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LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 222 Figure 8.8. Tangle operations. (a) Tangle addition, (b) tangle closure, and (c) N((â3,0) + (1)) =< 2 > THE TANGLE MODEL FOR SITE-SPECIFIC RECOMBINATION The fundamental observations underlying this model are that a pair of sites bound by an enzyme forms a tangle and that most of the products of recombination experiments performed on unknotted substrate are 4-
LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 223 plats. We will use tangles to build a model that will compute the topology of the pre- and post-recombination synaptic complex in a single recombination event, given knowledge of the topology of the substrate and product (Ernst and Sumners, 1990; Sumners, 1990, 1992; Sumners et al., 1994). In site-specific recombination on circular DNA substrate, two kinds of geometric manipulation of the DNA occur. The first is a global ambient isotopy, in which a pair of distant recombination sites are juxtaposed in space, and the enzyme binds to the molecule(s), forming the synaptic complex. Once synapsis is achieved, the next move is local and due entirely to enzyme action. Within the region occupied by the enzyme, the substrate is broken at each site, and the ends are recombined. We will model this local move. The aim of our mathematical model is, given the observed changes in geometry and topology of the DNA, to compute the topology of the entire synaptic complex, both before and after enzyme action. Within the region controlled by the enzyme, the enzyme breaks the DNA at each site and recombines the ends by exchanging them. We model the enzyme itself as a 3-ball. The synaptosome consisting of the enzyme and bound DNA forms a 2- string tangle. What follows is a list of biological and mathematical assumptions made in the tangle model (Ernst and Sumners, 1990; Sumners, 1992; Sumners et al., 1994). Most of these assumptions are implicit in the existing analyses of the results of enzyme experiments on circular DNA (Cozzarelli et al., 1984; Stark et al., 1989; Spengler et al., 1985; Wasserman and Cozzarelli, 1986; Wasserman et al., 1985; Kanaar et al., 1990; White et al., 1987; Kanaar et al., 1988; Abremski et al., 1986; Droge and Cozzarelli, 1986; Spengler et al., 1984). We make the following biological assumption: Assumption 1 The enzyme mechanism in a single recombination event is constant, independent of the geometry (supercoiling) and topology (knotting and catenation) of the substrate population. Moreover, recombination takes place entirely within the domain of the enzyme ball, and the substrate configuration outside the enzyme ball remains fixed while the strands are being broken and recombined inside and on the boundary of the enzyme. That is, we assume that any two pre-recombination copies of the synaptosome are identical, meaning that we can by rotation and
LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 224 translation superimpose one copy on the other, with the congruence so achieved respecting the structure of both the protein and the DNA. We likewise assume that all of the copies of the post-recombination synaptosome are identical. In a recombination event, we can mathematically divide the DNA involved into three types: (1) the DNA at and very near the sites where the DNA breakage and reunion are taking place; (2) other DNA bound to the enzyme, which is unchanged during a recombination event; and (3) the DNA in the synaptic complex that is not bound to the enzyme and that does not change during recombination. We make the following mathematical assumption about DNA types (1) and (2): Assumption 2 The synaptosome is a 2-string tangle and can be mathematically subdivided into the sum Ob + P of two tangles. One tangle, the parental tangle P, contains the recombination sites where strand breakage and reunion take place. The other tangle, the outside bound tangle Ob, is the remaining DNA in the synaptosome outside the P tangle; this is the DNA that is bound to the enzyme but that remains unchanged during recombination. The enzyme mechanism is modeled as tangle replacement (surgery) in which the parental tangle P is removed from the synaptosome and replaced by the recombinant tangle R. Therefore, our model assumes the following: pre-recombination synaptosome = Ob + P post-recombination synaptosome = Ob +R. In order to accommodate nontrivial topology in the DNA of type (3), we let the outside free tangle Of denote the synaptic complex DNA that is free (not bound to the enzyme) and that is unchanged during a single recombination event. We make the following mathematical assumption: Assumption 3 The entire synaptic complex is obtained from the tangle sum (Of + synaptosome) by the tangle closure construction.