## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 17, Number 1 (2007), 1-419

### Waiting for regulatory sequences to appear

Richard Durrett and Deena Schmidt

#### Abstract

One possible explanation for the substantial organismal differences between humans and chimpanzees is that there have been changes in gene regulation. Given what is known about transcription factor binding sites, this motivates the following probability question: given a 1000 nucleotide region in our genome, how long does it take for a specified six to nine letter word to appear in that region in some individual? Stone and Wray [*Mol. Biol. Evol.* **18** (2001) 1764–1770] computed 5,950 years as the answer for six letter words. Here, we will show that for words of length 6, the average waiting time is 100,000 years, while for words of length 8, the waiting time has mean 375,000 years when there is a 7 out of 8 letter match in the population consensus sequence (an event of probability roughly 5/16) and has mean 650 million years when there is not. Fortunately, in biological reality, the match to the target word does not have to be perfect for binding to occur. If we model this by saying that a 7 out of 8 letter match is good enough, the mean reduces to about 60,000 years.

#### Article information

**Source**

Ann. Appl. Probab. Volume 17, Number 1 (2007), 1-32.

**Dates**

First available: 13 February 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoap/1171377175

**Digital Object Identifier**

doi:10.1214/105051606000000619

**Mathematical Reviews number (MathSciNet)**

MR2292578

**Subjects**

Primary: 92D10: Genetics {For genetic algebras, see 17D92}

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Regulatory sequence population genetics Moran model Poisson approximation clumping heuristic

#### Citation

Durrett, Richard; Schmidt, Deena. Waiting for regulatory sequences to appear. The Annals of Applied Probability 17 (2007), no. 1, 1--32. doi:10.1214/105051606000000619. http://projecteuclid.org/euclid.aoap/1171377175.