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fix miswording
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jakebeal
jakebeal
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This sort of method is indeed quite useful and frequently used in synthetic biology: I've used a similar approach before to generate 5' insulators for promoters.

Calculating the exact theoretical likelihood of what you want, so far as I understand, is a complex combinatoric equation that I wasn't able to get a nice closed form on with a few minutes of work. For your purposes, however, I expect that a reasonable estimate will do as well, and that can be made much more easily by modeling the population of each combination as an independent series of biased coin flips, i.e., a binomial distribution.

Consider a situation with $n$ potential combinations that you attempt to cover with $k$ random mutants. If we focus on a particular combination, we can see that each random mutant has a $\frac{1}{n}$ chance of hitting that combination. The whole set of mutants is thus a series of $k$ biased coin flips, with a probability $p_0 = (1-\frac{1}{n})^k$ of getting no hits onmutants that mutanthit that combination. This means that the probability $p_c$ that the combination is covered is $p_c = 1-p_0$, expanding to: $$p_c = 1-(1-\frac{1}{n})^k$$

With a large number of potential combinations and a large number of mutants, we can make a reasonable approximate by treating each combination as independent. In this case, the probability that all combinations are covered is the product of the probabilities that each combination is covered independently. They aren't actually independent: each combination that gets covered slightly detracts from the ability of other combinations to be covered. An estimate using an independence assumption is thus an overestimate of the actual likelihood, and the higher that $k$ is the closer the estimate should be to the exact probability. In short: $$p_{all} < (1-(1-\frac{1}{n})^k)^n$$

This smells like the sort of equation there ought to be a nice identity for, but I don't know the identity, so I'll just give a couple of examples of calculating with it:

  • NNK = 32 combinations, 50 mutants (n=32, k=50): $p_{all}$ < 0.07%
  • NNK = 32 combinations, 100 mutants (n=32, k=100): $p_{all}$ < 25.5%
  • NNK = 32 combinations, 200 mutants (n=32, k=200): $p_{all}$ < 94.6%

As you can see there's a sharp transition as the coverage count increases.

Important caveat: the theoretical value may not give you what you want. In the paper I linked above, we used two degenerate primers with 18 N bases to inject 36bp of random material, which means we had $4.7x10^{21}$ possible combinations and should never have seen a repeat --- except that sequencing showed that we did indeed get several repeats out of only a few hundred colonies. Thus, biases in the biology may lead your statistics to be skewed from what the theory suggests.

This sort of method is indeed quite useful and frequently used in synthetic biology: I've used a similar approach before to generate 5' insulators for promoters.

Calculating the exact theoretical likelihood of what you want, so far as I understand, is a complex combinatoric equation that I wasn't able to get a nice closed form on with a few minutes of work. For your purposes, however, I expect that a reasonable estimate will do as well, and that can be made much more easily by modeling the population of each combination as an independent series of biased coin flips, i.e., a binomial distribution.

Consider a situation with $n$ potential combinations that you attempt to cover with $k$ random mutants. If we focus on a particular combination, we can see that each random mutant has a $\frac{1}{n}$ chance of hitting that combination. The whole set of mutants is thus a series of $k$ biased coin flips, with a probability $p_0 = (1-\frac{1}{n})^k$ of getting no hits on that mutant. This means that the probability $p_c$ that the combination is covered is $p_c = 1-p_0$, expanding to: $$p_c = 1-(1-\frac{1}{n})^k$$

With a large number of potential combinations and a large number of mutants, we can make a reasonable approximate by treating each combination as independent. In this case, the probability that all combinations are covered is the product of the probabilities that each combination is covered independently. They aren't actually independent: each combination that gets covered slightly detracts from the ability of other combinations to be covered. An estimate using an independence assumption is thus an overestimate of the actual likelihood, and the higher that $k$ is the closer the estimate should be to the exact probability. In short: $$p_{all} < (1-(1-\frac{1}{n})^k)^n$$

This smells like the sort of equation there ought to be a nice identity for, but I don't know the identity, so I'll just give a couple of examples of calculating with it:

  • NNK = 32 combinations, 50 mutants (n=32, k=50): $p_{all}$ < 0.07%
  • NNK = 32 combinations, 100 mutants (n=32, k=100): $p_{all}$ < 25.5%
  • NNK = 32 combinations, 200 mutants (n=32, k=200): $p_{all}$ < 94.6%

As you can see there's a sharp transition as the coverage count increases.

Important caveat: the theoretical value may not give you what you want. In the paper I linked above, we used two degenerate primers with 18 N bases to inject 36bp of random material, which means we had $4.7x10^{21}$ possible combinations and should never have seen a repeat --- except that sequencing showed that we did indeed get several repeats out of only a few hundred colonies. Thus, biases in the biology may lead your statistics to be skewed from what the theory suggests.

This sort of method is indeed quite useful and frequently used in synthetic biology: I've used a similar approach before to generate 5' insulators for promoters.

Calculating the exact theoretical likelihood of what you want, so far as I understand, is a complex combinatoric equation that I wasn't able to get a nice closed form on with a few minutes of work. For your purposes, however, I expect that a reasonable estimate will do as well, and that can be made much more easily by modeling the population of each combination as an independent series of biased coin flips, i.e., a binomial distribution.

Consider a situation with $n$ potential combinations that you attempt to cover with $k$ random mutants. If we focus on a particular combination, we can see that each random mutant has a $\frac{1}{n}$ chance of hitting that combination. The whole set of mutants is thus a series of $k$ biased coin flips, with a probability $p_0 = (1-\frac{1}{n})^k$ of getting no mutants that hit that combination. This means that the probability $p_c$ that the combination is covered is $p_c = 1-p_0$, expanding to: $$p_c = 1-(1-\frac{1}{n})^k$$

With a large number of potential combinations and a large number of mutants, we can make a reasonable approximate by treating each combination as independent. In this case, the probability that all combinations are covered is the product of the probabilities that each combination is covered independently. They aren't actually independent: each combination that gets covered slightly detracts from the ability of other combinations to be covered. An estimate using an independence assumption is thus an overestimate of the actual likelihood, and the higher that $k$ is the closer the estimate should be to the exact probability. In short: $$p_{all} < (1-(1-\frac{1}{n})^k)^n$$

This smells like the sort of equation there ought to be a nice identity for, but I don't know the identity, so I'll just give a couple of examples of calculating with it:

  • NNK = 32 combinations, 50 mutants (n=32, k=50): $p_{all}$ < 0.07%
  • NNK = 32 combinations, 100 mutants (n=32, k=100): $p_{all}$ < 25.5%
  • NNK = 32 combinations, 200 mutants (n=32, k=200): $p_{all}$ < 94.6%

As you can see there's a sharp transition as the coverage count increases.

Important caveat: the theoretical value may not give you what you want. In the paper I linked above, we used two degenerate primers with 18 N bases to inject 36bp of random material, which means we had $4.7x10^{21}$ possible combinations and should never have seen a repeat --- except that sequencing showed that we did indeed get several repeats out of only a few hundred colonies. Thus, biases in the biology may lead your statistics to be skewed from what the theory suggests.

Source Link
jakebeal
jakebeal
  • 7k
  • 24
  • 46

This sort of method is indeed quite useful and frequently used in synthetic biology: I've used a similar approach before to generate 5' insulators for promoters.

Calculating the exact theoretical likelihood of what you want, so far as I understand, is a complex combinatoric equation that I wasn't able to get a nice closed form on with a few minutes of work. For your purposes, however, I expect that a reasonable estimate will do as well, and that can be made much more easily by modeling the population of each combination as an independent series of biased coin flips, i.e., a binomial distribution.

Consider a situation with $n$ potential combinations that you attempt to cover with $k$ random mutants. If we focus on a particular combination, we can see that each random mutant has a $\frac{1}{n}$ chance of hitting that combination. The whole set of mutants is thus a series of $k$ biased coin flips, with a probability $p_0 = (1-\frac{1}{n})^k$ of getting no hits on that mutant. This means that the probability $p_c$ that the combination is covered is $p_c = 1-p_0$, expanding to: $$p_c = 1-(1-\frac{1}{n})^k$$

With a large number of potential combinations and a large number of mutants, we can make a reasonable approximate by treating each combination as independent. In this case, the probability that all combinations are covered is the product of the probabilities that each combination is covered independently. They aren't actually independent: each combination that gets covered slightly detracts from the ability of other combinations to be covered. An estimate using an independence assumption is thus an overestimate of the actual likelihood, and the higher that $k$ is the closer the estimate should be to the exact probability. In short: $$p_{all} < (1-(1-\frac{1}{n})^k)^n$$

This smells like the sort of equation there ought to be a nice identity for, but I don't know the identity, so I'll just give a couple of examples of calculating with it:

  • NNK = 32 combinations, 50 mutants (n=32, k=50): $p_{all}$ < 0.07%
  • NNK = 32 combinations, 100 mutants (n=32, k=100): $p_{all}$ < 25.5%
  • NNK = 32 combinations, 200 mutants (n=32, k=200): $p_{all}$ < 94.6%

As you can see there's a sharp transition as the coverage count increases.

Important caveat: the theoretical value may not give you what you want. In the paper I linked above, we used two degenerate primers with 18 N bases to inject 36bp of random material, which means we had $4.7x10^{21}$ possible combinations and should never have seen a repeat --- except that sequencing showed that we did indeed get several repeats out of only a few hundred colonies. Thus, biases in the biology may lead your statistics to be skewed from what the theory suggests.