DNA (deoxyribonucleic acid) is the blueprint for life. It can be viewed as
two very long curves that are intertwined millions of 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 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. This 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 determined by 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.
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 magnified by) the vital
cellular life processes of replication, transcription, and recombination.
For information retrieval and
cell viability, some geometric and topological features must be introduced
into the DNA, and others quickly removed. For example, the Crick-Watson
helical twist of duplex DNA may require local unwinding in order to make
room for a protein involved in transcription to attach to the DNA. The DNA
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 the process is finished, the original DNA conformation
must be restored. Some enzymes maintain 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. Other enzymes break the DNA
apart and recombine the ends by exchanging them. 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. This use of mathematics
as an analytical and computational tool is essential because there are few
ways to directly observe an enzyme in action.
In the experimental study of DNA structure and enzyme mechanism, biologists
developed the topological approach to enzymology shown schematically in
Figure 1, in the article "Lifting the Curtain: Using Topology to
Probe the Hidden Action of Enzymes,"
Notices of the
AMS, May 1995.
this approach, one performs experiments on circular substrate DNA
molecules; enzymes act on these DNA circles, creating an enzymatic
signature by changing the coiling of the DNA, and making and breaking knots
and links in the DNA. 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.
The topological approach to enzymology poses an interesting challenge for
mathematics: from the observed changes in DNA geometry and topology, how
deduce enzyme mechanisms? This requires the construction of mathematical
models for enzyme action and the use of these models to analyze the results
topological enzymology experiments. The entangled form of the product DNA
knots and links contains information about the enzymes that made them. In
to utility in the analysis of experimental results, the use of mathematical
models forces all of the background assumptions about the biology to be
carefully laid out.
At this point they can be examined and dissected, and their influence on
the biological conclusions drawn from experimental results can be