Natural Clones in the Republic/Empire

[Surlethe]

Given that there is a limited (albeit large) amount of information in the human genome, what are the chances that someone in the Star Wars galaxy has an identical twin walking around simply because there's not enough genetic information to make everyone unique? Since there's between 2e15 and 2e21 people in the SW galaxy, is such unintentional "cloning" possible?

[Isolder74]

Highly unlikely even on that large of a scale. Unless it was a Twin seperated at birth I'd highly doubt this could be possible.

[King Kong]

Assuming that SW humans and we are (basically) the same genetically (they have about 3 billion base pairs and individuals differ in their genomes by about 0.01%), then there are 4^300000 different combinations of base pairs in the changeable parts of the genome that you could make.

Of course, some of these mutations might prove to be fatal, but Jesus Christ, that's a huge number. Not gonna happen.

If the assumptions (or methodology) I made are wrong, feel free to point them out. But I highly doubt that any result will be small enough to have unintentional identical twins, even within the huge Star Wars population.

Out of curiosity, could there have been identical twins when we take into account all the people who have lived in the Star Wars galaxy ever? How many people have lived in the SW universe? Or how long would it take for there to be an unintentional identical twin?

[Isolder74]

Well it would be a massive number.

TOTAL POSSIBLE COMBINATIONS

N^number of permutations.

nCr where order matters

n!1!/(n-1)!n!

I think that's the formula but I'd have to look it up but it is going to be a huge change in the negative.

[Surlethe]

Assuming that SW humans and we are (basically) the same genetically (they have about 3 billion base pairs and individuals differ in their genomes by about 0.01%), then there are 4^300000 different combinations of base pairs in the changeable parts of the genome that you could make.

Of course, some of these mutations might prove to be fatal, but Jesus Christ, that's a huge number. Not gonna happen.

Okay, thanks; I wasn't sure how to calculate the total number of possible base pairs. On a completely unrelated note, I was trying to google-calculator the ratio of 4^3e6 and 5e21, and the calculator broke; the only links showing up were to DragonBall Z help forums. That is a really, really, big number.

If the assumptions (or methodology) I made are wrong, feel free to point them out. But I highly doubt that any result will be small enough to have unintentional identical twins, even within the huge Star Wars population.

I wonder if some of the universes larger than SW-verse (e.g., Xeelee-verse, Culture-verse, Darwinia-verse) would approach 4^3e6 people in a larger civilization?

Out of curiosity, could there have been identical twins when we take into account all the people who have lived in the Star Wars galaxy ever? How many people have lived in the SW universe? Or how long would it take for there to be an unintentional identical twin?

Hmm. To do this, one would have to calculate the total number of humans throughout human history in the SW-verse (duh). For a crude, lower estimate, there have been a thousand generations in the Republic, and if each of those generations has had 1e20 people, then we have a good 1e23 people throughout the history of the Republic. Of course, this isn't taking into account prehistory; however, the fact humans have been able to influence evolution means they must have been ubiquitous in the galaxy for nearly geological timeframes before the founding of the Republic, so 1e23 is a very low guess. I wonder what the population logistics curve would look like.

[Surlethe]

Well it would be a massive number.

TOTAL POSSIBLE COMBINATIONS

N^number of permutations.

nCr where order matters

n!1!/(n-1)!n!

I think that's the formula but I'd have to look it up but it is going to be a huge change in the negative.

Actually, C(n,r) = n!/(n-r)!r!. It's the total number of r-element subsets from an n-element set -- so, order doesn't matter. When order matters, you want to turn to permutations [P(n,r) = n!/(n-r)! -- the total number of r-element subsets out of an n-element set such that order does matter]; however, the question of how many combinations are in the genome requires neither C(n,r) nor P(n,r), because both of those assume you're irreplaceably choosing r elements out of a finite set with cardinality n, and that isn't what we're doing. Put more explicitly, King Kong's argument is as follows:

There are four possible base chemicals.
There are three billion locations in the genome to which one must assign one of the four base chemicals; multiply three billion by 0.0001 because that's the different places people will be different.
Thus:

Code: Select all

4*4*4*4* ... <three hundred thousand times> ... *4*4*4
^ ^ ^ ^
Each of these fours represents a given location out of the three million

and this gives the total number of possible combinations in the genome. Does that make sense?

Thanks to King Kong for the corrections.

[King Kong]

I wonder if some of the universes larger than SW-verse (e.g., Xeelee-verse, Culture-verse, Darwinia-verse) would approach 4^3e6 people in a larger civilization?

The only one that I'm familiar with is the Xeelee-verse, and only vaguely. I believe humans had colonized the entire universe, not just the observable portion, which would make the population mind-numbingly huge. Of course, there would be pretty low population densities, but the Xeelee-verse (from what I know of it) has a good chance to produce an unitentional genetic twin.

[Surlethe]

I wonder if some of the universes larger than SW-verse (e.g., Xeelee-verse, Culture-verse, Darwinia-verse) would approach 4^3e6 people in a larger civilization?

The only one that I'm familiar with is the Xeelee-verse, and only vaguely. I believe humans had colonized the entire universe, not just the observable portion, which would make the population mind-numbingly huge. Of course, there would be pretty low population densities, but the Xeelee-verse (from what I know of it) has a good chance to produce an unitentional genetic twin.

Wouldn't it also limit communication -- and, thus, genetic transfer -- between different portions, causing geographic speciation?

In any case, human evolution occurred over the course of about four million years, so if humans have been ubiquitous in the SW galaxy for some four million years (long enough to influence evolution toward pseudohumans; I'm making a stab in the dark, since I've no idea how long it would take for pseudohumans like Xizor's race to evolve), then we ought to have some 160,000 generations; this, in turn, ups our population estimate another two or three orders of magnitude, at best (since the human population density was probably much lower before the Republic).

[King Kong]

Wouldn't it also limit communication -- and, thus, genetic transfer -- between different portions, causing geographic speciation?

I'm pretty sure they must have had some sort of insane communication and transport technologies; humanity based all of its technological innovation off of stolen Xeelee artifacts, who definitely had the technology necessary to communicate and travel from one end of the universe to the other.

The one example that I can recall was a supersymmetry drive, which was used to travel ~14 billion light years in ten days. And this drive was apparently discarded in favor of Xeelee devices that could move ships faster.

I can't recall of any natural evolution of humanity into separate sub-species in the Xeelee-verse, though there were humans designed to live in unique environments, such as on the surfaces of neutron stars.

But anyway, that's not what this thread is about.

In any case, human evolution occurred over the course of about four million years, so if humans have been ubiquitous in the SW galaxy for some four million years (long enough to influence evolution toward pseudohumans; I'm making a stab in the dark, since I've no idea how long it would take for pseudohumans like Xizor's race to evolve), then we ought to have some 160,000 generations; this, in turn, ups our population estimate another two or three orders of magnitude, at best (since the human population density was probably much lower before the Republic).

It looks like humans would have to exist (at their present population sizes) until long after the universe becomes unbearably cold and all the black holes evaporate before an unintentional identical twin would be born.

I'm pretty sure that the entire SW population would have to engage in physically impossible amounts of rutting to produce an unintentional genetic twin before life in the universe becomes impossible for humans.

This is of course idle speculation and I have no numbers to back it up.

[King Kong]

I'm pretty sure that the entire SW population would have to engage in physically impossible amounts of rutting to produce an unintentional genetic twin before life in the universe becomes impossible for humans.

Strike that, I underestimated the human capacity to rut (a fun word).

Assuming an initial population of 1x10^21 reproductive individuals, doubling in size every generation by producing exactly two girls and two boys, and with a generation time of 20 years, it would be possible to produce a large enough population so that there would be an unintentional clone in around 24 million years.

Of course, this is all very hand-wavy and assumes an unnaturally ideal scenario (even if I got the initial necessary population estimate right), but the point is that a large, (relatively) stable, and expansive population (like the SW universe) could produce an unintentional clone in a few tens of millions of years.

[Kuroneko]

[Edit: I need to rethink this. I'll repost after dinner.]

[Isolder74]

Well it would be a massive number.

TOTAL POSSIBLE COMBINATIONS

N^number of permutations.

nCr where order matters

n!1!/(n-1)!n!

I think that's the formula but I'd have to look it up but it is going to be a huge change in the negative.

Actually, C(n,r) = n!/(n-r)!r!. It's the total number of r-element subsets from an n-element set -- so, order doesn't matter. When order matters, you want to turn to permutations [P(n,r) = n!/(n-r)! -- the total number of r-element subsets out of an n-element set such that order does matter]; however, the question of how many combinations are in the genome requires neither C(n,r) nor P(n,r), because both of those assume you're irreplaceably choosing r elements out of a finite set with cardinality n, and that isn't what we're doing. Put more explicitly, King Kong's argument is as follows:

There are four possible base chemicals.
There are three million locations in the genome to which one must assign one of the four base chemicals.
Thus:

Code: Select all

4*4*4*4* ... <three million times> ... *4*4*4
^ ^ ^ ^
Each of these fours represents a given location out of the three million

and this gives the total number of possible combinations in the genome. Does that make sense?

That is the formula I was looking for thanks.

Now since we are asking out of n possible now many will give us X combination. We would use N (your 4^3 million), into the nPr formula X 2(we want two of the same combination) so N!/(N-r)! R I think would be 3 million, That breaks my calculator the number is so huge....

(4^3 million!/(4^3 million - 3 million)!) * 2

That's the chance of having at random two unrelated Twins not at the same time. not sure how to set that part up.

The Chance of a natural twin is much lower.

[Kuroneko]

On further consideration, I'll have to amend my previous answer. The occurence of random pseudoclones is, in fact, extremely likely in this scenario. If something needs clarification, feel free to ask.

If the total variability is N of roughly uniform distribution, then given population size n, the probability of every individual being distinct has the usual birthday paradox form P = [N!/(N-n)!]/N^n. Needless to say, actual genetic behavior will not give a uniform distribution, but this should serve as an order of magnitude for the order of magnitude. Applying the first-order formal Stirling approximation, x! ≅ sqrt[2π][x^{x+1/2}][e^{-x}], we have, for N≫n≫1 and ε = n/N, log P ≅ -[N-n+1/2]log[1-n/N] - n ≅ [N-n+1/2][ε + ε²/2 + ε³/3 + ...] - n ≅ -n[ε/2 + ε²/6 + ε³/12 + ε⁴/20 + O(ε⁵+ε/N)].

As a sample calculation, assume 3.0e4 genes with an average of seven alleles each, giving N = [3.0e4]^7. For n = 2e21, ε = 9.1e-11, ε/N is negligible compared to ε, as well as ε², so all terms after ε/2 can be ignored, giving so log P ≅ -[2e21][ε/2] = -9.1e10. Dividing by log(10), this gives log10(P) ≅ -4.0e10. Thus, these sorts of 'pseudoclones' are very likely. If the average number of alleles is 8, then log10(P) ≅ -1.3e6, i.e., pseudoclones are slightly less likely, but still overwhelming so.

As for King Kong's original assumption of N = 4^{3.0e5} = 10^{1.8e5} and n = 1.0e21, then ε is incalculably small, making the occurance of clones similarly small (log P ≅ 0 to extreme precision).

[Surlethe]

Of course, this is all very hand-wavy and assumes an unnaturally ideal scenario (even if I got the initial necessary population estimate right), but the point is that a large, (relatively) stable, and expansive population (like the SW universe) could produce an unintentional clone in a few tens of millions of years.

This appears true. However, there's no indication that the SW-verse has an expanding population of humans; rather, I would assume the galaxy has held a fairly level population throughout the Old Republic, at least, because there's no way the galaxy's logistics could support a population that doubled in size from, say, the end of the Clone Wars to the Battle of Endor.

EDIT: I should point out that the exponential growth function is used to describe uncontrolled population growth as an approximation to the first half of a logistics curve.

[Anguirus]

All this math is accurate as far as I know, but after the initial matches in human fertilization you still get after that crossing over and random mutation. You are just NEVER going to get this. It's incalcuable, I think.

[King Kong]

This appears true. However, there's no indication that the SW-verse has an expanding population of humans; rather, I would assume the galaxy has held a fairly level population throughout the Old Republic, at least, because there's no way the galaxy's logistics could support a population that doubled in size from, say, the end of the Clone Wars to the Battle of Endor.

Stupid of me. I meant to say something more along the lines of that it would be possible for SW universe humans to expand in this manner, but, as you pointed out, I have probably overestimated their logistical capacity. The fact that the SW civilization has remained at roughly the same population size for many thousands of years, without even expanding into easily explored regions of its own galaxy, indicates that there must be some logistics aspect holding back SW humans from expanding at a sufficient rate.

All this math is accurate as far as I know, but after the initial matches in human fertilization you still get after that crossing over and random mutation. You are just NEVER going to get this. It's incalcuable, I think.

There must be a finite number of different genomes that still create a fully functioning human being. It may be insanely large, but it must be finite. We're just trying to find the insanely large population size necessary so that there must be an unintentional clone, simply because there are no other possible genomes available.

[Surlethe]

Stupid of me. I meant to say something more along the lines of that it would be possible for SW universe humans to expand in this manner, but, as you pointed out, I have probably overestimated their logistical capacity. The fact that the SW civilization has remained at roughly the same population size for many thousands of years, without even expanding into easily explored regions of its own galaxy, indicates that there must be some logistics aspect holding back SW humans from expanding at a sufficient rate.

That makes sense. Actually, the existence of the Yuuzhan Vong may indicate that SW humans have seeded other galaxies, which may, in turn, imply that humans in the SW-verse are much more widespread than thought. However, because the SW galaxy has maintained isolation by simply not exploring beyond its boundaries, humans might actually have speciated elsewhere; it's conceivable that (assuming SW humans have colonized extragalactically in the past) other galaxies enjoy regular contact with each other, thus creating even larger pools of genetic resources.

What could possibly be holding the Republic back from colonizing those well-explored worlds? Lack of interest?

[Kuroneko]

There must be a finite number of different genomes that still create a fully functioning human being. It may be insanely large, but it must be finite. We're just trying to find the insanely large population size necessary so that there must be an unintentional clone, simply because there are no other possible genomes available.

If you still uphold your definition of human as 0.01% variability of nucleotides over three billion pairs, presumably compared with some standard, there is a variability of N = 4^{3e5}, then your problem is completely trivial--by the pigeonhole principle, the requisite number of humans n is equal to N+1. If you're looking for a specific probability p of of there being at least one random clone, then the population n is given implicitly by 1-p = [N!/(N-n)!]/N^n. It has no exact solution.

Edit:
Again, if the number of human genomes is N≫1, the minimal population n required for the probability of at least one random clone to be over 1/2 can be approximated using the formula found above: log 1/2 ≅ -n[ε/2], ε = n/N, giving n ≅ [2N log(2)]^{1/2}. For N = 4^{3.0e5}, this gives log n = [log 2 + log(log 2) + log N]/2 = 2.1e5, or n = 10^{9.0e4}. Is this the kind of result you wanted?

[Anguirus]

Ah, ok, I misunderstood what your initial calculations were based on.

[Surlethe]

If something needs clarification, feel free to ask.

If the total variability is N of roughly uniform distribution, then given population size n, the probability of every individual being distinct has the usual birthday paradox form P = [N!/(N-n)!]/N^n.

What's the "birthday paradox"?

Needless to say, actual genetic behavior will not give a uniform distribution, but this should serve as an order of magnitude for the order of magnitude. Applying the first-order formal Stirling approximation, x! ≅ sqrt[2π][x^{x+1/2}][e^{-x}], we have, for N≫n≫1 and ε = n/N, log P ≅ -[N-n+1/2]log[1-n/N] - n ≅ [N-n+1/2][ε + ε²/2 + ε³/3 + ...] - n ≅ -n[ε/2 + ε²/6 + ε³/12 + ε⁴/20 + O(ε⁵+ε/N)].

This is just a series of algebraic manipulations and approximations? I suppose I'll look up the Stirling Approximation -- why is it necessary to use an approximation for x! here?

As for King Kong's original assumption of N = 4^{3.0e5} = 10^{1.8e5} and n = 1.0e21, then ε is incalculably small, making the occurance of clones similarly small (log P ≅ 0 to extreme precision).

As you stated in your most recent post, correct?

[King Kong]

What could possibly be holding the Republic back from colonizing those well-explored worlds? Lack of interest?

It must be something similar to this. There seem to be no technological limitations that could prevent expansion into these areas, but I don't know if the Unknown Regions have been described as having unusual properties that have made expansion using hyperdrive and FTL communication prohibitively expensive. The only other options are human limitations: lack of interest, cultural taboos, or simple fear of the unknown. I'm not sure if any reason is given in the EU.

EDIT: The Star Wars Technical Commentaries explains that the Unknown Regions are not part of the galactic disk and would produce little economical benefits if they were properly explored and colonized.

Few of the complacent people of the greater galaxy would bother to stray from their tens-of-millennia-old trade routes to visit these spaces beyond the disk, because the distances are so vast and the destinations so scarce. Chemical considerations make planets unlikely or uncommon in globular clusters, even though these associations of millions of stars must have high abundances of interesting power sources like exotic stellar corpses. In total the halo would still contain millions of interesting destinations but because they're spread across space much larger than the disk, it wouldn't be economical to establish trade routes so far out. This zone is probably what constitutes the Unknown Regions and Wild Space.

Is this the kind of result you wanted?

I was just looking for the completely trivial N+1 solution, as the OP seemed to ask (to me, at least) if the population of the SW universe was large enough so that at least one unintentional clone would be inevitable. So I was simply finding the population necessary for this to happen, using the most basic possible means.

I appreciate the reminder of probability math, though. Summer turns my brain to mush.

[Surlethe]

I was just looking for the completely trivial N+1 solution, as the OP seemed to ask (to me, at least) if the population of the SW universe was large enough so that at least one unintentional clone would be inevitable.

Bingo. I was implicitly invoking the pigeonhole principle in my question. As a reminder (as if it's not obvious ... ), the pigeonhole principle declares (essentially) that function from a set with n+1 elements to a set with n elements can not be injective. In layman's terms, it states that if you're trying to fit n+1 objects into n slots, you're going to have to double up somewhere.

[Kuroneko]

If the total variability is N of roughly uniform distribution, then given population size n, the probability of every individual being distinct has the usual birthday paradox form P = [N!/(N-n)!]/N^n.

What's the "birthday paradox"?

If you have population size n each assigned an integer in [1,N] with uniform distribution, the probability that there is no repetition is can be computed by multiplying the probability that the second is distinct from the first, 1-1/N, by the probability that the third is distinct from both the first and the second, 1-2/N, ..., that the nth is distinct from all previous, 1-(n-1)/N. Thus, P = Prod{k,1,n-1}[ 1-k/N ] = [N!/(N-n)!]/N^n. The probability of repetion is 1-P. The so-called birthday paradox is one needs a random sample of only 23 people for the probability of some pair having the same birthday to be over 1/2 (N = 365 for simplicity)--some find this unexpectedly low, and so it is called a paradox.

This is just a series of algebraic manipulations and approximations? I suppose I'll look up the Stirling Approximation -- why is it necessary to use an approximation for x! here?

Yes, it is a Stirling approximation followed by a Maclaurin expansion. It is necessary here because it allows the calculation of N! for N = 2.2e31, or rather its logarithm, without specialized software. Something like MATLAB can calculate this, but there is great rounding error when the logarithms are subtracted for large N. This is unacceptable if N is much larger, and N = 2.2e31 is really one of the smallest halfway-reasonable estimates for our clone condition.

[Wyrm]

As a sample calculation, assume 3.0e4 genes with an average of seven alleles each, giving N = [3.0e4]^7.

Um, shouldn't that be N = 7^[3.0e4]? If there's one locus, there are seven possible human genomes. If there are two loci, there are 49 (= 7²) human genomes. Three loci, 343 (= 7³), and so on.

R borks on 7^[3.0e4], so this exceeds the precision of the floating point numbers it uses. This is a lot of genomes.

[Kuroneko]

Um, shouldn't that be N = 7^[3.0e4]?

You are quite right. That was rather boneheaded. Mea culpa.

R borks on 7^[3.0e4], so this exceeds the precision of the floating point numbers it uses. This is a lot of genomes.

The probability is even worse. 7^{3.00e4} = 10^{2.53e4}. For n ≅ 2e21, the probability of no random clones has log P ≅ -n²/N, so that P ≅ 10^{-10^{-2.53e4}}, so that a computer would need about 10^{25300} digits to distinguish it from 1, give or take a few dozen factors of ten.