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LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 207 The topological approach to enzymology poses an interesting challenge for mathematics: from the observed changes in DNA geometry and topology, how can one mathematically deduce enzyme mechanisms? This requires the construction of mathematical models for enzyme action and the use of these models to analyze the results of topological enzymology experiments. The entangled form of the product DNA knots and links contains information about the enzymes that made them. Mathematics is required to extract mechanism information from the topological structure of the reaction products. In addition 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 determined. SITE-SPECIFIC RECOMBINATION Site-specific recombination is one of the ways in which nature alters the genetic code of an organism, either by moving a block of DNA to another position on the molecule (a move performed by transposase) (Sherratt et al., 1984) or by integrating a block of alien DNA into a host genome (a move performed by integrase). One of the biological purposes of recombination is the regulation of gene expression in the cell, because it can alter the relative position of the gene and its repressor and promoter sites on the genome. Site-specific recombination also plays a vital role in the life cycle of certain viruses, which utilize this process to insert viral DNA into the DNA of a host organism. An enzyme that mediates site-specific recombination on DNA is called a recombinase. A recombination site is a short segment of duplex DNA whose sequence is recognized by the recombinase. Site- specific recombination can occur when a pair of sites (on the same or on different DNA molecules) become juxtaposed in the presence of the recombinase. The pair of sites is aligned through enzyme manipulation or random thermal motion (or both), and both sites (and perhaps some contiguous DNA) are then bound by the enzyme. This stage of the reaction is called synapsis, and we will call this intermediate protein-DNA complex formed by the part of the substrate that is bound to the enzyme together with the enzyme
LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 208 itself the synaptosome (Benjamin and Cozzarelli, 1990; Heichman and Johnson, 1990; Pollock and Nash, 1983; Griffith and Nash, 1985; Kim and Landy, 1992). We will call the entire DNA molecule(s) involved in synapsis (including the parts of the DNA molecule(s) not bound to the enzyme), together with the enzyme itself, the synaptic complex. The electron micrograph in Figure 8.2 shows a synaptic complex formed by the recombination enzyme Tn3 resolvase when reacted with unknotted circular duplex DNA. In the micrograph of Figure 8.2, the synaptosome is the black mass attached to the DNA circle, with the unbound DNA in the synaptic complex forming twisted loops in the exterior of the synaptosome. It is our intent to deduce mathematically the path of the DNA in the black mass of the synaptosome, both before and after recombination. We want to answer the question: How is the DNA wound around the enzyme, and what happens during recombination? Figure 8.2 Tn3 synaptic complex. (Courtesy of N.R. Cozzarelli.)
LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 209 After forming the synaptosome, a single recombination event occurs: the enzyme then performs two double- stranded breaks at the sites and recombines the ends by exchanging them in an enzyme-specific manner. The synaptosome then dissociates, and the DNA is released by the enzyme. We call the pre-recombination unbound DNA molecule(s) the substrate and the post-recombination unbound DNA molecule(s) the product. During a single binding encounter between enzyme and DNA, the enzyme may mediate more than one recombination event; this is called processive recombination. On the other hand, the enzyme may perform recombination in multiple binding encounters with the DNA, which is called distributive recombination. Some site-specific recombination enzymes mediate both distributive and processive recombination. Site-specific recombination involves topological changes in the substrate. In order to identify these topological changes, one chooses to perform experiments on circular DNA substrate. One must perform an experiment on a large number of circular molecules in order to obtain an observable amount of product. Using cloning techniques, one can synthesize circular duplex DNA molecules, which contain two copies of a recombination site. At each recombination site, the base pair sequence is in general not palindromic (the base pair sequence for the site read left-to-right is different from the base pair sequence read right-to-left), and hence induces a local orientation (arrow) on the substrate DNA circle. If these induced orientations from a pair of sites on a single circular molecule agree, this site configuration is called direct repeats (or head-to-tail), and if the induced orientations disagree, this site configuration is called inverted repeats (or head-to-head). If the substrate is a single DNA circle with a single pair of directly repeated sites, the recombination product is a pair of DNA circles and can form a DNA link (or catenane) (Figure 8.3). If the substrate is a pair of DNA circles with one site each, the product is a single DNA circle (Figure 8.3 read in reverse) and can form a DNA knot (usually with direct repeats). In processive recombination on a circular substrate with direct repeats, the products of an odd number of rounds of processive recombination are DNA links, and the products of an even number of rounds of processive recombination are DNA knots. If the substrate is a single DNA circle with inverted repeats, the product is a single DNA circle and can form a DNA knot. In all figures where DNA is represented by a line drawing
LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 210 (such as Figure 8.3), duplex DNA is represented by a single line, and supercoiling is omitted. The experimental strategy in the topological approach to enzymology is to observe the enzyme-caused changes in the geometry and topology of the DNA and to deduce the enzyme mechanism from these changes, as in Figure 8.1a. The geometry and topology of the circular DNA substrate are experimental control variables. The geometry and topology of the recombination reaction products are observables. In vitro experiments usually proceed as follows: Circular substrate is prepared, with all of the substrate molecules representing the same knot type (usually the unknot, that is, a curve without knots). The amount of supercoiling of the substrate molecules (the supercoiling density) is also a control variable. The substrate molecules are reacted with a high concentration of purified enzyme, and the reaction products are fractionated by gel electrophoresis. DNA molecules are naturally Figure 8.3 A single recombination event: direct repeats.
LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 211 negatively charged, with the amount of negative charge proportional to the molecular weight. A gel is a resistive medium through which the DNA molecules can be forced to migrate under the influence of an electric field. The DNA sample is placed at the top of a gel column, and similar molecules migrate through the gel with similar velocities, forming discrete DNA bands in the gel when the electric field is turned off. Normally, gel electrophoresis discriminates among DNA molecules on the basis of molecular weight; given that all molecules are the same molecular weight (as is the case in these topological enzymology experiments), electrophoresis discriminates on the basis of subtle differences in the geometry (supercoiling) and topology (knot and link type) of the DNA molecules. For example, in unknotted DNA, gel electrophoresis discriminates on the basis of number of supercoils and can detect a difference of one in the number of supercoils. In gel electrophoresis of knotted and linked DNA, one must nick (break one of the two backbone strands of) the reaction products prior to electrophoresis in order to relax the supercoils in the DNA knots and links, because supercoiling confounds the gel migration of knotted and linked DNA. For nicked DNA knots and links, under the proper conditions gel velocity is (surprisingly) determined by the crossing number of the knot or link; knots and links of the same crossing number migrate with the same gel velocities (Dean et al., 1985); the higher the crossing number, the greater the gel mobility. After the gel is run, the gel bands are excised, and the DNA molecules are removed from the gel and coated with RecA protein. It is this new observation technique (RecAenhanced electron microscopy) (Krasnow et al., 1983) that makes possible the detailed knot-theoretic analysis of reaction products. RecA is an E. coli protein that binds to DNA and mediates general recombination in E. coli. Naked (uncoated) duplex DNA is approximately 20 angstroms in diameter, and RecA-coated DNA is approximately 100 angstroms in diameter. The process of RecA coating fattens, stiffens, and stretches (untwists) the DNA. This fattening and stiffening facilitates the unambiguous determination of crossings (nodes) in an electron micrograph of a DNA knot or link and reduces the number of extraneous crossings. After RecA coating, the DNA is shadowed with platinum for viewing under the electron microscope. Electron micrographs of the reaction products (Figure 8. 1b and c) are made, and frequency distributions of knot types of the products are prepared. This
