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Author Topic:   Peppered Moths and Natural Selection
RAZD
Member (Idle past 1395 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 241 of 350 (358117)
10-22-2006 11:32 AM
Reply to: Message 240 by MartinV
10-22-2006 3:50 AM


Preferential Predation still the best model
Mm and Mt morphs (75% M and 25%T alleles).
Don't confuse alleles with morphs. CT is a carbonaria morph with recessive typica allele, as is TC. TT is pure typica alleles and CC is pure carbonaria alleles. So in the population - without preferential predation pressure - there are as many typica alleles as there are carbonaria alleles, but there are 75% carbonaria morphs and 25% typica morphs.
Yet there will be as result one typica TT which in case of selective predation will be eaten, so T alleles are permanently spliting away from population. If first MM: Mt was 1:1, next generation will be MM: Mt ratio 9:6.
Remember I said that would be the population proportions in the absence of preferential predation.
Your introduction of preferential predation negates that condition, and thus any subsequent conclusion is not relevant to the argument since you changed the conditions.
What you are covering is the same as I covered with preferential predation of the typica variety.
{abe}
If first MM: Mt was 1:1, next generation will be MM: Mt ratio 9:6.
This is, of course the effect of preferential predation on the populations that forces a higher proportion of carbonaria to typica.
This IS natural selection
{/abe}
If carbonaria gene frequency in Leeds was truly 100% before 1970 ...
So immigration is needed to start change.
OR the carbonaria gene frequency was NOT truly 100%.
Seeing as typica alleles are recessive to carbonaria alleles it is much more likely that typica alleles continued to exist within the carbonaria population than that they HAD to be introduced from outside.
Anyway it seems to be neccesary to involve mutation as starting factor creating new morphs as well - question avoided by all of you neodarwinists here completely.
What? Message 124
JonF writes:
The statistics of the populations have been investigated in many experiments. The color differences are due to a mutation in a gene that causes melanin production. The "dark" allele is dominant, so a moth must have two "light" alleles to be light. The dark mutation is "recurrent", in that it arises anew once in a while; but, before industrialization, the dark moths were at such a disadvantage that the light moths were far in the majority.
That was a message that specifically replied to one of your messages.
Nobody has said that the difference between the two varieties was not due to previous mutations within the population. That is the basis for natural selection: that there is a variety of genetic variations within any and all populations of species such that some offer benefits while others are handicaps and others are neutral within the existing environment, but when the environment changes the balance between beneficial and handicap changes as well.
The existence of mutations within the population is taken as a given in this case.
Looks like you are trying to move the goalposts again.
You still have not provided any mechanism to explain the change in populations that even comes close to preferential predation, nor have you in any way shown that the result cannot be due to preferential predation.
You don't have an alternative mechanism that explains the observed evidence as well as or better than the natural selection mechanism.
You have not invalidated the natural selection mechanism.
Your argument fails.
Enjoy.
Edited by RAZD, : modified message link
Edited by RAZD, : No reason given.
Edited by RAZD, : added {abe} section

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This message is a reply to:
 Message 240 by MartinV, posted 10-22-2006 3:50 AM MartinV has replied

Replies to this message:
 Message 242 by MartinV, posted 10-24-2006 2:33 PM RAZD has replied

  
MartinV 
Suspended Member (Idle past 5819 days)
Posts: 502
From: Slovakia, Bratislava
Joined: 08-28-2006


Message 242 of 350 (358547)
10-24-2006 2:33 PM
Reply to: Message 241 by RAZD
10-22-2006 11:32 AM


Re: Preferential Predation still the best model
JonF writes:
The dark mutation is "recurrent", in that it arises anew once in a while; but,before industrialization, the dark moths were at such a disadvantage that the light moths were far in the majority.
This "reccurent" mutation is really interesting. If we assume, that typica has 2 alleles tt this "reccurent" mutation not only should create de novo always dominat(!) C allele but whats more it should create "de novo" instict that led moths to rest on proper black lichens/backround. This to believe suppose strong faiths in neodarwinism. I would say that mutation with so pleitropic effect cannot occurs "reccurently", but should be sitting somewhere in DNA and only be "derepressed".
RAZD writes:
OR the carbonaria gene frequency was NOT truly 100%.
Seeing as typica alleles are recessive to carbonaria alleles it is much more likely that typica alleles continued to exist within the carbonaria population than that they HAD to be introduced from outside.
Your claim contradicts the article I have given you in previous link:
Melanic Moth Frequencies in Yorkshire, an Old English Industrial Hot Spot | Journal of Heredity | Oxford Academic
But thinking about difficulties of peitropic effect of "reccurent" mutation from typica to melanica and vice versa it is underestandable that you stick on claim that "typica continued to exist within carbonaria" even if it seems in case of Leeds contradicts completely the basics of genetics and mendelian maths.

This message is a reply to:
 Message 241 by RAZD, posted 10-22-2006 11:32 AM RAZD has replied

Replies to this message:
 Message 243 by RAZD, posted 10-24-2006 10:50 PM MartinV has replied

  
RAZD
Member (Idle past 1395 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 243 of 350 (358663)
10-24-2006 10:50 PM
Reply to: Message 242 by MartinV
10-24-2006 2:33 PM


Re: Preferential Predation still the best model
I read the article the first time. It talked about nearly 100% carbonaria\melanic form at the top, and then in the discussion switched to 100% carbonaria genes at the part you quoted, which is a different matter. That is the first time he mentions genes. He does discuss dominant forms, but seems to ignore that hybrids would be considered carbonaria\melanic:
quote:
B. betularia was scored as the dominant melanic carbonaria Jordan or the dotted white typical.
One or the other, no consideration for 'mixed' genes.
An obvious conclusion is that IF carbonaria is a dominant gene (as all the evidence seems to show) that a population can be 100% carbonaria\melanic form and NOT be 100% carbonaria genes because that's comparing apples and oranges.
With 100% carbonaria\melanic forms the proportion of the carbonaria genes could be anywhere from 50% to 100%.
Consider an organism {BUG} that has two form of a gene, {SHOW} and {HIDE}; if the {BUG} has two copies of {SHOW} then it is visible to predators and gets eaten, while ones that have {SHOW}{HIDE} can hide from predators, but if it has two copies of {HIDE} that this is lethal and the individuals die before reaching reproductive age.
Only the ones that have {SHOW}{HIDE} live long enough to reproduce, so all the long term (ie - breeding) population will be {SHOW}{HIDE}, half of each new generation will be {SHOW}{HIDE}, 1/4 will be {SHOW}{SHOW} and get eaten before reproduction, and 1/4 will be {HIDE}{HIDE} and die before reaching reproduction, but the species as a whole will continue to provide new generations.
If we assume, that typica has 2 alleles tt this "reccurent" mutation not only should create de novo always dominat(!) C allele ...
There are some genes that are more susceptible to mutations than others. This has been shown by genetic studies. The same mutation recurring would also be dominant because it would be the same gene.
Another way the gene could recur is that there could be a secondary gene that protects against the carbonaria gene by disabling it or turning it off, but that it is subject to disruption by random mutations - which pretty much ensures repetitions of expression of the very same carbonaria gene.
Look at the evolution of wings in walkingstick bugs -- NOT to get into a discussion of walking sticks, but to show the recurrence of wings:

(Source: Nature article, need sign in to access more than this abstract)
quote:
These results suggest that wing developmental pathways are conserved in wingless phasmids, and that 're-evolution' of wings has had an unrecognized role in insect diversification.
Wing genes are probably much more complex in arrangement than this one color gene eh? And yet we see wings being lost and then regained.
This was discussed on Message 101 but that thread is now closed. If you want to discuss this further I can start another thread on walking sticks.
NOTE: Discussing walkingsticks will be off topic on this thread.
The issue that can be discussed, and stay on topic, is being able to 'recover' genes that had existed previously, and it looks like this is possible, depending on how they were {removed\disabled} from ancestral species.
Certainly if a dominant gene is as potentially harmful to a species as carbonaria normally appears to be (being almost totally suppressed by predation in normal woods), one could expect a mutation that suppresses or disables the carbonaria gene, leaving the moth with the expression of the other, typica, gene, and this would maintain the carbonaria gene in the population gene pool.
I would say that mutation with so pleitropic effect cannot occurs "reccurently", but should be sitting somewhere in DNA and only be "derepressed".
Exactly. But the expression of the gene would still be seen as "recurrent" - just that the method of the recurrence is a different mutation.
... but whats more it should create "de novo" instict that led moths to rest on proper black lichens/backround.
Only if you think that is really what is going on. As I recall that was one of your previous hypothesis, but I don't need to make that assumption.
I can assume either of two things: (1) the moths compare a visible part of their {wing\body} with potential resting places and choose ones that match (thus trying to match whatever shade they are against whatever backgrounds are available) or (2) that resting places are relatively randomly chosen but that the locations (parts of the tree) normally chosen have characteristics matching their camouflage pattern (that has evolved to match the "normal" habitat).
  1. leads to moths only being exposed when they run out of places to rest that allow them to benefit from their relative camouflage ability, thus needing change to the environment that destroys or significantly alters beneficial resting places (which pollution would do), while
  2. leads to moths being exposed whenever they chose a less advantageous background to rest on, which happen much more for dark moths in normally light forests and light moths in normally dark forests (also caused by pollution).

Thus (1) is a specific localized tree environment adaptation with generalized behavior, while (2) is a generalized neighborhood environmental adaptation with localized behavior, but BOTH have the same relationship of camouflage ability to preferential predation by birds in different exposure environments.
But thinking about difficulties of peitropic effect of "reccurent" mutation from typica to melanica ...
I think you mean pleiotropic: "the phenomenon of one gene being responsible for or affecting more than one phenotypic characteristic." But even that is not necessary, there just needs to be a mechanism that disables the dominant C gene. This could well mean that an individual that inherits two suppressed C genes may not be able to survive but that would still be an advantage to the population compared to have a lot of visible moths.
... and vice versa it is underestandable that you stick on claim that "typica continued to exist within carbonaria" ...
And carbonaria\melanics don't need to evolve back to typica for each generation, the typica gene is still there, just not expressed because it is recessive compared to the carbonaria gene.
... even if it seems in case of Leeds contradicts completely the basics of genetics and mendelian maths.
But it doesn't really. He says quite plainly "IF"
quote:
If carbonaria gene frequency in Leeds was truly 100% before 1970, then no change would have been possible without introduction of typicals by mutation or migration of immigrants from regions of lower frequency.
Not that it was 100% carbonaria gene. He then goes on to say:
quote:
Because typical is recessive, these processes would have introduced genes in heterozygotes that were not subject to selection, and the response to selection would necessarily be very slow.
So he rules out mutation to 'reintroduce' typica variety.
Then he concludes that:
quote:
It is likely that the response observed was in some part due to immigration.
BUT it is equally valid to consider that it WASN'T 100% carbonaria genes at Leeds. The fact that doing so invalidates the necessity of his conclusion that immigration must play some role in the changes is HIS problem, as he has not considered that alternative in his paper eh?
Whether that "some" is 1% or 50% is not stated. Certainly he does not say "mostly due to immigration" does he? And he certainly does not rule out preferential predation on Peppered moths, even with his conclusion for "some" immigration.
Enjoy.

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This message is a reply to:
 Message 242 by MartinV, posted 10-24-2006 2:33 PM MartinV has replied

Replies to this message:
 Message 244 by MartinV, posted 10-28-2006 3:02 PM RAZD has replied

  
MartinV 
Suspended Member (Idle past 5819 days)
Posts: 502
From: Slovakia, Bratislava
Joined: 08-28-2006


Message 244 of 350 (359512)
10-28-2006 3:02 PM
Reply to: Message 243 by RAZD
10-24-2006 10:50 PM


Biston betularia math.
An obvious conclusion is that IF carbonaria is a dominant gene (as all the evidence seems to show) that a population can be 100% carbonaria\melanic form and NOT be 100% carbonaria genes because that's comparing apples and oranges.
Not at all.There is strong corellation between genotype frequencies and allele frequecies in this case. If there is selective predation on typica, then carbonaria are cleaned out of typica allele after some time and the result is only homozygotous carbonaria. There is a little bit complicated equation of gene changes dependency on selective advantage. If carbonaria has 0,3 selective advantage over
typica then after 50 years the frequency of Carbonaria genes raised from almost 0% to more than 95% and it is obvious, that as selective predation persists, then the number will be soon almost 100%.
After all if we started as in case of Leeds with the fraction Melanica genes 0,999 the fraction of normal moths are only 0.001x0.001 = 0.000001. According professor Jeremy Tatum of the University of Victoria if we started with fraction of Melanica
genes from this 0,999 and put them under strong selective disatvantage -0,9 even after 50 gnerations M will make 0,9927!
If you see on table 2 of the mentioned article you will notice that in Leeds before rise of typica were numbers of caught moths:
--------C T
1967 47 0
1968 58 0
1969 27 0
1970 75 1
1971 41 0
1972 76 0
1973 40 1
1974 40 0
1975 3 0
Carbonaria/typica ration was 407/2 = 0,9951.
Take into consideration, that in Leeds there was - I suppose - pollution more than 80 years, so carbonaria was in 1967 necessary homozygous and gene frequency should be consequently almost 100%.
That is why scientists claim, that
oxfordjournal writes:
If carbonaria gene frequency in Leeds was truly 100% before 1970, then no change would have been possible without introduction of typicals by mutation or migration of immigrants from regions of lower frequency.
but you:
BUT it is equally valid to consider that it WASN'T 100% carbonaria genes at Leeds. The fact that doing so invalidates the necessity of his conclusion that immigration must play some role in the changes is HIS problem, as he has not considered that alternative in his paper eh?
HE proceeded correctly and scietifically using mathematics. So conclusions are correct until you present us some other mathematics.
Exactly. But the expression of the gene would still be seen as "recurrent" - just that the method of the recurrence is a different mutation.
This is very interesting and yet perplexing. Where do "derepressed" typica allele rest on your opinion in case of homozygous Biston betularia in which both locus are occupied by same carbonaria allele? Is after mutation that "release" typica gene also reshuffle of genes on locus?
--------------------
--------------------------------
Maths and moths by Jeremy Tatum:
Page not found | Pacific Institute for the Mathematical Sciences - PIMS
Edited by MartinV, : link added

This message is a reply to:
 Message 243 by RAZD, posted 10-24-2006 10:50 PM RAZD has replied

Replies to this message:
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 Message 246 by RAZD, posted 10-31-2006 9:18 PM MartinV has replied

  
RAZD
Member (Idle past 1395 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 245 of 350 (360003)
10-30-2006 10:47 PM
Reply to: Message 244 by MartinV
10-28-2006 3:02 PM


More Biston betularia math. tomorrow.
Had a long reply last night and lost it. I'll see if I can reconstruct it from the notes and calcs ... tomorrow night.

This message is a reply to:
 Message 244 by MartinV, posted 10-28-2006 3:02 PM MartinV has not replied

  
RAZD
Member (Idle past 1395 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 246 of 350 (360283)
10-31-2006 9:18 PM
Reply to: Message 244 by MartinV
10-28-2006 3:02 PM


More Biston betularia math. Continued.
Had a long reply last night and lost it. I'll see if I can reconstruct it from the notes and calcs ...
Not at all.There is strong corellation between genotype frequencies and allele frequecies in this case. If there is selective predation on typica, then carbonaria are cleaned out of typica allele after some time and the result is only homozygotous carbonaria.
The problem I have with this is that every heterozygous female lays eggs that are 1/2 heterozygous moths, whether mated with a homozygous male or a heterozygous male, and every heterozygous male fertilizes eggs that are 1/2 heterozygous moths, whether mated with a homozygous female or a heterozygous female.
To me that indicates that there would always be heterozygous moths, and as long as there are heterozygous moths there will be typica alleles in the population.
Remember this:
Pattern {C} - male carbonaria breeding with female carbonaria
  • CC+cc = Cc Cc Cc Cc - all four "carbonaria"
  • CC+ct = Cc Ct Cc Ct - all four "carbonaria"
  • CC+tc = Ct Cc Ct Cc - all four "carbonaria"
  • CT+cc = Cc Cc Tc Tc - all four "carbonaria"
  • CT+ct = Cc Ct Tc Tt - one "typica" and three "carbonaria"
  • CT+tc = Ct Cc Tt Tc - one "typica" and three "carbonaria"
  • TC+cc = Tc Tc Cc Cc - all four "carbonaria"
  • TC+ct = Tc Tt Cc Ct - one "typica" and three "carbonaria"
  • TC+tc = Tt Tc Ct Cc - one "typica" and three "carbonaria"
    4 typica out of 36
    11% typica
    89% carbonaria
Let's re-write that to show heterozygous and homozygous carbonaria proportions:
Pattern {C} - male carbonaria breeding with female carbonaria
  • CC+cc = Cc Cc Cc Cc = all four Cc "carbonaria"
  • CC+ct = Cc Ct Cc Ct = two Cc "carbonaria" and two Ct "carbonaria
  • CC+tc = Ct Cc Ct Cc = two Cc "carbonaria" and two Ct "carbonaria
  • CT+cc = Cc Cc Tc Tc = two Cc "carbonaria" and two Tc "carbonaria
  • CT+ct = Cc Ct Tc Tt = one tt "typica" one Cc "carbonaria and one Ct "carbonaria" and one Tc "carbonaria"
  • CT+tc = Ct Cc Tt Tc = one tt "typica" one Cc "carbonaria and one Ct "carbonaria" and one Tc "carbonaria"
  • TC+cc = Tc Tc Cc Cc = two Cc "carbonaria" and two Tc "carbonaria
  • TC+ct = Tc Tt Cc Ct = one tt "typica" one Cc "carbonaria and one Ct "carbonaria" and one Tc "carbonaria"
  • TC+tc = Tt Tc Ct Cc = one tt "typica" one Cc "carbonaria and one Ct "carbonaria" and one Tc "carbonaria"
    4 typica out of 36 = 11.1%
    16 Cc carbonaria out of 36 = 44.4%
    8 Tc carbonaria out of 36 = 22.2%
    8 Ct carbonaria out of 36 = 22.2%
If all typica are consumed before mating then those specific typica alleles are removed, but not all the ones in the heterozygous carbonaria moths.
We would then have 88.8% survival and still all 88.8% carbonaria, becomes 100% for next generation, half of them homozygous and half heterozygous.
The next generation would produce less typica moths and less heterozygous carbonaria moths, but before we get to that, let's simplify it a bit so we can compare the ratios of heterozygous carbonaria to homozygous carbonaria in the way we used for the typica ratios before:
Before I used ucase for one sex alleles and lcase for the other sex alleles, and we had Cc, Ct and Tc carbonaria moths. A male Ct and a male Tc moth are the same, as are female Ct and Tc moths, so lets use CC and Ct, where CC is homozygous and Ct is heterozygous. Now we have:
Pattern {C1} male CC carbonaria breeding with female CC carbonaria
  • CC + CC = CC CC CC CC - all four CC "carbonaria"
    4 CC out of 4 eggs
    100% CC
    0 Ct out of 4 eggs
    0% Ct
    0 tt out of 4 eggs
    0% tt
Pattern {C2a} male CC carbonaria breeding with female Ct carbonaria OR
  • CC+Ct = CC Ct CC Ct - two CC "carbonaria" and two Ct "carbonaria
    2 CC out of 4 eggs
    50% CC
    2 Ct out of 4 eggs
    50% Ct
    0 tt out of 4 eggs
    0% tt
Pattern {C2b} male Ct carbonaria breeding with female CC carbonaria
  • Ct+CC = CC CC Ct Ct - two CC "carbonaria" and two Ct "carbonaria
    2 CC out of 4 eggs
    50% CC
    2 Ct out of 4 eggs
    50% Ct
    0 tt out of 4 eggs
    0% tt
The result of Pattern {C2a} and Pattern {C2b} are the same, so we can call this Pattern {C2}
Pattern {C3} male Ct carbonaria breeding with female Ct carbonaria
  • Ct+Ct = CC Ct Ct tt - one CC "carbonaria" and two Ct "carbonaria and one tt "typica" that is eaten before reproduction.
    1 CC out of 4 eggs
    25% CC
    2 Ct out of 4 eggs
    50% Ct
    1 tt out of 4 eggs
    25% tt
and to reproduce pattern {C} from the original we have Pattern {C} - male carbonaria breeding with female carbonaria
  • CC+cc = C1 = 4xCC
  • CC+ct = C2 = 2xCC + 2xCt
  • CC+tc = C2 = 2xCC + 2xCt
  • CT+cc = C2 = 2xCC + 2xCt
  • CT+ct = C3 = 1xCC + 2xCt + 1xtt
  • CT+tc = C3 = 1xCC + 2xCt + 1xtt
  • TC+cc = C2 = 2xCC + 2xCt
  • TC+ct = C3 = 1xCC + 2xCt + 1xtt
  • TC+tc = C3 = 1xCC + 2xCt + 1xtt
    4 tt typica out of 36 = 11.1% but consumed before reproduction
    16 CC carbonaria out of 36 = 44.4% as before
    16 Ct carbonaria out of 36 = 44.4% Tc and Ct combined
As before, if all typica are consumed before mating then those specific typica alleles are removed, but not all the ones in the heterozygous carbonaria moths.
We would then have 88.8% survival and still all 88.8% carbonaria, becomes 100% for next generation, half of them homozygous and half heterozygous.
Now we can look at the overall population similar to what we did with typica in part of the equation, but now including preferential predation of homozygous tt (typica) moths.
Starting with the proportions is Pattern {C} above, CC is 33% of the breeding population and Ct is 67% of the breeding population -- this gives us a 3x3 grid - with females across the top axis (CC, Ct, Ct), and males on the left side axis (CC, Ct, Ct), and then within this grid we can place each of the three (3) cases above
   | CC | Ct | Ct |
-------------------
CC | C1 | C2 | C2 |
-------------------
Ct | C2 | C3 | C3 |
-------------------
Ct | C2 | C3 | C3 |
-------------------
As before, we notice the proportions: we have Ct at 2:1 to CC in both male and female populations, and this results in (2+1)^2 = 9 mating patterns with 1 mating pattern {C1}, 2x2 = 4 mating pattern {C2}, and 2^2 = 4 mating pattern {C3}, for an overall mating result:
CC = (1x100% + 4x50% + 4x25%)/9 = 400%/9 = 44.4%
Ct = (1x0% + 4x50% + 4x50%)/9 = 400%/9 = 44.4%
tt = (1x0% + 4x0% + 4x25%)/9 = 100%/9 = 11.1%
Still 11.1% typica (consumed) and 88.8% survival and still all 88.8% carbonaria, becomes 100% for next generation.
We can also show the above table as:
    |  CC | nCt |
-----------------
CC | C1 | nC2 |
-------------------
nCt | nC2 |n^2C3|
-------------------
Where n is the proportion of heterozygous Ct carbonaria moths to homozygous CC carbonaria moths
And that we can generalize this as (1xC1 + 2nC2 + n^2C3)/(n+1)^2:
CC = (100% + 2n50% + n^225%)/(n+1)^2 = (100% + n100% + n^225%)/(n+1)^2 = CC%
Ct = (0% + 2n50% + n^250%)/(n+1)^2 = (n100% + n^250%)/(n+1)^2= Ct%
tt = (0% + 2n0% + n^225%)/(n+1)^2 = (n^225%)/(n+1)^2= tt%
If we assume that homozygous tt (typica) moths are always consumed before reproduction, then the astute observer will see from this that Ct will never mathematically be zero no matter how small n is, although it is possible to reach a practical limit. When that occurs depends on original population size and makeup and the numbers of generations involved.
Further, if we assume that homozygous tt (typica) moths are always consumed before reproduction, then the astute observer will see that this becomes an exponential decay curve. After 10 generations (years) there is still 1 Ct moth for every 5 CC moths and after 20 generations (years) there is still 1 Ct moth for every 9 CC moths.
After all if we started as in case of Leeds with the fraction Melanica genes 0,999 the fraction of normal moths are only 0.001x0.001 = 0.000001. According professor Jeremy Tatum of the University of Victoria if we started with fraction of Melanica genes from this 0,999 and put them under strong selective disatvantage -0,9 even after 50 gnerations M will make 0,9927!
But we don't know what the fraction was for homozygous versus heterozygous. That is the issue here eh? From the above equations we can calculate the numbers of homozygous tt (typica) moths produced each generation (from the heterozygous Ct carboanaria moths) even if we do not include them (homozygous tt (typica) moths) in the mating and reproduction calculations. This would generate regular samples of typica moths to collect.

C T
-----------
1967 47 0
1968 58 0
1969 27 0
1970 75 1
1971 41 0
1972 76 0
1973 40 1
1974 40 0
1975 3 0
Carbonaria/typica ration was 407/2 = 0,9951.
The explanation for the singular homozygous tt typica moths in this data is either (a) they were bred there from heterozygous Ct carbonaria moths, or (b) they immigrated.
Calculating the numbers of generations needed for typica to be produced at 0.5% (to match the above data) only takes about 12 generations.
But a bigger problem arises from this data: at least two typica (likely male) moths survived long enough to be captured. This means we cannot assume 100% lethal consumption of homozygous tt typica moths before reproduction.
We know from the moth sample rates that we are not dealing with moths at genetic equilibrium, as we are NOT capturing homozygous tt typica moths at anything close to 25% of the population, and we know that we cannot assume total consumption of homozygous tt typica moths before reproduction, therefore we will be somewhere between the genetic equilibrium and total consumption models.
To see how this affects the numbers game, lets assume that all male typicas mate and all female typicas are consumed before laying eggs.
This re-introduces
Pattern {B} - male typica breeding with female carbonaria
  • TT+cc = Tc Tc Tc Tc - all four "carbonaria"
  • TT+tc = Tt Tc Tt Tc - two "typica" and two "carbonaria"
  • TT+ct = Tc Tt Tc Tt - two "typica" and two "carbonaria"
    4 typica out of 12
    33% typica
    67% carbonaria
Let's re-write that to show heterozygous and homozygous carbonaria proportions as we did for Pattern {C}:
Pattern {B1} - male tt typica breeding with female CC carbonaria
  • tt+CC = Ct Ct Ct Ct = all four Ct "carbonaria"
    0 tt typica out of 4 = 0%
    0 CC carbonaria out of 4 - 0%
    4 Ct carbonaria out of 4 = 100%
Pattern {B2} - male tt typica breeding with female Ct carbonaria
  • tt+Ct = Ct tt Ct tt = two tt "typica" and two Ct "carbonaria
    2 tt typica out of 4 = 50%
    2 Ct carbonaria out of 4 = 50%
Pattern {B} is made up of 1x Pattern {B1} + 2x Pattern {B2}
We can now introduce the male homozygous tt moths into the above table as:
    |  CC | nCt |
-----------------
CC | C1 | nC2 |
-----------------
nCt | nC2 |n^2C3|
-----------------
mtt | mB1 | mnB2|
-----------------
Where n is the proportion of heterozygous Ct carbonaria moths to homozygous CC carbonaria moths
And m is the proportion of homozygous tt typica male moths to homozygous CC carbonaria male moths
And that we can generalize this as (1xC1 + 2nC2 + n^2C3 + mB1 + mnB2)/(n+1)(n+m+1):
CC = (100% + 2n50% + n^225% + m0% + mn0%)/((n+1)(n+m+1)) = (100% + n100% + ^225%)/((n+1)(n+m+1)) = CC%
Ct = (0% + 2n50% + n^250% + m100% + mn50%)/((n+1)(n+m+1)) = (n100% + n^250% + m100% + mn50%)/((n+1)(n+m+1)) = Ct%
tt = (0% + 2n0% + n^225% + m0% + mn50%)/((n+1)(n+m+1)) = (n^225% + mn50%)/((n+1)(n+m+1)) = tt%
If all female typica are consumed before mating then those specific typica alleles are removed, but not all the ones in the heterozygous carbonaria moths OR the the male typica moths.
For the next generation n2 = Ct%/CC% and m2 = tt%/CC% (assuming 50% males and females in both varieties).
Going back to your chart above we calculated 12 generations to match the data with total pre-breeding consumption of homozygous tt typica moths:

C T
-----------
1967 47 0
1968 58 0
1969 27 0
1970 75 1
1971 41 0
1972 76 0
1973 40 1
1974 40 0
1975 3 0
Carbonaria/typica ration was 407/2 = 0,9951.
Running these same calculations with male homozygous tt typica moths surviving to breed the numbers of generations needed for typica to be produced at 0.5% (to match the above data) now takes about 26 generations (years).
We still see the same effect of the exponential decay of the typica allele in the population, but now we no longer need to assume that all males are consumed before mating, thus generating occasional males in the traps without having to assume immigration into the area of lonely homozygous tt typica moths.
Immigration could be a co-factor, but it is not necessary to explain the data.
You could also have males travelling to mate and females staying and end up with the same results.
We could do the same thing again while assuming that female homozygous tt typica moths survived long enough to lay half their eggs and we would see a similar relaxing of the exponential curves, and still see the predominance of the CC alleles in the population rise as a result of the predation of the homozygous tt typica moths.
This is what the natural selection is about - the change in the proportions of alleles over time. It isn't about it becoming 100% {X} or 100% {Y}, but about a rarer allele becoming more popular or a more popular allele becoming rarer.
And as I have said before, it does not take ALL the moths being consumed by preferential predation to have this effect, it just requires that SOME are.
Enjoy.
Edited by RAZD, : corrected math in last grid formulas for Ct and tt proportions where males survive to breed.
Edited by Admin, : Fix formatting.

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This message is a reply to:
 Message 244 by MartinV, posted 10-28-2006 3:02 PM MartinV has replied

Replies to this message:
 Message 247 by MartinV, posted 11-02-2006 3:23 PM RAZD has replied

  
MartinV 
Suspended Member (Idle past 5819 days)
Posts: 502
From: Slovakia, Bratislava
Joined: 08-28-2006


Message 247 of 350 (360858)
11-02-2006 3:23 PM
Reply to: Message 246 by RAZD
10-31-2006 9:18 PM


Re: More Biston betularia math. Continued.
Let us start with this your table that describe situation exactly:
......| CC | nCt |
-----------------
CC | C1 | nC2 |
-------------------
nCt | nC2 |n^2C3|
-------------------
where C1 = 4*CC
C2 = 2*CC + 2*Ct
C3 = 1*CC + 2*Ct + 1*tt
Outcome in the next generation will be - if initial
ratio CC/CT is 1/n:
CC = CC from (C1 + nC2 + nC2 + n^2C3)
= 4 + n*2 + n*2 + n^2
= 4 + 4n + n^2
Ct = Ct from (nC2*2 + nC2*2 + n^2C3)
= n*2 + n*2 + 2*n^2
= 4n + 2n^2
tt = n^2
So if tt are eaten before mating, astute observer will notice that if ratio CC/Ct is 1/n, in next generation it will be
(2n^2 + 4n) / (4 + 4n + n^2).
If n is going to infinity then in next generation there will be ratio CC/Ct 1/2, then 1/1, 3/2, 2/1, 5/2, 3/1, 7/2, 4/1 or - I would say that ratio CC/Ct in case, that there was no CC at beginning, only Ct, the ratio in n-th generation will be n/2.
If there was strong pollution in Leeds say 1875-1975 that is 100 years and in 1875 there were no typica only Ct than the ratio CC/Ct after 100 years should be 50/1.
Probability, that Ct find partner Ct to mate and subsequently express tt genotype in form of typica phenotype is very low - 1/50^2 = 0,0004.
Yet I do not know where I do mistake, because according graph in Maths and moths after 10 years of flawless predation on typica there should be CC/Ct ratio 90%:10%, but on my computation it is only 80%:20%. It is however very important, because my result after 100 years will be very inaccurate and whats more, it is not as strong decline of Ct population as I thought before.
Do you think, that profesor Tatum started with different preconditions?
My formula for CC/Ct ratio after n generation in case that it started from clear Ct population and tt are eaten before mating is: n/2.
Yet I am still not sure on your view - can you send also some explanation as to the change of ratio in subsequent generations, or do you held opininin, that CC/Ct ratio will be always 50%:50% in each generation?

This message is a reply to:
 Message 246 by RAZD, posted 10-31-2006 9:18 PM RAZD has replied

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 Message 248 by RAZD, posted 11-02-2006 10:55 PM MartinV has replied

  
RAZD
Member (Idle past 1395 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 248 of 350 (361004)
11-02-2006 10:55 PM
Reply to: Message 247 by MartinV
11-02-2006 3:23 PM


Re: More Biston betularia math. Continued. Again.
If there was strong pollution in Leeds say 1875-1975 that is 100 years and in 1875 there were no typica only Ct than the ratio CC/Ct after 100 years should be 50/1.
That (2%) is what I get for the assumption that 100% of all typica are consumed prior to reproduction.
When I assume that all male typica reproduce but all female typica are consumed prior to laying their eggs, then I get 4% (25/1).
This is still very strong predation effect.
Probability, that Ct find partner Ct to mate and subsequently express tt genotype in form of typica phenotype is very low - 1/50^2 = 0,0004.
Agreed, but this is still much lower than the percentage of typica alleles in the population - they are still being carried by the Ct moths, and IF male typica survive to breed then their chances of mating with a Ct moth increase the probability of another typica generation.
... according graph in Maths and moths after 10 years of flawless predation on typica there should be CC/Ct ratio 90%:10%, but on my computation it is only 80%:20%.
I get ~84% CC, ~15% Ct and ~1% tt - in round numbers.
Do you think, that profesor Tatum started with different preconditions?
Where you start affects it to some degree. I started with genetic equilibrium as an arbitrary point - 25% CC 50% Ct and 25% tt. The next generation was 44% CC 44% Ct and 11% tt.
He could also be assuming less than 100% consumption of both sexes.
It is however very important, because my result after 100 years will be very inaccurate ...
You should be getting an exponential decay type curve, like half-life on radioactive material.
... and whats more, it is not as strong decline of Ct population as I thought before.
It is stronger than I expected. The conclusion that I reach is that the predation does not need to be that aggressive to still have the necessary impact - perhaps the female typicas on average lay half their eggs before predation?
One could work in predation rates with different rates for males and females into the formulas .... (oh boy, what fun complications eh?) . This reintroduces Pattern {A}
Pattern {A} - male typica breeding with female typica
  • tt+tt = tt tt tt tt - all four "typica"
    0 CC out of 4 eggs
    0% CC
    0 Ct out of 4 eggs
    0% Ct
    4 tt out of 4 eggs
    100% tt
Now shown as all tt in the manner consistent with later posts, and this generates this grid:
     |  CC  |  nCt  |  rmtt |
-----------------------------
CC | C1 | nC2 | rmB1 |
-----------------------------
nCt | nC2 | n^2C3 | nrmB2 |
-----------------------------
pmtt | pmB1 | pmnB2 | pmrnA |
-----------------------------

where p and r are ratios between 0 and 1 relating to the degree of predation prior to reproduction, and this gives us (C1+2nC2+n^2C3+(p+r)mB1+(p+r)mnB2+pmrnA)/((1+n+pm)(1+n+rm)):
CC = (100%+2n50%+n^225%+(p+r)m0%+(p+r)mn0%+prmn0%)/((1+n+pm)(1+n+rm)) = (100%+n100%+n^225%)/((1+n+pm)(1+n+rm)) = CC%
Ct = (0%+2n50%+n^250%+(p+r)m100%+(p+r)mn50%+prmn0%)/((n+1)(n+m+1)) = (n100%+n^250%+(p+r)m100%+(p+r)mn50%)/((1+n+pm)(1+n+rm)) = Ct%
tt = (0%+2n0%+n^225%+(p+r)m0%+(p+r)mn50%+prm^2100%)/((n+1)(n+m+1)) = (n^225%+(p+r)mn50%+prm^2100%)/((1+n+pm)(1+n+rm)) = tt%
With these formulas and with p=1.0 (full male survival until reproduced) and r=0.5 (half female egg laying completed prior to predation) I get:
after 10 years = 54.1% CC: 38.9% Ct: 7.0% tt
after 100 years = 92.0% CC: 7.8% Ct: 0.2% tt
This still gives a very strong result of typica predation after 100 years that would be consistent with the data from the studies, even though only 1/4 of the typica moths are consumed prior to reproduction.
And this still has typica alleles within the population to allow reversal without needing to assume immigration into the polluted areas.
I can go further and assume there is only 10% predation on typica moths due to preferential predation and I get:
after 10 years = 37.5% CC: 47.5% Ct: 15.0% tt
after 100 years = 81.0% CC: 18.0% Ct: 1.0% tt
With subsequent predation of typica after reproductive success, this could still give a result of typica predation after 100 years that would be consistent with the data from the studies, even though only 10% of the typica moths are consumed prior to reproduction.
The other option is to assume immigration into the area to replace lost t alleles. You could assume that males fly further to mate and so would disperse across {pollution\non-pollution} boundaries.
They would not have to reach far, as if we assume 1 mile a year then in 10 years the population 10 miles from non-polluted areas would be mixing local population with the gene flow population that has 84% CC: 15% Ct: 1% tt assuming total predation.
If we use 100 years for the age of the local population, then I get a local population proportion of 98% CC: 2% Ct: ~0% tt and averaging those, I get 92% CC: 8.5% Ct: 0.5% tt.
In any case we still see that the t alleles are depressed by preferential predation and that this causes a decrease in t alleles and an increase in C alleles.
This change is still basic evolutionary natural selection due to preferential predation of moths by birds.
All this change in predation rates and gene flow rates accomplishes is to help explain the recovery of typica moths in a relatively short period after the pollution was cleaned up and preferential predation focused back on the carbonaria variety moths instead of the typica ones.
The mechanism of change is still preferential bird predation first on carbonaria (before pollution) then on typica (during pollution) and then on carbonaria (after pollution).
Natural Selection.
Enjoy.

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This message is a reply to:
 Message 247 by MartinV, posted 11-02-2006 3:23 PM MartinV has replied

Replies to this message:
 Message 249 by MartinV, posted 11-05-2006 11:48 AM RAZD has replied

  
MartinV 
Suspended Member (Idle past 5819 days)
Posts: 502
From: Slovakia, Bratislava
Joined: 08-28-2006


Message 249 of 350 (361820)
11-05-2006 11:48 AM
Reply to: Message 248 by RAZD
11-02-2006 10:55 PM


Re: More Biston betularia math. Continued. Again.
Math will be even little bit more complicated assuming small population of Biston betularia - frequency of heterozygotes should be affected substantialy by genetic drift by formula 2qn(t) = 2pq(0)*(1-1/N)^t where N is the number of individuals, t is nuber of generations. I have read that N is about 400 individuals of peppered moths on square kilometer. If we assume strong cline as in Leeds I suppose that in that case "population" can be make by square kilometre. But then ratio between heterozygotes without any selective predation after 100 years will be 1:0,36.
Then I do not know where melanic alleles sit - if on sex chromozomes, than their expression would be different for males and females and our computations are inaccurate.
And we do not know when and where mating occurs and how are selection effective before/after mating.
In any case selective predation seems to have no dramatic influence as to the typica, while it recovers own population in short time after change of environment, so selection seems to be incapable to reverse population into typica and vice versa. Typica alleles should have been created by mutation during industrialization. I would like see if the pollution would have made green or blue sooths, it there would be also mutation in this color. If not, biston would be eaten and it would be again proof of selection as in case if blue would have arousen and survive. In any case neodarwinism as to the selection would be right, no? Yet mechanism of hidden alleles which were previously succesfull in heteroyzgotes seems to me be very good device how not to react headless to change of environment.
*****************
Is it possible to change the format - lines are too long and it is difficult to rad responses.
Edited by MartinV, : remark on formatting of this page

This message is a reply to:
 Message 248 by RAZD, posted 11-02-2006 10:55 PM RAZD has replied

Replies to this message:
 Message 250 by AdminNWR, posted 11-05-2006 12:54 PM MartinV has not replied
 Message 257 by RAZD, posted 11-06-2006 9:52 PM MartinV has not replied

  
AdminNWR
Inactive Member


Message 250 of 350 (361834)
11-05-2006 12:54 PM
Reply to: Message 249 by MartinV
11-05-2006 11:48 AM


Re: More Biston betularia math. Continued. Again.
Is it possible to change the format - lines are too long and it is difficult to rad responses.
The length of lines is set by your browser. If you set your browser to be wide, you will have long lines. Make the browser window narrow, and you will have narrow lines.
There is an exception to this. If there is a very-long-word (often a url), or a wide image, the browser will use that to set the line length. Admins usually notice these problems, and shorten the url (or its appearance), and use a thumbnail to reduce the image size. If you notice a message that needs shortening, post a note to the moderation thread (with a reference to the wide post).


This message is a reply to:
 Message 249 by MartinV, posted 11-05-2006 11:48 AM MartinV has not replied

Replies to this message:
 Message 251 by Faith, posted 11-05-2006 1:23 PM AdminNWR has replied

  
Faith 
Suspended Member (Idle past 1434 days)
Posts: 35298
From: Nevada, USA
Joined: 10-06-2001


Message 251 of 350 (361840)
11-05-2006 1:23 PM
Reply to: Message 250 by AdminNWR
11-05-2006 12:54 PM


Re: More Biston betularia math. Continued. Again.
This isn't a browser problem. Something in the posts is creating impossibly long lines. I just clicked on this thread to give it a read and found I can't.

This message is a reply to:
 Message 250 by AdminNWR, posted 11-05-2006 12:54 PM AdminNWR has replied

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AdminNWR
Inactive Member


Message 252 of 350 (361845)
11-05-2006 1:48 PM
Reply to: Message 251 by Faith
11-05-2006 1:23 PM


Re: More Biston betularia math. Continued. Again.
I am seeing reasonable lines on the last page (msgs 241-251). I also checked 221-240, and did not see a problem. If there is a problem, can one of you point to the message (or approx message) so I know where to look. A response in the moderation thread would be preferred.

This message is a reply to:
 Message 251 by Faith, posted 11-05-2006 1:23 PM Faith has not replied

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 Message 253 by MartinV, posted 11-05-2006 1:56 PM AdminNWR has not replied

  
MartinV 
Suspended Member (Idle past 5819 days)
Posts: 502
From: Slovakia, Bratislava
Joined: 08-28-2006


Message 253 of 350 (361850)
11-05-2006 1:56 PM
Reply to: Message 252 by AdminNWR
11-05-2006 1:48 PM


Re: More Biston betularia math. Continued. Again.
It occurs when message from RAZD was added.
Edited by MartinV, : No reason given.

This message is a reply to:
 Message 252 by AdminNWR, posted 11-05-2006 1:48 PM AdminNWR has not replied

  
Admin
Director
Posts: 12993
From: EvC Forum
Joined: 06-14-2002
Member Rating: 2.1


Message 254 of 350 (361954)
11-05-2006 4:40 PM


Too-long lines problem fixed

--Percy
EvC Forum Director

Replies to this message:
 Message 255 by RAZD, posted 11-06-2006 8:44 PM Admin has seen this message but not replied

  
RAZD
Member (Idle past 1395 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 255 of 350 (362258)
11-06-2006 8:44 PM
Reply to: Message 254 by Admin
11-05-2006 4:40 PM


Re: Too-long lines problem fixed
was it one of the formulas? I never saw it so don't know what you had to change.

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 Message 254 by Admin, posted 11-05-2006 4:40 PM Admin has seen this message but not replied

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