
0,11111 |
0,06944 |
0,13889 |
0,08333 |
0,06944 |
0 |
0,01389 |
|
7/72 |
14/72 |
16/72 |
8/72 |
5/72 |
10/72 |
6/72 |
5/72 |
0 |
1/72 |
|
→0n→0a |
→0n→1a |
→0n→2a |
→0n→3a |
→0n→4a |
→0n→5a |
→0n→6a |
→0n→7a |
→0n→8a |
→0n→9a |
|
7/72 |
0 |
0 |
0 |
0 |
|
|
|
|
|
|
→1n→0a |
→1n→1a |
→1n→2a |
→1n→3a |
→1n→4a |
→1n→5a |
→1n→6a |
→1n→7a |
→1n→8a |
→1n→9a |
|
7/72 |
7/72 |
0 |
|
|
|
|
|
|
|
|
→2n→0a |
→2n→1a |
→2n→2a |
→2n→3a |
→2n→4a |
→2n→5a |
→2n→6a |
→2n→7a |
→2n→8a |
→2n→9a |
|
4/72 |
8/72 |
4/72 |
0 |
|
|
|
|
|
|
|
→3n→0a |
→3n→1a |
→3n→2a |
→3n→3a |
→3n→4a |
→3n→5a |
→3n→6a |
→3n→7a |
→3n→8a |
→3n→9a |
|
1/72 |
3/72 |
3/72 |
1/72 |
0 |
|
|
|
|
|
|
→4n→0a |
→4n→1a |
→4n→2a |
→4n→3a |
→4n→4a |
→4n→5a |
→4n→6a |
→4n→7a |
→4n→8a |
→4n→9a |
|
0,3125/72 |
1,25/72 |
1,875/72 |
1,25/72 |
0,3125/72 |
0 |
|
|
|
|
|
→5n→0a |
→5n→1a |
→5n→2a |
→5n→3a |
→5n→4a |
→5n→5a |
→5n→6a |
→5n→7a |
→5n→8a |
→5n→9a |
|
0,3125/72 |
1,5625/72 |
3,125/72 |
3,125/72 |
1,5625/72 |
0,3125/72 |
0 |
|
|
|
|
→6n→0a |
→6n→1a |
→6n→2a |
→6n→3a |
→6n→4a |
→6n→5a |
→6n→6a |
→6n→7a |
→6n→8a |
→6n→9a |
|
0,09375/72 |
0,5625/72 |
1,40625/72 |
1,875/72 |
1,406256/72 |
0,5625/72 |
0,09375/72 |
0 |
|
|
|
→7n→0a |
→7n→1a |
→7n→2a |
→7n→3a |
→7n→4a |
→7n→5a |
→7n→6a |
→7n→7a |
→7n→8a |
→7n→9a |
|
5,42535E-4 |
3,79774E-3 |
0,01139332 |
0,0189887 |
0,0189887 |
0,01139332 |
3,79774E-3 |
5,42535E-4 |
0 |
|
|
→8n→0a |
→8n→1a |
→8n→2a |
→8n→3a |
→8n→4a |
→8n→5a |
→8n→6a |
→8n→7a |
→8n→8a |
→8n→9a |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
→9n→0a |
→9n→1a |
→9n→2a |
→9n→3a |
→9n→4a |
→9n→5a |
→9n→6a |
→9n→7a |
→9n→8a |
→9n→9a |
|
2,71267E-5 |
2,44141E-4 |
9,765625E-4 |
2,278646E-3 |
3,417969E-3 |
3,417969E-3 |
2,278646E-3 |
9,765625E-4 |
2,44141E-4 |
2,71267E-5 |
|
∑ 0a |
∑
1a |
∑
2a |
∑
3a |
∑
4a |
∑
5a |
∑
6a |
∑
7a |
∑ 8a |
∑9a |
|
0,274442 |
0,300917 |
0,198568 |
0,121962 |
0,06798 |
0,026964 |
0,007378 |
0,001519 |
0,000244 |
0,000027 |
|
In this distribution ∑p=1 en
∑a=1,527775=3,05555x0,5 |
|
|
|
|
|
||||
|
Poisson
λ=1,527778 |
|
|
|
|
|
|
|
|
|
|
0,217017 |
0,331554 |
0,253271 |
0,12898 |
0,049263 |
0,015053 |
0,003833 |
0,000837 |
0,00016 |
0,000027 |
|
|
|
|
|
|
|
|
|
|
|
At Table 9c1 is made another
distribution with the historical population of 72 parents. The data are somewhat
mutated, so that the parents now have in total 216 children, thus exact 3 on
the average. Further is superposed for
this calculation an asymmetric binomium over the distribution of the
reproduction. The change the allele is transferred or not is posed here not on
ex aequo ½ as in the Tables 9 and 9c but on respectively
⅓ yes and ⅔ no. So the binomium is (⅓+⅔)^q. The
distribution further is described with the arrows. The result here is an
hypothetic allele distribution with ∑p=1 and so the average intensity of
the reproductive distribution with λ=3 is reversed into
an allele distribution λ=1. So the allele
distribution in this natural growing population is reversed into a population
with constant size and the larger variation in the parities can now be compared
with the neutral population dynamics by λ=1. In this natural population
the distribution of the alleles can be regarded as a distribution with
different Poisson intensities for any of the quanta (values of i=n) and these
different Poisson intensities can also be calculated for comparison. So we see
in the tables 9c and 9c1 a shift to smaller averages and
larger extremes compared with the random distribution, but there is a
limitation in the ultimate large values especially in Table 9c1.
Table 9c1
|
Poisson
λ=3 |
|
|
|
|
|
|
|
|
|
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
0,04979 |
0,14361 |
0,22404 |
0,22404 |
0,16803 |
0,10082 |
0,05041 |
0,0216 |
0,0081 |
0,0027 |
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
→7 |
→8 |
→9 |
|
Populatie
n=72 gemiddeld 3,0 kinderen per ouder |
|
|
|
|
|
|
|||
|
0,09722 |
0,194444 |
0,22222 |
0,11111 |
0,06944 |
0,152778 |
0,08333 |
0,06944 |
0 |
0 |
|
7/72 |
14/72 |
16/72 |
8/72 |
5/72 |
11/72 |
6/72 |
5/72 |
0 |
|
|
→0n→0a |
→0n→1a |
→0n→2a |
→0n→3a |
→0n→4a |
→0n→5a |
→0n→6a |
→0n→7a |
|
|
|
7/72 |
0 |
0 |
0 |
0 |
|
|
|
|
|
|
→1n→0a |
→1n→1a |
→1n→2a |
→1n→3a |
→1n→4a |
→1n→5a |
→1n→6a |
→1n→7a |
|
|
|
7/54 |
7/108 |
0 |
|
|
|
|
|
|
|
|
→2n→0a |
→2n→1a |
→2n→2a |
→2n→3a |
→2n→4a |
→2n→5a |
→2n→6a |
→2n→7a |
|
|
|
0,098765432 |
0,098765432 |
0,024691358 |
0 |
|
|
|
|
|
|
|
→3n→0a |
→3n→1a |
→3n→2a |
→3n→3a |
→3n→4a |
→3n→5a |
→3n→6a |
→3n→7a |
|
|
|
0,03292181 |
0,049382716 |
0,024691358 |
0,004115226 |
0 |
|
|
|
|
|
|
→4n→0a |
→4n→1a |
→4n→2a |
→4n→3a |
→4n→4a |
→4n→5a |
→4n→6a |
→4n→7a |
|
|
|
0,013717421 |
0,027434842 |
0,020576131 |
0,006858711 |
0,000857339 |
0 |
|
|
|
|
|
→5n→0a |
→5n→1a |
→5n→2a |
→5n→3a |
→5n→4a |
→5n→5a |
→5n→6a |
→5n→7a |
|
|
|
0,020118884 |
0,05029721 |
0,05029721 |
0,025148605 |
0,006287151 |
0,000628715 |
0 |
|
|
|
|
→6n→0a |
→6n→1a |
→6n→2a |
→6n→3a |
→6n→4a |
→6n→5a |
→6n→6a |
→6n→7a |
|
|
|
0,007315958 |
0,021947873 |
0,027434842 |
0,018289894 |
0,006858711 |
0,001371742 |
0,000114312 |
0 |
|
|
|
→7n→0a |
→7n→1a |
→7n→2a |
→7n→3a |
→7n→4a |
→7n→5a |
→7n→6a |
→7n→7a |
|
|
|
0,004064421 |
0,014225473 |
0,02133821 |
0,017781842 |
0,008890921 |
0,002667276 |
0,000444546 |
3,17533E-05 |
0 |
0 |
|
∑ 0a |
∑
1a |
∑
2a |
∑
3a |
∑
4a |
∑
5a |
∑
6a |
∑
7a |
|
|
|
0,403755771 |
0,326868354 |
0,16902911 |
0,072194278 |
0,022894122 |
0,00466774 |
0,000558858 |
3,17533E-05 |
∑P=1 |
|
|
Populatie;
verdeling over de quanta met onderstaand de intensiteiten van de quanta. |
|
|
|
|
|||||
|
λ=0,90694511 |
λ=0,5891883 |
λ=0,9219 |
λ=1,087057 |
λ≈1,146831 |
λ≈1,112461 |
λ≈1,018122 |
λ≈0,871774 |
|
|
|
|
λ=1,568 |
|
|
|
|
|
|
|
|
|
0,36787944 |
0,36787944 |
0,18393972 |
0,06131324 |
0,0153283 |
0,00306566 |
0,000510944 |
0,000072992 |
Poiss
λ=1 |
|
|
Poisson
λ=1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
The Table 1a; 1b; 2 and 3 that describe
random or in the case non random and than thus probably selective distributions
of the reproduction can be converted in a simple way into allele distributions,
as described in the tables 9; 9b and
9c(1). The differences between these
tables 9b and 9c(1) now are prompt an indication for an important trend. These
simple data indicate namely the probability of a natural selection by people
until the recent past, which is almost suddenly disappeared with the entrance
with the modern industrial society. The existence of the natural selection in
the past is however not proven by these data. It is yet obvious that the
historical data of table 2 and 9c(1) are not representative for all
the people of for instance the
Some other specific situations:
Asexual reproduction
Other conditions for Table 4 are diploid genes
and sexual reproduction. By asexual reproduction the genes are transferred 1on
1 to the next generation. In this way of reproduction there is another
reproductive population, or in other words another effective size of the
population. The replacement ratio r is yet the ratio that defines the
intensity of the reproduction en so describes at which average parity the
population of reproduction and the total number of the alleles remains
constant. The inverse 1/r indicates which part of their genome individuals pass
to individuals of the next generation. Furthermore is by the asexual
reproduction the distribution at the reproduction alone is decisive for the
distribution at the allele transfer, because the endowment of the alleles to
the children is 1 on 1 and so no random distribution is at the endowment here
as it does contrary at the sexual reproduction. Now have the various species
different ways of asexual reproduction. In accordance with this the
distribution of the asexual reproduction will take shape different at the
various species and the differences of these distributions with the Poisson
distribution be larger in some cases than generally at the sexual reproduction[10].
This expectation however is not right in some cases. The unicellular organisms
are for instance particular in this respect, because they always have two descendants
in the actual reproduction. During the start of the growing phase of a
bacterial colony there is an exponential increase in size for the actual
reproduction as well as for the effective one In the phase with a constant
number of the bacteria and an equilibrium between the bacteria that die off by
internal and external factors and the bacteria that can split anymore is a
Poisson distributed effective reproduction of the individuals. The condition
for this is however that the bacteria all have the same change on dividing and
survival. If the bacteria do have different changes on survival in the
environment, as can happen after administration of antibiotics, they are
different susceptible to, the variation in the distribution of their effective
reproduction will be larger than according to the Poisson distribution. So than
there will be more genetic differences between the generations than Poisson
expected.
By many plants is a combination of sexual and asexual reproduction. The
aspect of the distribution of the total reproduction is more complicated by
this. Specially by plants the asexual part of the reproduction induces
generally augmentation the aspect of the total reproduction is complicated. In
instance of this are the tuberous plants like the potatoes. Without asexual
augmentation, because just one bulb in made generally, as in the tulip, you can
regard this as a prolonged survival of the organism, reproducing only sexually.
It is however to be taken into account that not only plants but also animals
and even people do have the combination of sexual and asexual augmentation! Also
the birth of uniovular twins, triplets etc yet is a form of asexual
augmentation. The distribution of the descendants and the transferred alleles
has been made different by this from the aspect of the exclusive sexual
reproduction. Although these differences may be small, as in people, they are
of fundamental importance.
Haploid genes
The distributions of Table 4 and 9 have of course also as a condition
diploid (twofold) genes. Some genes however are haploid (singular) and are
transferred 1 on 1 to the next generation. In many animal species and in people
these are a (large) part of the genes on the Y-chromosome and the genes of the
mitochondrial DNA. The genes on the Y-chromosome only are transferred from
fathers to sons and the mt-DNA is only transferred through mothers. That is why
the distribution of the transfer of these genes should be specified to the
gender, so that only the masculine, respectively the feminine descendants are
counted at the effective reproduction. The replacement factor here also is r=1 and for this part the whole genome
is transferred to the next generation. Also here is no endowment of the alleles
superposed upon the distribution of the reproduction. It is an important point
that the transfer of the non recombinant alleles is in neutral population
dynamics distributed with the Poisson intensity λ=1 with only source of variation in this the variation in the
reproduction. As pointed out at Table 9 has the neutral allele transfer of the
recombinant diploid genes the same intensity
λ=1. At the recombinant genes
this intensity λ=1 however is
constructed of a part reproduction with intensity λ=2 and a part
endowment of the alleles to the offspring. Only the first part of this, the
reproduction, may have a non random distribution as a consequence of the
selection, but the endowment always is at random and buffers the influence of
the reproductive selection on the allele transfer. So it is obvious that the consequences
of the selective differences are not buffered for the transfer of the non
recombinant haploid genes at the sexual reproduction. That is why the non
recombinant genes are 2x as susceptible for selective differences than the
diploid genes. So for general stochastic reasons the non recombinant genes will
differ more in place and time through the populations than the “normal”
recombinant genes. This furthermore is much more the case at the genes of the
Y-chromosome than the mt-DNA on account of another reason: The alleles of the
Y-chromosome will have of course an extra large variation in the transfer
caused by the sexual selection in evolutionary populations. This is very
important if you will use these genes for studying genetic changes in the whole
genome.
Inbreeding
Inbreeding in the population is a complex datum. The most important
reason for this properly is that the conception inbreeding is ill-defined in
the literature. This seems curious and even incredible, because inbreeding is
yet a simple idea, already known in the grey antiquity. The problems at the
quantification of incest are however the differences in its forms and aspects,
which are delicate and difficult to be defined. The aspect of the inbreeding
does yet determine the size of the reproductive population and the effective
reproduction. These aspect are: allele transfer in a defined population at an
average inbreeding; allele transfer at the descendants of related parents;
allele transfer at the descendants of the common ancestors of the related
parents; allele transfer at lines of inbreeding with accumulating relationship
between the parents, etc An essential aspect is the study of inbreeding at
allele transfer out of the common ancestors. Study of inbreeding in this aspect
results in the important conclusion that inbreeding is in the population
genetics a hybrid of the sexual and the asexual reproduction. The ratio r, the
replacement ratio, is in this aspect of inbreeding smaller than in out
breeding. In inbreeding is yet a larger part of the whole genome transferred to
the next and further generations than in out breeding. This implies also that
the size of the population of the effective reproduction decreases in this
aspect of inbreeding. An example as an illustration: In a family is a marriage
between first cousins. The common grandparents of these partners will get on
the average (Table 3) 2 children and
4 grandchildren in neutral population dynamics. In this situation however did 2
of these grandchildren become partners of each other and so have these
grandparents now to expect 6 great-grandchildren on the average instead of the
The picture of Table 10
affirms this idea indeed. Table 10
shows the allele transfer by selfing in an unlimited population. This
population is composed of an unlimited number of populations of reproduction
with selfing, so with n=1. The extinction within this selfing population n=1 is
given on Table 5c n=1, page… The total of an (relative) unlimited number n
of such populations is for instance a population of n plants, all having
different, so 2n unique alleles. The reproduction by selfing is started at F0.
The descendants of the plants now are Poisson distributed with the intensity λ=1, on account of r=1. The plants do have yet one
descendant on the average at selfing in the neutral population. The endowment
of the alleles to the offspring is according to an incomplete binomium. The
possibilities here are different from the situation in out breeding (Table 9). In the case of for instance 1
descendant now the allele a of the heterozygote parent will be singular
transferred in ¼ of the cases, in ½ it will be transferred in twofold and in ¼
it will not be transferred, etc. Shortly the allele transfer now is described
by the binomium (a+b)^2n, superposed on the distribution of the reproduction (λ=1). In Table
10 the average intensity of the distribution remains λ=1, but the distribution now has become irregular and
has no more a constant Poisson intensity. The P0 or extinction now is in
accordance with a distribution at λ=0,75 and the
multiples are corresponding more increased as compared with the out breeding,
with regular Poisson λ=1. There is
obvious extra extinction and in proportion extra multiplication of the alleles
by the increase in the homozygosis. The elaboration of this distribution in the
further generations is complicated by the increasing homozygosis, but can be
deduced in this case (n=1 in the part population) from the random increase in
the homozygosis. Table 10a shows
this extinction according to –σ(Fg-1)=μν(Fg-2) –μ, in this is for F0→F1
μ=0,75; for F1→F2 μ=0,875; for F2→F3 μ=0,9375,etc. In comparison with the situation by out breeding goes the
extinction in this aspect of the inbreeding at first faster, but is gained up
later on by the out breeding, so that both by the development of total
homozygosis are going equal after a number of generations. In the complete
homozygosis the total population here is of course not genetic identical. In
the situation of homozygosis is similarity with the out breeding in the random
extinction but not in the potential non random extinction by selection on
account of the smaller average parity at inbreeding. The buffer by the
endowment of the alleles disappears at total homozygosis. The random and non
random extinction than go both similar to the situation in the asexual
reproduction.
Table 10
|
|
|
|
|
|
|
|
|
|
F0 |
|
|
|
|
|
|
|
|
Q=1 λ=1 |
|
|
|
|
|
|
|
|
F1
descendants |
|
|
|
|
|
|
|
|
1→0 |
1→1 |
1→2 |
1→3 |
1→4 |
1→5 |
1→6 |
|
|
℮^-1 |
℮^-1 |
1/2.℮^-1 |
1/6.℮^-1 |
1/24.℮^-1 |
1/120.℮^-1 |
1/720.℮^-1 |
|
|
Alleles
a |
|
|
|
|
|
|
|
|
→0n→0a |
→0n→1a |
→0n→2a |
→0n→3a |
→0n→4a |
→0n→5a |
→0n→6a |
|
|
℮^-1 |
0 |
|
|
|
|
|
|
|
→1n→0a |
→1n→1a |
→1n→2a |
→1n→3a |
→1n→4a |
→1n→5a |
→1n→6a |
|
|
1/4.℮^-1 |
1/2.℮^-1 |
1/4.℮^-1 |
0 |
|
|
|
|
|
→2n→0a |
→2n→1a |
→2n→2a |
→2n→3a |
→2n→4a |
→2n→5a |
→2n→6a |
|
|
1/32.℮^-1 |
1/8.℮^-1 |
3/16.℮^-1 |
1/8.℮^-1 |
1/32℮^-1 |
0 |
|
|
|
→3n→0a |
→3n→1a |
→3n→2a |
→3n→3a |
→3n→4a |
→3n→5a |
→3n→6a |
|
|
1/384.℮^-1 |
1/64.℮^-1 |
5/128.℮^-1 |
10/192.℮^-1 |
5/128.℮^-1 |
1/64.℮^-1 |
1/384.℮^-1 |
|
|
→4n→0a |
→4n→1a |
→4n→2a |
→4n→3a |
→4n→4a |
→4n→5a |
→4n→6a |
|
|
1/6144.℮^-1 |
1/768.℮^-1 |
7/1536.℮^-1 |
3/768.℮^-1 |
35/3072.℮^-1 |
3/768.℮^-1 |
7/1536.℮^-1 |
|
|
→5n→0a |
→5n→1a |
→5n→2a |
→5n→3a |
→5n→4a |
→5n→5a |
→5n→6a |
|
|
1/122880.℮^-1 |
1/12288.℮^-1 |
3/8192.℮^-1 |
1/1024.℮^-1 |
21/12288.℮^-1 |
63/30720.℮^-1 |
21/12288.℮^-1 |
|
|
→6n→0a |
→6n→1a |
→6n→2a |
→6n→3a |
→6n→4a |
→6n→5a |
→6n→6a |
|
|
1/2949120.℮^-1 |
1/245760.℮^-1 |
11/491520.℮^-1 |
11/147456.℮^-1 |
11/65536.℮^-1 |
11/40960.℮^-1 |
77/245760.℮^-1 |
|
|
∑P=0 |
∑P=1 |
∑P=2 |
∑P=3 |
∑P=4 |
∑P=5 |
∑P=6 |
∑a=0,9841 |
|
0,4723665 |
0,2361832 |
0,1771370 |
0,0669690 |
0,0307483 |
0,0080384 |
0,0033785 |
∑p=0.996 |
|
|
|
|
|
|
|
|
|
|
1,2840254.℮^-1 |
0,6420127.℮^-1 |
0,4815095.℮^-1 |
0,1820407.℮^-1 |
0,0835825.℮^-1 |
0,0218506.℮^-1 |
0,0091837.℮^-1 |
x
℮^-1 |
|
℮^-1 |
℮^-1 |
0,5.℮^-1 |
0,1667.℮^-1 |
0,04167.℮^-1 |
0,0083.℮^-1 |
0,0013889.℮^-1 |
Poiss. λ=1 |
|
℮^-0,75 |
2/4.℮^-0,75 |
12/32.℮^-0,75 |
0.14177.℮^-0,75 |
0,06502.℮^-0,75 |
0,01702.℮^-0,75 |
0,00715.℮^-0,75 |
x
℮^-0,75 |
|
℮^-0,75 |
3/4.℮^-0,75 |
9/32.℮^-0,75 |
0,07031.℮^-0,75 |
0.01318.℮^-0,75 |
0,00198℮^-0,75. |
0,00025.℮^-0,75 |
Poiss.
λ=0,75 |
|
λ=0,75 |
λ≈0,3278 |
λ≈0,96365 |
λ≈1,045 |
λ≈1,275 |
λ≈1,283 |
λ≈1,485 |
Intensities
of |
|
|
|
|
|
|
|
|
the
selfing dist |
|
1→7 |
1→8 |
1→9 |
|
|
|
1/5040.℮^-1 |
1/40320.℮^-1 |
2,76.10^-6
℮^-1 |
|
|
|
|
|
|
|
|
|
→0n→7a |
→0n→8a |
→0n→9a |
|
|
|
|
|
|
|
|
|
→1n→7a |
→1n→8a |
→1n→9a |
|
|
|
|
|
|
|
|
|
→2n→7a |
→2n→8a |
→2n→9a |
|
|
|
|
|
|
|
|
|
→3n→7a |
→3n→8a |
→3n→9a |
|
|
|
0 |
|
|
|
|
|
→4n→7a |
→4n→8a |
→4n→9a |
|
|
|
1/768.℮^-1 |
1/6144.℮^-1 |
0 |
|
|
|
→5n→7a |
→5n→8a |
→5n→9a |
→5n→10a |
|
|
1/1024.℮^-1 |
3/8192.℮^-1 |
1/12288.℮^-1 |
1/122880.℮^-1 |
|
|
→6n→7a |
→6n→8a |
→6n→9a |
→6n→10a |
|
|
11/40960.℮^-1 |
11/65536.℮^-1 |
11/147456.℮^-1 |
11/491520.℮^-1 |
|
|
∑P=7 |
∑P=8 |
∑P=9 |
∑P=10 |
∑a=0,9841 |
|
0,0009371 |
0,0002563 |
0,0000574 |
0,0000112 |
∑p=0.996 |
|
|
|
|
|
|
|
0,0025472.℮^-1 |
0,0006968.℮^-1 |
0,000156.℮^-1 |
0,0000305.℮^-1 |
x
℮^-1 |
|
0,000189.℮^-1 |
0.000025.℮^-1 |
2,76.10^-6
℮^-1 |
|
Poiss. λ=1 |
|
0,00198.℮^-0,75 |
0,00054.℮^-0,75 |
0,00012.℮^-0,75 |
|
x
℮^-0,75 |
|
0,00003.℮^-0,75 |
0.0000006.℮^-0,75 |
0,0000000 |
|
Poiss.
λ=0,75 |
|
λ≈1,56 |
λ≈1,645 |
λ≈1,69 |
λ≈ |
Intensities
of |
|
|
|
|
|
the
selfing dist |
Table 10a
|
Selfing |
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
|
σ=0,75 |
σ=0.4617 |
σ=0,3467 |
σ=0,2838 |
σ=0,24322 |
σ=0,2142 |
σ=0,1921 |
σ=0,1741 |
σ=0,1594 |
σ=0,1472 |
|
P0=0,4724 |
P0=0,6302 |
P0=0,7070 |
P0=0,7529 |
P0=0,7841 |
P0=0,8071 |
P0=0,8252 |
P0=0,8402 |
P0=0,8526 |
P0=0,8631 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
Outbreeding |
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
|
λ=1 |
σ=0,6321 |
σ=0,4685 |
σ=0,3741 |
σ=0,3121 |
σ=0,2681 |
σ=0,2352 |
σ=0,2095 |
σ=0,1890 |
σ=0,1723 |
|
P0=0,368 |
P0=0,531 |
P0=0,626 |
P0=0,688 |
P0=0,732 |
P0=0,765 |
P0=0,790 |
P0=0,811 |
P0=0,828 |
P0=0,842 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
These calculations at selfing are relative simple because here is a
close subpopulation of reproduction with n=1 and the population also is close in
the case of mating between 1st degree relatives, than with n=2.
Furthermore the mating parents in n=1 also are the common ancestors. The
populations with continued inbreeding in the lower degrees of relationship are
open and that is why the calculations in
these populations of reproduction are much more complicated. The population of
the descendants than is namely not constant, but is increasing through the
generations and the increase is slower than in out breeding. This increase of
the descendants in neutral dynamics in out breeding is according the series
2-4-8-16-..and this is in half-brother half-sister mating in a linear genealogy
the series 2-3-4-5-.. Furthermore accumulates the factor of inbreeding f
in closed populations always to complete f=1 and in this ultimate situation
the population of reproduction is genetic identical. Within such an identical
population no extinction is possible anymore. So within the population of
reproduction the extinction is limited by the chance of fixation 1/n. From the
aspect of a collection populations of reproduction the extinction is yet
continued, because populations can dye out, see Tables 9 and 9a. In the
open populations with inbreeding in the
lower degrees the f accumulates in the half-brother
half- sister according 1/8 – (1/8+1/16) – (1/8+1/16+1/32) - ..etc according the
geometric series to f=1/4. So the f accumulates at the first cousin continued in the generations in
the neutral dynamics from f=1/16 to f=1/8, etc. So in the
open populations the completed inbreeding does arise and the extinction remains
to be continued in this hybrid of sexual and asexual reproduction with 1>r>2. This makes these calculation
too much complicated.
So it is easy to conclude that the factor f of inbreeding
accumulates in selfing fast and in the larger close populations slower to
complete, f=1, in the ultimate population with identical individuals. The
existence of these identical sexual subpopulations in a population of
reproduction is effectively equal to asexual reproduction in the subpopulations.
So asexual reproduction properly is the highest form of inbreeding with f=1.
If in a large population are various lines of inbreeding or of asexual
reproduction, this has globally no influence on the random changes in the genes of the total population. The random
changes in the different lines will annul each other. This however is not the
case for the influence of inbreeding in the subpopulations on the non random changes in the population. The smaller average numbers of
descendants in the further generation by inbreeding makes a population more
susceptible for selection, if there is inbreeding in the subpopulations. So the
genetic changes by selection are accelerated by inbreeding. By inbreeding and
asexual reproduction the same alleles remain random in the whole population,
but their presence in the population is less homogeneous. Inbreeding is the
accumulation of alleles. The population by inbreeding is divided up into groups
with different genotypic and phenotypic characteristics. Those groups also will
react different on factors in their environment determining their fitness. So
the results of the reproduction in the groups with the various lines of
inbreeding will be different by selection and some lines may dye out. It is
obvious that the selection of recessive
alleles for some more reasons is more
effective by the inbreeding. Namely because the expression of recessive alleles
mainly is possible by inbreeding, so that only than the positive or negative
fitness of these alleles can come to light. Furthermore the selection by
inbreeding is more effective by the
distribution of the alleles with the multiple lines of descent over smaller
numbers of descendants in the neutral dynamics. Yet if an ancestor has fewer or
more descendants by the selection the consequences of it for the allele
transfer are amplified by the multiple descent. That is why a population is by
inbreeding more susceptible for selection than a population in panmixia of
alleles. This selection strengthened by inbreeding so gives important and fast
non random genetic changes in a population.
So by these indirect consequences
of inbreeding a population is by
inbreeding more susceptible for selection than a population in panmixia of
alleles. Furthermore inbreeding may have a direct
effect on the selection, so that the inbreeding itself is selective. This
is the case if the related parents do have more children than on the average
and the common ancestors so have more descendants than at random, as described
in the neutral dynamics. A systematically larger than random expected success
in the procreation at inbreeding in fact is inbreeding selection. This should
increase the amplifying force of inbreeding on the selection even more.
Indications a-priori for the selection on inbreeding are: Inbreeding has been
for various reasons opportune in the human societies and is this also generally
in the nature, if both parents have to cooperate at the care for the next
generation. It brings a better oneness and team-spirit between the parents if
they are relatives or at least acquaintances of each other. To this latter has
been developed a deep-rooted custom of
inbreeding by people and some animal species: namely monogamy[11].
At inbreeding are fewer territorial problems within the reproductive
population, etc. So it is interesting to investigate if indeed is affirmed by
observation that inbreeding gives larger numbers of effective children.
How inbreeding is present in a human population, the allele transfer
influences and interferes with the selection can also be studied on the basis
of genealogical data as they are on this site and more complete on the geneanet
site http://gw.geneanet.org/wschot. Particularly in our branch of
the family there were many marriages between relatives and that often during
some generations. Consanguinity was by the way in the past much more frequent
in the autochthon Dutch population. Not only by noblemen but also by common
citizens ware often practical and material circumstances that facilitate
consanguine marriages, as there were in the serfdom and later in the system of
the guilds. The anchovies fishery within the guild in
Interesting thus is that consanguinity, or inbreeding, the companion of
the selection, has been nearly disappeared out of the modern human populations
together with the selection. The inbreeding is as pointed out an important
amplifier of the efficiency of the selection and it appears also a frequent
companion of it. Inbreeding in different measures do occur yet very often in
natural populations, although not every
species reproduces with some inbreeding. The species that don’t inbreed are active,
mobile organisms that all come together all at the same place, at the same
time, where in a gigantic orgy the total
species reproduces at once. So the squids are doing it for instance. This
policy of reproduction giving panmixia of the genes can decrease and possibly
stop the selection and will preserve the properties of the species. The species
doing so however are in the minority and this policy of reproduction is
impossible for many species and it is not evolutionary. Mostly the species do
reproduce with some measure of inbreeding, that together with the sexual
selection determines the intensity of the species’ non random genetic changes
and so the velocity of their micro-evolution. So it come to the point that the
sexual and social behaviour of the animals and the plants[12]
determines the measure of inbreeding the species has and so its evolution. That
was also what Charles Darwin had in view when he wrote that the nature
procreated the species in the way that people did breed their domestic animals
and their improved plants. The intuitive knowledge of the people in the past about
the consequences of inbreeding is functioning
also as a system throughout the nature. The hypothesis that the speed of
the micro-evolution of the species may be well ruled in the chain: genetic
decided reproductive behaviour → measure of inbreeding and sexual
selection → genetic changes → behaviour at the reproduction etc,
however is yet a bridge further. This hypothesis is not generally known I think
and it seems worthy to be studied.
This concerning the interaction between inbreeding and selection, which
should be of great importance according to global observation, but about which
only can be concluded after analysis on the basis of comparison with the random
changes by inbreeding in neutral dynamics. In these random changes Sewall Wright did provide clarity in
the different aspects of inbreeding by turning the matter in that he let decide
the size of the population the measure of the inbreeding. This construction of
the random inbreeding by itself is a really good one, which has provided its
value for many years. The genetic drift is described in this theory as random
inbreeding and is defined as: H(g+t) =
Hg x [1−1/2n]^t and f
= 1/2n. In this f is the inbreeding factor in the
primary, original generation g (or F0).
So this formula describes the measure of the heterozygosis as an
exponential function in the time in relation to the population size. The great
benefit of this formula is its efficiency, giving in a simple way good insight
in the most important genetic changes in a close population. The restrictions
however are an disadvantage: It is not applicable in variable population size
or in non random inbreeding, but the most important problem is that it only
describes the heterozygosis extinction[13]
and not the primary and more essential allele extinction. The description of
the allele extinction is yet indeed more complicated and in general not used as
a theory for genetic drift and random genetic changes. Nevertheless the allele
extinction is necessary in the neutral theory and so as a basis for the
micro-evolution theory. The use of the Poisson distributions does simplify
substantially the application of the theory of the allele extinction. Also in
limited and even in very small populations the allele extinctions are to be
calculated in different ways and may give important information besides to the
well-known heterozygosis extinction.
In the population with inbreeding the transfer of the singular allele
follows the superposed Poisson distributions with intensities μ=1+f, as
described at Table 9c. This is the
case in the open population with relative (more than average) inbreeding as
well as in the close population with random inbreeding because of the limited
size n. The inbreeding factor f
is not constant in (random) continued
inbreeding, but increases through the generations because of the increase of the consanguinity and the
number of identical alleles of the parents in the population. In the open and
unlimited H-W population the inbreeding factor in the same line increases from f=1/2n in the F0 to f=1/n
in the F∞, but within the close, limited populations this will increase
from f=1/2n
in the F0 to complete, f=1 in the F∞, when the
population is at last complete homozygote and genetically identical. In a
population of somewhat larger size n f=1/2n is small in relation to 1
and so the chance on survival (=1-F0) in the F∞ will be about 1/n
according to Table 5b. Then it will
be about 2x as large as the a-priori chance on fixation, 1/2n. This seems
curious but in Table 5 and 5b extinctions are calculated of one
allele on one locus of one individual in F0 and the complement of it is the
survival on one locus in a proportion of individuals in F∞. This all in
the H-W population. Within the limited population is survival in F∞ also complete fixation. This then is
however a survival on 2 loci within the population n. So the chance on fixation
in a limited population is about the half of the chance on ultimate survival in
an unlimited population.
As described on page 19 the extinction in the primary population is 1/℮
in the (relative) unlimited
population, because P0= (1−1/2n)^2n →1/℮, if n→∞.
In the limited population the
primary extinction depends than however on the population size and is to be
calculated as P0= (1−1/2n)^2n. The general negative exponential intensity
can be calculated here in the limited population as well by μ=−ln(1-1/2n)^2n, or μ = −2n.ln(1−1/2n).
So this is μ=1, if n→∞, because
(1−1/2n)^2n →1/℮, if n→∞. If n=3, as in the
example of Table 5c 2n=6,
it is obvious that the μ in
the general recurrence formula −σ(Fg)=μν-μ is simple to be calculated
as P0=
(1-1/6)^6 en μ=−6.ln(5/6)=
1,093929.. The extinction calculated with the accumulated exponential
distribution as in Table 5c does
give the a-priori expected ultimate survival and fixation. In this example the
fixation is 1/6, the complement of the extinction, p0 in F∞, thus 1-5/6.
So if the right intensity is used the extinction can be calculated in any
population. For a population of unlimited
and limited size the intensity simply is to be calculated as μ=−2n.ln(1-1/2n).
The
practical proof of the rightness of this formula follows from the application
of it with trials by observation of drawings, as described on page
Because
there is just one individual in each generation, there is also only one
possibility to heterozygosis and describes the heterozygosis extinction in n=1 the complete course of the
allele distributions through the generations. From Wrights’ formula namely is
direct deducible that the genotype ab occurs in the generations with proportions
or chances following the series 1 - 0,5 – 0,25 -0,125 etc and both homozygotic
types aa and bb each following 0 - 0,25
– 0,375 – 0,4375 etc. However if n>1 there are more possibilities of
heterozygosis and are the courses of the genotypes and the allele extinctions
no longer deducible from the heterozygosis extinction. Already at n=2 there are
unlimited many possibilities of distributions that comply with the data, that
for instance the heterozygosis is decreased in two generations from complete,
Het=1 to Het=(3/4)^2=0,5625. To be able to calculate average expected
distributions of the original 4 alleles you must return to the complete
statistical distribution, of which the Wrights’ formula has been deduced.
In Table 5 and 5c only is showed the accumulating
exponential distribution and not the superposed total Poisson distributions.
The Tables 5b and 5c are the same in the way of the calculation, at both the
extinction is calculated with the intensities
μ and σ of the cumulating exponential
distribution, as ℮^−σ. The particularity of Table 5c however is that these
exponential intensities here are determined by the limited or small population
size and in this is the number of the parities in the distribution of course
also maximized to 2n. Because of this
limited population size the accumulating exponential distribution can not be
extended to the complete Poisson distributions in superposition. At the Tables
5c P0 again is the allele extinction and the complement of it the survival is
indicated as Su, so that Su=1-P0; α is de survival van
of all the alleles, so that α =2n.Su. Het is the
heterozygosis of the old and the new generations calculated with the Wright’s
formula. The generations are indicated as F0; F1, etc. The population with 51
alleles, Table 5c 2n=51, is not quit exact, but an
approach. This table has been made with some data of Kimura, see Table 5b, but 51ln(50/51)=−1,009934
and so gives μ=1.009934, here this is now rounded to μ=1.01 and with this
intensity was already made a table.
In the example of table 5c 2n=6
is indicated: In a population with 3 individuals ( in both genders fertile) are
vanished of the 6 allelic variations that (possibly) are in F0 on the loci,
after 1 generation about 2, after 3 generations nearly 3, after 5 generations
nearly 4, after 70 generations nearly 5 and after ∞ generations exact 5
of the 6 alleles on all the loci.
So the exponential accumulation and recurrence seems excellent
applicable in small populations. Yet the extinction P0 converges ever to the
a-priori expected value, the complement of the fixation., so that in F∞
always rests one allele, α=1.
Table 5c 2n=6
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
|
μ=1,093929 |
σ=0,72757 |
σ=0,56548 |
σ=0,47247 |
σ=0,41192 |
σ=0,36933 |
σ=0,33781 |
σ=0,31360 |
σ=0,29447 |
σ=0,27903 |
|
P0=0,334898 |
P0=0,48308 |
P0=0,56809 |
P0=0,62346 |
P0=0,66238 |
P0=0,6912 |
P0=0,71333 |
P0=0,73081 |
P0=0,74493 |
P0=0,75652 |
|
Su=0,6651 |
Su=0,5169 |
Su=0,4319 |
Su=0,3765 |
Su=0,3376 |
Su=0,3088 |
Su=0,2867 |
Su=0,2692 |
Su=0,2552 |
Su=0,24348 |
|
α=3,99 |
α=3,10 |
α=2,59 |
α
=2,26 |
α
=2,026 |
α
=1,853 |
α
=1.72 |
α
=1,62 |
α
=1,531 |
α
=1,461 |
|
Het=1 |
Het=0,8333
|
Het=0,6944
|
Het=0,5787
|
Het=0,4823
|
Het=0,4019
|
Het=0,3349
|
Het=0,2791
|
Het=0,2326
|
Het=0,1938
|
|
Het=0,8333 |
Het=0.6944
|
Het=0,5787
|
Het=0,4823
|
Het=0,4019
|
Het=0,3349
|
Het=0,2791
|
Het=0,2326
|
Het=0,1938
|
Het=0,1615
|
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
|
|
|
|
|
|
|
|
|
|
|
|
F10 |
F11 |
F12 |
F13 |
F14 |
F15 |
F16 |
F17 |
F18 |
F19 |
|
|
σ=0,26635 |
σ=0,25580 |
σ=0,24690 |
σ=0,23934 |
σ=0,23284 |
σ=0,22723 |
σ=0,22236 |
σ=0,21809 |
σ=0,21436 |
σ=0,21107 |
|
|
P0=0,76617 |
P0=0,77430 |
P0=0,78122 |
P0=0,78715 |
P0=0,79228 |
P0=0,79674 |
P0=0,80063 |
P0=0,80405 |
P0=0,80706 |
P0=0,80972 |
|
|
F11 |
F12 |
F13 |
F14 |
F15 |
F16 |
F17 |
F18 |
F19 |
F20 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F20 |
F21 |
F22 |
F23 |
F24 |
F25 |
F26 |
F27 |
F28 |
F29 |
|
|
σ=0,20814 |
σ=0,20556 |
σ=0,20082 |
σ=0,20121 |
σ=0,19938 |
σ=0,19774 |
σ=0,19627 |
σ=0,19496 |
σ=0,19377 |
σ=0,19269 |
|
|
P0=0,81209 |
P0=0,81419 |
P0=0,81806 |
P0=0,81774 |
P0=0,81924 |
P0=0,82058 |
P0=0.82179 |
P0=0,82287 |
P0=0,82385 |
P0=0,82474 |
|
|
F21 |
F22 |
F23 |
F24 |
F25 |
F26 |
F27 |
F28 |
F29 |
F30 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F30 |
F31 |
F32 |
F33 |
F34 |
F35 |
F36 |
F37 |
F38 |
F39 |
|
|
σ=0,19173 |
σ=0,19086 |
σ=0,19007 |
σ=0,18936 |
σ=0,18871 |
σ=0,18813 |
σ=0,18760 |
σ=0,18713 |
σ=0,18669 |
σ=0,18629 |
|
|
P0=0,82553 |
P0=0,82625 |
P0=0,82690 |
P0=0,82749 |
P0=0,82803 |
P0=0,82851 |
P0=0,82895 |
P0=0,82934 |
P0=0,82970 |
P0=0,83003 |
|
|
F31 |
F32 |
F33 |
F34 |
F35 |
F36 |
F37 |
F38 |
F39 |
F40 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F40 |
F41 |
F42 |
F43 |
F44 |
F45 |
F46 |
F47 |
F48 |
F49 |
|
|
σ=0,18593 |
σ=0,18561 |
σ=0,18532 |
σ=0,18504 |
σ=0,18480 |
σ=0,18458 |
σ=0,18438 |
σ=0,18419 |
σ=0,18402 |
σ=0,18387 |
|
|
P0=0,83033 |
P0=0,83060 |
P0=0,83084 |
P0=0,83107 |
P0=0,83127 |
P0=0,83145 |
P0=0,83162 |
P0=0,83178 |
P0=0,83192 |
P0=0,83204 |
|
|
F41 |
F42 |
F43 |
F44 |
F45 |
F46 |
F47 |
F48 |
F49 |
F50 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F50 |
F51 |
F52 |
F53 |
F54 |
F55 |
F56 |
F57 |
F58 |
F59 |
|
|
σ=0,18373 |
σ=0,18361 |
σ=0,18349 |
σ=0,18339 |
σ=0,18330 |
σ=0,18321 |
σ=0,18313 |
σ=0,18306 |
σ=0,18300 |
σ=0,18294 |
|
|
P0=0,83216 |
P0=0,83226 |
P0=0,83236 |
P0=0,83244 |
P0=0,83252 |
P0=0,83259 |
P0=0,83266 |
P0=0,83272 |
P0=0,83277 |
P0=0,83282 |
|
|
F51 |
F52 |
F53 |
F54 |
F55 |
F56 |
F57 |
F58 |
F59 |
F60 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
F60 |
F61 |
F62 |
F63 |
F64 |
F65 |
F66 |
F67 |
F68 |
F69 |
F∞ |
|
σ=0,18287 |
σ=0,18283 |
σ=0,18278 |
σ=0,18273 |
σ=0,18270 |
σ=0,18266 |
σ=0,18264 |
σ=0,18261 |
σ=0,18258 |
σ=0,18255 |
σ=0,18232 |
|
P0=0,83287 |
P0=0,83291 |
P0=0,83295 |
P0=0,83299 |
P0=0,83302 |
P0=0,83305 |
P0=0,83307 |
P0=0,83309 |
P0=0,83312 |
P0=0,83314 |
P0=0,83333 |
|
F61 |
F62 |
F63 |
F64 |
F65 |
F66 |
F67 |
F68 |
F69 |
F70 |
F∞ |
|
|
|
|
|
|
|
|
|
|
|
|
Table 5c 2n=2
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
|
μ=1,386294361 |
σ=1,03972 |
σ=0,89617 |
σ=0,82050 |
σ=0,77603 |
σ=0,74828 |
σ=0,73033 |
σ=0,71845 |
σ=0,71046 |
σ=0,70505 |
|
P0=0,25 |
P0=0,35355 |
P0=0,40813 |
P0=0,44021 |
P0=0,46023 |
P0=0,47318 |
P0=0,48175 |
P0=0,48751 |
P0=0,49142 |
P0=0,49409 |
|
α
=1,5 |
α
=1,2929 |
α
=1,1837 |
α
=1,1120 |
α
=1,0795 |
α
=1,0536 |
α
=1,0365 |
α
=1,0250 |
α
=1,0172 |
α
=1,0118 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
|
|
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
|
Het=1
|
Het=0,5
|
Het=0,25
|
Het=0,125
|
Het=1/16
|
Het=1/32
|
Het=1/64
|
Het=1/128
|
Het=1/256
|
Het=1/512
|
|
Het=0,5 |
Het=0,25
|
Het=0,125
|
Het=0,0625
|
Het=1/32
|
Het=1/64
|
Het=
1/128 |
Het=1/256
|
Het=1/512
|
Het=1/1024
|
|
P0=0,25 |
P0=0,375 |
P0=0,4375 |
P0=0,46875 |
P0=0.48438 |
P0=0,49219 |
P0=0,49609 |
P0=0,49805 |
P0=0,49902 |
P0=0,49951 |
|
α
=1,5 |
α
=1,25 |
α
=1,125 |
α
=1,0625 |
α
=1,0312 |
α
=1,0156 |
α
=1,0078 |
α
=1,0039 |
α
=1,0020 |
α
=1,0010 |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
Table 5c 2n=51
|
F0 |
F4 |
F9 |
F19 |
F49 |
F99 |
F199 |
F299 |
F399 |
F499 |
F∞ |
|
μ=1,01 |
σ=0,32215 |
σ=0,18241 |
σ=0,10197 |
σ=0,04919 |
σ=0,03117 |
σ=0,02298 |
σ=0,02096 |
σ=0,02029 |
σ=0,02006 |
σ=0,01993 |
|
P0=0,36422 |
P0=0,72459 |
P0=0,83326 |
P0=0,90306 |
P0=0,95200 |
P0=0,96931 |
P0=0,97728 |
P0=0,97926 |
P0=0,97991 |
P0=0,98014 |
P0=0,98027 |
|
α
=32,4 |
α
=14,0 |
α
=8,5 |
α
=4,9 |
α
=2,45 |
α
=1,57 |
α
=1,16 |
α
=1,06 |
α
=1,0246 |
α
=1,013 |
α
=1,006 |
|
Het=1
|
Het=0,9238
|
Het=0,8368
|
Het=0,6864
|
Het=0,3790
|
Het=0,1408
|
Het=0,0194
|
Het=0,0027
|
Het=0,0004
|
Het=0,00005
|
Het=0
|
|
Het=0,9804
|
Het=0,9057
|
Het=0,8203
|
Het=0,6730
|
Het=0,3715
|
Het=0,1380
|
Het=0,0191
|
Het=0,0026
|
Het=0,0004
|
Het=0,00005
|
Het=0
|
|
F1 |
F5 |
F10 |
F20 |
F50 |
F100 |
F200 |
F300 |
F400 |
F500 |
F∞ |
The binomial superposition and
extinction
The binomial distribution will describe the random changes in a small
population better than the Poisson distribution does. Like in the example of
the 6 drawings under replace in the bag with 6 marbles and followed by the
composition of the new bag by the result of the former bag. The binomial
distribution will describe the total distribution of all the drawing events
within this small space or population and not only the P0, the complement of
the drawing as does the exponential distribution. The Poisson, binomial and
normal distributions however are continuous in each other and compose properly
one stochastic distribution. One can pose that the Poisson distribution is a
limit case in the binomial distribution for n→∞ and that the
binomial distribution is a specific case in the Poisson distribution for the
limited n. In the Tables 11 the superposition of the binomial distribution is
made. This could be for instance the chances to cast in 2n=2 throws 0x; 1x or
2x six with the dice, or the chances to cast in 2n=6 throws 0x; 1x; 2x…6x six
with the dice or as well to draw so many times the right marble or the right
allele. In the well known primary binomial distribution this distribution of
chances is calculated as the product of the binomial coefficient and the chance
to draw the right allele and the complement of this. At the second and further
degree binomial distributions the proportions of the former distribution are
further distributed, following the arrows. It is obvious that the chances will
change in this: So for instance the right white marble is at first singular in
the bag of 6 that white marble is drawn with the change 1/6 and so it is drawn
twice in the proportion of 0,2009.. following the distribution. In the second
turn this proportion is not drawn with the chance 1/6 but with 1/3, because it
now occurs in twofold. Of course marbles can not be drawn that were not drawn
in the former turn (extinction) and always is drawn 6x the right marbles from
bags that contain exclusively the right marbles (fixation).
Table 11 2n=2 shows the binomial superposition at
2n=2, so in selfing. Mind the increase of the extinction under∑=0 and of
the fixation under ∑=2. Also the alternative allele has the same
distribution and so the percentage of identical populations increases here as
2x the fixation. The binomial allele extinction is in 2n=2 also to be
calculated direct from the heterozygosis extinction following the Wrights’
formula as is done on Table 9c 2n=2. This Table 9c also shows the
difference between the binomial and exponential extinction.
Table 11 Binomial 2n=2
|
→0 |
→1 |
→2 |
|
|
|
|
|
|
F0 |
|
|
|
|
|
|
|
|
0 |
1 |
0 |
|
|
|
|
|
|
1→0 |
1→1 |
1→2 |
|
F3 |
|
|
|
|
1.0,5^0.0,5^2 |
2.0,5^1.0,5^1 |
1.0,5^2.0,5^0 |
Binomial |
→0→0 |
→0→1 |
→0→2 |
|
|
0,25 |
0,5 |
0,25 |
F1 |
0,4375 |
0 |
0 |
|
|
F1 |
|
|
50%pop
id |
→1→0 |
→1→1 |
→1→2 |
|
|
→0→0 |
→0→1 |
→0→2 |
|
0,03125 |
0,0625 |
0,03125 |
|
|
0,25 |
0 |
0 |
|
→2→0 |
→2→1 |
→2→2 |
|
|
→1→0 |
→1→1 |
→1→2 |
|
0 |
0 |
0,4375 |
|
|
0,125 |
0,25 |
0,125 |
|
∑0 |
∑1 |
∑2 |
|
|
→2→0 |
→2→1 |
→2→2 |
|
0,46875 |
0,0625 |
0,46875 |
F4 |
|
0 |
0 |
0,25 |
|
F4 |
|
|
93,75%
pop id |
|
∑0 |
∑1 |
∑2 |
|
→0→0 |
→0→1 |
→0→2 |
|
|
0,375 |
0,25 |
0,375 |
F2 |
0,46875 |
0 |
0 |
|
|
F2 |
|
|
75%
pop id |
→1→0 |
→1→1 |
→1→2 |
|
|
→0→0 |
→0→1 |
→0→2 |
|
0,015625 |
0,03125 |
0,015625 |
|
|
0,375 |
0 |
0 |
|
→2→0 |
→2→1 |
→2→2 |
|
|
→1→0 |
→1→1 |
→1→2 |
|
0 |
0 |
0,46875 |
|
|
0,0625 |
0,125 |
0,0625 |
|
∑0 |
∑1 |
∑2 |
|
|
→2→0 |
→2→1 |
→2→2 |
|
0,484375 |
0,03125 |
0,484375 |
F5 |
|
0 |
0 |
0,375 |
|
F5 |
|
|
96,9%
pop id |
|
∑0 |
∑1 |
∑2 |
|
|
|
|
|
|
0,4375 |
0,125 |
0,4375 |
F3 |
|
|
|
|
|
F3 |
|
|
87,5%pop
id |
|
|
|
|
|
|
|
|
|
|
|
|
|
The table 11 2n=4, so as by inbreeding of first degree relatives in the
F1? These distributions always are without mixture of the generations, so here
is in the first generation the possibility of full brother and sister indeed,
but also is possible the random selfing in this model with random mating of two
individuals, fertile in the both genders. In the further generations the
genetic relationship increases fast as is evident yet more in the next Table 12 2n=4. Distributions of populations with mixture of the generations
and separate genders are more complicated. Although the picture of these tables
give general insight in the principle the distributions in very small mammal
populations will deviate substantially from these numbers. Mind the increase of
the extinction at ∑0 and the fixation at ∑4. The percentage
identical populations will increase with 4x the fixation.
Table 11 Binomial 2n=4
|
Binomial 2n=4. Distribution alleles, gametes of F0 four times singular. |
|
|
|
|
|
|
||||
|
→0 |
→1 |
→2 |
→3 |
→4 |
|
|
|
|
|
|
|
0 |
1 |
0 |
0 |
0 |
|
|
|
|
|
|
|
1→0 |
1→1 |
1→2 |
1→3 |
1→4 |
|
F3 |
|
|
|
|
|
1.(1/4)^0.(3/4)^4 |
4.(1/4)^1.(3/4)^3 |
6.(1/4)^2.(3/4)^2 |
4.(1/4)^3.(3/4)^1 |
1.(1/4)^4.(3/4)^0 |
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
|
|
0,3164063 |
0,421875 |
0,2109375 |
0,046875 |
0,0039063 |
|
0,5484354 |
0 |
0 |
0 |
0 |
|
F1 |
|
|
|
1,6%
pop id |
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
|
0,0465369 |
0,0620492 |
0,0310246 |
0,0068944 |
0,0005745 |
|
0,3164063 |
0 |
0 |
0 |
0 |
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
|
0,0084586 |
0,0338345 |
0,0507517 |
0,0338344 |
0,0084586 |
|
0,13348389 |
0,1779785 |
0,0889893 |
0,0197754 |
0,00164795 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
|
0,0003685 |
0,0044224 |
0,01990092 |
0,03980184 |
0,0298514 |
|
0,0131835 |
0,0527344 |
0,0791016 |
0,0527344 |
0,0131835 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
|
0 |
0 |
0 |
0 |
0,07480275 |
|
0,0001831 |
0,0021973 |
0,0098876 |
0,0197754 |
0,01483154 |
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
|
0,6037994 |
0,1003061 |
0,10167722 |
0,08053064 |
0,11368725 |
|
0 |
0 |
0 |
0 |
0,0039063 |
|
F4 |
|
|
|
45,5%
pop id |
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
|
0,4632567 |
0,2329102 |
0,1779785 |
0,0922852 |
0,03356929 |
|
0,6037994 |
0 |
0 |
0 |
0 |
|
F2 |
|
|
|
13,4%
pop id |
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
|
0,0317375 |
0,0423166 |
0,0211583 |
0,0047018 |
0,0003918 |
|
0,4632567 |
0 |
0 |
0 |
0 |
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
|
0,0063548 |
0,0254193 |
0,038129 |
0,0254193 |
0,0063548 |
|
0,07369442 |
0,098259 |
0,0491295 |
0,0109177 |
0,0009098 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
|
0,0003146 |
0,0037749 |
0,0169869 |
0,0339738 |
0,0254804 |
|
0,0111237 |
0,0444946 |
0,0667419 |
0,0444946 |
0,0111237 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
|
0 |
0 |
0 |
0 |
0,11368725 |
|
0,0003605 |
0,0043259 |
0,0194664 |
0,03893282 |
0,02919996 |
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
|
0,6422063 |
0,0715108 |
0,0762742 |
0,0640949 |
0,14591425 |
|
0 |
0 |
0 |
0 |
0,03356929 |
|
F5 |
|
|
|
58,4%
pop id |
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
|
|
|
|
|
|
|
0,5484354 |
0,1470795 |
0,1353378 |
0,09434512 |
0,07480275 |
|
|
|
|
|
|
|
F3 |
|
|
|
29,9%
pop id |
|
|
|
|
|
|
In Tabel 12 2n=4 Binomial homozygosis the increase
in the homozygosis through the generations is calculated direct from the superposed
binomial distribution. Its complement the heterozygosis survival than is of
course also given. The results are equal to the calculations with the well
known Wrights’ formula. That is no wonder because Wright deduced his formula from
the binomial distribution. The niceness of this inductive derivation however is
that it gives easier insight. It also gives a specification of the homozygosis
over the fixation parity and the other parities. The calculations are simple:
as is described in Table 11 the singular allele in the F0 is in the next
generation also singular, or in twofold, in threefold, in fourfold or it is
vanished. The average expected proportions of the n-folds of all the four alleles are of course found by
multiplication with 4.
So the principle is simple: The n-folds or parities to for instance the
F1 are calculated with the binomial distribution. These are than properly the
alleles in the gametes of F0, which are transferred to the F1. Out of this are
calculated the genotypes of F1 in the sense of heterozygote or homozygote, as
is showed in this Table 12 at
F0→F1. From F1 is for any generation to be calculated in the stochastic
process the expected measure of homo and heterozygosis. The structure of the F0
however is defined as starting population and is not determined in the
stochastic process. The potential homozygosis starts from a free stochastic
cause of F0 and so can have another
value than the real one. The heterozygosis is of course complete (Het=1) in the
starting population with its 4 unique alleles. The potential homozygosis of
In F1 the mating individuals are from a global view full brothers and
sisters, but their genetic relationship is much larger and they procreate much
more homozygotes (0,44) than these relatives do in an open population (0,25).
This here is obvious caused by the chances on homozygosis by random selfing in
F0 and F1. In these of course are differences with the situation at the actual
mating in the nature or in the laboratory in very small populations. In larger
populations this random selfing is much smaller and in the Poisson
distributions it is fallen off. It also appears from this table 12 2n=4 that the
random selfing decreases fast in the further increasing generations. The global
estimation thus is that the allele and heterozygosis extinction is some
generations slower in real populations with separated gender in 2n=4. It
however is obvious that the possibilities of such population are else and so
the distributions are. So it yet is a challenge to make specific distributions
for these natural populations.
Tabel 12 2n=4F0 singular.
Binomial homozygosis
|
Binomial 2n=4. Distribution alleles, gametes van F0
four times singular |
|
|
|||||
|
→0 |
→1 |
→2 |
→3 |
→4 |
∑ |
|
|
|
0 |
1 |
0 |
0 |
0 |
|
F0 distribution Q=1 |
|
|
0 |
4 |
0 |
0 |
0 |
4 |
F0
distrubution all 4 alleles |
|
|
|
0,25 |
|
|
|
0,25 |
Hom.
in F1 |
|
|
According to Wrights’ formula is the Homozygosis in
F1 1-0,75=0,25 |
|||||||
|
|
|
|
|
|
|
|
|
|
0,31640625 |
0,421875 |
0,2109375 |
0,046875 |
0,00390625 |
1 |
F1
distribution Q=1 |
|
|
1,2656252 |
1,6875 |
0,84375 |
0,1875 |
0,0156252 |
4,0000004 |
F1
distribution all 4 alleles |
|
|
x(0/4)^1 |
x(1/4)^2 |
x(2/4)^2 |
x(3/4)^3 |
x(4/4)^4 |
|
|
|
|
0 |
0,10546875 |
0,2109375 |
0,10546875 |
0,0156252 |
0,4375002 |
Hom.
in F2 |
|
|
According to Wrights’ formula is the Homozygosis in
F2 1-(0,75)^2=0,4375 |
|||||||
|
|
|
|
|
|
|
|
|
|
0,4632567 |
0,2329102 |
0,1779785 |
0,0922852 |
0,03356929 |
0,99999989 |
F2
distribution Q=1 |
|
|
1,8530268 |
0,9316408 |
0,711914 |
0,3691408 |
0,1342772 |
3,9999996 |
F2
distribution all 4 alleles |
|
|
0 |
0,05822755 |
0,1779785 |
0,2076417 |
0,1342772 |
0,57812495 |
Hom.
in F3 |
|
|
According to Wrights’ formula is the Homozygosis in
F3 1-(0,75)^3=0,578125 |
|||||||
|
|
|
|
|
|
|
|
|
|
0,5484354 |
0,1470795 |
0,1353378 |
0,09434512 |
0,07480275 |
1,00000057 |
F3
distribution Q=1 |
|
|
2,1937416 |
0,588318 |
0,5413512 |
0,3773804 |
0,2992108 |
4,000002 |
F3
distribution all 4 alleles |
|
|
0 |
0,03676988 |
0,1353378 |
0,21227648 |
0,2992108 |
0,68359495 |
Hom.
in F4 |
|
|
According to Wrights’ formula is the Homozygosis in
F4 1-(0,75)^4=0,68359375 |
|||||||
|
|
|
|
|
|
|
|
|
|
0,6037994 |
0,1003061 |
0,10167722 |
0,08053064 |
0,11368725 |
1,00000061 |
F4
distribution Q=1 |
|
|
2,4151976 |
0,4012244 |
0,4067088 |
0,3221224 |
0,4547488 |
4,000002 |
F4
distribution all 4 alleles |
|
|
0 |
0,02507653 |
0,10167722 |
0,18119385 |
0,4547488 |
0,7626964 |
Hom.
in F5 |
|
|
According to Wrights’ formula is the Homozygosis in
F5 1-(0,75)^5=0,7626953 |
|||||||
|
|
|
|
|
|
|
|
|
|
0,6422063 |
0,0715108 |
0,0762742 |
0,0640949 |
0,14591425 |
1,00000045 |
F5
distribution Q=1 |
|
|
2,5688252 |
0,2860432 |
0,3050968 |
0,2563796 |
0,5836568 |
4,0000016 |
F5
distribution all 4 alleles |
|
|
0 |
0,0178777 |
0,0762742 |
0,14421353 |
0,5836568 |
0,82202223 |
Hom.
in F6 |
|
|
According to Wrights’ formula is the Homozygosis in
F6 1-(0,75)^6=0,8220215 |
|||||||
Starting from a population with 6 singular alleles is made Tabel 11 2n=6 six times singular. Mind the increasing extinction at ∑0
and the increasing fixation at ∑6.
In this table the Poisson distributions are given for comparison at the F1, F2
and F3. With these binomial distributions only the neutral population dynamics
can be described. As for this aspect the Poisson distribution with λ=1 is comparable with the binomial distribution. So
mind the differences between the Poisson and binomial distributions are not
large, even in this vary small population and they decrease in the further
increasing generations. Also interesting is comparison with the Poisson
distribution with intensity λ=6ln(5/6)=-1,093929,
because the P0 (extinction) of this intensity is equal to the binomial
extinction.
Tabel 11 2n=6 six times singular
|
Binomial 2n=6. Distribution alleles, gametes of F0 six times singular |
|
|
|
|
|||||
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
∑ |
|
|
|
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
F0
distribution Q=1 |
|
|
F0 |
|
|
|
|
|
(1/6)^5
pop id |
|
|
|
|
1→0 |
1→1 |
1→2 |
1→3 |
1→4 |
1→5 |
1→6 |
|
|
|
|
1.(1/6)^0(5/6)^6 |
6.(1/6)^1(5/6)^5 |
15.(1/6)^2.(5/6)^4 |
20.(1/6)^3.(5/6)^3 |
15.(1/6)^4.(5/6)^2 |
6.(1/6)^5.(5/6) |
1.(1/6)^6.1 |
Binomiaal |
|
|
|
0,33489797 |
0,40187757 |
0,20093878 |
0,05358368 |
0,00803755 |
0,000643 |
0,0000214 |
0,99999995 |
F1 Binomial Q=1 |
|
|
0,367879 |
0,367879 |
0,1839397 |
0,06131324 |
0,0153283 |
0,00306566 |
0,00051094 |
0,99991584 |
F1 Poiss λ=1
Q=1 |
|
|
0,33489797 |
0,3663547 |
0,2003831 |
0,0730683 |
0,0199829 |
0,004372 |
0,000797 |
0,99985597 |
F1
Poiss λ=6ln(5/6) |
|
|
F1 |
|
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,33489797 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,13458798 |
0,16150558 |
0,08075279 |
0,02153408 |
0,00323011 |
0,00025841 |
8,613E-06 |
0,40187756 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,01764072 |
0,05292215 |
0,06615269 |
0,04410179 |
0,01653817 |
0,00330735 |
0,00027564 |
0,2009385 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,00083725 |
0,00502347 |
0,01255868 |
0,01674491 |
0,01255867 |
0,00502347 |
0,00083725 |
0,05358369 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
1,10E-05 |
0,00013231 |
0,00066153 |
0,00176408 |
0,00264612 |
0,0021169 |
0,00070563 |
0,0080376 |
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
1,38E-08 |
4,13E-07 |
5,17E-06 |
0,0000345 |
0,0001292 |
0,00025841 |
0,00021534 |
0,000643 |
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,0000214 |
0,0000214 |
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
0,66510175 |
|
|
|
0,48797495 |
0,21958392 |
0,16013086 |
0,08417931 |
0,03510228 |
0,01096453 |
0,00206387 |
0,99999972 |
F2 Binomial Q=1 |
|
|
∑0
Poiss |
∑1Poiss |
∑2Poiss |
∑3Poiss |
∑4Poiss |
∑5Poiss |
∑6Poiss |
∑≥7
Poiss |
|
|
|
0,53146305 |
0,19551454 |
0,13372015 |
0,07295863 |
0,03614535 |
0,01697346 |
0,00763095 |
0,00503485 |
F2 Poisson Q=1
λ=1 |
|
|
F2 |
|
|
|
|
|
1,24%
pop id |
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,48797495 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,0735382 |
0,08824585 |
0,04412293 |
0,01176612 |
0,00176493 |
0,00014119 |
4,699E-06 |
0,21958391 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,01405813 |
0,04217439 |
0,05271799 |
0,03514533 |
0,0131795 |
0,0026359 |
0,00021966 |
0,1601309 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,0013153 |
0,0078918 |
0,0197295 |
0,026306 |
0,0197295 |
0,0078918 |
0,0013153 |
0,0841793 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
4,82E-05 |
0,00057782 |
0,00288908 |
0,00770421 |
0,01155631 |
0,00924505 |
0,00308168 |
0,0351023 |
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
2,35E-07 |
7,05E-06 |
8,81E-05 |
5,88E-04 |
0,00220319 |
0,00440639 |
0,00367169 |
0,01096421 |
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,00206387 |
0,00206387 |
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
0,51202448 |
|
|
|
0,57693497 |
0,13889692 |
0,11954765 |
0,08150921 |
0,04843345 |
0,02432034 |
0,01035689 |
0,99999944 |
F3 Binomial Q=1 |
|
|
∑0
Poiss |
∑1Poiss |
∑2Poiss |
∑3Poiss |
∑4Poiss |
∑5Poiss |
∑6Poiss |
∑≥7
Poiss |
|
|
|
0,625917694 |
0,122378031 |
0,095642736 |
0,062799424 |
0,038774185 |
0,023203445 |
0,013552238 |
0,01647518 |
F3 Poisson Q=1
λ=1 |
|
|
F3 |
|
|
|
|
|
6,2%
pop id |
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,57693497 |
0 |
0 |
0 |
0 |
0 |
0 |
0,57693497 |
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,0465163 |
0,0558195 |
0,0279098 |
0,0074426 |
0,0011164 |
0,0000893 |
0,000003 |
0,1388969 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,0104953 |
0,0314858 |
0,0393572 |
0,0262382 |
0,0098393 |
0,0019679 |
0,000164 |
0,1195476 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,0012736 |
0,0076415 |
0,0191037 |
0,0254716 |
0,0191037 |
0,0076415 |
0,0012736 |
0,0815092 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
0,0000664 |
0,0007973 |
0,0039863 |
0,0106301 |
0,0159452 |
0,0127561 |
0,004252 |
0,0484335 |
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
5,21E-07 |
0,0000156 |
0,0001955 |
0,0013032 |
0,0048869 |
0,0097738 |
0,0081448 |
0,0243203 |
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,01035689 |
0,01035689 |
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
|
|
|
|
0,63528706 |
0,09575972 |
0,0905525 |
0,07108568 |
0,0508915 |
0,03222859 |
0,02419431 |
0,99999936 |
F4 Binomial Q=1 |
|
|
F4 |
|
|
|
|
|
14,5%
pop id |
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,63528706 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,0320697 |
0,0384837 |
0,0192418 |
0,0051312 |
0,0007697 |
0,0000616 |
0,0000021 |
0,0957597 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,0079497 |
0,0238492 |
0,0298115 |
0,0198743 |
0,0074529 |
0,0014906 |
0,0001242 |
0,0905525 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,0011107 |
0,0066643 |
0,0166607 |
0,0222143 |
0,0166607 |
0,0066643 |
0,0011107 |
0,0710857 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
0,0000698 |
0,0008377 |
0,0041886 |
0,0111696 |
0,0167544 |
0,0134035 |
0,0044678 |
0,0508915 |
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
6,91E-07 |
2,07E-05 |
0,000259 |
0,0017269 |
0,006476 |
0,012952 |
0,0107933 |
0,0322286 |
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,02419431 |
0,02419431 |
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
0,3647123 |
|
|
|
0,67648775 |
0,06985562 |
0,07016171 |
0,06011632 |
0,04811364 |
0,0345719 |
0,04069242 |
0,99999935 |
F5 Binomiaal Q=1 |
|
|
F5 |
|
|
|
|
|
24,4%
pop id |
|
|
|
On Tabel 12 2n=6 F0 singular the homozygosis also is
calculated out of the binomial distribution and the results here are also
compared with the method of Wright. The character of this calculus is showed in
the F1. The equality here also supports evidence that Wrights’ formula is
compatible with the binomial distribution and extinction, but not with the
exponential extinction.
Tabel 12 2n=6 F0 singular. Binomial homozygosis
|
2n=6. Distribution alleles, gametes of F0 6x
singular. Heterozygosis F0, Het=1 |
|
Real heterozygosis Het=1 |
|||||||
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
∑ |
|
|
|
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
F0
distribution Q=1 |
|
|
0 |
6 |
0 |
0 |
0 |
0 |
0 |
6 |
F0
distribution all 6 alleles |
|
|
0 |
0,166666 |
0 |
0 |
0 |
0 |
0 |
0,166666 |
Potential
Hom. in F0 |
|
|
By binomial allele distribution in F0→ Het in
F0=1-0,166..=0,833.. By Wrights’
formula Het in F0=1x(5/6)^1=
0,8333..= Potential Het. |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,33489797 |
0,40187757 |
0,20093878 |
0,05358368 |
0,00803755 |
0,000643 |
0,0000214 |
0,99999995 |
F1 verdeling Q=1 |
|
|
x6 |
x6 |
x6 |
x6 |
x6 |
x6 |
x6 |
|
|
|
|
2,00938786 |
2,4112656 |
1,2056328 |
0,3215022 |
0,0482256 |
0,003858 |
0,0001284 |
6,00000046 |
F1
distribution all 6 alleles |
|
|
x0 |
x(1/6)^2 |
x(1/3)^2 |
x(1/2)^2 |
x(4/6)^2 |
x(5/6)^2 |
x1 |
|
|
|
|
0 |
0,0669796 |
0,1339592 |
0,08037555 |
0,0214336 |
0,00267917 |
0,0001284 |
0,30555552 |
Hom.
in F1 |
|
|
By binomial allele distribution in F1→ Het in
F1=1-0,30555=0,69444. By Wrights’
formula Het in F1=1x(5/6)^2=0,6944..
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,48797495 |
0,21958392 |
0,16013086 |
0,08417931 |
0,03510228 |
0,01096453 |
0,00206387 |
0,99999972 |
F2
distribution Q=1 |
|
|
2,92785 |
1,3175034 |
0,9607854 |
0,5050758 |
0,2106138 |
0,065787 |
0,0123834 |
5,9999988 |
F2
distribution all 6 alleles |
|
|
0 |
0,03659732 |
0,10675393 |
0,1262690 |
0,09360613 |
0,04568542 |
0,0123834 |
0,42129515 |
Hom.
in F2 |
|
|
By binomial allele distribution in F2→ Het in
F2=1-0,4212951=0,5787049. By Wrights’
formula Het in F2=1x(5/6)^3=0,5787037 |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,57693497 |
0,13889692 |
0,11954765 |
0,08150921 |
0,04843345 |
0,02432034 |
0,01035689 |
0,99999944 |
F3
distribution Q=1 |
|
|
3,46161 |
0,8333814 |
0,7172856 |
0,4890552 |
0,290601 |
0,1459218 |
0,0621414 |
5,9999964 |
F3
distribution all 6 alleles |
|
|
0 |
0,02314948 |
0,0796984 |
0,1222638 |
0,129156 |
0,10133458 |
0,0621414 |
0,51774367 |
Hom.
in F3 |
|
|
By binomial allele distribution in F3→ Het in
F3=1-0,5177437=0,4822563. By Wrights’
formula Het in F3=1x(5/6)^4=0,4822531 |
|||||||||
|
0,62346 |
|
|
|
|
|
|
|
|
|
|
0,63528706 |
0,09575972 |
0,0905525 |
0,07108568 |
0,0508915 |
0,03222859 |
0,02419431 |
0,99999936 |
F4
distribution Q=1 |
|
|
3,8117226 |
0,5745582 |
0,543315 |
0,4265142 |
0,305349 |
0,1933716 |
0,1451658 |
5,9999964 |
F4
distribution all 6 alleles |
|
|
0 |
0,01595995 |
0,06036833 |
0,10662855 |
0,13571067 |
0,13428583 |
0,1451658 |
0,59811913 |
Hom.
in F4 |
|
|
By binomial allele distribution in F4→ Het in
F4=1-0,5981191=0,4018809. By Wrights’
formula Het in F4=1x(5/6)^5=0,4018776 |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,67648775 |
0,06985562 |
0,07016171 |
0,06011632 |
0,04811364 |
0,0345719 |
0,04069242 |
0,99999935 |
F5
distribution Q=1 |
|
|
4,0589262 |
0,4191336 |
0,4209702 |
0,3606978 |
0,2886816 |
0,2074314 |
0,2441544 |
5,9999952 |
F5
distribution all 6 alleles |
|
|
0 |
0,0116426 |
0,04677447 |
0,09017445 |
0,12830293 |
0,14404958 |
0,2441544 |
0,66509843 |
Hom.
in F5 |
|
|
By binomial allele distribution in F5→ Het in
F5=1-0,6650984=0,3349016. By Wrights’
formula Het in F5=1x(5/6)^6=0,334898 |
|||||||||
In Table 11 2n=6 in F0 two times in threefold is a symmetric distribution with allele
frequencies
Table 11 2n=6 in F0 two times in threefold
|
Binomial 2n=6. Distribution alleles, gametes of F0 two
times in threefold |
|
|
|||||||
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
∑ |
|
|
|
0 |
0 |
0 |
3 |
0 |
0 |
0 |
3 |
F0 binomial Q=3 |
|
|
F0 |
|
|
|
|
|
|
|
|
|
|
3→0 |
3→1 |
3→2 |
3→3 |
3→4 |
3→5 |
3→6 |
|
|
|
|
0,015625 |
0,09375 |
0,234375 |
0,3125 |
0,234375 |
0,09375 |
0,015625 |
1 |
F1 binomial Q=3 |
|
|
F1 |
|
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,015625 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,0313966 |
0,03767602 |
0,018838 |
0,00502347 |
0,00075352 |
0,00006028 |
2,009E-06 |
0,0937499 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,02057613 |
0,0617284 |
0,07716049 |
0,05144033 |
0,0192901 |
0,00385802 |
0,0003215 |
0,23437497 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,0048828 |
0,02929688 |
0,07324219 |
0,09765625 |
0,07324219 |
0,02929688 |
0,0048828 |
0,31249998 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
0,0003215 |
0,00385802 |
0,0192901 |
0,05144033 |
0,07716049 |
0,0617284 |
0,02057613 |
0,23437497 |
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
2,009E-06 |
0,00006028 |
0,00075352 |
0,00502347 |
0,018838 |
0,03767602 |
0,0313966 |
0,0937499 |
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,015625 |
|
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
0 |
|
|
|
0,07280404 |
0,13261959 |
0,1892843 |
0,21058385 |
0,1892843 |
0,13261959 |
0,07280404 |
0,99999971 |
F2 binomial Q=3 |
|
|
F2 |
|
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,07280404 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,044414 |
0,05329684 |
0,02664842 |
0,00710625 |
0,00106594 |
8,5275E-05 |
2,8425E-06 |
0,13261956 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,01661755 |
0,04985266 |
0,06231582 |
0,04154388 |
0,01557895 |
0,00311579 |
0,00025965 |
0,18928429 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,00329037 |
0,01974224 |
0,0493556 |
0,06580747 |
0,0493556 |
0,01974224 |
0,00329037 |
0,21058389 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
0,00025965 |
0,00311579 |
0,01557895 |
0,04154388 |
0,06231582 |
0,04985266 |
0,01661755 |
0,18928429 |
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
2,8425E-06 |
8,5275E-05 |
0,00106594 |
0,00710625 |
0,02664842 |
0,05329684 |
0,044414 |
0,13261956 |
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,07280404 |
|
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
0 |
|
|
|
0,13738845 |
0,1260928 |
0,15496473 |
0,16310772 |
0,15496473 |
0,1260928 |
0,13738845 |
0,99999967 |
F3 binomial Q=3 |
|
|
F3 |
|
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,13738845 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,04222823 |
0,0506739 |
0,0253369 |
0,00675652 |
0,00101348 |
0,0000811 |
2,7026E-06 |
0,12609283 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,01360458 |
0,04081375 |
0,05101718 |
0,03401146 |
0,0127543 |
0,00255086 |
0,00021257 |
0,1549647 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,00254856 |
0,01529135 |
0,03822837 |
0,0509712 |
0,03822837 |
0,01529135 |
0,00254856 |
0,16310775 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
0,00021257 |
0,00255086 |
0,0127543 |
0,03401146 |
0,05101718 |
0,04081375 |
0,01360458 |
|
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
2,7026E-06 |
0,0000811 |
0,00101348 |
0,00675652 |
0,0253369 |
0,0506739 |
0,04222823 |
|
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,13738845 |
|
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
|
|
|
|
0,19598509 |
0,10941096 |
0,12835023 |
0,13250716 |
0,12835023 |
0,10941096 |
0,19598509 |
0,99999971 |
F4 binomial Q=3 |
|
|
F4 |
|
|
|
|
|
|
|
|
|
|
→0→0 |
→0→1 |
→0→2 |
→0→3 |
→0→4 |
→0→5 |
→0→6 |
|
|
|
|
0,19598509 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
→1→0 |
→1→1 |
→1→2 |
→1→3 |
→1→4 |
→1→5 |
→1→6 |
|
|
|
|
0,0366415 |
0,04396983 |
0,0219849 |
0,00586264 |
0,0008794 |
0,00007035 |
2,3451E-06 |
0,10941096 |
|
|
|
→2→0 |
→2→1 |
→2→2 |
→2→3 |
→2→4 |
→2→5 |
→2→6 |
|
|
|
|
0,01126806 |
0,0338042 |
0,04225521 |
0,02817014 |
0,0105638 |
0,00211276 |
0,00017606 |
0,12835023 |
|
|
|
→3→0 |
→3→1 |
→3→2 |
→3→3 |
→3→4 |
→3→5 |
→3→6 |
|
|
|
|
0,00207043 |
0,01242255 |
0,03105638 |
0,0414085 |
0,03105638 |
0,01242255 |
0,00207043 |
0,1325072 |
|
|
|
→4→0 |
→4→1 |
→4→2 |
→4→3 |
→4→4 |
→4→5 |
→4→6 |
|
|
|
|
0,00017606 |
0,00211276 |
0,0105638 |
0,02817014 |
0,04225521 |
0,0338042 |
0,01126806 |
|
|
|
|
→5→0 |
→5→1 |
→5→2 |
→5→3 |
→5→4 |
→5→5 |
→5→6 |
|
|
|
|
2,3451E-06 |
0,00007035 |
0,0008794 |
0,00586264 |
0,0219849 |
0,04396983 |
0,0366415 |
|
|
|
|
→6→0 |
→6→1 |
→6→2 |
→6→3 |
→6→4 |
→6→5 |
→6→6 |
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
0,19598509 |
|
|
|
|
∑0 |
∑1 |
∑2 |
∑3 |
∑4 |
∑5 |
∑6 |
|
|
|
|
0,24614348 |
0,09237969 |
0,10673968 |
0,10947406 |
0,10673968 |
0,09237969 |
0,24614348 |
0,99999977 |
F5 binomial Q=3 |
|
|
F5 |
|
|
|
|
|
|
|
|
|
At table 12 F0 in threefold the calculation of the
homozygosis with the binomial distribution again are compared with the
calculation in accordance with Wright. Also here the results are equal. This
now is more interesting than at the former table, because the start here is at F0 in a population with
already some measure of homozygosis. The homozygosis in the F1 now is larger
than in the former population, that started with singular alleles and the
homozygosis in all the generations here is not exclusively caused by identity
of the alleles by descent but also by general identical alleles. That is
different in all the populations that start with singular alleles (Q=1). So
this supports evidence for the point that Wrights’
formula describes the binomial extinction of the heterozygosis in general and not only for alleles that are identical by descent, as it is
suggested by some people. Different from the population with singular alleles
here are many genotypes possible for F0, all with 2x3 alleles, or frequency
Table 12 2n=6 F0 2x in threefold. Binomial homozygosis
|
2n=6. Distribution alleles, gametes of F0 2x in
threefold. |
Real Het is unknown |
||||||||
|
→0 |
→1 |
→2 |
→3 |
→4 |
→5 |
→6 |
∑ |
|
|
|
0 |
0 |
0 |
3 |
0 |
0 |
0 |
3 |
F0 distribution Q=3 |
|
|
0 |
0 |
0 |
6 |
0 |
0 |
0 |
6 |
F0
distribution all 6 alleles |
|
|
0 |
0 |
0 |
0,5 |
0 |
0 |
0 |
0,5 |
Gemiddelde
Het. = 0,5 |
|
|
|
0,5x1/6 |
|
0,5 |
|
|
|
0,5833… |
Potential
Hom. in F0 |
|
|
By the Wrights’ formula Het in F0=0,5x(5/6)^0=0,5 |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,015625 |
0,09375 |
0,234375 |
0,3125 |
0,234375 |
0,09375 |
0,015625 |
1 |
F1
distribution Q=3 |
|
|
x2 |
x2 |
x2 |
x2 |
x2 |
x2 |
x2 |
|
|
|
|
0,03125 |
0,1875 |
0,46875 |
0,625 |
0,46875 |
0,1875 |
0,03125 |
2 |
F1
distribution all 6 alleles |
|
|
x0 |
x(1/6)^2 |
x(1/3)^2 |
x(1/2)^2 |
x(4/6)^2 |
x(5/6)^2 |
x1 |
|
|
|
|
0 |
0,005208 |
0,052083 |
0,15625 |
0,208333 |
0,130208 |
0,03125 |
0,583333 |
Hom.
in F1 |
|
|
By the binomial distribution in F1→ Het in
F1=1-0,58333=0,4166667. By the Wrights’ formula Het in
F1=0,5x(5/6)^1=0,416667 |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,07280404 |
0,13261959 |
0,1892843 |
0,21058385 |
0,1892843 |
0,13261959 |
0,07280404 |
0,99999971 |
F2
distribution Q=3 |
|
|
0,14560808 |
0,26523918 |
0,3785686 |
0,4211677 |
0,3785686 |
0,26523918 |
0,14560808 |
1,99999943 |
F2
distribution all 6 alleles |
|
|
0 |
0,00736776 |
0,04206318 |
0,10529193 |
0,16825271 |
0,18419389 |
0,14560808 |
0,65277754 |
Hom.
in F2 |
|
|
By the binomial distribution in F2→ Het in
F2=1-0,6652777=0,3472222. By the Wrights’ formula Het in
F2=0,5x(5/6)^2=0,347222. |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,13738845 |
0,1260928 |
0,15496473 |
0,16310772 |
0,15496473 |
0,1260928 |
0,13738845 |
0,99999967 |
F3
distribution Q=3 |
|
|
0,2747769 |
0,2521856 |
0,30992945 |
0,32621544 |
0,30992945 |
0,2521856 |
0,2747769 |
1,99999934 |
F3
distribution all 6 alleles |
|
|
0 |
0,00700516 |
0,03443661 |
0,08155385 |
0,13774644 |
0,17512889 |
0,2747769 |
0,71064785 |
Hom.
in F3 |
|
|
By the binomial distribution in F3→ Het in
F3=1-0,7106478=0,2893522. By the Wrights’ formula Het in
F3=0,5x(5/6)^3=0,289352. |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,19598509 |
0,10941096 |
0,12835023 |
0,13250716 |
0,12835023 |
0,10941096 |
0,19598509 |
0,99999971 |
F4
distribution Q=3 |
|
|
0,39197018 |
0,21882191 |
0,25670046 |
0,26501432 |
0,25670046 |
0,21882191 |
0,39197018 |
1,99999942 |
F4
distribution all 6 alleles |
|
|
0 |
0,00607839 |
0,02852228 |
0,06625358 |
0,11408911 |
0,15195965 |
0,39197018 |
0,75887318 |
Hom.
in F4 |
|
|
By the binomial distribution in F4→ Het in
F4=1-0,7588732=0,2411268. By the Wrights’ formula Het in
F4=0,5x(5/6)^4=0,2411265 |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
0,24614348 |
0,09237969 |
0,10673968 |
0,10947406 |
0,10673968 |
0,09237969 |
0,24614348 |
0,99999977 |
F5
distribution Q=3 |
|
|
0,49228696 |
0,18475938 |
0,21347937 |
0,21894813 |
0,21347937 |
0,18475938 |
0,49228696 |
1,99999954 |
F5
distribution all 6 alleles |
|
|
0 |
0,00513221 |
0,02371993 |
0,05473703 |
0,09487997 |
0,12830514 |
0,49228696 |
0,79906123 |
Hom.
in F5 |
|
|
By the binomial distribution in F5→ Het in
F5=1-0,7990612=0,2009388. By the Wrights’ formula Het in
F5=0,5x(5/6)^5=0,2009388. |
|||||||||
Binomial and exponential
extinction
As pointed out there is difference between the binomial extinction and
the exponential extinction. The binomial extinction is found by complete
elaborating of the superposition in the binomial distributions, but the
exponential extinction can be found much easier and is calculated here in two
ways. The first way of calculation is described at Table 5c. In this the
primary exponential extinction is calculated from the primary intensity μ by P0=℮^-μ and μ as μ=−2n.ln(1-1/2n). The recurrence of it is
followed by the exponential accumulation and the intensities in the following
generations are calculated by−σ(Fg)=μν-μ and the extinctions by P0=℮^−σ. Different from Table 5b is the accumulating
exponential distribution at Table 5c not
a part of the Poisson superposition. This calculation only can be applied (by
me) at the singular alleles at the F0 (Q=1). The extinctions of the multiple
alleles (Q>1) cannot be summed in the recurrence in the way pointed out in
the text at table 7 of the Poisson
distributions. In small populations the courses of the extinctions and
multiplications of the various alleles are not independent of each other, as it
does in the infinite population. The second way of calculation starts from the
complementary chance. This is the chance that a marble, an allele will not been
drawn in one drawing. This calculation simply takes this chance as a base and
the allele survival, α, as an exponent. If for instance there are 6 different coloured
marbles in the bag at the first drawing (2n=6 Q=1), the extinction chance for
the total first drawing turn is (5/6)^6, because the 6 original marbles are in
the bag. At the second drawing turn the extinction chance is (5/6)^3,9906 now,
because on the average for drawing were left 3,9906 of the original 6, namely
6{1-(5/6)^6}. Also if we start with the multiple alleles in the F0 (Q>1), for
instance 2x3 marbles (2n=6 Q=3) the resting marbles do decide the exponent of
the recurrence. So the formula is {Q(1-1/2n)}^α.. Both
calculations of the exponential extinction are giving the same results, but I
only can apply the second formula generally.
The differences between the exponential and binomial values are obvious
not very large in the first generations, but they increase and will decrease
afterwards, because they will converge to the same points. These differences
probably are due to differences in taking the averages in the total line at the
superposition of the distributions. The successive drawing turns or generations
are described in the binomial superposition as separate processes and the problem then is that they
are not independent of each other. In the binomial superposition the average of
the events in one generation is ever taken as the basis for the calculations
for the next generation. In this is not taken into account how deviations from
the averages within the separate generations may change the total average
through the course of all the generations as is the extinction and fixation
process. This binomial extinction so is a compound process. In the exponential
approach on the contrary the events through all the turns or generations are
described within one uniform process, according to the fact the exponential
accumulating distribution just is one distribution through the generations (see
at Table 5). The deviations from the
average within the uniform exponential distribution also are not described
here, but they are simply determined within this distribution itself. The
uniformity of the exponential approach will describe probably better what can
be measured as extinction velocity in computer simulations.
Besides of the exponential extinctions also can calculated the
exponential (or recurrence) fixations. These exponential fixations probably are
easy to be calculated by Q.(1/2n)^α. Also the exponential heterozygosis extinctions can be
calculated. The values of the exponential heterozygosis extinctions will differ
somewhat from the binomials of Wright. These exponential values may have a
better approach, but they probably are to be calculated more difficult than the binomial ones do with
the simple Wrights formula. I did not yet found a way to calculate the
exponential extinctions of the heterozygosis. So this remains yet as a challenge
as well as the proofs for the exponential fixations and a good uniform
description of the total exponential theory for the small population.
Table 5d
|
Table
5d
2n=2 |
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
|
|
μ=1,38629 |
σ=1,03972 |
σ=0,89617 |
σ=0,82050 |
σ=0,77603 |
σ=0,74828 |
σ=0,73033 |
σ=0,71845 |
σ=0,71046 |
|
|
P0=0,25 |
P0=0,35355 |
P0=0,40813 |
P0=0,44021 |
P0=0,46023 |
P0=0,47318 |
P0=0,48175 |
P0=0,48751 |
P0=0,49142 |
exponentiial |
|
0,25 |
0,375 |
0,4375 |
0,46875 |
0,484375 |
0,49219 |
0,49609 |
0,49805 |
0,49902 |
binomial |
|
F1 |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Table
5d 2n=4
Q=1 |
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
|
|
|
|
|
|
μ=1,15072829 |
σ=0,78663 |
σ=0,62671 |
σ=0,53584 |
σ=0,47735 |
accumulating
exponential intensity |
|
|
||
|
(3/4)^4 |
(3/4)^2,7344 |
(3/4)^2,1785 |
(3/4)^1,8626 |
(3/4)^1,6593 |
accumulating
exponential extincton |
|
|
||
|
P0=0,31641 |
P0=0,45538 |
P0=0,53434 |
P0=0,58518 |
P0=0,62043 |
accumulating
exponential extincton |
|
|
||
|
0,3164063 |
0,4632567 |
0,5484354 |
0,6037994 |
0,6422063 |
binomial
extincton |
|
|
||
|
F1 |
F2 |
F3 |
F4 |
F5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Table
5d
2n=6 Q=1 |
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
|
|
|
|
|
|
μ=1,093929 |
σ=0,72757 |
σ=0,56548 |
σ=0,47247 |
σ=0,41192 |
accumulating
exponential intensity |
|
|
||
|
(5/6)^6 |
(5/6)^3,9906 |
(5/6)^3,1015 |
(5/6)^2,5915 |
(5/6)^2,2593 |
accumulating
exponential extincton |
|
|
||
|
P0=0,334898 |
P0=0,48308 |
P0=0,56809 |
P0=0,62346 |
P0=0,66238 |
accumulating
exponential extincton |
|
|
||
|
0,334898 |
0,487975 |
0,576935 |
0,635287 |
0,676488 |
binomial
extincton |
|
|
||
|
F1 |
F2 |
F3 |
F4 |
F5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Table
5d 2n=6
Q=3 |
|
|
|
|
|
|
|
|
|
|
F0 |
F1 |
F2 |
F3 |
F4 |
|
|
|
|
|
|
(3/6)^6 |
(3/6)^3,9906 |
(3/6)^3,1015 |
(3/6)^2,5915 |
(3/6)^2,2593 |
accumulating
exponential extincton |
|
|
||
|
0,015625 |
0,062909 |
0,116506 |
0,165918 |
0,208878 |
accumulating
exponential extincton |
|
|
||
|
0,015625 |
0,072804 |
0,1373885 |
0,1959851 |
0,2461435 |
binomial
extincton |
|
|
||
|
F1 |
F2 |
F3 |
F4 |
F5 |
|
|
|
|
|
Some Practical conclusions
It is obvious that -for the absolute numbers Q- the total sum of the
extinctions and fixations together of the alleles in a population of limited size n converges through the
generations to the values the extinction has in the unlimited population according to table 5 and as described further on in the application of the
recurrence if Q≠1. In somewhat larger populations (ca n>50), so with
small fixation chances the extinctions do differ little from the extinction by Table
Suppose an allele occurs homogeneous in the world population, n=5.10^9
and once a year one homozygous child is born on 10^8 births with the lethal
marks of the recessive allele. The absolute quantum of the allele now is easy
to be calculated, it is 500000, by the allele frequency (10^-8)^0,5. By the selection every year will
vanish perhaps 2 alleles, which is practical nothing. So globally the
extinction will last about 5.10^7 generations to reach a change of 99%. So from
the perspective of the survival times of individuals, species or even total
planets such an allele will never disappear.
In large homogenous populations extinction only is to be expected for
alleles with very small quanta if they are neutral or recessive. In large populations the very small quanta do occur only
the case of new mutations and as dominant
alleles and further observable genetic changes are excluded in large
populations at homogeneous reproduction. Important
changes in frequencies of genes with limited or none expression are excluded in
large homogeneous populations, this concerning changes by selection as well as
at random. Total recessive alleles only are sensitive for selection, if
they occur in the population in such high numbers, that the presence of homozygosis is evident. Changes in the
frequencies of genes with an obvious expression in the functions of the
organisms at their changes on survival and behaviour in the reproduction are
yet possible, but this concerns a very limited number of genes, so that there
is no talk of a general micro-evolution in large homogeneous populations. That
is why evolution within large populations only is possible if they are not homogeneous
by place and so exist out of demes, or by time and so are fluctuating strong in
number.
The flexible small quanta do occur of course in small populations. In
this is the extinction of the singular neutral allele within the limited
population only is a little slower than in the unlimited H–W population and
there only are observable differences in very small populations. So by the neutral theory it easily is to be
seen how fast genetic changes (by selection or random) can take place within
small populations and demes or by inbreeding. So the extinction is of great importance in small populations and the
study of it as the zero-hypothesis of the selection is indispensable. The allele extinction are to be calculated very
well within limited and small populations and at non random inbreeding, but why
this is not be done? Why is the extinction not described in the manuals and why
she is considered as irrelevant? Are even the sciences for a big part
fashionable parroting and a product of a narrow-minded and dependent way of
thinking?!
The conclusion that micro and so also macro-evolution only can take
place by reproduction within small demes or small populations, by reproduction
with non random inbreeding and with selection by sexual behaviour leads also to
the conception that the living organisms themselves actively are concerned in
their evolution. So in the evolution does not exist something like a sieve that
is screening the organisms in their struggle for live and decides which
organisms or alleles can go through the meshes and which can not. No, the
struggle for live is not suffered by the organisms, but is pursued by them and
they decide how it is executed. The evolution is an integral part of the
existence of the organisms. The chemical functioning of the DNA in joining with
all the processes of live let the organisms themselves decide and regulate how
their evolution is, yet:
An organism exists, because it can survive in its life surroundings,
otherwise it could not exist.
An organism exists, because it can procreate, otherwise a mortal
organism could not exist.
An organism exists, because it can change itself through the
generations, otherwise it could not exist in the ever changing life surroundings[15].
So there is evolution because the organisms can change[16]
their genes, for if they could not do so….
So the organisms do participate actively in the evolutionary changes and
these are regulated within the processes of the living organisms, in their
organic functions. The micro-evolution is an integral part of the life
functions by joining as: genetic decided behaviour at the procreation →
the measure of inbreeding and sexual selection → genetic changes →
behaviour at the procreation, etc. The evolution takes place within the many
small and large ecological unities, so the small and large syntheses of the
many species of organisms on the one side and
on the other hand the changing existence of the vehicle of their life,
the planet Earth.
That active participation of the living organisms in their evolution
makes the study of the macro-evolution so very fascinating. Although there are
yet many challenges in the micro-evolution I now will make a study of
especially the macro-evolution to find out the possibilities to get here also somewhat
more clarity in the vagueness.
Some notions
Out breeding: The complement of
inbreeding. So this can be relative but often the meaning here is absolute in
the sense of: Random mating in the infinite Hardy-Weinberg population, or
absolute no common ancestors.
Inbreeding, absolute (and
relative): Relationship (more than average) between the parents, because they have
common ancestor or they themselves are genetically identical.
Effective size population of
reproduction: The number of individuals participating effectively in the
reproduction. So they have children that get children of themselves. Individuals
having no children are not counted and parents that have children but get no
grandchildren are parents with 0 children etc. Elucidation: Inbreeding
determines the average expected number of descendants of the common ancestors,
So inbreeding determines the effective population size. However, different from
S. Wright and M. Kimura I think that a variation in the parities larger of
smaller than random expected can not determine the size of the effective
population. This should be than an ideal or virtual population, but you can
only work in science with perceptible
populations, so real or possible physically existent populations. Non random
differences in the results by individuals participating at the procreation is
yet in the picture as the only possible source of the selection. Also the
inequality men and women in the participation of the procreation is usually
equalized in the formula for the effective reproduction. In this study this
also is not a matter of course or necessity. The population of reproduction
here is the concrete or possible progeny of one or more individuals.
Parity: In obstetrics the
number of deliveries an (expectant) mother has passed through. In this context
the number of (effective) children of one parent and also the number of
exemplars of one allele, transferred from one individual in one
generation.
Replacement factor: This ratio r gives the number of descendants that
replaces one individual in the following generation(s) in neutral population
dynamics. The value of r per
generation is 1≤r≤2.
This value of r is decided by the
form of reproduction. In asexual reproduction r=1. In absolute out breeding r=2.
In inbreeding 1≤r<2.
Elucidation at the
literature.
So I tried to develop a simple uniform system that can easily be
followed and that give some insight in the figures at the complicated steps of
the biologic substrate. This elementary system that uses the versatile
abilities of the transcendent number ℮ can extend to more applications.
The sciences and mathematics are uniform so there are of course many ways
leading to the same results. In this it is not necessary and sometimes even not
desirable to consult and pursue always the ideas of other people in the
literature. So I did not do this, but if I try to compare afterwards this
calculus with the literature I get the idea this is a particular way and it
describes new possibilities. I did make some study of the literature of
particular Kimura after a useful advise of dr Gerdien de Jong of the
Literature
Classical approach:
Fisher, RA. On the Dominance ratio. Proceedings
of the royal society of Edinburgh 1921-22, Vol 42: 321-342, especially 325-326. and for many Fisher publications: http://digital.library.adelaide.edu.au/coll/special//fisher/
Fisher, RA. The distribution of gene ratio’s for rare mutations, Proceedings of the royal society of
Freeman, S and Herron, JC. Evolutionary analysis.
Haldane, JBS. A mathematical theory of natural and artificial selection,
Proceedings of the
Kimura, M. The neutral theory of molecular evolution. Page 195
Kimura, M and Crow, JF. An introduction to population genetics theory
(1970). Page 421-423
Sarkar, S. Evolutionary theory in the 1920s: The nature of a synthesis, preliminary draft on the internet, 2004.
Philosophy of sciences Vol 71, 1215-122 .
and: http://www.journals.uchicago.edu/PHILSCI/journal/issues/v71n5/710527/710527.web.pdf
Wright, S. Evolution in Mendelian populations. Genetics 1931, Vol 16, 97-158. also http://www.esp.org/foundations/genetics/classical/holdings/w/sw-31.pdf
Some alternative modern
approaches:
Buss, SR and Clote, P. Solving the Fisher-Wright and coalescent problems
with a discrete Markov chain analysis. Advances
in applied probability 2004, Vol 36,
1175-1197, and: http://citeseer.ist.psu.edu/cache/papers/cs/32196/http:zSzzSzeuclid.ucsd.eduzSz~sbusszSzResearchWebzSzmarkovzSzpaper.pdf/buss04solving.pdf
Bustamante, CD. Population genetics of molecular evolution. Springer issn 1431-8776 see also http://bustamantelab.cb.bscb.cornell.edu/docs/Bustamante_05.pdf
Cambell, RB. A logistic branching process alternative to the
Wright-Fisher model, internet publication,
http://cns2.uni.edu/~campbell/evol01.pdf
Gordo, I and Dionio, F. Nonequilibrium model for estimating parameters
of deleterious mutations, Physical review
2005, E71 031907 and: http://eao.igc.gulbenkian.pt/EB/PRE_nonequilibrium.pdf
Heylighen, F. Evolutionary cybernetics, complexiteit en evolutie, etc.
This professor of
Hoppe, FM, The sampling theory of neutral alleles and an urn model in
population genetics. Journal of
mathematical bioilogy, 1987, Vol 25, 123-159. and: http://deepblue.lib.umich.edu/bitstream/2027.42/46946/1/285_2004_Article_BF00276386.pdf
Jagers, P and Sagitov, S. Coalescent process reversed branching. In
branching process: variation, growth and extinction of populations. Page 200-208 , also http://www.math.leidenuniv.nl/~verduyn/ndns-leiden/section71.pdf
Joyce, P, Krone SM and Kurtz TG, Gaussian limits associated with the
Poisson-Dirichlet distribution and the Ewens sampling formula. The annuals of applied probability 2002, Vol
12, 1-24 and: http://citeseer.ist.psu.edu/cache/papers/cs/15801/http:zSzzSzkleene.math.wisc.eduzSz~kurtzzSzpaperszSzjoyce.pdf/gaussian-limits-associated-with.pdf
Wakeley, J. The limits of theoretical population genetics. Genetics 2005, Vol 169, 1-7 and with many links: http://www.genetics.org/cgi/content/full/169/1/1?etoc#BIB28
Statistics of human
procreation:
Jain, SK and McDonald, PF, Fertility of Australian birth cohorts,
components and differentials. Journal of
the Australian population association, 1997, Vol 14,no 1. and http://dspace.anu.edu.au/bitstream/1885/41454/2/fertility.pdf
Kalabikhna, IE, Fertility in
Vlaams agentschap zorg en
gezondheid, pariteit naar etniciteit, http://www.zorg-en-gezondheid.be/topPage.aspx?id=3228#etniciteit
Zakharov, SV and Ivanova, EI. Fertility decline and recent chances in
[1] ©Willem Schot, ’s-Hertogenbosch,
[2] This is to be described more exactly as the
total heterozygosis of all the allelic variations on the gene loci. This
makes it impossible to derive the allele frequencies from this parameter. See
also table 11 and 12.
[3] See Table 9
[4] In this the meaning of ‘or’ is inclusive in the sense of or/and.
[5] The selection on these genes must have been very
strong because they mostly are recessive, were probably seldom present in the source populations that came
from the South, while they now are nearly fixed in many populations in the
North of Europe.
[6] The n individuals of F0 do have 2 children on the
average together with their sexual partners in the population of constant size.
[7] Increase in population size has the same effect as
positive selection on the absolute
numbers of alleles. In this there is
increase on specific alleles by the selection and on all the alleles by
the increase in population size.
[8] Kimura: an introduction to population genetics
theory blz 422
[9] On the dominance ratio blz 326
[10] So in those cases the asexual reproduction is
more susceptible for selection than then the sexual one.
[11] This affinity is an indirect cause of
inbreeding: descendants of full brothers and sisters have a higher relationship
and will give a larger degree of inbreeding than descendants of half-brothers
and half-sisters.
[12] Also plants do have sexual behaviour or sexual
communication by their flowers mostly in the symbiotic relations with insects.
[13] Properly its complement the heterozygosis
survival.
[14]In this the homozygosis in F0 and the further
generations is not exclusively by common descent.
[15] By a lot of micro and macro factors the live
surroundings and so the life conditions are changing ever: By the annual
alternation of summer and winter or wet and dry seasons and on the longer term
the periodical climate is changed by astronomical factors as the precession of
the earth’s axis and the variable elliptic form of the earth’s orbit. There
also are sudden changes in the surrounding by irregular and unpredictable
factors as are droughts and floods, but also by global factors as an impact of
a meteor and a mega volcanic outburst. For all the living organisms on earth
the circumstances of live are changing ever more and they will have to change
by this, but they are affecting each other in this process, so giving a
snowball effect to the changing surroundings..
[16] This is a global conclusion on the basis of
generalisations, but the fact that we do not know yet how the organisms can change their genes nowise is an guilty
negation of this lemma that they do so, because ignorance never is an argument.