Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology (1995)

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 225 If one deproteinizes the pre-recombination synaptic complex, one obtains the substrate; deproteinization of the post-recombination synaptic complex yields the product. The topological structure (knot and catenane types) of the substrate and product yields equations in the recombination variables {Of, Ob, P, R}. Specifically, a single recombination event on a single circular substrate molecule produces two recombination equations in four unknowns: substrate equation: N(Of +Ob + P) = substrate product equation: N(Of +Ob + R ) = product. The geometric meaning of these recombination equations is illustrated in Figure 8.3. In Figure 8.3, Of = (0), Ob = (−3,0), P = (0), and R = (1). With these values for the variables, our recombination equations become: substrate equation: N((0) + (−3, 0) + (0)) =< 1 > product equation: N((0) + (−3,0) + (1)) = < 2 >. THE TOPOLOGY OF TN3 RESOLVASE Tn3 resolvase is a site-specific recombinase that reacts with certain circular duplex DNA substrate with directly repeated recombination sites (Wasserman et al., 1985). One begins with supercoiled unknotted DNA substrate and treats it with resolvase. The principal product of this reaction is known to be the DNA 4-plat < 2 > (the Hopf link, Figures 8.4a and 8.5a) (Wasserman and Cozzarelli, 1985). Resolvase is known to act dispersively in this situation—to bind to the circular DNA, to mediate a single recombination event, and then to release the linked product. It is also known that resolvase and free (unbound) DNA links do not react. However, once in 20 encounters, resolvase acts processively—additional recombinant strand exchanges are promoted prior to the release of the product, with yield decreasing exponentially with increasing number of strand exchanges at a single binding encounter with the enzyme. Two successive rounds of processive recombination produce the DNA 4-plat 

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 226 < 2,1,1 > (the figure eight knot, Figures 8.4b and 8.5b); three successive rounds of processive recombination produce the DNA 4-plat <1,1,1,1,1> (the Whitehead link, Figures 8.4c and 8.5c), whose electron micrograph appears in Figure 8.1b; four successive rounds of recombination produce the DNA 4-plat < 1,2,1,1,1 > (the knot , Figures 8.4d and 8.5d), whose electron micrograph appears in Figure 8.1c. The discovery of the DNA knot < 1,2,1,1,1> substantiated a model for Tn3 resolvase mechanism (Wasserman et al., 1985). In processive recombination, it is the synaptosome itself that repeatedly changes structure. We make the following biologically reasonable mathematical assumption in our model: Assumption 4 In processive recombination, each additional round of recombination adds a copy of the recombinant tangle R to the synaptosome. More precisely, n rounds of processive recombination at a single binding encounter generate the following system of (n + 1) tangle equations in the unknowns {Of, Ob, P, R}: substrate: N(Of + Ob +P) = substrate rth round: N(O + 0b + rR )= rth round product, 1 £ r £ n. For resolvase, the electron micrograph of the synaptic complex in Figure 8.2 reveals that Of = (0), since the DNA loops on the exterior of the synaptosome can be untwisted and are not entangled. This observation from the micrograph reduces the number of variables in the tangle model by one, leaving us with three variables {Ob, P, R} . One can prove (Sumners, 1990, 1992; Ernst and Sumners, 1990) that there are four possible tangle pairs {Ob, R}, which can produce the experimental results of the first two rounds of processive Tn3 recombination. The third round of processive recombination is then used to discard three of these four pairs as extraneous solutions. The following theorems can be viewed as a mathematical proof of resolvase synaptic complex structure: the model proposed in (Wasserman et al., 1985) is the unique 

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 227 explanation for the first three observed products of processive Tn3 recombination, assuming that processive recombination acts by adding on copies of the recombinant tangle R. The process of obtaining electron micrographs of RecA-enhanced DNA knots and catenanes is technically difficult and requires a relatively large amount of product due to the extensive work-up required for RecA coating and microscopy. Gel electrophoresis is not only technically much easier to do, but it detects vanishingly small amounts of DNA product. For these reasons, biologists prefer to use gel electrophoresis as the assay from which experimental conclusions are to be drawn. For relaxed DNA knots and links, the gel determines the crossing number of the (relaxed) products, and comparison to gel ladders for known knot and catenane structures can be used to obtain more information than crossing number alone. As an aid to the analysis of topological enzymology experiments, a table of possible (and biologically reasonable!) tangle mechanisms has been prepared (Sumners et al., 1994) for each possible sequence of crossing numbers of reaction products that can be read from the gel. This tangle table should make the mathematical analysis of topological enzymology experiments easier to do. We now come to the rigorous mathematical proof of Tn3 mechanism. The proofs of the following two theorems can be skipped without detriment to the continuity of the exposition. Theorem 8.1 Suppose that tangles Ob,P, and R satisify the following equations: (i) N(Ob + P) = < 1 the unknot (ii) N(Ob + R) = < 2 > the Hopf link (iii) N(Ob + R + R) = < 2,1,1 > the figure 8 knot. Then {Ob ,R} = {(−3,0),(1)}, {(3,0),(−l)}, {(−2,− 3,− ),(1)}, or {(2,3,1),(−1)}. Proof: In this proof we use the following notation: Rn denotes Euclidean n-space, Bn denotes the unit ball in Rn (the set of all vectors in Rn of length < 1), and Sn−1 denotes the boundary of Bn (the set of all vectors in Rn of length 1). The first (and mathematically most 

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 228 interesting) step in the proof of this theorem is to argue that solutions {Ob, R} must be rational tangles. Now Ob, R, and (Ob + R) are locally unknotted, because N(Ob + R) is the Hopf link, which has two unknotted components. Any local knot in a tangle summand would persist in the Hopf link. Likewise, P is locally unknotted, because N(Ob + P) is the unknot. Let A' denote the 2-fold branched cyclic cover of the tangle A; then ∂A' = S1 × S1. If A is a prime tangle, then the inclusion homeomorphism injects π1(∂A') =Z Z into π1(A') (Lickorish, 1981). If both A and B are prime tangles, and N(A + B)' denotes the 2-fold branched cyclic cover, then π1,(N(A + B)') contains a subgroup isomorphic to Z Z. If K is any 4-plat, then π1(K') is a cyclic group, since K' is a lens space (Burde and Zieschang, 1985). Since no cyclic group contains Z Z, no 4-plat has two prime tangle summands. This means that if A and B are locally unknotted tangles, and N(A + B) is a 4-plat, then at least one of A and B must be a rational tangle. From equation (ii) above, we conclude that at least one of {Ob, R} is rational. Suppose that Ob is rational and that R is prime. Given that N((Ob + R)+ R) is a knot, one can argue (Lickorish, 1981) that Ob + R is also a prime tangle. From equation (iii), we then have that the 4-plat < 2,1,1 > admits two prime tangle summands, which is impossible. We therefore conclude that R must be a rational tangle. The next step is to argue that Ob, is a rational tangle. Suppose that Ob is a prime tangle. Then P must be a rational tangle, because N(Ob + P) is the unknot (equation (i)). Passing to 2-fold branched cyclic covers, we have that N(Ob + P)' = S3, and P' is homeomorphic to S1 × B2 (since P is rational), so is a bounded knot complement in S3. We know that R is a rational tangle and can argue that equation (iii) implies that (R + R) is likewise rational. Again passing to the 2-fold branched cyclic covers of equations (ii) and (iii), we obtain the equations N(Ob + R)' = the lens space L(2,1) and N(Ob+(R+R))'= the lens space L(5,3). Since R' and (R + R)' are each homeomorphic to a solid torus S1 × B2 , this means that there are two attachments of a solid torus to along ∂ =S1 × S1, yielding the lens spaces L(2,1) and L(5,3). The process of adding on a 

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 229 solid torus along its boundary is called Dehn surgery, and the Cyclic Surgery Theorem (Culler et al., 1987) now applies to this situation to imply that, since the orders of the cyclic fundamental groups of the lens spaces differ by more than one, the only way this can happen is for to be a Seifert fiber space and hence a torus knot complement. Fortunately, the results of Dehn surgery on torus knot complements are well understood, and one can argue that in fact must be a complement of the unknot (a solid torus) (Ernst and Sumners, 1990), which means that Ob is a rational tangle. The proof now amounts to computing the rational solutions to equations (ii) and (iii), exploiting the classifying schemes for rational tangles and 4-plats. In Ernst and Sumners (1990), a ''calculus for rational tangles" was developed to perform such calculations. One can use this calculus of classifying vectors to solve equations (ii) and (iii), obtaining the four solution pairs {Ob,R} = {(−3,0),(1)}, {(3,0),(−1)}, {(−2, −3, −1),(1)} , and {(2,3,1), (−1)}. Because each of the unoriented 4-plat products in equations (ii) and (iii) is achiral, given any solution set {Ob, R} to equations (ii) and (iii), its mirror image {−Ob, −R} must also be a solution. So the mathematical situation, given equations (i) through (iii), is that we have two pairs of mirror image solution sets for {Ob, R}. In order to decide which is the biologically correct solution, we must utilize more experimental evidence. The third round of processive resolvase recombination determines which of these four solutions is the correct one. Theorem 8.2 Suppose that tangles Ob, P, and R satisfy the following equations: (i) N(Ob + P) = < 1 > (the unknot) (ii) N(Ob + R) = < 2 > (the Hopf link) (iii) N(Ob + R + R) = < 2,1,1 > (the figure 8 knot). (iv) N(Ob + R + R + R) = < 1,1,1,1,1 > (the (+) Whitehead link). Then Ob = −3,0, R = (1), and N(Ob + R + R + R + R)=< 1,2,1,1,1 >.