Spring 2007

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IMA Outcomes

Outcomes, impacts, highlights, nuggets... whatever name they go by, these short, novice-friendly research descriptions are just one of the ways the IMA shares the achievements of its visitors with the scientific community and strives to increase public appreciation of applied mathematics. We've recently added links to some of our most intriguing stories to our homepage to give visitors a glimpse of some of the fascinating research connected with the IMA. If your research has been influenced by a visit to the IMA, whether as a workshop participant, long-term visitor, or postdoc, please share your story with us!

EvoGeo

Can polyhedral geometry and commutative algebra—usually regarded as pure mathematics—help biologists? We at the IMA certainly think so, and the emerging applications of these mathematical areas to evolutionary biology was a major theme during a workshop bringing 135 mathematicians, statisticians, biologists, and computer scientists to the IMA in March 2007 as part of our year-long thematic program on Applications of Algebraic Geometry.

The evolutionary and developmental biology (EvoDevo) community apparently agrees as well. The article "Analysis of epistatic interactions and fitness landscapes using a new geometric approach" by Niko Beerenwinkel, Lior Pachter, Bernd Sturmfels, Santiago Elena and Richard Lenski, was the most accessed article in May 2007, with over 2,000 online accesses in the first three weeks of its appearance. This paper studies the reproductive fitness of 37 genetic variants of E. coli, based on data from long-term experimental studies of evolution in bacteria at Michigan State made in Lenski's laboratory at Michigan State.

Beerenwinkel presented this work, which uses computations with the algebraic geometry software system Macauley 2 to find and interpret the fitness landscape for E. coli, at the IMA workshop. Pachter, an organizer for the workshop, Sturmfels, a former IMA postdoc, a member of the IMA Board of Governors, and a lead organizer of the IMA 2006-2007 thematic program were participants at the workshop as well.


Fitness landscapes in the simple situation of two biallelic loci (negative, zero, and positive epistasis).

Related material

Alimentary math

Two separate groups of long-term visitors to the IMA 2005–2006 thematic program on imaging are harnessing their expertise and creativity for the improved production of two important foodstuffs: beef and soybeans.

beef rib eye (435x348) For beef, the area of the rib eye is an important indicator of the meat quality of an animal. Technology allows ultrasound imaging of cattle on the field, providing a potentially very useful tool for estimating the rib eye area. Deriving accurate estimates from such images is difficult, however, relying on experts who laboriously trace the outline of the rib eye and measure the enclosed area. Moreover, this process cannot be carried out in the field, and is subject to errors due to fatigue, image quality variation, and other factors.

ultrasound image (371x284)

Gregory Randall—who spent a year at the IMA for the imaging program—and other IMA participants, all from Uruguay, have developed a method which enables accurate automatic in-the-field rib eye measurement. Their algorithm processes the ultrasound image data in combination with stored statistical information about rib eye shapes derived from a bank of curves traced by experts. Their method, which has been extensively tested agrees well with the traditional expert marking and measurement method.

Mathematics and imaging science help with the vegetable course as well. Farmers use population estimates obtained by counting the aphids on sample soybean leaves in planning their crop dusting schedules, and so adjust yield and the use of pesticides. As with rib eye tracing, a manual approach to aphid counting is slow, off-line, and error-prone. However, this low-tech problem presents formidable challenges to even the most sophisticated automatic image segmentation methods. The colors of aphids change as the leaves age, with the color of the aphids on young leaves nearly matching that of leaf veins. The intricate structure of the veins and the tendency of aphids to cluster near veins further complicate automated image segmentation.

This problem was brought to the IMA's attention by a USDA researcher, and tackled by Korean mathematician Chang-Ock Lee and his graduate student Jooyoung Hahn during their year stay at the IMA. They started with a segmentation algorithm they had developed which utilizes geometric attraction-driven flow and edge regions, and optimized it for soybean leaf images. The resulting method requires neither interaction with end users nor mid-process parameter manipulations. The method succeeds in providing highly accurate, efficient aphid counts.