Randomness and evolution

Here’s a simple experiment one can actually try. Take a bag of M&M’s, and without peeking reach in and grab one. Eat it. Then grab another and return it to the bag with another one, from a separate bag, of the same colour. Give it a shake. I guarantee (and if you tell me how big your bag is I’ll have a bet on how long it’ll take) that your bag will end up containing only one colour. Every time. I can’t tell you which colour it will be, but fixation will happen.

This models the simple population process of Neutral Drift. Eating is death, duplication is reproduction, and the result is invariably a change in frequencies, right through to extinction of all but one type. You don’t have to alternate death and birth; choose any scheme you like short of peeking in the bag and being influenced by residual frequencies (ie: frequency-dependent Selection), and you will end up with all one colour.

Is Chance a cause here? Well … yes, in a sense it is, in the form of sample error. Survival and reproduction are basically a matter of sampling the genes of the previous generation. More random samples are a distortion of the larger population than aren’t, so, inexorably, your future populations will move away from any prior makeup, increasing some at the expense of others till only one variant remains.

Selection is a consistent bias upon this basic process. If different colours also differed a little in weight, say, more of some would be at the bottom of the bag than others, so you’d be more likely to pick one type than another. In more trials, the type more likely to be picked would be picked more often, to express it somewhat tautologously. You’d get a sampling bias.

Both of these processes are random – or stochastic, to use the preferred term. In reality, they are variations of the same process, with continuously varying degrees of bias from zero upwards. It makes no sense to call selection nonrandom, unless by ‘random’ you mean unbiased. Where there is no bias, all is Drift. But turning up the selective heat does not eliminate drift – sample error – and so does not eliminate stochasticity.

With a source of new variation, these processes render evolution inevitable. Even with a brand new mutation, with no selective advantage whatsoever, 1/Nth of the time (where N is the population size) it will become the sole survivor. That’s the baseline. If there is a selective advantage, it will be more likely and quicker to fix, on the average. If at a selective disadvantage, it will be less likely and slower.

Conversely, without a source of new variation, all existing variation would be squeezed out of the population, and evolution would stop.

650 thoughts on “Randomness and evolution

  1. phoodoo: You don’t see a logical problem with claiming that you can go from zero in a population of 4 to a novel mutation becoming fixed in the population in ONE generation? That makes sense to you? That’s how your math works?

    No, there is no problem here, phoodoo. One generation is the shortest, not typical time for a mutation to get fixed. The random nature of the process guarantees that the typical time will be longer.

    I have just run my code with 4 MMs, starting with 3 blue and 1 black MM (the novel mutation has just appeared). Here is the output, with the successful runs in bold:

    Run 1: {3, 1} at t=0; {4, 0} at t=2;
    Run 2: {3, 1} at t=0; {4, 0} at t=2;
    Run 3: {3, 1} at t=0; {0, 4} at t=57;
    Run 4: {3, 1} at t=0; {0, 4} at t=16;
    Run 5: {3, 1} at t=0; {0, 4} at t=6;
    Run 6: {3, 1} at t=0; {4, 0} at t=3;
    Run 7: {3, 1} at t=0; {0, 4} at t=3;
    Run 8: {3, 1} at t=0; {4, 0} at t=7;
    Run 9: {3, 1} at t=0; {0, 4} at t=11;
    Run 10: {3, 1} at t=0; {4, 0} at t=6;

    The shortest time to fixation was in run 7: 3 steps from 3 blue and 1 black to 4 black. Adding the first step makes it 4 units of time, or 1 generation.

    Again, that is the shortest time. You can see that it is not typical. Other successful runs had durations 57, 16, 6, and 11 units of time, or 14, 4, 1.5, and 3 generations.

  2. olegt: No, there is no problem here, phoodoo. One generation is the shortest, not typical time for a mutation to get fixed. The random nature of the process guarantees that the typical time will be longer.

    I have just run my code with 4 MMs, starting with 3 blue and 1 black MM (the novel mutation has just appeared). Here is the output, with the successful runs in bold:

    Run 1: {3, 1} at t=0; {4, 0} at t=2;
    Run 2: {3, 1} at t=0; {4, 0} at t=2;
    Run 3: {3, 1} at t=0; {0, 4} at t=57;
    Run 4: {3, 1} at t=0; {0, 4} at t=16;
    Run 5: {3, 1} at t=0; {0, 4} at t=6;
    Run 6: {3, 1} at t=0; {4, 0} at t=3;
    Run 7: {3, 1} at t=0; {0, 4} at t=3;
    Run 8: {3, 1} at t=0; {4, 0} at t=7;
    Run 9: {3, 1} at t=0; {0, 4} at t=11;
    Run 10: {3, 1} at t=0; {4, 0} at t=6;

    The shortest time to fixation was in run 7: 3 steps from 3 blue and 1 black to 4 black. Adding the first step makes it 4 units of time, or 1 generation.

    Again, that is the shortest time. You can see that it is not typical. Other successful runs had durations 57, 16, 6, and 11 units of time, or 14, 4, 1.5, and 3 generations.

    It has nothing to do with it being typical or not, your reasoning is completely fallacious, you can’t see that? It is not ONE generation when FOUR new births and FOUR deaths need to occur in order for a population to switch from all of one color to all of another. Your premise is incorrect, so every conclusion from there after does not make sense.

  3. phoodoo: It has nothing to do with it being typical or not, your reasoning is completely fallacious, you can’t see that? It is not ONE generation when FOUR new births and FOUR deaths need to occur in order for a population to switch from all of one color to all of another. Your premise is incorrect, so every conclusion from there after does not make sense.

    I am not sure what you are trying to say, phoodoo. Come again?

  4. olegt: : With 4 MMs,
    4 deaths = 1 generation.
    1 death = 1/4 of a generation.

    Yes. 4 deaths = 1 generation.

    I am not sure what you are trying to say, phoodoo. Come again?

    I’m pretty sure the fractions confused him again.

  5. olegt: I am not sure what you are trying to say, phoodoo. Come again?

    Four new deaths and four new births of one lineage is not one generation. Get it?

  6. phoodoo: Four new deaths and four new births of one lineage is not one generation. Get it?

    It’s a matter of convention, phoodoo.

    I defined one generation as N deaths in this comment. Each death is accompanied by exactly one birth. So 1 generation passes when N deaths and N births occur. Exactly as in this example.

  7. phoodoo: It has nothing to do with it being typical or not, your reasoning is completely fallacious, you can’t see that? It is not ONE generation when FOUR new births and FOUR deaths need to occur in order for a population to switch from all of one color to all of another. Your premise is incorrect, so every conclusion from there after does not make sense

    Slow down. Think.
    In a human (or other animal) family, what does a “generation” mean? Well, it sure doesn’t mean one and only kid, and every kid after that first one is a whole new generation.

    A “generation” is the total number of the replacements (first generation descendants )for the original number of parents. Yes, in human society the next generation is a greater number than their parents’ generation, but in a stable ecology each generation, on average, is the same number, replacing the parents, again on average, one-to-one. And in this test, it’s explicitly defined that the next generation is exactly the same number as the previous.

    True, you could count the start of yet another generation whenever you have the specified number of births, rather than the deaths of each of the originals; in this test it makes no difference as long as you’re willing to understand.

  8. thorton: I’m pretty sure the fractions confused him again.

    No, this time it’s a matter of definitions. I think phoodoo wants to have a convention different from mine, calling N deaths and N births 2 generations.

    That would be fine by me. Under phoodoo’s convention, the shortest time to start from N blue MMs and to end with N black MMs would be 2 generations. Under mine, 1 generation. Both correspond to the same number of deaths, N.

    Sadly, he seems tangled up in definitions and can’t move past that stage.

  9. olegt

    Sadly, he seems tangled up in definitions and can’t move past that stage.

    He certainly seems like he wants to stay confused. It’s a wonderful defense mechanism against reality.

  10. olegt:
    phoodoo, let me ask you. What is the level of your education?

    Oh I am just a dumb rice farmer, that never went to school. I taught myself how to read simple stick figures in the dirt, and I am dyslexic, and I never learned to count.

    But EVEN I know that to get four deaths and four births from one lineage of one new novel mutation spontaneously arisen in one offspring is not one generation-not in my world, not in your world, and pretty much not in any world where the world generation has any meaning at all.

    Maybe if I would have studied physics like you, I also could convince myself that this was one generation.

  11. phoodoo: Maybe if I would have studied physics like you, I also could convince myself that this was one generation.

    I am still not sure what you are trying to accomplish, phoodoo. Do you want to call N deaths and N births 2 generations? That would be fine by me. As I said above, this is a matter of convention.

    Different unit of time, that’s all, Mr. Rice Farmer. 🙂

  12. olegt: No, this time it’s a matter of definitions. I think phoodoo wants to have a convention different from mine, calling N deaths and N births 2 generations.

    That would be fine by me. Under phoodoo’s convention, the shortest time to start from N blue MMs and to end with N black MMs would be 2 generations. Under mine, 1 generation. Both correspond to the same number of deaths, N.

    Sadly, he seems tangled up in definitions and can’t move past that stage.

    Well, the definition matters when you start to extrapolate from this toy example to time-to-fixation in real population. We already saw that phoodoo wants to go for “not enough time, too many generations needed for mutation to spread through population”. Xe mistook every death/birth event in the 1000s examples as a whole new generation. No wonder xe feels that we can’t have evolved from monkeys! Under that generation assumption, praise the lard, the Interior Decorator has a vital role after all.

    Then momentarily we seemed to have cleared that up, but now it looks like phoodoo is heading back to the “every death/birth event is a brand new generation”.
    Too bad.

  13. hotshoe: but now it looks like phoodoo is heading back to the “every death/birth event is a brand new generation”.

    Maybe, maybe not.

    phoodoo, if my attempt to guess your definition of a generation failed, feel free to give your definition of a generation.

  14. phoodoo: from one lineage of one new novel mutation spontaneously arisen in one offspring is not one generation

    If you stop and recollect, the “one black” mutation was assigned in exactly one of the first “parents” at your own insistence, phoodoo. Not in “one offspring”.

  15. hotshoe: Well, the definition matters when you start to extrapolate from this toy example to time-to-fixation in real population.We already saw that phoodoo wants to go for “not enough time, too many generations needed for mutation to spread through population”.Xe mistook every death/birth event in the 1000s examples as a whole new generation.No wonder xe feels that we can’t have evolved from monkeys!Under that generation assumption, praise the lard, the Interior Decorator has a vital role after all.

    Then momentarily we seemed to have cleared that up, but now it looks like phoodoo is heading back to the “every death/birth event is a brand new generation”.
    Too bad.

    Is that right Hotshoe?

    If you have no existence of a certain mutation in a population of 4, and ONE individual is born with this novel mutation, then they give birth to one offspring which also has this mutation, how many generations are we up to so far? And we still need two more to make the whole population with this new mutation.

    The answer is ONE?

    I guess the problem is I need to study physics more because perhaps one needs an understanding of quantum entanglement and the Copenhagen interpretation to describe how this is one generation.

  16. phoodoo: If you have no existence of a certain mutation in a population of 4, and ONE individual is born with this novel mutation, then they give birth to one offspring which also has this mutation, how many generations are we up to so far?

    phoodoo,

    You have not yet told us what you think we should call a generation. So let me ask you again.

    In a population of N MMs (or people), how many deaths occur during one generation?

    By my definition of a generation, N.

    What do you think?

  17. Hilarious!

    Darwinists actually testing the details of their precious theory that they’ve known to be true all along.

    ahahahaha…

  18. olegt: phoodoo,

    You have not yet told us what you think we should call a generation. So let me ask you again.

    In a population of N MMs (or people), how many deaths occur during one generation?

    By my definition of a generation, N.

    What do you think?

    You are the one that stated that it is ONE generation to go from 0 in a population of four without the novel mutation, to everyone in the population carrying the mutated gene. YOU say that is ONE generation. And you have the nerve to say that I am so incorrect to say that if you start with a population of 1000 individuals and only 1 with the the new mutation, the odds of drifting into the entire population is pretty much impossible.

    Oh, but you showed that if you run the experiment THOUSANDS of times, with MILLIONS of birth and death substitutions, it could statistically happen once. In other words it would pretty much never happen. This is for one lousy little neutral drift which has to happen about a zillion times for all the predicted incidents of neutral drift to occur throughout the animal kingdom, and we haven’t even started to address the problem of what happens when you get a second mutation which wipes out the first one, or which turns the mutation deadly, or, or…a million other things that can go wrong.

    That is my whole problem with your little computer model problem, you just gloss over every problem with it, and you can’t even get that four new births in a lineage is not equal to ONE generation.

  19. phoodoo,

    I am already aware that you don’t like my convention for a generation. That’s fine, offer your own and we will discuss it. If it makes sense, we will adopt your convention.

    However, if you cannot even tell us what you would like to count as a generation, phoodoo, there can be no further discussion.

    I am turning out for the night. Think this through and let me know how you would like to define a generation.

  20. And furthermore, this is the odds problem with this minuscule population of 1000 which is way below what is need for a population to thrive. When you ramp up the numbers to populations of 2000, 5000, 10,000 suddenly your odds go into the stratosphere. But you still want to say, well, its possible.

    Yea right, and four new births and deaths equals one generation, if you study M theory and 19 dimensions that is.

  21. petrushka:
    The question is why do bacteria not succumb to genetic meltdown?[plus from subsequent post] my point is that The Lenski experiment involved hundreds of mutations that did not affect the viability of the colonies. Where’s the meltdown?

    As has already been stated and stipulated, most mutations are initially neutral. That being said, I maintain that once mutations accrue to the point that they are either beneficial or harmful, they are more likely to be harmful. And I accept that these harmful mutations would then be less likely to be fixed in the population.
    In regards to the Lenski experiment, the populations of bacteria were forced into an unnatural environment where they did not ordinarily do well. After thousands of generations, several developed simple mutations that gave them the ability to thrive in this unnatural environment. If placed back into a regular natural e.coli environment, which do you think would do better – the original unmutated bacteria or the “new and improved” mutated version? If this were tested, I believe the original unmutated bacteria would fare better.

  22. Rumraket: Absolutely correct, the effect of mutations depend on environment. But that’s how evolution works, there are no super-organisms that are perfectly adapted to all environments.

    This is a strawman – no mention was made of perfectly adapted super-organisms. Organisms would never become extinct if this were the case. I did suggest that organisms seem to be equipped to adapt to changing environments, but this is by no means the same thing.

    Moving from one niche to another, organisms for the most part trade fitness in their previous environment for fitness to their new environment.

    To the extent they can adapt. But the manner(s) in which the adaptation occurs is still being debated.

    You will notice that despite being a mammal, a dolphin have a terrible set of legs.

    Now who’s being irrelevant?

    Lenski experiment proves that beneficial mutations happen and get fixed in populations. In so doing, it also demonstrates a key feature of the theory of evolution: adaptation through natural selection to the extant environment.

    Still remains to be seen the long term viability of the mutated bacteria and the scalability of this mechanism to more complex organisms if more than one or two mutations were required to enable adaptation to the new environment.

  23. phoodoo: And furthermore, this is the odds problem with this minuscule population of 1000 which is way below what is need for a population to thrive. When you ramp up the numbers to populations of 2000, 5000, 10,000 suddenly your odds go into the stratosphere.

    The odds go to 1/N (where N is the population number). 1/2000; 1/5000, 1/10000 …. not excellent odds, but I’d play the megamillions lottery at those odds, wouldn’t you?
    Go ahead and say the odds go “into the stratosphere” if you like. Shows you’re not too good with simple arithmetic, but suit yourself.

    But you still want to say, well, its possible.

    The test examples show fixation happening, including starting with 1000, and including starting with just 1 member of the population having the “black” mutation, as you insisted. Fixation happened right before your eyes. Which are you gong to believe, your own eyes or your cloudy preconceptions?

  24. phoodoo: That is my whole problem with your little computer model problem, you just gloss over every problem with it, and you can’t even get that four new births in a lineage is not equal to ONE generation.

    Exactly how do you choose to define one generation, Phoodoo?

    Just tell us. No need to go on and on about how we did it wrong, just tell what you think would be the right way to define it.

  25. Allan Miller:
    Piltdown2,

    First, thanks for your patience as I may have “drifted” from your OP and the fun of modeling M&M ratios.

    -In the event that a deleterious mutation did become fixed by drift, that is not the end of the story. The population has moved a little down the adaptive hill, which opens up the way for a beneficial mutation to take it back up again.

    I confess I do not have the same faith in beneficial mutations when all the variables of real life and complex organisms are added in, but I do acknowledge the possibility on a small scale.

    If, at the same time, it becomes large, the tenure of this fixed deleterious mutation is even more likely to be brief.

    By “even more likely to be brief”, I assume you are speaking of extinction. Or are you referring to some other mechanism that repairs the large mutation which has fixed in the population?

    But still, the illustration was to get across the baseline – the thing you can’t turn off: loss of variation with or without selection, unless new mutations arise (and then, the same process fixes them). Diversity arises from the lag, with simultaneous progressions at various stages of their unfolding.

    Thanks again.

  26. A comment on what a the length in time of a “generation” is. Here is my way of defining it. I had to work this out back 42 years ago when I worked on effective population number in a population with overlapping generations.

    Suppose we have a population, reproducing somehow. We paint a number 0 on everyone’s forehead. Then we go forward in time with the population reproducing itself. As each individual is born, take its mother (or designate one parent as the mother) and copy its mother’s number, plus 1, and paint that on the offspring’s forehead.

    At any given time there will be a distribution of the forehead numbers. After a time this distribution will stabilize, except that its mean will increase linearly through time. The time it then takes to for the mean increase by 1 is the length of a (maternal) generation.

    Alternatively we could copy the fathers’ numbers, plus 1. Then we get the length of a paternal generation.

    It turns out that this definition makes the generation time the average age of everyone’s mother (father) when they were born. I should note that this is consistent with the second of the three definitions given in the Wikipedia article on “Generation time”.

    Under this definition, with N M&M’s or Smarties, the time it takes to replace N of them turns out to be one generation. It is easy to show that replacing 2 M&M’s/Smarties increases the population mean of the generation number by 2/N.

  27. Piltdown2: As has already been stated and stipulated, most mutations are initially neutral.That being said, I maintain that once mutations accrue to the point that they are either beneficial or harmful, they are more likely to be harmful.And I accept that these harmful mutations would then be less likely to be fixed in the population.
    In regards to the Lenski experiment, the populations of bacteria were forced into an unnatural environment where they did not ordinarily do well.After thousands of generations, several developed simple mutations that gave them the ability to thrive in this unnatural environment.If placed back into a regular natural e.coli environment, which do you think would do better – the original unmutated bacteria or the “new and improved” mutated version?If this were tested, I believe the original unmutated bacteria would fare better.

    That is correct. Just like if we threw a dolphin back onto dryland and compared it to it’s terrestrial ancestor (supposing we could go back in time), the dolphin would make a terrible terrestrial mammal and would die rather quickly. That’s how evolution works. Things change, they don’t become perfectly adapted to all environments simultaneously. Becoming well adapted for one thing usually entails being worse off in another thing. Polar bears are best around the polar regions too.

    So yes, absolutely, moving an organism from one environment to which it is well adapted, into another different environment and giving it time to adapt, will usually entail that if you move it back to the first environment again, it will be worse off there compared to it’s ancestors. This isn’t an argument against evolution, this is how the process works.

  28. Joe Felsenstein:
    A comment on what a the length in time of a “generation” is.Here is my way of defining it.I had to work this out back 42 years ago when I worked on effective population number in a population with overlapping generations.

    Suppose we have a population, reproducing somehow. We paint a number 0 on everyone’s forehead.Then we go forward in time with the population reproducing itself.As each individual is born, take its mother (or designate one parent as the mother) and copy its mother’s number, plus 1, and paint that on the offspring’s forehead.

    At any given time there will be a distribution of the forehead numbers.After a time this distribution will stabilize, except that its mean will increase linearly through time.The time it then takes to for the mean increase by 1 is the length of a (maternal) generation.

    Alternatively we could copy the fathers’ numbers, plus 1.Then we get the length of a paternal generation.

    It turns out that this definition makes the generation time the average age of everyone’s mother (father) when they were born. I should note that this is consistent with the second of the three definitions given in the Wikipedia article on “Generation time”.

    Under this definition, withNM&M’s or Smarties, the time it takes to replace N of them turns out to be one generation.It is easy to show that replacing 2 M&M’s/Smarties increases the population mean of the generation number by 2/N.

    There is a population of all 4 blue M&M’s. A blue M&M gives birth to a black M&M and then promptly dies. So we now have 3 blue M&M’s and one black. One blue gets eaten and the black M&M has a baby which is also black. Another blue gets eaten, another baby M&M is born, and another blue is eaten and another black M&M is born.

    The entire population is now black. We can assume that each blue represents the offspring of another blue, and that it is not a blue grandfather having sex with a black infant. How many generations did it take?

  29. foxtraitor:
    Hilarious!

    Darwinists actually testing the details of their precious theory that they’ve known to be true all along.

    ahahahaha…

    I’m not quite sure what you mean. What is being “tested” (demonstrated) here is not Darwin’s theory, but the remarkable effect of neutral drift.

    In which there is no selection, only random death and reproduction, yet “all heads” reliably turns up.

  30. phoodoo: There is a population of all 4 blue M&M’s.A blue M&M gives birth to a black M&M and then promptly dies.So we now have 3 blue M&M’s and one black.One blue gets eaten and the black M&M has a baby which is also black.Another blue gets eaten, another baby M&M is born, and another blue is eaten and another black M&M is born.

    The entire population is now black.We can assume that each blue represents the offspring of another blue, and that it is not a blue grandfather having sex with a black infant.How many generations did it take?

    I’m not sure of your point, phoodoo. It’s very easy to model a scenario where you start with all one “colour”, and let them randomly replicate or die, but have the replication occasionally spawn an M&M of a different colour. What happens is that very rapidly, the colours are well-distributed through the population, and sometimes a new colour will come to dominate. This is without selection.

    If the population is small, from time to time all the M&Ms will be the same colour, but not the colour they started.

  31. Lizzie: I’m not sure of your point, phoodoo.It’s very easy to model a scenario where you start with all one “colour”, and let them randomly replicate or die, but have the replication occasionally spawn an M&M of a different colour.What happens is that very rapidly, the colours are well-distributed through the population, and sometimes a new colour will come to dominate.This is without selection.

    If the population is small, from time to time all the M&Ms will be the same colour, but not the colour they started.

    In the scenario I proposed, how many generations does that count as?

  32. phoodoo: In the scenario I proposed, how many generations does that count as?

    Well, at least one of your starting population had a grandchild, so three, I guess.

    But you can only really count generations precisely within one lineage – but you could work out the average number of generations per unit time. It doesn’t make a lot of sense in a population of four, though.

    In some simulations, all organisms replicate simultaneously, so it’s easy. I tend to a proportion replicating every iteration of the model, so it’s staggered, as in real life. But you can still average generation time.

    The youngest great grandparent I know personally is about my age. But I’m not even a grandparent yet!

  33. phoodoo,

    I have defined what I mean by one generation. So have Lizzie and Joe.

    How do you define a generation? Until we agree on what constitutes one generation, we can’t discuss how many generations it takes for a mutation to fixate.

  34. phoodoo,

    If the population is sufficiently large, the mean time-to-fixation of a novel mutation equates to 4N2 individual ‘evolutionary events’. Thus, one can ignore ‘what-is-a-generation’. Your dispute involves only a factor of 2 anyway; N is far the most significant term.

    It might matter if you wanted to count how many ancestors separate a particular copy in the fixed population from the original mutant ancestor. And it will differ for each one you take. As there is not a fixed number of generations for each individual anyway, and only a twofold difference in the mean if you choose deaths-or-births or deaths-and-births, not worth fretting about.

    Your point would appear to be that with inflated (realistic) population sizes, fixation of ancestry (the ‘novel mutation’) cannot happen. You forget the role of mutation rate. If a neutral mutation happens every 1000 individuals (say), then there will be 1 per ‘generation’ (qv) in a population of 1000, but 100 pg in a population of 100,000. Each individual mutation has only a 1 in 100,000 chance of ultimately becoming fixed, but there is a pool of 100 such mutations, rendering a probability that a mutation will be fixed as 100/100,000, or 1/1000 – exactly the same.

  35. phoodoo: There is a population of all 4 blue M&M’s. A blue M&M gives birth to a black M&M and then promptly dies. So we now have 3 blue M&M’s and one black. One blue gets eaten and the black M&M has a baby which is also black. Another blue gets eaten, another baby M&M is born, and another blue is eaten and another black M&M is born.

    The entire population is now black. We can assume that each blue represents the offspring of another blue, and that it is not a blue grandfather having sex with a black infant. How many generations did it take?

    Let’s use Joe’s definition to count generations.

    We start with all blue MMs having little zeros painted on their backs (the original generation). The population is
    blue 0, blue 0, blue 0, blue 0.

    One of them gets eaten and replaced with a black child, so now we have
    blue 0, blue 0, blue 0, black 1.

    At the next step, one more blue MM is eaten and black 1 produces child black 2:
    blue 0, blue 0, black 1, black 2.

    One more blue MM dies, black 1 produces another child, also black 2:
    blue 0, black 1, black 2, black 2.

    Finally, the last blue blue MM gets eaten and black 1 makes yet another black 2:
    black 1, black 2, black 2, black 2.

    The average generation number is now 1.75.

    We can implement a similar scenario of quick fixation with 1000 MMs. As each of the original blue MMs gets eaten, the mutant black 1 produces a child black 2. At the end of the day, we will have
    black 1, black 2, black 2, … black 2.

    The average generation will be close to 2.

    We can randomize which of the blacks give births. We can also consider longer scenarios. But first we need to hear from you: how do you define a generation?

  36. phoodoo: Well, Olegt made the point that you can take the number of your births replacing deaths exchanges (one round of eating an M&M and replacing an M&M), divide it by the size of your population, and that was the number of generations. That was clearly logical nonsense, (and as you have just stated you don’t agree with that formula) and yet, your normal cast of effervescent cheerleaders like Hotshoe and Thorton couldn’t resist cheering on Olegt as being so clever (you know he is a scientist and all, how dare you question him) because well, they just can’t be bothered to have a independent thought (or any thought at all from what I can tell).

    First of all, phoodoo, if what I say is “clearly nonsense,” you should be easily able to demonstrate that I am wrong. So far you haven’t. So perhaps it isn’t nonsense.

    Second, if you don’t like my definition, for the umpteenth time, give yours. Don’t throw temper tantrums.

  37. Piltdown2,

    Me: -In the event that a deleterious mutation did become fixed by drift, that is not the end of the story. The population has moved a little down the adaptive hill, which opens up the way for a beneficial mutation to take it back up again.

    Piltdown: I confess I do not have the same faith in beneficial mutations when all the variables of real life and complex organisms are added in,

    Yep, I kind of guessed that from your sig! 😉

    but I do acknowledge the possibility on a small scale.

    That’s a step, at least. The small scale must, in some way, provide a component to more complex processes at the large scale unless we can identify countervailing forces not accounted for in the model. It’s not just creationists that attempt to locate such countervailing processes. People want better models. This is simply an entry-level guide to the basics (and that, to be honest, is about as far as my population genetics stretches!).

    But it is important to be aware of this essential quality to birth and death: that they are sampling processes. The mathematics they entail is closely related to the mathematics of sampling. The idea that it applies to beans in a bag but somehow does not apply to ‘real’ populations can certainly be held without further support, but if one is looking to convince others …

    The deleterious/beneficial distinction is an important one, worth reiterating. If a population is poorly adapted, more of its random mutations will be beneficial, and fewer deleterious, than when it is well adapted. So one could look at a well-adapted population and convince oneself that the deleterious nature of nearly all its mutations disbars evolution. But when it was poorly-adapted, there was more scope. The population climbs the ladder then kicks it away. Even the mutation beneficial on the way up has become deleterious now that everything is ‘down’.

    Me: If, at the same time, it becomes large, the tenure of this fixed deleterious mutation is even more likely to be brief.

    Piltdown: By “even more likely to be brief”, I assume you are speaking of extinction. Or are you referring to some other mechanism that repairs the large mutation which has fixed in the population?

    Sorry, I confused matters with sloppy use of ‘large’. I meant the population (becomes large), not the mutation. As to extinction, then yes, but not necessarily of the entire population. I see fixation of A as extinction of not-A.

    If a deleterious mutation becomes fixed (by drift in a small population), then the proportion of random mutations that are potentially beneficial must grow, and that which is deleterious must reduce. The more deleterious mutations dominate against the rub of the green, the more this happens. But the population can do little with these mutations, because it is small and subject to drift. If it grows, benefit and detriment more clearly follow their sign, the beneficial mutations cause re-adaptation and the detrimental ones are purged. This thus provides a potential, limiting counter to the effects of drifting detriment. If a deleterious mutation has drifted to fixation, the back-mutation can still occur. It’s never secure on its perch.

  38. Joe Felsenstein: Suppose we have a population, reproducing somehow. We paint a number 0 on everyone’s forehead. Then we go forward in time with the population reproducing itself. As each individual is born, take its mother (or designate one parent as the mother) and copy its mother’s number, plus 1, and paint that on the offspring’s forehead.

    Under this definition, with N M&M’s or Smarties, the time it takes to replace N of them turns out to be one generation. It is easy to show that replacing 2 M&M’s/Smarties increases the population mean of the generation number by 2/N.

    If I read this correctly, Joe’s definition is equivalent to mine in the limit of large N.

    As phoodoo is still figuring his next move, I added Joe’s method of keeping track of generations to my code. Here is sample output for N=100.

    Run 181: {36, 31, 32, 1} at t=0; {100, 0, 0, 0} at t=6978; 64.69 generations.
    Run 182: {37, 27, 35, 1} at t=0; {100, 0, 0, 0} at t=4641; 49.08 generations.
    Run 183: {31, 34, 34, 1} at t=0; {0, 0, 100, 0} at t=21585; 191.13 generations.
    Run 184: {29, 39, 31, 1} at t=0; {0, 0, 0, 100} at t=9636; 89.28 generations.
    Run 185: {38, 30, 31, 1} at t=0; {0, 100, 0, 0} at t=10403; 104.03 generations.
    Run 186: {37, 32, 30, 1} at t=0; {0, 0, 100, 0} at t=4089; 37.28 generations.
    Run 187: {41, 29, 29, 1} at t=0; {0, 0, 100, 0} at t=3889; 44.43 generations.

    One can see that Joe’s definition and mine give rather similar generation counts. My recipe is to divide the number of deaths t by N. So, in run 181 my method yields 70 generations and Joe’s 65.
    Run 182: 46 and 49 generations.
    Run 183: 216 and 191.
    Run 184: 96 and 89.
    And so on.

    The agreement isn’t perfect, but as I said, the two methods only agree in the limit of large N.

    Here are numbers for N=1000:

    Run 1170: {344, 321, 334, 1} at t=0; {0, 0, 1000, 0} at t=145398; 141.961 generations.
    Run 1171: {318, 342, 339, 1} at t=0; {0, 0, 1000, 0} at t=2663251; 2707.75 generations.
    Run 1172: {332, 331, 336, 1} at t=0; {0, 0, 0, 1000} at t=1556280; 1524.76 generations.
    Run 1173: {326, 332, 341, 1} at t=0; {1000, 0, 0, 0} at t=299787; 316.458 generations.
    Run 1174: {360, 324, 315, 1} at t=0; {1000, 0, 0, 0} at t=301662; 317.315 generations.

    This time, the agreement is even closer. Dividing t by 1000 gives 145 generations in Run 1170, whereas painting numbers 142.
    Run 1171: 2663 and 2708.
    Run 1172: 1556 and 1525.
    And so on.

    So indeed Joe’s method of counting generations and mine give similar answers. Great minds think alike!

  39. phoodoo: So if I have four M&M’s and it takes two turns to go from a 50-50 population to 100% blue, that happened in 1/2 a generation?

    The example shows why the effective population is half the census population when using the Moran Model, as opposed to the Wright-Fisher Model, wherein every member of the population would have a chance to reproduce.

  40. phoodoo: No, I said a starting population of 1000.And then try adding in a virus which kills off thirty percent of the population randomly.

    Have you had a chance to review the data above? We would like your response.

  41. Allan Miller:
    Piltdown2,

    The small scale must, in some way, provide a component to more complex processes at the large scale unless we can identify countervailing forces not accounted for in the model.

    The small scale mutations found in the lab after several exhaustive studies (i.e. Lenski) point as much to the limits of evolutionary theory as they do to the possibilities. Just because we have not yet identified other forces or processes responsible for large scale changes that have occurred doesn’t mean we have to accept evolutionary theory as the cause. I think it is perfectly acceptable (and scientific) to say we are not yet sure how the large scale changes occurred.

    Sorry, I confused matters with sloppy use of ‘large’. I meant the population (becomes large), not the mutation. As to extinction, then yes, but not necessarily of the entire population.I see fixation of A as extinction of not-A.

    Thanks for clarifying my misinterpretation here.

  42. I think it is perfectly acceptable (and scientific) to say we are not yet sure how the large scale changes occurred.

    I’m waiting until a few complete revolutions of Neptune and Pluto have actually been observed before I accept orbiting the sun as an explanation for their behavior.

    Because I’m a skeptic.

    And because micro-orbits don’t necessarily imply macro-orbits.

  43. Rumraket: That is correct. Just like if we threw a dolphin back onto dryland and compared it to it’s terrestrial ancestor (supposing we could go back in time), the dolphin would make a terrible terrestrial mammal and would die rather quickly. That’s how evolution works. Things change, they don’t become perfectly adapted to all environments simultaneously. Becoming well adapted for one thing usually entails being worse off in another thing. Polar bears are best around the polar regions too.

    So yes, absolutely, moving an organism from one environment to which it is well adapted, into another different environment and giving it time to adapt, will usually entail that if you move it back to the first environment again, it will be worse off there compared to it’s ancestors. This isn’t an argument against evolution, this is how the process works.

    The dolphin story is quite a stretch from bacteria grown in a lab, or M&Ms mixed up in a bag :). Although you seem to be fairly confident this is how it all works, I think there have to be other forces or processes at work here to get from simple demonstrated mutations to changes in whole body plans.

  44. Allan Miller:
    Joe Felsenstein,

    That link appears to be to the European, or lesser, Smartie.

    http://en.wikipedia.org/wiki/Smarties_Candy_Company

    Yes, but I suggest that “lesser” is a misnomer. I was trying to make any Europeans who were reading understand what we were talking about. I guess I am out of date — M&M’s appear to be available everywhere (and Elizabeth, who is in UK, chose them in her example). But Nestlé’s Smarties are sold in many countries, just not in the U.S. because the name is already taken here. I suggest that the Smarties sold in the U.S. are the ones that should be called “lesser” Smarties, because they (a) sell less overall, worldwide and (b) aren’t as tasty as international (Nestlé) Smarties, because they don’t even contain chocolate.

    Now if we had an example involving both M&M’s and the international Smarties, the two would distribute themselves nonrandomly in the bag;. As has been noted, that would lead to natural selection.

  45. petrushka: I’m waiting until a few complete revolutions of Neptune and Pluto have actually been observed before I accept orbiting the sun as an explanation for their behavior.

    Because I’m a skeptic.

    And because micro-orbits don’t necessarily imply macro-orbits.

    Just keep in mind, those arguing that planets orbited the sun were challenging the scientific consensus of the day. Other than circular logic and minor small scale variation in species, what is the evidence for macro-evolution?

  46. I think there have to be other forces or processes at work here to get from simple demonstrated mutations to changes in whole body plans.

    Why?

  47. olegt:

    One can see that Joe’s definition and mine give rather similar generation counts. My recipe is to divide the number of deaths t by N.

    If you count the number of M&M’s/(Nestlé)Smarties drawn from the bag, you will be in agreement with the definition I am using. The difference is, I think, that when you draw a piece and replace it with a copy of itself, you don’t count a death. I do. Am I right about that?

    PS I don’t think “my” definition is only mine. An equivalent definition is given by the Wikipedia page on “generation time”. They cite it to Brian Charlesworth’s great monograph on the population genetics of models with overlapping generations, I suspect demographers had already discussed an equivalent notion. My illustration of assigning generation numbers to newborns is what is mine. Although I do not actually use that illustration in any of my published papers.

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