National Academies Press: OpenBook
« Previous: Top-down
Suggested Citation:"Bottom-up." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
×
Page 125
Suggested Citation:"Bottom-up." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
×
Page 126

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 125 rate (m − 1) / 2 to each of the m genes and wait for one to ring. The probability that a mutation clock rings first is θ / (θ + m − 1), and, given that a mutation occurs first, the gene that mutates is chosen uniformly and at random. Similarly, the probability that a split occurs first is (m − 1) / (θ + m − 1), with the splitting gene being chosen at random from the m possibilities. The only wrinkle left is to describe the rule that tells us when to stop generating splits or mutations. In order to have the right distribution for the numbers of mutations when the sample has n ancestors, we must run until the first split after n, discard the last observation, and then stop. This simple scheme can be used effectively to simulate observations from extremely complex mutation mechanisms using only Bernoulli random variables, and provides a way of generating and storing the effects of each of the mutations. Some examples are given in the following sections. Bottom-up The second scheme, which proves very useful for deriving recurrence relations for the distribution of allele configurations, is the "bottom-up" method. In this case, the idea is to use the exponential alarm clocks from the bottom of the tree (that is, beginning at the sample) and run up to the common ancestor at the top. If we look up from the sample of size n toward the root, the probability that we will encounter a mutation before a coalescence is θ / (θ + n − 1), and the probability that a coalescence will occur first is (n − 1) (θ + n − 1). The probability distribution of the configuration at the tips may then be related to the distribution of the configuration at the mutation or coalescence time. To illustrate how this works, consider the infinitely-many-alleles mutation structure. Suppose that the current configuration consists of counts a = (a1,a2,. . .,an) with an = 0, and let Pn(a) denote the probability of this configuration. If the first event in the past is a coalescence, then the configuration of n − 1 genes must have been (al,. . .,aj−1,aj + 1, aj+1 − 1,aj+2,. . .,an−l)

CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 126 for some j = 1, 2,. . ., n − 2, and a gene in class j must be chosen to have an offspring. Since this last event has probability jaj + 1) / n − 1), the contribution to Pn a from such terms is (5.9) If, on the other hand, the first event in the past was a mutation, then the configuration must have been either (a1− 1,a2,. . .,aj−2, aj−1 − 1,aj + 1,aj+l,. . .,an−1,0) and the mutation occurred to a gene in a j class, j = 3,4,. . .,n-1 (probability j(aj + 1) / n ), or (a1− 2,a2 + 1,a3,. . .,an−1,0) and the mutation occurred to a gene in the 2 class (probability 2(a2 + 1) / n),or (a1,. . .,an−1,0) and the mutation occurred to a singleton gene (probability a1 / n). Finally, the configuration could have been (a − 1,a2,. . .,an−2,an −1−1,1) and the mutation occurred in the n class (probability 1). Combining all these possibilities and adding the term in (5.9) gives

Next: The Infinitely-Many-Sites Model »
Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology Get This Book
×
Buy Paperback | $80.00
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

As researchers have pursued biology's secrets to the molecular level, mathematical and computer sciences have played an increasingly important role—in genome mapping, population genetics, and even the controversial search for "Eve," hypothetical mother of the human race.

In this first-ever survey of the partnership between the two fields, leading experts look at how mathematical research and methods have made possible important discoveries in biology.

The volume explores how differential geometry, topology, and differential mechanics have allowed researchers to "wind" and "unwind" DNA's double helix to understand the phenomenon of supercoiling. It explains how mathematical tools are revealing the workings of enzymes and proteins. And it describes how mathematicians are detecting echoes from the origin of life by applying stochastic and statistical theory to the study of DNA sequences.

This informative and motivational book will be of interest to researchers, research administrators, and educators and students in mathematics, computer sciences, and biology.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!