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Author
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Topic: Structure of Scientiifc Anlalysis
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warren_bergerson
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Member # 262
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posted 14. March 2003 09:16
The ‘mathematics’ used in evolutionary biology, or so it is claimed, supports the claim that a two factor mutation-selection model is compatible with or explains evolutionary change. Actuarial multiple decrement analysis and current knowledge of genetics suggests, as has been discussed here, that two factor mutation-selection are compatible with and/or predict the absence of evolutionary change. Since actuarial multiple decrement analysis appears to produce results which directly contradict the firmly held belief of 99% of evolutionary biologists and a similar portion of ID supporters it may be useful to try to explain the underlying concepts.
To begin, predictive models or algorithms have the general form F(Gt)=Gt+k. For the discussion here, Gt and Gt+k are the probability distributions of allele in a population at times t and t+k.
A two factor mutation-selection process suggests an algorithm of the general form Gt +M(Gt)+ S(Gmt)=Gt+k. where M(Gt) is the impact of mutation operations on the initial population of genes and S(Gmt) is the impact of selection operations on the gene pool modified by mutation. By tradition, the change from Gt to Gt+k is depicted as an iterative process involving successive generations or cycles of mutation and selection.
The iterative two factor process defines what will be referred to here as the structure of predictive model or algorithm. It will be noted that any complex change process can potentially be modeled using the two factor structure. To use this structure we need to be able to precisely define and quantify the beginning and ending states and one of the two change factors. The second change factor then becomes the balance item representing all change processes other than those specified by the precisely defined process. A two factor mutation-selection process is logically equivalent to a ‘mutation-everything else’ two factor change process.
Multiple decrement mathematics suggests that analyzing changes in a large complex distributions can be reduced to or divided into analysis of the components of the population. The analysis of the composition of a gene pool can thus be reduced to the analysis of changes in the frequency of individual alleles. For an individual allele Ax, the two factor change model becomes- Axt + M(Axt) + S(Axmt)=Axt+k.
For the vast majority of ‘potential alleles Ax’, we know that 1)the probability of Axt and Axt+k (i.e. both the beginning and ending probabilities) are 0%. We also know that the impact of ‘raw mutatiion rates’ M(Axt) is a positive incremental increase in the frequency of the allele in the population. If Axt and Axt+1 are zero percent, then S(Axmt), as the balance item, must be a force of decrement capable of offsetting the incremental increase produced by M.
A common misconception is that because the incremental process M is weak- produces only a very small increase in the probability of Ax- the offsetting decrement process can also be weak. Contrary to what might at first seem intuitive, it requires a very strong 100% force of decrement to offset even a weak force of increment and maintain a 0% end of cycle distribution. Even a very strong force of decrement, a 99% decrement for example, result in a small but steady increase in the frequency of Ax in the population. The absence of Ax in the population therefore leads to the conclusion that the force of decrement applicable to Ax is 100%. Either 1)the allele Ax is fatal, 2)the allele from which Ax mutates is essential to existence, or 3)the biological system contains some mechanism for completely suppressing Ax.
It might be argued that it is difficult to prove that the frequency of a specific Ax is 0%. However, there are millions of potential Ax’s whose frequency would be increasing if the rate of selection/decrement were not 100%. If the selection or decrement rate is not 100% the increase in the frequency of these divergent alleles would be readily apparent in any measure of the distribution of genes.
Using the observed 100% decrement rate for the vast majority of alleles which are not present in the population, and a 0% decrement rate for the commonly observed alleles, we can create a predictive model. The model accurately predicts that despite know mutation processes which increase the frequency of divergent alleles, the strong forces of selection produce a distribution Gt+k at t+k which is identical, except for chance fluctuations, to the distribution Gt at t.
The use of multiple decrement mathematics demonstrates that :
1) Using engineering standards and existing knowledge it is possible to construct predictive models of genetic change. The models produce accurate predictions under a wide range of ‘normal’ conditions. The ‘too complex to model’ claim is not valid.
2) The rates or forces of selection, or more accurately the rates or forces of decrement operating in biological systems are very powerful. Relative to these powerful forces of expected change, ‘chance fluctuation’ processes, so called drift and founders effect, are trivial and largely irrelevant.
3) If we accept that gene pools do change under certain conditions and if we accept that evolutionary change can occur, then we must accept the existence of as yet unidentified processes for ‘suppressing the normal forces of selection’ and ‘generating complex non-random genetic changes’.
The subject of this thread is proposals for standards for rigorous scientific analysis. The demonstration above shows that 1)if rigorous engineering standards are applied, 2)then it is possible to construct predictive models of genetic change processes. The conclusions generated by this rigorous engineering analysis directly contradict the conclusions generated by more conventional mathematical and scientific analysis. From this we can conclude that either 1)engineering standards and actuarial multiple decrement analysis are fundamentally flawed, or 2) the scientific standards and mathematical techniques of conventional biology and genetics are fundamentally flawed.
Rex,
Just read your post. I believe the above addresses the issues you raise.
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brauer
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posted 14. March 2003 11:49
Um, I'm not sure why you feel the need to introduce your novel terminology to an established field of mathematical modeling. I think you drasticallly underestimate the complexity and utility of population genetics models.
For example, quantitative population genetics can account for the size of the population. Your model seems to assume an infinite population. Current genetics models can incorporate diploidy, while your model seems to assume a haploid organism. Population genetics can deal with subdivision of a population, and migration, and epistasis, etc.
All in all, your model seems to be a restatement of the most simple special case of general population genetics models.
It becomes problematic to make grand sweeping conclusions about a model when you're examining only a specific set of the model's parameters.
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Rex Kerr
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posted 14. March 2003 18:00
Warren,
I'm afraid your model is much too simplistic, and doesn't address my points. For example, you say that
quote:
1)the probability of Axt and Axt+k (i.e. both the beginning and ending probabilities) are 0%
But we don't know this. We can't distinguish experimentally between 0.000000% and 0.00001%. In fact, the distribution of SNPs in humans, observed rates of spontaneous mutations in bacteria, and a host of other factors (i.e. those which are used to estimate the mutation rate "M(Axt)" as you call it) suggest that recently-arising alleles should be present in frequencies of 0.00001% or lower.
Both 0% and 0.00001% are consistent with the observed distribution of alleles. However, what is manifestly not consistent with observed data is a instant-"decrement" mechanism that absolutely prevents almost every possible allele from arising.
And when you say
quote:
It might be argued that it is difficult to prove that the frequency of a specific Ax is 0%. However, there are millions of potential Ax?s whose frequency would be increasing if the rate of selection/decrement were not 100%. If the selection or decrement rate is not 100% the increase in the frequency of these divergent alleles would be readily apparent in any measure of the distribution of genes.
you are ignoring population bottlenecks (and bulk population change) yet again. This is a real effect with clearly measurable results in extant populations, and absolutely cannot be ignored.
In an infinite population, if M(Axt)=-S(Axmt), then the frequency of Ax doesn't change over time. However, in a finite population, even if M(Axt)=-S(Axmt), the frequency does change over time via random fluctuation. With small populations of sexually reproducing organisms, this tends to lead to a single allele taking over--simply by chance--with the others being removed from the population.
Unfortunately, the math required to do this properly is a little more extensive than I want to bother with or is appropriate to post here. Consult a population genetics textbooks for details. Intuitively, you can imagine that if an allele occurs at frequency p, it will stochastically disappear from the population about (1-p)^2N of the time in each generation, since there are 2N copies of that gene, and the probability is (1-p) that any given copy will not contain that allele. This provides a "decrement" mechanism that you haven't considered.
In summary,
(1) Your model is inconsistent with experimental data regarding the possibility of generating various alleles.
(2) The model is vastly oversimplified and neglects important effects known to be present in real populations of finite size that have a dramatic effect on allele frequency.
(3) The model is hypersensitive to the difference between zero and almost-but-not-exactly-zero rates of change of allele frequency, a difference which is too small to measure experimentally. The conclusions of the model are radically different if the change in allele frequency is not precisely zero.
From this I conclude that (1) engineering standards and actuarial multiple decrement analysis, as applied here, is fundamentally flawed.
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warren_bergerson
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posted 15. March 2003 10:22
Rex,
All two factor mutation-selection or ‘mutation-all other decrements’ models are too simplistic to model and simulate genetic change. However, the two factor model is useful in analyzing some of the selection/decrement processes which are part of genetic change processes. The topic here is standards for scientific analysis. Comparing the ‘multiple decrement-engineering standards’ models to ‘genetic algorithm-academic standards’ models provides specific and dramatic examples of the differences produced by different standards.
Two points are central to this discussion- 1)’multiple decrement-engineering’ models and ‘genetic algorithm-academic’ models produce, or appear to produce conflicting and contradictory predictions and 2)there are objective, verifiable, procedures for determining which of the predictions are valid.
CONTRARY PREDICTIONS
MULTIPLE DECREMENT PREDICTION- Preventing ‘destructive genetic divergence’, maintaining observable distributions of alleles, and maintaining ‘species identity’ requires the existence of forces of selection/decrement of a magnitude which would eliminate all but trivial genetic/evolutionary changes. Material evolutionary/genetic change will not occur in biological systems unless there exists some process or mechanisms other than ‘mutation or selection’ with the ability to override or modify normal mutation and selection/decrement processes.
APPARENT GENETIC ALGORITHM PREDICTION- Genetic/evolutionary change in biological systems can be produced by the interaction of mutation and natural selection without the need for higher level processes to control mutation rates and selection rates.
You are welcome to clarify or refine exactly what predictions conventional genetics makes or does not make on this subject.
OBJECTIVE BASIS FOR RESOLVING VALIDITY OF PREDICTIONS
Multiple decrement models make specific testable predictions regarding changes in the frequency of alleles in a gene pool for given rates of mutation(increment) and decrement(selection). Conversely, given rates of mutation, multiple decrement techniques can be used to measure/calculate/estimate the rates of decrement applicable to alleles in a system. You raised a number of practical issues regarding the practicality of testing certain predictions. If the practical issues you raise can be addressed, and as I will discuss, there are well established techniques for addressing these issues, then the predictions made by multiple decrement models are testable.
Testing the validity of genetic algorithm models also involves a number of technical issues. It is, for example, difficult to determine if genetic algorithm models actually claim to ‘model and simulate genetic change based only the types of mutation and selection processes observed in biological systems’. It would appear that at times GA models utilize techniques other than mutation and selection. It also appears at times that GA models use assumptions that are not compatible with what is observed in biological systems. If these technical problems can be addressed, then the claims/predictions associated with GA models should also be testable.
Again, multiple decrement models and engineering standards produce testable predictions which ‘appear’ to directly conflict with predictions generated by convention ‘mutation selection’ or GA models. While there may be technical problems to be addressed, if the technical problems can be addressed, then it can be determined which ‘models and standards’ generate the correct predictions.
TECHNICAL ISSUES
Quote: But we don't know this. We can't distinguish experimentally between 0.000000% and 0.00001%. In fact, the distribution of SNPs in humans, observed rates of spontaneous mutations in bacteria, and a host of other factors (i.e. those which are used to estimate the mutation rate "M(Axt)" as you call it) suggest that recently-arising alleles should be present in frequencies of 0.00001% or lower.
As you point out, it is technically difficult to distinguish between a probability of 0% and a probability of .00001%. As I am sure you are also aware, there are all sorts of techniques available to address the issue. I know of at least two basic techniques that be used to address the issue.
First, rather than searching for the presence of a specific allele we could look for the presence of ‘any members of the set of rare alleles’. If for a specific gene and a specific species, there are two common types of alleles A1 and A2 and 3000 possible or likely mutations, then we calculate the probability of A1, probability of A2, and probability of all alleles other than A1 or A2. If the probability of a specific rare allele is .00001%, then the probability of any member of the set of rare alleles would be .03%. Still challenging, but much easier than the original problem.
Second, if the rate of generating rare mutations from a given starting point allele is constant over time, and if the rate of decrement is some constant value other than 100%, then the frequency (or expected frequency) of rare alleles will increase steadily and predictably over time. The expected percentage of rare alleles in a population is a function of 1)mutation rates, 2)decrement rates and 3)amount of elapsed time (number of cell divisions) since the common genes became common. You are welcome to do the math, but unless decrement rates are very high for rare alleles, the expected frequency of rare alleles should be high enough to be readily detected.
Quote: However, what is manifestly not consistent with observed data is a instant-"decrement" mechanism that absolutely prevents almost every possible allele from arising.
This is an interesting comment. First of all we are aware of ‘error correction’ mechanisms which dramatically reduce the likelihood of rare alleles. Error correction may not ‘absolutely prevent every possible rare allele from arising’ but it apparently does represent a major force of decrement.
It is interesting to model the impact of error correction on the distribution of rare alleles. Assume for the moment that the raw mutation rate is 1 in 10^8 cell divisions and error correction eliminates 99% or these mutations. The mutation rate has in effect been reduced to 1 in 10^10. But this decrement would only apply in the generation the mutation occurs. In future generations the error correction process would in effect protect the rare mutation. Error correction by itself, would slow the rate at which rare alleles accumulate. By itself, simple error correction would not prevent rare alleles from becoming common, even dominant.
Quote: you are ignoring population bottlenecks
I am not sure what you mean by bottleneck. I assume you are referring to something which ‘could possibly explain why a distribution is different from expected.
Quote: In an infinite population, if M(Axt)=-S(Axmt), then the frequency of Ax doesn't change over time. However, in a finite population, even if M(Axt)=-S(Axmt), the frequency does change over time via random fluctuation. With small populations of sexually reproducing organisms, this tends to lead to a single allele taking over--simply by chance--with the others being removed from the population.
The above statement contains several misconceptions or inaccuracies.
First, M and -S represent the number of increments and decrements respectively, then Ax would remain constant no matter what the size of the population(assuming total population size doesn’t change".
Second, if M and S are forces or rates with probability distributions, then the percentage changes in Ax would get smaller as the size of the population increased toward infinity, but the size of the variance expressed in number of alleles would be expected to increase.
Third, the reduction to a single allele only occurs if the mutation rate is zero. If you have positive mutation rates, and the decrement rates on these mutation rates are not 100%, then the a population will not move toward a single allele. If you start with a single allele, positive mutation rates, and selection rates less than 100%, then the distribution will diverge from the single gene starting point. In the absence of very strong forces of decrement, the divergence is quite rapid and quite substantial. If there are 1000 possible alleles for a given gene, equal positive mutation rates and decrement rates are equal for all possible alleles, then a population starting with one dominant allele, will over time move to a flat distribution where the initial dominant allele is no more likely than the other 999 possible alleles.
As you pointed out, we typically see a limited number of common alleles for a gene(at least in more complex types of organisms). Multiple decrement analysis indicates that given known mutation rates, and the set of ‘likely mutations’, the observed distributions represent a major departure from the ‘expected distributions’. The observed distributions can only be explained by assuming very powerful forces of selection or decrement applied to ‘rare alleles’.
Quote: From this I conclude that (1) engineering standards and actuarial multiple decrement analysis, as applied here, is fundamentally flawed.
Your opinion based on your initial analysis is that the conclusions generated from multiple decrement model are flawed. Fine. I suggest that your initial analysis is flawed. Again fine. The point that is important is that the issue of who is right and who is wrong can be reduced to objective, verifiable mathematical analysis. You claim that a limited number of common alleles is a normal or expected phenomena. I claim that such a distribution will not occur unless there are very strong forces of selection or decrement to offset the forces of mutation. If we try to resolve these claims by subjective opinion, I will undoubtedly loose. In the realm of rigorous mathematics, my chances are much better.
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Rex Kerr
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posted 15. March 2003 23:51
I'd like to start by pointing out that most genetic models are designed to establish principles and make statistical predictions, not a detailed model of evolution-in-process. In this discussion, I will stick to mathematical and/or computational models with assumptions that are compatible with what is observed in biological systems. (Sometimes it is useful to introduce incompatability in order to diminish the complexity of a problem and demonstrate a principle--but since your statements were not specific, I can't really agree or disagree about biological compatability in genetic algorithm and population biology models.)
With respect to the frequency of rare alleles, in fact, the results I've seen look a lot like what you predicted for the set of rare allelesif the rates were very low but nonzero.
For example, in Li et al., "Diversity of mutations and distribution of single nucleotide polymorphic alleles in the human alpha-L-iduronidase (IDUA) gene", Genet Med 2002 Nov-Dec;4(6):420-6, the authors sequence 22 patients (44 alleles) (who are known to be mutant) and look at the IDUA gene; they find a variety of different mutations.
In Zamber et al., "Natural allelic variants of breast cancer resistance protein (BCRP) and their relationship to BCRP expression in human intestine.", Pharmacogenetics 2003 Jan;13(1):19-28, the authors identify nine different variants, of which two are in introns, three are synonymous mutations, and four actually change the protein. Of the four that change the protein, the distribution in allele frequency among Caucasians is: 0.14 (Q141K), 0.02 (V12M), 0.01 (N590Y), and 0.00 (I206L). (The last was detected in Hispanics, but not in the 100+ Caucasians who were sequenced.) Distributions for the noncoding alleles was similar: 0.03 (intron 2), 0.02 (pre-start-site), 0.01 (synonymous mutation exon 2), 0.00 (syn. mut. exon 4), 0.00 (intron 5).
Peterson et al., "Novel mutations and SNPs identified in CCR2 using a new comprehensive denaturing gradient gel electrophoresis assay.", Hum Mutat 2002 Oct;20(4):253-9, reports the detection of 11 new alleles of the CC chemokine receptor 2 gene. Again, the frequencies are all over the place, with lots at very low frequencies (e.g. R233Q was observed only twice out of 492 alleles; G355E was observed three times).
Rare alleles are, in fact, detected. Any specific rare allele may not be, but rare alleles definitely exist and are detected all the time.
Starting from a single common allele, we expect that on the order of 10^-6 of the next generation will lose the allele. Thus after about 10^6 generations, we might expect that the frequency of the original allele was down to only 1/e (about 37%). For humans, that's on the order of ten million years.
quote:
I am not sure what you mean by bottleneck.
A population bottleneck is when a species goes through a period where there is a reduced effective population size. This leads to a loss of allelic diversity. From [a href="http://arnica.csustan.edu/biol3020/population_genetics/population_genetics.htm"]a handout on population biology[/a]:
quote:
a. Intense natural selection or a disaster can cause a population bottleneck, a severe reduction in population size which reduces the diversity of a population. The survivors have very little genetic variability and little chance to adapt if the environment changes.
By the 1890's the population of northern elephant seals was reduced to only 20 individuals by hunters. Even though the population has increased to over 30,000 there is no genetic variation in the 24 alleles sampled. A single allele has been fixed by genetic drift and the bottleneck effect. In contrast southern elephant seals have wide genetic variation since their numbers have never reduced by such hunting.
b. Bottleneck effect, combined with inbreeding, is an especially serious problem for may endangered species because great reductions in their numbers has reduced their genetic variability. This makes them especially vulnerable to changes in their environments and/or diseases. The Cheetah is a prime example.
c. Sometimes a population bottleneck or migration event can cause a founder effect. A founder effect occurs when a few individuals unrepresentative of the gene pool start a new population.
E.g., a recessive allele in homozygous condition causes Dwarfism. In Switzerland the condition occurs in 1 out of 1,000 individuals. Amongst the 12,000 Amish now living in Pennsylvania the condition occurs in 1 out of 14 individuals. All the Amish are descendants of 30 people who migrated from Switzerland in 1720. The 30 founder individuals carried a higher than normal percentage of genes for dwarfism.
Here is your very powerful decrement mechanism. Note that it doesn't have to act very often--those northern elephant seals, will take millions of years to regain their distributions of alleles.
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warren_bergerson
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posted 17. March 2003 09:17
Rex,
The studies you list confirm that techniques exist to measure/estimate the frequency of uncommon alleles. Just for convenience lets use say that an allele is common if its frequency in a population is 10% or greater, uncommon if the frequency is 10% to .1%, and rare if the frequency is below .1%. While the probability of an individual rare allele may be small, very large numbers of such allele are possible. While the probability of an individual rare alleles may be very small, the probability of ‘some member of the set of rare alleles’ would be expected to be much higher and much easier to detect.
Given techniques for measuring uncommon and rare alleles, it is possible to test hypothesis of the form ‘the average rate of decrement/selection applicable to rare alleles is greater than X%’. If we start with 1)the assumption of a single dominant allele at some point in time, 2)an measure of the raw mutation rates, 3)a measure of the time elapsed since the last bottleneck, and 4)assumptions as to average rates of selection/decrement, then you can calculate an ‘expected distribution of alleles’ and test to determine if the observed distributions is consistent with the expected distribution. By repeating this process, you can calculate the range of rates of decrement compatible with observed distributions.
The important point here is that both techniques and information are available to calculate decrement rates per ‘potential allele’. The aggregate multi-decrement model of genetic change I propose is both possible and practical. The claim that genetic change is too complex to model is invalid.
IMPLICIT ASSUMPTIONS
It is useful to note that genetic models, such as those used to demonstrate drift, are based on simplifying assumptions. Implicitly included are the assumption that the decrement rates for all uncommon and rare alleles are 100%. If they did not bury this assumption in their models, the result obtained would be divergence not drift.
The assumption that the decrement rates associated with uncommon and rare alleles is 100% is not inappropriate. As I have been attempting to explain, this is the decrement assumption which appears to be supported by observed distributions. However, if decrement rates are 100% there are no ‘random mutations’. There is no source for the variation needed if evolution to occur.
As I have stated repeatedly, the subject here is standards. Current academic/scientific standards allow the use of models to demonstrate ‘drift’ using assumptions that make evolutionary change impossible. There is no requirement to disclose that the assumptions being used preclude the possibility of anything other than trivial evolutionary change. Two of the key requirements of the engineering standards are that 1)the assumptions used must be logically consistent with observed conditions, and 2)the assumptions used must be logical consistent among the various models and demonstrations.
RATE OF DIVERGENCE
Quote: Starting from a single common allele, we expect that on the order of 10^-6 of the next generation will lose the allele. Thus after about 10^6 generations, we might expect that the frequency of the original allele was down to only 1/e (about 37%). For humans, that's on the order of ten million years.
Given that your rate of divergence is correct, in a mere 10 million years 63% of the population would have rare and uncommon alleles. Since there are lots of organisms that have been fairly stable over such periods of time, organisms with 60% plus distributions of rare alleles must be quite common. Right?
Second, if you start with an assumed raw mutation rate of 10^-8 per potential allele per cell division, you get raw expected rates of divergence that are much higher than 10^-6. The raw expected divergence rate would be greater than 1% per generation.
Actual observed rates of divergence for the vast majority of alleles in complex organisms, are well below 1% and probably well below 10^-6. I suspect, or if you prefer I predict, that for most genes with a single dominant gene or with a handful of common genes, there is no detectable divergence. Complex organisms have an almost complete ability to avoid genetic divergence and maintain species identify.
The ‘problem’ is that this ability to avoid divergence also equates into an ability to avoid genetic change or evolution. The solution to this issue would appear to be variable rates of divergence produced by variable or dynamic rates of ‘allele specific decrement rates’. If we accept that allele specific decrement rates are dynamic then the next logical question is "What process controls changes in rates of decrement?".
Traditional theory suggests that changes in decrement are changes in natural selection. Changes in the rates of natural selection are due to changes in the fitness landscape. Probably beyond the scope of the discussion here, but there are techniques to determine if required changes in decrement rates can be explained by fitness landscape.
To briefly summarize, divergence rates are determined by raw mutation rates and allele specific decrement rates. Explaining very low observed divergence rates requires very high decrement rates applicable to rare alleles. To explain both ‘high decrement rates under normal stable conditions’ and ‘much lower decrement rate during periods of change’ suggests that decrement rates must by highly dynamic, and that there must, as my hypothesis suggested, by some mechanism other than mutation-natural selection responsible for controlling changes in the decrement rates. The mutation-natural selection model is therefore inadequate to explain evolutionary change.
VARIANCE
Quote: By the 1890's the population of northern elephant seals was reduced to only 20 individuals by hunters. Even though the population has increased to over 30,000 there is no genetic variation in the 24 alleles sampled. A single allele has been fixed by genetic drift and the bottleneck effect.
I can’t resist pointing out how misleading the above statement is. To begin, if the initial population had allele distributions of 50% dominant allele and 50% all others for each of the 24 genes, then the probability that all 20 surviving seals had the same set of allele is (10^-40)*(10^-24) or 10^-64. This seems to be more an example of a scientific or statistical miracle than of genetic drift or the bottleneck effect. It sounds even more like intentional fraudulent analysis.
Genetic drift, bottleneck effect, and founders effect are all fancy terms to express the impact or potential impact of variance or statistical fluctuations. Where I studied the subject it was called risk theory. It is important to remember that statistical fluctuations are a function of ‘mean, standard deviation, and type of distribution’. There are lots of demonstrations showing that probability distributions can drift or vary ‘assuming the mean is zero and the probability distribution is constant over time’. What is not explained, and apparently not generally recognized, is that drift or statistical fluctuation will not counteract strong forces of expected change. As is being discussed, the evidence suggests that the forces of decrement/selection applicable to most potential alleles are very strong, probably a 100% under normal conditions. The evidence would also suggest these forces are dynamic and with changes controlled by a process other than mutation-natural selection
SUMMARY
You have provided evidence that it is currently both possible to measure probability distributions for genes with a reasonable degree of accuracy. If we can measure probability distributions than we can develop and test hypothesis which will define the range of ‘rates of decrement applicable to rare and uncommon alleles. This in turn means that whole systems multiple decrement models of genetic change processes are both possible and practical. The claim that genetic change is too complex to model is not valid.
The initial conclusion from multiple decrement analysis is that ‘genetic change/evolution’ and ‘avoiding divergence and loss of species identity’ require very different forces of decrement. If we accept that biological systems are capable of both stability and change, then we must conclude there exist processes and mechanisms other than ‘mutation and natural selection’ capable of producing dramatic changes in decrement rates.
This conclusion is not the result of new math or new analytical techniques. This conclusion is the result of standards that requires the use the logically consistent decrement assumptions in explaining both stability and change.
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brauer
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posted 17. March 2003 11:14
Warren said:
quote:
I can’t resist pointing out how misleading the above statement is. To begin, if the initial population had allele distributions of 50% dominant allele and 50% all others for each of the 24 genes, then the probability that all 20 surviving seals had the same set of allele is (10^-40)*(10^-24) or 10^-64. This seems to be more an example of a scientific or statistical miracle than of genetic drift or the bottleneck effect. It sounds even more like intentional fraudulent analysis.
His incredulity at this reinforces a couple of important points:
- He assumes that two alleles are present in equal proportions in a population. This is theoretically unlikely, and such a situation is rarely observed. It is not hard to understand why: any allele conferring a marginal improvement in fitness will become nearly fixed in a popoulation. Equal allele frequencies is in fact a very unusual case that must be explained by esoteric mechanisms such as frequency-dependent selection.
- He ignores population structure. Given that most animals in close proximity are likely to be related to each other, a population bottleneck is not likely to be the same as a random draw of alleles from throughout the population.
- He ignores the fact that population bottlenecks have been observed many times in natural populations, and that the genetic variance behaves just as Rex Kerr has described.
The upshot of this is that Warren would rather cling to a demonstrably false theoretical framework for how populations change than to admit any degree of ignorance in a field that is not his own. Furthermore, he would sooner ascribe observed results to a miracle or to fraud than concede that he is not familiar with the population genetics models. As a population geneticist, I find this more than a little offensive.
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warren_bergerson
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posted 17. March 2003 11:54
Matt,
Quote: Furthermore, he would sooner ascribe observed results to a miracle or to fraud than concede that he is not familiar with the population genetics models. As a population geneticist, I find this more than a little offensive.
This is an interesting discussion and I don’t want to see it reduced to ‘I’m an expert and I am offended because your findings don’t agree with mine".
Rex cited the elephant seal study as a real life illustration of the impact of a bottleneck and of drift. The clear implication of the statement as quoted is the distribution of alleles for the 24 genes was much different before the population was reduced by hunting. The original statement is misleading, and probably intentionally misleading, with some information on the distribution of alleles for the 24 genes before the population was reduced by hunting.
Rex earlier suggested that for humans something like 40% of genes have a single dominant allele. If that were true for the elephant seals, then picking 24 genes at random and finding no-variations for any of them in a population of 20 individuals would be an extreme statistical rarity. To have discovered such an extreme rarity and published the results without providing some logical or mathematical explanation for the rarity would be misleading in the extreme.
If you are a population geneticist and you can provide the logical mathematical explanation for the elephant seal findings then such information would be useful. [Rex may have unintentionally quoted the study out of context.] Comments to the effect that you are offended without providing an explanation for what appears on the surface to be a misleading statement does not contribute to the discussion.
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Rex Kerr
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posted 18. March 2003 01:11
Warren, I'm afraid I'm getting tired of being a population biology instructor. There are a bunch of points to make (e.g. that species very often cluster into a small enough effective population size to decrease heterozygosity), and I'll let you consult a population genetics text to learn more.
One of the reasons I am not bothering is that you have too high of an error rate with simple calculations. I have better things to do that fix simple calculation mistakes, or oversimplifications in theories based on not having sufficiently studied current knowledge on the subject.
For example, assuming that the distribution was initially 50/50, the chance that 20 offspring would have identical alleles at each position is, in fact, something like ((0.5)^40)^24. This would seem to help your case, because ((0.5)^40)^24 = 2^-960 ~= 10^-289. I'm not sure what you're doing, but it's wrong.
Furthermore, your calculation appears to assume that there is exactly one generation at 20 seals. This is pretty unlikely, given what we know about hunting. And it likewise assumes that all alleles are preserved on expansion--also unlikely. And that all 20 seals actually reproduced (again unlikely).
quote:
What is not explained, and apparently not generally recognized, is that drift or statistical fluctuation will not counteract strong forces of expected change.
That completely depends on the population size. You've made this claim far too many times. I've tried to demonstrate why it is not true mathematically, and apparently you didn't listen or believe me or whatever. So I'll try again.
Sorry for the length, but I simply don't know how else to get the point across.
Here is ANSI C, in case you wish to run the experiments yourself:
code:
// pop.c -- quick and dirty random drift simulation
// Code by Rex Kerr, 2003
#include <stdio.h>
// Constants
int POP_SIZE = 20;
int NUM_GEN = 401;
int NUM_REPORT = 10;
int INITIALIZE = 1;
// Quick and dirty random number generatior from Numerical Recipes in C
unsigned long seed = 0;
unsigned long rng()
{
return seed = 1664525L*seed + 1013904223L;
}
unsigned long rng_n(unsigned long n)
{
return rng()/(0xFFFFFFFFL/n);
}
// Mutate an allele with frequency 1/4096 to any of 256 possibilites
int mutate(int allele)
{
if ( rng_n(4096)==0 ) { return rng_n(256); }
else return allele;
}
// Count the number of alleles in the population and state the most frequent
int count(int *population,double *freq,int *which)
{
int hist[256],i,j,k;
for (i=0;i<256;i++) hist[i]=0;
for (i=0;i<POP_SIZE;i++) hist[population[i]]++;
for (i=j=k=0;i<256;i++)
{
if (hist[i]>j) { j = hist[i]; *which = i; }
if (hist[i]>0) k++;
}
*freq = j / ((double)POP_SIZE);
return k;
}
// Create a new generation by picking alleles at random from the old
// generation and applying mutation.
void newgen(int *old_pop,int *new_pop)
{
int i,j,k;
for (i=0;i<POP_SIZE;i++) new_pop[i] = mutate( old_pop[ rng_n(POP_SIZE) ] );
}
// Run generations and tell us the results.
int main(int argc,char *argv[])
{
int *new_pop,*old_pop,*temp;
double freq;
int i,j,k;
if (argc<=1) { printf("Usage: pop POP_SIZE NUM_GEN NUM_REPORT\n"); return 1; }
if (argc>1) { i = atol(argv[1]); if (i>0) POP_SIZE = i; }
if (argc>2) { i = atol(argv[2]); if (i>0) NUM_GEN = i; }
if (argc>3) { i = atol(argv[3]); if (i>0) NUM_REPORT = i; }
if (argc>4) { i = atol(argv[4]); if (i==0 || i==1) INITIALIZE = i; }
new_pop = (int*) calloc(POP_SIZE,sizeof(int));
old_pop = (int*) calloc(POP_SIZE,sizeof(int));
if (INITIALIZE)
for (i=0;i<POP_SIZE;i++) new_pop[i] = rng_n(256);
else
for (i=0;i<POP_SIZE;i++) new_pop[i] = 0;
for (i=0;i<NUM_GEN;i++)
{
temp=new_pop; new_pop=old_pop; old_pop=temp;
newgen(old_pop,new_pop);
if ( (i%NUM_REPORT) == 0 )
{
j = count(old_pop,&freq,&k);
printf("Generation %4d has %3d alleles, max freq %3.1f% (allele %03d)\n",i,j,100.0*freq,k);
}
}
return 0;
}
As may be obvious, this explicitly simulates a population of POP_SIZE alleles by picking alleles at random and either propagating them unchanged to the next generation or mutating them with a frequency of 1/4096 (much higher than real rates, but I don't want to wait all day). There are 256 possible alleles here. What does this program produce?
Population size: 200 alleles (100 animals):
code:
Generation 0 has 138 alleles, max freq 2.0% (allele 148)
Generation 50 has 6 alleles, max freq 33.0% (allele 250)
Generation 100 has 3 alleles, max freq 45.5% (allele 243)
Generation 150 has 3 alleles, max freq 47.5% (allele 250)
Generation 200 has 3 alleles, max freq 52.5% (allele 250)
Generation 250 has 3 alleles, max freq 47.5% (allele 224)
Generation 300 has 3 alleles, max freq 44.5% (allele 224)
Generation 350 has 2 alleles, max freq 74.0% (allele 250)
Generation 400 has 4 alleles, max freq 48.5% (allele 250)
Generation 450 has 2 alleles, max freq 62.0% (allele 224)
Generation 500 has 2 alleles, max freq 61.5% (allele 250)
Generation 550 has 3 alleles, max freq 67.0% (allele 224)
Generation 600 has 5 alleles, max freq 73.5% (allele 224)
Generation 650 has 5 alleles, max freq 76.0% (allele 224)
Generation 700 has 2 alleles, max freq 96.5% (allele 224)
Generation 750 has 2 alleles, max freq 70.5% (allele 224)
Generation 800 has 2 alleles, max freq 91.5% (allele 224)
Generation 850 has 1 alleles, max freq 100.0% (allele 224)
Generation 900 has 2 alleles, max freq 98.0% (allele 224)
Generation 950 has 1 alleles, max freq 100.0% (allele 224)
Generation 1000 has 1 alleles, max freq 100.0% (allele 224)
. . .
Generation 3200 has 1 alleles, max freq 100.0% (allele 224)
Generation 3250 has 1 alleles, max freq 100.0% (allele 224)
Generation 3300 has 1 alleles, max freq 100.0% (allele 224)
Generation 3350 has 2 alleles, max freq 97.5% (allele 224)
Generation 3400 has 2 alleles, max freq 87.5% (allele 224)
Generation 3450 has 3 alleles, max freq 90.5% (allele 224)
Generation 3500 has 2 alleles, max freq 60.0% (allele 224)
Generation 3550 has 2 alleles, max freq 88.0% (allele 224)
Generation 3600 has 2 alleles, max freq 58.5% (allele 224)
Generation 3650 has 2 alleles, max freq 78.0% (allele 224)
Generation 3700 has 2 alleles, max freq 58.0% (allele 194)
Generation 3750 has 2 alleles, max freq 72.5% (allele 194)
Generation 3800 has 1 alleles, max freq 100.0% (allele 194)
Generation 3850 has 1 alleles, max freq 100.0% (allele 194)
Generation 3900 has 1 alleles, max freq 100.0% (allele 194)
Generation 3950 has 1 alleles, max freq 100.0% (allele 194)
Generation 4000 has 1 alleles, max freq 100.0% (allele 194)
Well, this is odd. We seem to have only one allele out of 200, starting from a completely random mishmash of every possible allele. Every now and then, a new single allele replaces the old one, but for most of the time it's constantly a single allele.
Elephant seals: population size 40 alleles (20 animals):
code:
Generation 0 has 37 alleles, max freq 5.0% (allele 087)
Generation 5 has 13 alleles, max freq 22.5% (allele 098)
Generation 10 has 5 alleles, max freq 32.5% (allele 087)
Generation 15 has 4 alleles, max freq 47.5% (allele 181)
Generation 20 has 3 alleles, max freq 45.0% (allele 098)
Generation 25 has 3 alleles, max freq 45.0% (allele 098)
Generation 30 has 3 alleles, max freq 52.5% (allele 181)
Generation 35 has 3 alleles, max freq 35.0% (allele 181)
Generation 40 has 3 alleles, max freq 47.5% (allele 098)
Generation 45 has 2 alleles, max freq 72.5% (allele 181)
Generation 50 has 2 alleles, max freq 97.5% (allele 181)
Generation 55 has 1 alleles, max freq 100.0% (allele 181)
Generation 60 has 1 alleles, max freq 100.0% (allele 181)
Generation 65 has 1 alleles, max freq 100.0% (allele 181)
Generation 70 has 1 alleles, max freq 100.0% (allele 181)
Generation 75 has 1 alleles, max freq 100.0% (allele 181)
Generation 80 has 1 alleles, max freq 100.0% (allele 181)
Generation 85 has 1 alleles, max freq 100.0% (allele 181)
Generation 90 has 1 alleles, max freq 100.0% (allele 181)
Generation 95 has 1 alleles, max freq 100.0% (allele 181)
Generation 100 has 1 alleles, max freq 100.0% (allele 181)
Well, what do you know, we have but one allele. Not in a single generation, but then the seals had reduced population for a while.
How about a large population: population size 20000 alleles (10k population):
code:
Generation 0 has 256 alleles, max freq 0.5% (allele 109)
Generation 1000 has 75 alleles, max freq 8.5% (allele 250)
Generation 2000 has 90 alleles, max freq 16.2% (allele 045)
Generation 3000 has 71 alleles, max freq 11.5% (allele 085)
Generation 4000 has 72 alleles, max freq 17.9% (allele 085)
Generation 5000 has 68 alleles, max freq 10.2% (allele 127)
. . .
Generation 36000 has 69 alleles, max freq 14.1% (allele 121)
Generation 37000 has 68 alleles, max freq 22.1% (allele 121)
Generation 38000 has 60 alleles, max freq 30.5% (allele 121)
Generation 39000 has 59 alleles, max freq 17.0% (allele 121)
Generation 40000 has 72 alleles, max freq 17.8% (allele 199)
Here we seem to have come to some kind of equilibrium; if we start off with a single allele:
code:
Generation 0 has 1 alleles, max freq 100.0% (allele 000)
Generation 1000 has 45 alleles, max freq 87.2% (allele 000)
Generation 2000 has 65 alleles, max freq 73.3% (allele 000)
Generation 3000 has 51 alleles, max freq 63.9% (allele 000)
Generation 4000 has 62 alleles, max freq 42.1% (allele 000)
Generation 5000 has 52 alleles, max freq 60.4% (allele 000)
Generation 6000 has 64 alleles, max freq 53.3% (allele 000)
Generation 7000 has 58 alleles, max freq 35.0% (allele 000)
Generation 8000 has 64 alleles, max freq 37.3% (allele 000)
Generation 9000 has 74 alleles, max freq 37.6% (allele 000)
Generation 10000 has 70 alleles, max freq 18.5% (allele 000)
Generation 11000 has 73 alleles, max freq 27.6% (allele 093)
Generation 12000 has 67 alleles, max freq 17.5% (allele 203)
Here the model behaves sort of as you and I predicted for large populations: under mutation pressure, the original allele 000 is no longer dominant after around ten thousand generations. Note that the population is significantly larger than the inverse of the mutation rate.
Bottom line: if the effective population size is small, drift is a very effective means of limiting appearance of alleles. What's this effective population size thing? Well, animals have a habit of reproducing with their neighbors. Even if they're not actually completely isolated from each other, this can effectively lead to clearing of rare alleles, and when that small population gets lucky and spreads into a new area, they start off with only a few common alleles per gene (unless selection has intervened).
Is this blatantly obvious enough yet?
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