National Academies Press: OpenBook
« Previous: Chapter 8 Lifting the Curtain: Using Topology to Probe the Hidden Action of Enzymes
Suggested Citation:"THE TOPOLOGY OF DNA." 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 203
Suggested Citation:"THE TOPOLOGY OF DNA." 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 204
Suggested Citation:"THE TOPOLOGY OF DNA." 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 205
Suggested Citation:"THE TOPOLOGY OF DNA." 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 206

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.

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 203 Experimental techniques such as X-ray crystallography and nuclear magnetic resonance provide ways to infer precise distances between atoms. However, these methods are not well suited to studying the dynamic mechanism by which enzymes act. Interestingly, topology can shed light on this key issue. The topological approach to enzymology is an experimental protocol in which the descriptive and analytical powers of topology and geometry are employed in an indirect effort to determine the enzyme mechanism and the structure of active enzyme-DNA complexes in vitro (in a test tube) (Wasserman and Cozzarelli, 1986; Sumners, 1987a). Once the enzyme structure and mechanism are understood in a controlled laboratory situation, this knowledge can be extrapolated to enzyme mechanism in vivo, that is, in a living cell. Topology is a branch of mathematics related to geometry. It is often characterized as "rubber-sheet geometry," because topological equivalence of spaces allows stretching, shrinking, and twisting of an object in order to make it congruent to another object. Topology is the study of properties of objects (spaces) that are unchanged by allowable elastic deformations. When a given topological property differs for a pair of spaces, then one can be sure that one space cannot be transformed into the other by elastic deformation. Changes that can produce nonequivalent spaces include cutting the space apart and reassembling the parts to produce another space. It is precisely this topological breakage and reassembly of DNA that characterizes the mechanism of some life- sustaining cellular enzymes, enzymes that facilitate replication, transcription, and transposition. Chapter 6 describes aspects of the geometry and topology of DNA and points out various topological transformations that must be performed on DNA by enzymes in order to carry out the life cycle of the cell. In the present chapter, we describe how recent results in three-dimensional topology (Culler et al., 1987; Ernst and Sumners, 1990; Sumners, 1990, 1992) have proven to be of use in the description and quantization of the action of these life-sustaining enzymes on DNA. THE TOPOLOGY OF DNA The DNA of all organisms has a complex and fascinating topology. It can be viewed as two very long curves that are intertwined millions of

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 204 times, linked to other curves, and subjected to four or five successive orders of coiling to convert it into a compact form for information storage. If one scales the cell nucleus up to the size of a basketball, the DNA inside scales up to the size of thin fishing line, and 200 km of that fishing line are inside the nuclear basketball. Most cellular DNA is double-stranded (duplex), consisting of two linear backbones of alternating sugar and phosphorus. Attached to each sugar molecule is one of the four bases (nucleotides): A = adenine, T = thymine, C = cytosine, G = guanine. A ladder whose sides are the backbones and whose rungs are hydrogen bonds is formed by hydrogen bonding between base pairs, with A bonding only with T, and C bonding only with G. The base pair sequence for a linear segment of duplex DNA is obtained by reading along one of the two backbones, and is a word in the letters {A,T,C,G}. Due to the uniqueness of the bonding partner for each nucleotide, knowledge of the sequence along one backbone implies knowledge of the sequence along the other backbone. In the classical Crick-Watson double helix model for DNA, the ladder is twisted in a right-hand helical fashion, with an average and nearly constant pitch of approximately 10.5 base pairs per full helical twist. The local helical pitch of duplex DNA is a function of both the local base pair sequence and the cellular environment in which the DNA lives; if a DNA molecule is under stress, or constrained to live on the surface of a protein, or is being acted upon by an enzyme, the helical pitch can change. Duplex DNA can exist in nature in closed circular form, where the rungs of the ladder lie on a twisted cylinder. Circular duplex DNA exists in the mitochondria of human cells, for example. Duplex DNA in the cell nucleus is a linear molecule, one that is topologically constrained by periodic attachment to a protein scaffold in order to achieve efficient packing. The packing, twisting, and topological constraints all taken together mean that topological entanglement poses serious functional problems for DNA. This entanglement would interfere with, and be exacerbated by, the vital life processes of replication, transcription, and recombination (Cozzarelli, 1992). For information retrieval and cell viability, some geometric and topological features must be introduced into the DNA, and others quickly removed (Wang, 1982, 1985). For example, the Crick-Watson helical twist of duplex DNA may require local unwinding in order to make room for a protein involved in

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 205 transcription to attach to the DNA. The DNA sequence in the vicinity of a gene may need to be altered to include a promoter or repressor. During replication, the daughter duplex DNA molecules become entangled and must be disentangled in order for replication to proceed to completion. After introduction of these life-sustaining changes in DNA geometry and topology, and after the process that these changes make possible is finished, the original DNA conformation must be restored. Some enzymes maintain the proper geometry and topology by passing one strand of DNA through another by means of a transient enzyme-bridged break in one of the DNA strands, a move performed by topoisomerases. Other enzymes break the DNA apart and recombine the ends by exchanging them, a move performed by recombinases. The description and quantization of the three-dimensional structure of DNA and the changes in DNA structure due to the action of these enzymes have required the serious use of geometry and topology in molecular biology. Geometry and topology provide ways of inferring the dynamic process of topological transformation carried out by an enzyme. This use of mathematics as an analytic tool for the indirect determination of enzyme mechanism is especially important because there is no experimental way to observe the dynamics of enzymatic action directly. In the experimental study of DNA structure and enzyme mechanism, biologists developed the topological approach to enzymology (Wasserman and Cozzarelli, 1986; Sumners, 1987b). In this approach, one performs experiments on circular substrate DNA molecules. These circular substrate molecules are genetically engineered by cloning techniques to contain regions that a certain enzyme will recognize and act upon. The circular form of the substrate molecule traps an enzymatic topological signature in the form of DNA knots and links (catenanes). Trapping such a topological signature is impossible if one uses linear DNA substrate. These DNA knots and links are observed by gel electrophoresis and electron microscopy of the reaction product DNA molecules. By observing the changes in geometry (supercoiling) and topology (knotting and linking) in DNA caused by an enzyme, the enzyme mechanism can be described and quantized. Figure 8.1 a gives the schematics of the topological enzymology protocol; the black box represents the dynamic reaction in which the enzyme attaches to the DNA substrate, breaks it apart and reconnects as necessary, and then releases the DNA products. Typical results of this experimental protocol

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 206 are the reaction products displayed in Figures 8.1b and 8.1c. Figure 8. 1b shows the electron micrograph of a DNA (+) figure eight catenane (Krasnow et al., 1983), and Figure 8.1c shows a micrograph of the DNA knot (Wasserman et al., 1985). Both are products of processive Tn3 recombination and are explained in detail below. Figure 8.1 (a) Topological approach to enzymology. (b) DNA (+) figure eight catenane, (c) DNA knot. Figure 8.1b reprinted, with permission, from Krasnow et al. (1983). Copyright 1983 by Macmillian Magazines Limited. Figure 8.c reprinted, by permission, from Wasserman et al. (1985). Copyright 1985 by the American Association for the Advancement of Science.

Next: SITE-SPECIFIC RECOMBINATION »
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!