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.
Yes, I’ve modelled just that, in MatLab. In fact it’s one of the things that made me realise that there’s huge value in the steady drip of new mutations into the pool, even if they are all neutral – keeps the population robust to change.
Oops! I failed to follow the experimental procedure at this step. As the add says, you can’t eat just one!
Seriously though, I didn’t know this, so like Lizzie I put together a simple program to verify this. Here’s a typical run:
1: 29
0: 26
3: 26
2: 19
— 1000
1: 50
3: 29
0: 21
— 2000
0: 57
3: 25
1: 18
— 3000
0: 70
3: 20
1: 10
— 4000
0: 53
3: 47
— 5000
3: 32
0: 68
— 6000
0: 94
3: 6
— 7000
0: 100
(The population was 100 with 4 uniformly distributed alleles initially, numbered 0 through 3. The numbers after ‘:’ are populations, the numbers after ‘—‘ are iterations.)
{IDiot Mode} But M&Ms are designed!! ID is proven!! {/IDiot Mode}
Here’s one of my runs:
eta: damn it won’t load
eta2
Or you too can do your own simulation. Program here. Only the two colors are called “alleles” and half of the number of M&M’s in the bag is the “population size”.
Actually, my program models the Wright-Fisher discrete-generations model, in which at each step the whole population is replaced, while Lizzie’s M&M process models a haploid population undergoing a Moran model (in which at each step one haploid individual dies and one of the remailing ones reproduces.
They will behave almost exactly the same except for a factor or 2 faster genetic drift in the Moran model, if you count enough steps to each as many M&M’s as were in the bag as one generation.
Shame that round M&Ms were beyond me!
It’s certainly a lovely example of something you can predict with certainty that it will happen eventually, but you can’t predict which outcome of N outcomes will occur, nor how long it will take (though you can make a good guess, and I’m guessing it will have a poisson distribution).
clear
f = getframe;
f.colormap=’jet’;
Sides=8;
N=Sides^2;
Bag=randsample(1:5,N,true)’;
Bag=reshape(Bag,Sides,Sides);
k=1;
while std(reshape(Bag,N,1))>0
k=k+1
Eaten=randsample(1:N,2,true);
Doubled=randsample(1:N,2,true);
Bag(Eaten)=Bag(Doubled);
figure(1)
image(Bag,’cdatamapping’, ‘scaled’)
caxis([1 5])
drawnow
f(k) = getframe;
end
Time to fixation is proportional to the size of the population, so fixation will occur quickly in a small population, but net change will occur only slowly in a large population.
The origin of the analysis was the Hardy-Weinberg Principle (1908), which showed that variants will remain in equilibrium only if certain conditions are met. One of those conditions is an infinite population. Consequently, a finite population of variants is not {generally} stable.
Ah, your facility with chocolates is amazing!
Also I can’t get the gif to loop for some reason.
eta done it!
Good old Gimp.
The distribution centres on expected Neutral Time to Fixation at 4N generations if the population is large enough (and diploid, ie N glued-together M&M pairs, with one ‘randomly’ selected from each of a pair for the reproduction step – 2N generations if each M&M is separately counted, ie haploid). Either way, I only need to know N to know how to wager.
[eta: though I’d also have to decide whether Wright-Fisher or Moran was in charge!]
What would the results be if there were more than one person doing the sampling (and the samplers were not communicating with each other)?
timothya,
If the two (or more) people were each very quick in eating and replacing, and they accessed the bag at slightly different times, the result would be no different than one person accessing the bag the same total number of times.
A generation would be N events, whether they were done by one person or more,
I need to attach real numbers to discussions such as this to get a feel for things.
Feel free to correct any mistakes. ( not that I actually need to say that)
Larry Moran posted on Sandwalk a while ago that every person has about 100-150 genetic changes relative to their parents and I think this was determined by direct sequencing. So, for an animal species with about the same size genome, a total population of 10 million, and a generation time of about 10 years that means there will be about 150 billion mutations (alleles) to sort through every century. I’ll put off considerations of ‘junk DNA’ for now.
Even if 1/10,000,000 of these are fixed that seems like a lot to work with! I believe the equations on fixation rate assume the entire species of 10 miilion is in Hardy-Weinberg equilibrium. It seems to me more likely that the population will be effectively partitioned into many smaller populations. This shouldnt effect the chances of an allele becoming fixed in the species but it might mean that slightly-deleterious mutations might hang around longer and contribute to genetic diversity.
My intuition is bad on these things so I’m just throwing this out there!
Joe
You may be right, but I don’t see why two independent samplers would necessarily be “fixing” the same colour.
Absent selection, the rate of fixation is equal to the rate of novel mutation. So if there are 150 novel mutations per individual per generation, then there will be 150 fixed each generation. The other one hundred forty-nine billion nine hundred ninety-nine million, nine hundred ninety-nine thousand, eight hundred fifty mutations make up the genetic diversity of the population.
Ah. The point is that the single sampler isn’t fixing any one colour.
If the sampler picked two M&Ms each time, ate the one she fancied, and put the other one back with a match, then we’d have biased sampling – darwinian adaptation: the survivor breeds.
But this person doesn’t do that. She picks one M&M and eats it, creating a vacancy. Then she picks another and puts it back, with a match (offspring). Rinse and repeat. There’s no selection.
In my implementation I simply took one M&M at random, deleted it, and then took a second one at random, and put a copy in the first one’s place. Rinse and repeat.
Even without selection, eventually, all the M&Ms in the bag are the same colour as in my gif.
Watch it carefully. There is no selection.
You may have misunderstood the experiment. There is no selection. You eat whatever color candy you pull from the bag. It makes no difference how many people take turns.
With the graphic example above, there are 64 M&Ms in the bag. You can expect the game to take about 64*64*2 = 8192 turns. There are three calories in a chocolate M&M, ten for a peanut M&M. You can burn five calories by walking briskly for one minute. Add the nought and carry the ought, you just might want to play with friends.
Hmm. That’s a bit off. Our quick simulation took an average of 3544 turns over a hundred runs. That may be because the population is fairly small. (Also, the simulation only used four colors, instead of five as in Lizzie’s example.)
Of course. The 2N (haploid) assumes that the initial frequency of the variant is negligible compared to population size. In this case, each color starts as a significant portion of the population.
It took 748, with five colours.
Which was longer than some.
2666
1185
2893
1541
1444
2757
1054
2809
3777
1727
894
1011
1642
1441
2490
3192
1044
990
1870
1017
2407
2184
2173
1909
4276
1292
1436
1324
1874
765
908
1526
570
1610
1944
760
1673
1246
1731
1096
578
738
2264
2196
1376
1156
1331
2911
2154
1764
3678
3051
982
1324
2324
1962
968
2165
2356
2822
1167
714
1641
2048
1273
1247
1346
2748
1034
1141
1443
2785
1276
701
728
1370
888
1470
3135
1752
1090
3580
1289
730
2726
3059
2030
1571
1537
2763
1383
3397
2575
1855
1452
1016
610
1629
1518
1256
A Poisson, as of course it should be.
Yep, in mine I started with an equiprobable random draw from the five colours.
Zachriel,
The rate of substitution equals the rate of mutation for a population in equilibrium, which is certainly not true for the human population; our population expanded massively much too recently to be in equilibrium. It takes a long time for rare alleles to drift up to the point of fixation, a very long time when the population is large. At this point new mutations are accumulating much faster in the population than they are fixing. Since it would take many billions of years for the allele frequency distribution to reach equilibrium, that situation isn’t going to change.
(Also, there are roughly 75 new mutations per birth, not 150.)
Because they aren’t independent. They are drawing from the same bag. Whatever one does to the bag, that is the starting point for the next person who draws from the bag (and replaces). If they each had a separate bag then the stochastic processes are indeed independent, and they are carrying out independent replicates.
I get, for a bag of 64 M&M’s (or Smarties), that the average fixation time should be 64 x 64, so 4096. The logic is that (1) each generation is 64 draws. (2) The population is haploid with N = 64, and (3) it is also true that, as this is a Moran model, the effective population size is half the population size. In a haploid population the time to coalescence of a largish sample is near 2N*N for a Wright-Fisher model, but half that for a Moran model. So N*N would be close. The coalescent is approximate and is most valid when the sample size is much smaller than the population size, so the use of the coalescence time will be approximate. Still, 4096 is much closer to Lizzie’s simulation results.
New mutations are rarely, if ever, beneficial in real-life experiments I’m familiar with. Take the fruit fly experiments (http://www.unc.edu/depts/our/hhmi/hhmi-ft_learning_modules/fruitflymodule/phenotypes.html): mutations result in either a dead fruit fly, a deformed fruit fly, or an apparently unaffected fruit fly. So the “even if they are all neutral” phrase is quite optimistic, not a concession. I do agree that maintaining diversity, not mutations, in the gene pool is beneficial for the population, so this m&m experiment is interesting since it shows a natural tendency against diversity, at least for asexual reproduction.
hmm. Can you tell what I’ve done from my script?
Overlapping generations explains the difference. Thanks.
Your average is 1752, which is about half of ours. Notice that there seem to be two changes between frames of your gif, though.
Reading Lizzie’s code more carefully, I see that
(1) she chooses two M&Ms from the bag, and records their colors and returns them to the bag. Then
(2) she chooses two from the bag. She replaces those two by ones of the colors that she chose the previous time.
So 32 of these events will be one generation, and as her process is still close to (but not identical to) a haploid Moran model with N=64, we expect not 4096 events until fixation but about half that, 2048, since it will be about 32*64. The 100 runs she posted took an average of 1752.21.
BTW, there is no expectation that the distributions of times is Poisson.
Huh? No, you can’t use a children’s page teaching about types of mutant flies deliberately bred by scientists as evidence to reject the fact that nearly all mutations are neutral.
That most mutations are not harmful is not news. Hasn’t been news for 50 years.
What do you think “diversity” means? It means a pool of mutations! Some of that pool of diverse mutations produce observable effects, some don’t. The mutations not currently producing a visible beneficial effect may be as important as the ones that do, because as Lizzie points out, they make the population more “robust to change”.
I don’t think you know what any of the words “natural tendency against diversity” mean. Certainly they don’t mean what you think they mean when you string them together.
As individual “neutral” mutations build up, a point can be reached where they are no longer neutral. My contention is that at that time the mutations are much more likely to be harmful than beneficial to the organism.
Is it possible the population starts with a built-in amount of beneficial diversity?
In your world, are all mutations either neutral or beneficial?
From the OP: 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. So what started as a bag of M&M’s of several different colors became a bag of all one color M&M’s. How is this not less diverse? Of course this is only the case when no variation is introduced, but again, my contention is that any new variation (mutation), although possibly/probably neutral at first, would eventually be more likely to be harmful than beneficial.
Piltdown2,
I don’t think you have any particularly strong basis for that supposition. Accelerated mutagenesis in lab populations can’t easily be extrapolated to disbar evolution in the entirety of life.
If a new mutation is deleterious, that is in respect of the other alleles in the population. And the consequence is that it has less than the baseline 1/N chance of being fixed. So such mutations are more likely to be lost. Neutral alleles that become deleterious likewise have an increased tendency to elimination. Even beneficial alleles can be lost. But you are rounding your expectation up to 100%. Because many alleles are not beneficial, and many beneficial alleles are lost, this does not mean that no beneficial mutations will become fixed. Continued adaptation in a fixed environment requires a constant input of beneficial mutations, which requires only that the proportion be non-zero. There will come a point when no more adaptation is available in that fixed environment, and the proportion of beneficial mutations will drop to zero. But environments aren’t fixed. Not least because there is an ecosphere of adapting organisms. Populations track adaptive peaks, but they switch around, so adaptation is more like a kitten chasing a torch beam than an army entrenched.
Piltdown2,
In that simple model, sex (manifest only as diploidy) makes no difference. You could have either N haploid M&M’s or N/2 diploid pairs. The contribution of sex to diversity comes with reciprocal recombination between (or within) alleles, creating many more genotypes for the same N. But even a sexual population would lose its diversity without new mutation. It would become equivalent to a clone, or a highly inbred line.
The ‘loss of diversity’ point is an important one. Finite populations lose allelic diversity. This is true whether alleles are under selection/drift or drift alone – the latter is just slower. Only novel mutations restore it. If you wanted to deny the possibility that new mutations ever become fixed, you’d need to address that matter.
In the Lenski long term evolution experiment, hundreds of beneficial mutations were fixed. Hundreds. It doesn’t matter that the proportion of deleterious to beneficial is very large, since carriers of deleterious mutations more frequently die. You don’t get an accumulation of deleterious mutations, you get dead carriers. What accumulates is neutral and beneficial ones.
If what piltdown2 imagines was really true, the outcome of Lenski’s experiment should be impossible. Yet after 50.000 generations, we have a population with significantly superior fitness compared to the ancestral one and over 600 mutations have fixed in the population. Many of those directly contributed to an increase in fitness, i.e. they are beneficial mutations.
That some of you think that this is even remotely close to being analogous to what happens in life is just beyond ridiculous. You people actually believe you are being scientific.
There are so many problems with your model its overwhelming, but let’s just start with the most obvious. Improve your model so first, you have an assortment of many colors, say ten, and then add in only ONE black one into your little bag of 1000 MM’s. Now run your little experiment and see how long it takes for your new novel mutation of a black one, to become an entire bag of only black ones.
But let’s improve it even more. Every once in a while throw in a disease that wipes out 30% of your population at random instantly. Then throw in a lion attack that eats 10% of your lightest colored MM,s and 10 % of your darkest colored MM’s. Then every once in a while introduce a brand new color into your mix, but it must be a color which causes a war between the blue colors the red colors and the black colors.
Then finally give a new mutation to all the black ones that not only let’s it be black, but also makes it slower, so that it is often the first one picked when deciding which one to eat.
Go ahead, run your program again, now let’s see how long it takes for the whole bag to be black.
This is precisely the reason so many simple minded people think that evolution must be true, because they are so persuaded by bad science without second guessing anything that they think oh it sounds reasonable to me. Its the exact opposite of skepticism.
I don’t think you have the details of the Lenski experiment down very well. Tell me about some of these beneficial mutations that you think are novel, caused an increase in fitness, and got fixed into the population.
I don’t think this simulation was meant to represent evolution. It was just given as an insight. Even sophisticated models cannot simulate evolution as there are too many variables for which we don’t know the probability distribution.
But insight into what? How a neutral mutation can become fixed? So then why not include the factors I mentioned.
This is the problem with all these BS models, they all are designed to give the easiest chance of success possible, and still claim to show something. Its the same with these so called models of eye evolution. If you want to take out every variable which would make it hard to achieve, but which is a part of reality,of course you will get the result you want.
phoodoo,
Uh-huh.
This will happen on average in every 1000 replicates of the experiment. But the number of times a single added mutation becomes fixed depends upon the number of mutations that occur, so you need to factor that in too. You can imagine situations where nothing much happens, but this illustration is breaking things down so one can focus upon the detail. Reality is much more complex. Well spotted. But it is not, in every respect, ‘more complex’ in the direction of nothing ever happening, which is clearly your fervent wish.
What is important to note is that this process is one that fixes ancestry. In all of the replicates that do not result in fixation of ‘black’, a different individual becomes the ancestor of all. Rather than colours, you could individually number every square. For each individual, the baseline chance is 1/N. There is nothing more remarkable about the fixation of black than the fixation of any other individual square. But one will fix, depend upon it.
OK, you have population bottlenecks. Simply a few more rounds of the ‘death’ process.
Indeed. I noted the role of selection – in the experiment, I suggested making some colours larger or heavier, but a lion would serve just as well.
Struggling to think of a biological equivalent, but … do go on.
Yes. Selection again. Black is deleterious, which gives a premium to non-black.
Longer and less likely, of course. A mutation that is selected against indeed has less than the ‘neutral’ baseline chance of 1/N. Are you saying that ALL mutations are ‘black? On what grounds?
Simple minded people? Well, I wouldn’t want to accuse you of underthinking the issue.
phoodoo,
Don’t like the MM model? Come up with yours. Code it. Share the results.
Join in the fun!
phoodoo,
The point is to break the process – population resampling – into its simplest form. Even that has consequences. These consequences are inescapable, unless selection acts to counter them. But selection more typically simply brings them on more quickly. By building upon this basic process, one can get a better handle on how the evolutionary process works.
The original point was not to ‘prove evolution’ but to show the relationship between selection, drift and randomness in a simple, graphic manner (without the graphic – thanks Lizzie!). Even the baseline neutral process is evolutionary, and mathematically supportable. How does it happen in simple models but not in the wild? That’s the point of model building. Add in factors and see if the basic process is submerged. In your previous post the posited factors did not succeed.
Then do it like you think would happen in the wild. You have a population of 5000 individuals, none with the coloration of black. Through one into your little game, and let’s see how long it takes for black to be in your entire bag. We don’t even need to make it complicated, if you put in all the variables that happen in real life black will never be the only color in the bag, but heck just ADD one piece of reality and it won’t be. Why are you starting with all the colors in equal numbers to begin with, that is already a silly proposition. So start with just one. Try it, and show us the results.
RodW,
That’s probably the case. One thing that bugs me about simple pop-gen models is that they frequently scale N independently of anything else, and so retain the assumption of random assortment in the teeth of something that clearly acts against it – the geometric difficulty of keeping larger and larger populations effectively mixed. (There are models that deal with this, naturally, but since I start blinking at equations with more than 2 terms in ’em, I leave that to the experts! 🙂 ).
A population with sluggish mixing effectively divides into a set of smaller populations, so drift is more dominant at this local level, but with migration and introgression serving to distribute genes more widely. Lizzie’s model plonks newborns in an available space on the grid, but the 64 grid could be subdivided – ‘demes’ 2×32, 4×16, 8 x 8 etc. We’d see more rapid fixation in each region, and then these ‘bridgeheads’ would suffer incursions as the segments exchange genes. This localises diversity – there is less diversity in a given geographic region. I think, though, that the overall variation would diminish at the same rate. A ‘real’ population geneticist may be able to confirm or deny.
We may have misread RodW’s question. You make a very good point, though.
Speaking as a lay reader of this forum, I have to say I get a kick from the fun the experts seem to be having in exploring this simple model.
More seriously, the recent series of entries on chance started, I think, with comments on null hypothesis testing. I thought I recalled reading comments somewhere on drift as the null hypothesis when testing for the effects of selection.
It was Larry Moran’s blog:
Null Hypothesis in Evolution
phoodoo,
For fixed alleles, yes it will. Are you saying that every locus is polymorphic? Selection or neutrality, all fix eventually (ignoring frequency dependent effects, the exception not the rule, and – obviously – further mutation and change in the selective milieu. Also known as evolution.).
I’m not. I’m starting with a fresh unopened packet of M&M’s. Lizzie’s simulation started with equal numbers. Had it started with any other frequencies, you’d have jumped on that instead! It simply does not matter. p(fixation) = current frequency. If current frequency is 1/N (new mutation), p(fixation) = 1/N. If 0.25, then 0.25. etc. You pick the frequencies. And the best way to do that would be to write your own version and play with the parameters. You seem to think that nobody has actually put anything more into this than you see on this page.
As I said, you can shortcut this fixation (heh) with colours by simply individually numbering the members of your starting population. It’s gone up to 5000 now? OK. One number will become fixed. Guaranteed. You try it. If 4757 happens to be black, then once every 5000 replicates (on average) 4757 (ie black) will be the only number left. The common ancestor.