A model to justify the mechanism responsible for the preservation of duplicate genes is presented. The dynamical system in the gene duplication model is asymptotically stable, and the process to obtain quantities of interest is discussed.

Abstract:

The mechanisms responsible for the preservation of duplicate genes have been debated for decades. Recently a new explanation has been proposed according to which after duplication two gene copies specialize to perform complementary functions. The quantities of interest are the probability that the separation of functionality occurs, and the amount of time after duplication that it takes for this occur.
Mathematically this problem can be described by a diffusion in six dimensional space in the simplest example when the duplicated gene is in charge of two functions. The diffusion describes the stochastic behavior of genotypic frequencies in the population. Simulations show that this diffusion spends most of its time near a particular one dimensional curve in the six dimensional space. This degenerate diffusion limit is an example of a solution to a stochastic differential equation that is forced to a lower dimensional manifold by a very srong drift. One can show that if the drift is a vector field whose deterministic flow has an asymptotically stable manifold of fixed points, then this drift term forces the stochastic process to stay close to and any limitting process must actually stay on the stable manifold.

We show that the dynamical system in the gene duplication model is asymptotically stable, and use the limitting diffusion process to give results for the quantities of interest.

Keywords:

Gene duplication; probability; stochastic behavior.

Lecture #13239

Need help? Visit our help pages at http://www.msri.org/communications/vmath/hints

Streaming Video

This is a high quality streaming video encoded with MPEG-4 and with 640x480 resolution.

Windows and Mac users, QuickTime 6.5 or later required
Linux users, please see our Linux Help Page on how to view our streaming videos

Follow this link to --- Watch the Video Now Via Streaming Video ---

Download QuickTime File

You can download the QuickTime file here. Right click on the link and "Save As..." to save to your local computer.

• 13239-13239-QuickTime.mov (182 MB)

Create a DVD

You can download the video and audio files here. Please note that you need both files to create a DVD. Right click on the link and "Save As..." to save to your local computer. You can find instructions on how to create a DVD on our help page at http://www.msri.org/communications/vmath/author

• 13239-13239-DVD PCM Audio.aiff (344 MB - Audio Only)
• 13239-13239-MPEG-2 120min High Quality Encode.m2v (785 MB - Video Only)

Buy the DVD

If none of the options work for you, you can always buy the DVD of this lecture.

If you would like to purchase a copy of this video for $15+shipping, please Click Here!

See more of our Streaming Videos on our main VMath - Streaming Video page.