Mutation Rates from Genome Resequencing

Case 1: Mutations with many identical sites

Assumptions:

1. The number of mutations is small compared to the number of sites.
2. There are no back mutations (reversions).
3. Mutations rates are constant over time and across sites.

Example: Single-base substitutions

Calculation:

1. If you know the number of sites at risk for the mutation (s), then you can calculate a per-site mutation rate. Example: Assume these 22 point mutations are A to G substitutions and there are 1,342,726 A bases in the original genome.
   

   >s = 1342726
   >rate$estimate/(T*s)
     event rate 
   2.730763e-10 
   >rate$conf.int/(T*s)
   [1] 1.711355e-10 4.134408e-10
   attr(,"conf.level")
   [1] 0.95
   

Case 2: One-time mutations

Assumptions:

Calculation:

1. Count the number of independent genomes that have the mutation (m) and total number of genomes analyzed (n) at a given time (T). Example: 5 of 12 independently evolved genomes have the mutation after 10,000 generations.
   

   > m = 5
   > n = 12
   > T = 10000
   

2. Calculate a maximum likelihood value and 95% exact (Clopper-Pearson) confidence limit for the fraction of independently evolved lineages that do not have the mutation from your observations.
   

   p = binom.test(n - m, n)
   >p
   
      Exact binomial test
   
   data:  n - m and n 
   number of successes = 7, number of trials = 12, p-value = 0.7744
   alternative hypothesis: true probability of success is not equal to 0.5 
   95 percent confidence interval:
    0.2766697 0.8483478 
   sample estimates:
   probability of success 
                0.5833333
   

3. If the mutations happen at a constant rate per unit time, then you can calculate the rate that gives this fraction of independent lineages without a mutation up to the given time point using the zero event term from a Poisson process:
   

   > -log(p$estimate) / T
   probability of success 
             5.389965e-05
   > -log(p$conf.int) / T
   [1] 1.284931e-04 1.644646e-05
   attr(,"conf.level")
   [1] 0.95
   

This is a particularly simple type of survival analysis.

Mutation Rates from Genome Resequencing

Motivation: You have re-sequenced several genomes after a mutation accumulation or adaptive evolution experiment. How do you infer the rates of different types of mutations from these data? What are the 95% confidence intervals on these values?

Case 1: Mutations with many identical sites

Assumptions:

1. The number of mutations is small compared to the number of sites.

1. Mutations rates are constant over time and across sites.

Example: Single-base substitutions

Calculation:

1. If you restrict your data to one genome per experimental population, then you can calculate the maximum likelihood value and 95% confidence limits from a Poisson distribution. Count the total number mutation (m) and the total elapsed generations or time of independent evolution (T). Example: 22 point mutations found in 6 genomes that each evolved for 10,000 generations.
   

   >m = 22
   >T = 10000 * 6
   >rate = poisson.test(m)
   >rate$estimate/T
     event rate 
   0.0003666667 
   >rate$conf.int/T
   [1] 0.0002297880 0.0005551377
   attr(,"conf.level")
   [1] 0.95
   

2. If you know the number of sites at risk for the mutation (s), then you can calculate a per-site mutation rate. Example: Assume these 22 point mutations are A to G substitutions and there are 1,342,726 A bases in the original genome.
   

   >s = 1342726
   >rate$estimate/(T*s)
     event rate 
   2.730763e-10 
   >rate$conf.int/(T*s)
   [1] 1.711355e-10 4.134408e-10
   attr(,"conf.level")
   [1] 0.95
   

Case 2: One-time mutations

Assumptions:

1. The mutation can only happen once per genome.
2. The mutation rate is constant per unit time or generation

Example: Deletion of an unstable chromosomal region. Once deleted, it can never be deleted again.

Calculation:

1. Count the number of independent genomes that have the mutation (m) and total number of genomes analyzed (n) at a given time (T). Example: 5 of 12 independently evolved genomes have the mutation after 10,000 generations.
   

   > m = 5
   > n = 12
   > T = 10000
   

2. Calculate a maximum likelihood value and 95% exact (Clopper-Pearson) confidence limit for the fraction of independently evolved lineages that do not have the mutation from your observations.
   

   p = binom.test(n - m, n)
   >p
   
      Exact binomial test
   
   data:  n - m and n 
   number of successes = 7, number of trials = 12, p-value = 0.7744
   alternative hypothesis: true probability of success is not equal to 0.5 
   95 percent confidence interval:
    0.2766697 0.8483478 
   sample estimates:
   probability of success 
                0.5833333
   

3. If the mutations happen at a constant rate per unit time, then you can calculate the rate that gives this fraction of independent lineages without a mutation up to the given time point using the zero event term from a Poisson process:
   

   > -log(p$estimate) / T
   probability of success 
             5.389965e-05
   > -log(p$conf.int) / T
   [1] 1.284931e-04 1.644646e-05
   attr(,"conf.level")
   [1] 0.95
   

This is a particularly simple type of survival analysis.

Issues: Pseudo-replication

Issues: Different mutation rates in different lineages

Mutation Rates from Genome Resequencing

Case 2: One-time mutations

Mutation Rates from Genome Resequencing

Motivation: You have re-sequenced several genomes after a mutation accumulation or adaptive evolution experiment. How do you infer the rates of different types of mutation rates from these data? What are the 95% confidence intervals on these values?

Case 1: Single-base substitutions

Assumptions: The number of mutations is small compared to the number of sites.

If you restrict your data to one genome per experimental population, then you can calculate the 95% confidence limits by assuming this is a Poisson process (poisson.test in R).

If you take multiple genomes from one experimental population, this is a type of pseudo-replication (they may have a shared evolutionary history). This makes calculating the 95% confidence intervals more complicated.

Case 2: One-time mutations

Assumptions: A mutation can only happen once per genome.

Example: Deletion of a chromosomal region. Once deleted, it can never be deleted again.

This is a type of "survival analysis". You can calculate the fraction of genomes that have and do not have your mutation. Then consider this a binomial process, to calculate a 95% confidence interval Then, convert this to a per-generation rate by dividing by the number of mutations.