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6 8 Number of substitutions N(t)

Time

Figure 8.15 Substitution patterns under a Poisson process. The top panel shows the probability distribution for the number of substitutions that might occur during one time interval. N(t) between 0 and 9 all have probabilities of greater than 0.01. The bottom panel shows the cumulative number of substitutions under a Poisson process for five independent trails. Each trail is akin to an independent lineage experiencing substitutions. The average number of substitutions is approximately 40 (four multiplied by the number of time intervals) but there is variation among the lineages. In both panels the rate of substitution is the same at X = 4.

Time

Figure 8.15 Substitution patterns under a Poisson process. The top panel shows the probability distribution for the number of substitutions that might occur during one time interval. N(t) between 0 and 9 all have probabilities of greater than 0.01. The bottom panel shows the cumulative number of substitutions under a Poisson process for five independent trails. Each trail is akin to an independent lineage experiencing substitutions. The average number of substitutions is approximately 40 (four multiplied by the number of time intervals) but there is variation among the lineages. In both panels the rate of substitution is the same at X = 4.

molecular clock is a Poisson process, a stochastic process which is defined in terms of the count of events, N(t), since time was equal to zero. In a Poisson process the expected number of events between two times follows a Poisson distribution. Assuming that all substitutions are independent events, the probability of a substitution is very small, and the number of time intervals is very large, a Poisson clock gives the probability of observing some number of substitutions after a time period has elapsed as

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