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LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 230 Proof: The unoriented (+) Whitehead link is chiral and {ObR}= {(â3,0),(1)} is the unique solution to equations (i) and (iv). The correct global topology of the first round of processive Tn3 recombination on the unknot is shown in Figure 8.3. Moreover, the first three rounds of processive Tn3 recombination uniquely determine N(Ob + R + R + R + R), the result of four rounds of recombination. It is the 4-plat knot <1,2,1,1,1>, and this DNA knot has been observed (Figure 8.1c). We note that there is no information in either Theorem 8.1 or Theorem 8.2 about the parental tangle P. Since P appears in only one tangle equation (equation (i)), for each fixed rational tangle solution for Ob, there are infinitely many rational tangle solutions to equation (i) for P (Ernst and Sumners, 1990). Most biologists believe that P= (0), and a biomathematical argument exists for this claim (Sumners et al., 1994). SOME UNSOLVED PROBLEMS 1. How does TOPO II recognize knots? E. coli contains circular duplex DNA molecules. In wild-type E. coli, no knotting has been observed for these molecules. However, in a mutant strain of E. coli where the production of Topoisomerase II (the enzyme that performs strand passage via an enzyme-bridged transient double-stranded break in the DNA) can be blocked by heat shock, a small fraction (about 7 percent) of knotted DNA has been observed (Shishido et al., 1987). All observed knots have the gel mobility of trefoil knots. The observed knots are presumably the by-products of other cellular processes (such as recombination). This experiment shows that TOPO II is able to detect DNA knots and kill them in wild-type E. coli. How does the enzyme (which can act only locally) detect the global topology of a DNA knot and then make just the right combination of passages to kill the knot? It must be the energy minimization of the DNA itself that detects the knotting. The enzyme has only to detect when two DNA strands are being pushed together in space by the DNA itself in an effort to attain a lower energy state, whence the enzyme can operate, allowing one DNA strand to pass through another to reach a lower- energy configuration. If one ties a knot in a short stiff rubber tube (an elastic tube) and seals up the ends to form a knotted circle, the tube will touch itself, trying to pass through itself to
LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 231 relieve strain and minimize energy. For circular elastica in R3, minimization of the bending energy functional occurs when the elasticum is the round planar unknot (Langer and Singer, 1984, 1985). This means that a knotted elasticum has at least one point of self-contact and that the elasticum is pushing at that point of self-contact to get through to a lower energy state. Does this elasticum model adequately explain the ability of Topoisomerase II to detect and selectively kill DNA knots in vivo? 2. What is the topology of the kDNA network? The kinetoplast DNA (kDNA) of the parasite trypanosome forms a link of some 5,000 to 10,000 unknotted DNA circlesâthe DNA equivalent of chain mail (Marini et al., 1980; Englund et al., 1982; Rauch et al., 1994). Work is ongoing (Rauch et al., 1994) in which the topological structure of kDNA is being studied by means of partial digest of the network, electrophoresis, and electron microscopy of the characteristic fragments, in which the large kDNA link is being randomly broken up into small sublinks, and the frequency of occurrence of these sublink units is being used (statistically) to reconstruct the large link itself. The kDNA network consists of small minicircles and a few large maxicircles. The minicircles are known to be unknotted, and it is known that neighbors link in the fashion of the Hopf link (Figure 8.4a) (like the links in a chain). Moreover, it is believed that the kDNA network has a fundamental region that is repeated in space to generate the entire structure. This gives rise to a knot theory problem: classify the links that allow a diagram in which each component has no self-crossings (and hence is unknotted) and in which each component links another component simply (like the links in a chain) or not at all, and in which the linking structure is periodic in space. The spatial periodicity amounts to drawing the link diagram on a torus (or some other compact, orientable 2-manifold), from whence the entire diagram is reproduced by taking the universal cover. The algebraic classification of such "chain mail links" should be interesting and obtainable with off-the-shelf topological invariants. Another topological problem has arisen in this biological system. A trefoil knotted minicircle has been observed as an intermediate to the replication process on the kDNA network (Ryan et al., 1988). What is the mechanism that produces this knotted minicircle? Does the topology of the network naturally generate knots as replication intermediates? 3. Why is the figure eight knot faster than the trefoil knot? The phenomenon of gel mobility of relaxed knotted duplex DNA circles (Dean et al., 1985) has no adequate theoretical explanation. The gel velocity of
