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 1 km with books and magazines with data relevant for the evolution theory about genetics, biochemistry, and palaeontology that ware unknown by Darwin, his book “the origin of the species” yet is in our days an important source which often is cited in manuals. So this does indicate the scientific and nature-philosophical grandiosity of Darwin. On the other hand this also indicates that we after 150 years do not know much more about the fundaments of the evolution theory than Darwin did, in spite of  all the details of the genetics, the molecular properties of the DNA, the cell structure, the physiology etc. We do have now much more evidence from the palaeontology and by the molecular biology the genetic similarities and differences can be studied by DNA research very good now, so that the descent of the species, their common ancestors and thus their evolution can be followed directly. There now is from the different disciplines a lot of evidence for the existence of the evolution in the way it has been described in essential by Darwin. So now we do know very well that new species do arise, but how and why the genetic changes arise and how different species do arise, we do not know much more than Darwin did: hazard and selection by the survival of the fittest. Many researchers in our time are thinking that is all and there is nothing more essential to be discovered. Probably the scientists of the 19th century had a more optimistic view on this in the sense of: evolution is a natural process and thus a regular process and the natural laws that underlie it will soon also been discovered. The very important discovery of Georg Mendel however, the discrete transfer of genetic variations in pairs with their segregation in the gametes in 1866, did not make the theory of evolution more clear initially, on the contrary. When this discovery was at last accepted in 1900 within the official biological science as an important tenet, after the research of Hugo de Vries, Carl Correns and Erich von Tschermak, it was difficult to unify this law of Mendel with the evolution theory. The difficulty was that the material steady genes could not change apropos of nothing or as a direct consequence of the circumstances in the life surroundings of the plants and animals, like Darwin thought as did Lamarck. Nevertheless science has accepted now both the laws of Mendel and the evolution theory of Darwin. The idea now is that the discrete genes change by accidental mutations. These mutated genes are passed through the generations as genetic variations or alleles. These changed genes can increase or decrease within the populations by hazard or by selection because they may be advantageous or unfavourable in the circumstances of the living individual. In the accepted view the accidental mutations are the primary and the only possible causes for changes in the genetic DNA of the organisms. It  now is again disputed that all the changes and mutations of the genes do arise in events that are hazardous, so not logical lied with biological functions of living individuals. For the micro-evolution however the starting point is the observed variations (alleles) on the loci of the genes and their changes in number and disappearances within populations.     

 

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 Ghana for development aid. They are handling careless their malaria prophylaxis and 6 of them acquire malaria. So now is measured here an average disease and infection prevalence of 0,6. But if 60% of the population has been infected  more infections than 60 on 100 will be present in this population, because some people did probably acquire more than one infection. So what is here the infection incidence or infection load, or how many mosquito stings with the parasite plasmodium has been distributed in average to this population. Following this formula the time t is given and no infection in 4 on 10 people is given, so P0 =0,4 =℮^-q and in this the negative quantity –q is to be calculated as –q= ln 0,4=−0,9163..So there were more than 9 infections in 10 persons. So here is the quantity q=0,9163 the proportion of events, or drawings, or hits per population. This proportion can also be expressed as a ‘chance’ or a ‘probability’, but than it is essential different from the change to be hit, to be infected, etc. The exponential distribution does however not describe how the events in the time, as infections, or drawings, or hits further are distributed into the space. In this example how many people do have in average 1; 2; 3, or more infections. The complete description of the distribution however is given in the Poisson distribution. Already in 1838 this complete theory has been posed by Siméon-Dénis Poisson, but only after 60 years (!) it was for the first time applied by Ladislaus Bortkiewiecz in the practical calculus of probabilities. This complete theory is given in the simple notion: P(i) = ℮^-q . q^i/i!  In this formula P(i) is the expected, or average proportion of event in number i, in this i always is a natural number as 0; 1; 2; 3 etc and i! means i factorial. In the example it is from the observation certain that 60% has been infected and 40% has been not and  with aid of the formula of Poisson than is to be calculated from this that on the average 36,7% did acquire one infection; 16,8% two infections; 5,1% three infections; 0,2% four etc. This generally well-known Poisson distribution is often describe by the formula Pt(i) = ℮^-λt . λt^i/i! In this λt is the parameter of the intensity of the Poisson distribution. The t in this is the variable period of time in which the events take place. In a system with a constant standard time like the generation time the t can be let away.

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 1 in the calculations here) or the absolute numbers of alleles in F0 that are transferred to the following generations. The further quantities of the absolute numbers in the generations from F1 are indicated with q. The factor indicating the change in the size of the population is p. If p=1 the size is constant; if p>1 the size is increasing; if p<1 the size is decreasing. The s faces the theoretical selection on account of the supposed fitness of the allele. Alleles with s=1 are neutral, with s>1 are advantageous and with s<1 are noxious for the survival and the reproduction. The factor r is the replacement ratio in neutral population dynamics, so if the population remains constant in size and if the alleles are neutral, so that the only changes are by hazard. At the reproduction in random mating with absolute out breeding r=2 and by inbreeding 1≤r<2; at the allele transfer always r=1. The validity of these factors determining the average numbers and so the Poison intensities in this way is pointed out further on here, but a-priori it is self-evident. 

   

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

Table H2.  Distribution of Women 40 to 44 Years Old by Number of Children Ever Born and Marital Status:  Selected Years, 1970 to 2004

 

 

 

 

 

 

 

 

 

 

 

Source:  U.S. Census Bureau

 

 

 

 

 

 

 

 

Internet release date:

 

 

 

 

 

 

 

 

 

(leading dots indicate sub-parts)

 

 

 

 

 

 

 

 

(Years ending in June.  Numbers in thousands)

 

 

 

 

 

 

Year

 Women 40-44 yr x1000

Women by number of children ever born in %

Children ever born per woman

Total

None

One

Two

Three

Four

Five and six

Seven or more

 

 

 

 

 

 

 

 

 

 

 

All Marital Classes

 

 

 

 

 

 

 

 

 

.1976

5684

100

10,2

9,6

21,7

22,7

15,8

13,9

6,2

3,091

Poisson λ=3,091

100

4,546

14,051

21,715

22,374

17,289

16,195

3,827

 

.1982

6336

100

11

9,4

27,5

24,1

13,8

10,4

3,9

2,783

Poisson λ=2,783

100

6,185

17,214

23,953

22,22

15,46

12,596

2,373

 

.2004

11535

100

19,3

17,4

34,5

18,1

7,4

2,9

0,5

1,895

Poisson λ=1,895

 

100

15,032

28,485

26,99

17,049

8,077

4,028

0,34

 

 

 

 

 

 

 

 

 

 

 

 

Women Ever Married

 

 

 

 

 

 

 

 

 

.1970

5815

100

8,6

11,8

23,8

21,4

14,6

12,9

6,8

3,096

Poisson λ=3,096

100

4,523

14,003

21,168

22,371

17,315

16,254

3,858

 

.1976

5455

100

7,5

9,6

22,4

23,4

16,4

14,4

6,3

3,19

Poisson λ=3,190

100

4,117

13,134

20,948

22,275

17,764

17,359

4,396

 

.1982

6027

100

7,6

9,6

28,7

25,1

14,3

10,8

4

2,885

Poisson λ=2,885

100

5,585

16,114

23,245

22,354

16,123

13,776

2,804

 

.1985

6836

100

8

12,9

34,2

24,1

11,4

7,4

2

2,548

Poisson λ=2,548

100

7,823

19,935

25,397

21,571

13,741

9,976

1,557

 

.1988

7543

100

10,2

14,7

37,3

22,1

9,5

5,2

0,9

2,28

Poisson λ=2,280

100

10,228

23,321

26,586

20,205

11,517

7,247

0,896

 

.1998

9995

100

13,7

18,1

38,7

19,6

6,2

3,2

0,6

2,002

Poisson λ=2,002

100

13,506

27,04

27,067

18,063

9,04

4,828

0,456

 

.2004

10036

100

13,2

17,4

38

19,9

8,2

2,9

0,4

2,046

Poisson λ=2,046

 

100

12,925

26,445

27,053

18,45

9,437

5,179

0,513

 

             

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 2 in the average number of children a women of 40 to 44 years of age had in those years and also, what is important here, how this was distributed over the numbers of children, the parities, the have. Under a row with the data of the US Census Bureau I did ever make for comparison a Poisson distribution of the parities with, of course, the average number of children as its intensity λ. It concerns here figures of women that nearly all did complete their family (99%). In this way however are compared in Table 1 the figures of the actual reproduction with the Poisson distribution and only the effective reproduction is the basis of the genetic transfer. There are indications that in this population the figures of the juvenile mortality are relative low and that the infertility, the younger generation will have later, should be distributed about equal over the parities. So the expectation here is that the failure by using the actual reproduction instead of the effective is not large, but there will be some differences. The point of interest is that these figures of the reproduction, despite these are actual here, give some direct indication for answers to the questions: are the genetic differences between the generations here larger, smaller, or as large as they are in at random expectation as the Poisson distribution indicates. Another problem is that the various differences may strengthen each other, but may also weaken and even neutralize each other. So further analysis and the calculation of the effective reproduction will give more specific evidence, but these raw figures already may give a realistic indication, which is described here. 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

Table 1. Women by Number of Children Ever Born by Race, Hispanic Origin, Nativity Status, Marital Status, and Age: June 2004

 

 

(Numbers in thousands.  For meaning of symbols, see table of contents.)                

 

 

 

 

--------------------------------------------------------------------------------------------------

 

 

 

 

 

(leading dots indicate sub-parts)

 

 

 

 

 

 

 

 

 

(Column B is in persons, all others are percents)

 

 

 

 

 

 

 

                                                 Women by number of children ever born

 

 

 

 

 

    age

total x1000

  Total 

  None 

  One 

  Two 

  Three 

  Four 

  5 and 6 

 ≥ 7

 children 

coëf MP

 

women

 women %

 %

 %

 %

 %

 %

 %

 %

ever born

 

ALL RACES

 

 

 

 

 

 

 

 

per woman

All Marital Classes

 

 

 

 

 

 

 

 

 

 

.40 to 44 

11.535

100

19,3

17,4

34,5

18,1

7,4

2,9

0,5

1,895

0,868

Poisson λ=1,895

100

15,032

28,485

26,99

17,049

8,077

4.028

0,34

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ALL RACES

 

 

 

 

 

 

 

 

 

 

.Women Ever Married

 

 

 

 

 

 

 

 

 

..40 to 44 

10.036

100

13,2

17,4

38

19,9

8,2

2,9

0,4

2,046

0,723

Poisson λ=2,046

100

12,925

26,445

27,053

18,45

9,437

5,179

0,513

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ALL RACES

 

 

 

 

 

 

 

 

 

 

.Women Never Married

 

 

 

 

 

 

 

 

 

40 to 44 

1.498

100

59,8

17

11,2

6,2

2,1

2,9

0,9

0,88

1,973

Poisson λ=0,88

100

41,478

36,501

16,06

4,711

1,036

0,042

0,001

 

 

 

 

 

 

 

 

 

 

 

 

 

 

WHITE ONLY

 

 

 

 

 

 

 

 

 

 

.Women Ever Married

 

 

 

 

 

 

 

 

 

 

40 to 44

8289

100

13,4

16,8

39,3

19,7

8

2,4

0,3

2,02

0,694

Poisson λ=2,02

100

13,266

26,796

27,064

18,223

9.203

4,97

0,478

 

 

 

 

 

 

 

 

 

 

 

 

 

 

WHITE ONLY, NOT HISPANIC

 

 

 

 

 

 

 

 

 

.Women Ever Married

 

 

 

 

 

 

 

 

 

 

40 to 44

7206

100

14,1

17,2

39,8

19,6

7,1

1,9

0,2

1,959

0,718

Poisson λ=1,959

100

14,1

27,622

27,056

17,667

8,653

4,498

0,406

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HISPANIC (of any race)

 

 

 

 

 

 

 

 

 

.Women Ever Married

 

 

 

 

 

 

 

 

 

 

40 to 44

1179

100

8,1

14,3

36,1

20,5

14,6

5,2

1,2

2,437

0,854

Poisson λ=2,437

100

8,742

21,305

25,96

21,088

12,848

8,8056

1,251

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BLACK ONLY

 

 

 

 

 

 

 

 

 

 

.Women Ever Married

 

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ASIAN ONLY

 

 

 

 

 

 

 

 

 

 

.Women Ever Married

 

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Source: U.S. Census Bureau, Current Population Survey, June 2004.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

          

 

 

 

 

 

 

 

 

 

 

 

 

 

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 Netherlands in the 18th and 19th century. They are 72 parents with 220 effective children, all descendants of one person. As is shown on Table 2 the variations of the parities, so the observed divergence in the distribution, are larger than the divergence of the Poisson distribution. This is an indication that this people, different from the modern one, did have unequal chances on reproduction and survival and that by this the genetic differences between the generations must be larger than random expected. Thus this is an evolutionary population. If larger and more of these populations are inquired in this way and these results should be confirmed, this should give direct evidence for the evolutionary character of these populations and also parameters of the evolutionary intensity of the different populations are easily to be deducted by experts in this study. It is probable, I think, that the populations of the animals and the plants in nature and also of people mostly are evolutionary, but some populations or species may be not evolutionary and we as modern living people are one of them. The non-evolutionary species may be exceptions, but possibly they occur more often than we think. There may be  indeed different reasons by what animals can be non- or very slow evolutionary.           

 

Tabel 2

→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

populatie n=72

gemiddeld 3,0555 kinderen per ouder

 

 

 

 

→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 USA that do so, see the literature. These diagrams of the parities do also show obvious how was the development in Russia from more variation to less variation than at random in the parities. 

 

Figure 2.9--Distribution of Women by Number of Children Born by Age 50 in Different Birth Cohorts in Russia (estimates for birth cohorts of 1945 and 1955)

 

 

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 1970-2000. In the data of Table 2 from some small number of Dutch people getting children in about 1740-1890 is a larger variation in the parities than the Poisson distribution gives. This does not mean that the Poisson distribution gives invalid information, or it should be not appropriate here. It does mean that people in both populations did not procreate by accident. This point of course is very well known, but interesting to this may be the exact specification of the total different deviation from the random. This complement of the accident is what makes a researcher curious in many fields. The modern women with the smaller variation in the parities appears to have little chance (smaller than average) to get anymore children, if they have already 2 or 3 children. If they have only one child their chance to become mother again is larger than of the average other women. As we know the cause of it is a strong selection on the numbers of children itself. In the social context of their society modern American people do prefer to have 2 or 3 children. They do have possibilities to do so and of the children that have been born only about 3% dies before adulthood. However more than 100 years ago people lived in total different situations in most countries, as did my ancestors, the people of Table 2. They did live in another social context, with also less possibilities and a large (ca 50%) and variable number of their children died before adulthood. So especially by the high infant mortality  they also had their ‘family planning’, so determining the number of their adult (or effective) children, but effectuate it more by caring the children well and preventing their dead than by contraception. They had much more differences in the numbers of their adult, effective children than the modern people do, by difference in good luck, but also by their different talents, their motivations and possibilities to keep their children alive. By these large variations in their effective parities and also by inbreeding in combination with more competition our ancestors were biologic evolutionary and we are not anymore. More evidence for this conclusion is pointed out further here.  By this conclusion however is no reason to worry about our modern way of family planning as being non-evolutionary and bad for our genes, on the contrary we have to worry about the many ill-functioning genes acquired into our genomes by the evolution of our ancestors! Interesting also are the US figures of birth in the parities, collected in the former years, so the women getting their children in the years about 1940-1970. In those years the infant mortality was a bit higher than now, but still so low that it already was no more a substantial part of the effective reproduction. The women in this time did differ more from each other in their family planning. The culture was not yet so uniform in this as it is now and this issue may be more important than the lesser possibilities for contraception, because this already was sufficient possible. These more different choices in the family planning are showed by the variety in the distribution of the actual children over the parities they had, being nearly at random, or somewhat larger. 

 

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 Europe do have fair or red  hair and white skins[5] chiefly because once that was the most sexy trend for our ancestors. Also the invisible interference of the ‘neuronic’ and genetic information may increase the selection. For instance this situation: an epidemic comes over 3 groups consisting of a few genetic resistant people and a variable number of people that also are able to survive because they are smart. The 1st group with none smart people may easily dye out, because the resistant ones probably die of starvation. The 2nd group with some smart people will survive, because they have more and smarter survivors of the epidemic. The 3rd group has only smart people, so that the total group survives. By the interference only in the second group is selection on the resistance gene and this is the situation at the primitive people. The first group is more the situation at the animals and the third group is at the modern people, both without selection. Also the archaeological data indicate a fast evolution. That data shows a picture of the different homo species the arose and died out, following each other in a short time in relation to the evolution of many animal species. On the other hand apparently is against the fast evolution that the genetic differences between in our time living people are small, so that the people of the different continents did not yet grow much different in such a long time, which is not consistent with the fast evolution. It however is possible in two cases that the human and particularly the sapiens evolution has been  much faster than average in the nature in spite of the small genetic differences between people now. Than there should have 1st been much migration between the continents, or 2nd the evolution should have been in the same direction at the continents, so that the same genetic differences are transferred by prefer in the different continents. Both motives seem improbable to me, but the later is of course consistent with the multiple regionality theory. In this theory the homo sapiens should have arisen not only in Africa but in different places on the world and the different stocks of these sapiens like mutated  former homo erectus people must than have converged to our modern people, with their relative small genetic differences. This multiple regionality theory, with some things more, will indicate than also there must be something like an internal organisation of the genes that is self-regulating them. That this exists is not totally evident by what is known in the molecular biology, I guess, although there may be some hints in this. Nevertheless there is evidence for the existence of self-regulation in the physiology of the organisms and broader in their biological functions and by different reasons this implicates imperatively also self-regulation within the genome. This is a logic a-priori premise, which yet is to be checked and specified a-posterior by the molecular characteristics of the DNA and some primary biologic functions. By self-regulation the allelic characteristics will not only diverge, but also converge in the evolution and this is a very important issue. Self-regulation in the bio-evolution is not generally taught, but it is by Prof. Francis Heyligen in Brussels.        

 

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

 

 

 

 

 

 

 

 

 

Descendants of  F0 in F1

 

 

 

 

 

 

→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

 

 

 

 Descendants of  F0 in F2

 

 

 

 

 

→0→0

 

 

 

 

 

 

 

 

℮^-2

 

 

 

 

 

 

 

 

→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

 

 

 

→→

→→

→7→6

→7→7

→7→8

0,135342

 

 

 

 

 

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

 

 

 

 

 

 

 

 

0,177412

 

 

 

 

 

 

 

 

[(℮^2^(℮^-2)]/℮^2

 

 

 

 

 

 

 

X 0

 

 

 

 

 

 

 

 

 

→9

 

 

 

 

 

 

 

 

0,00141093℮^-2

 

 

 

 

 

 

 

0,00019

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

→1→9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

→2→9

→2→10

→2→11

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

→4→18

 

 

 

 

 

1,876℮^-10

 

 

 

 

 

→5→18

→5→19

→5→20

→5→21

 

 

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 4. In this secondary distribution F1→F2 there is no uniform λ for the different ways of descent, because of the variable quantities, q. However there can be a constant μ=prs through the generations. Than you can use λ*=qμ by the substitution in the formula P(i) = ℮^- λ* . λ*^i/i! of the Poisson distribution. There is also a sum Poisson intensity ς up to the level of the distributions, here generations. This ς  here is simply the average number of descendant in a generation Fg to an ancestor in the F0. If μ is constant through the generations ς=μ^g  for  F0→Fg. The calculations in Table 3 are restricted. It is stopped with 8 descendants in the F1 and with 23 in the F2. So there has been made a calculation over about 99% of the descendants over the two generations. Notice as a conclusion of Table 3: there are great differences in the efficiency of the reproduction even if there are equal changes. Even if any individual has the same reproductive possibilities and there are no differences between the organisms themselves or their living surroundings you see that 21,2% of the organisms has the half of the progeny in the F2 and that 17,7% has no descendants in the F2. 

 

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 1 in Table 4. For instance in the 2nd distribution, F1→F2, the total intensity, the sum of all the different λ*, so μ= 0x(℮^-1) + 1x(℮^-1) + 2x[½(℮^-1)] + 3x[1/6(℮^-1)] + .. The sums of all these distribution form than the distribution for the new generation, always in proportions of the alleles in F0. Because the total intensity here keeps constant μ=1, is valid λ=μ=ς=1. Notice that only the sums under ∑0, the proportions of alleles that disappear in the further generations can be expressed as a function of ℮. The cause of this is that the Poisson distribution with the formula P(i) = ℮^-λ . λ ^i/i! gets the particularity of the complementary exponential distribution, if i=0 than P(i)=^-λ. So in the F3 62,6..% of the singular alleles in F0 has been vanished. In this continuing distribution with intensity 1 there are of course as many alleles disappeared as there have come alleles to in the two folds, three folds etc, so that the total number of the alleles in all the generations is equal. The number of the alleles in the F0, 1x2n, is of course equal to the number of the alleles in the F3, so 2n x  (0 x 0,626.. + 1 x 0,122 + 2 x 0,134 etc).   

 

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-1. In this is σ the accumulated intensity of generation Fg and ν is the P(i=0) or the extinction accumulated up to the former generation F(g-1). The extinction for generation g than is P(i=0)=℮^(ν-1). This is the most convenient formula for making the recurrence table, as given here on Table 5. This is easy to be done with a simple calculator. The allele survival in the generations or degrees of the exponential distribution goes as an exponential ladder with the steps t, so t1=1  t2=1-℮^-1  t3=1-℮^-(1-℮^-1)  t4=1-℮^-[1-℮^-(1-℮^-1)]  t5=1-℮^-{1-℮^-[1-℮^-(1-℮^-1)]}, etc.

 

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